Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 36 (2024) 173002 (26pp) https://doi.org/10.1088/1361-648X/ad2159 Topical Review Anharmonic theory of superconductivity and its applications to emerging quantum materials Chandan Setty1,∗, Matteo Baggioli2,3 and Alessio Zaccone4,5 1 Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice University, Houston, TX 77005, United States of America 2 Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China 3 Shanghai Research Center for Quantum Sciences, Shanghai 201315, People’s Republic of China 4 Department of Physics ‘A. Pontremoli’, University of Milan, via Celoria 16, 20133 Milan, Italy 5 Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, United Kingdom E-mail: settychandan@gmail.com Received 20 April 2023, revised 11 November 2023 Accepted for publication 22 January 2024 Published 1 February 2024 Abstract The role of anharmonicity on superconductivity has often been disregarded in the past. Recently, it has been recognized that anharmonic decoherence could play a fundamental role in determining the superconducting properties (electron–phonon coupling, critical temperature, etc) of a large class of materials, including systems close to structural soft-mode instabilities, amorphous solids and metals under extreme high-pressure conditions. Here, we review recent theoretical progress on the role of anharmonic effects, and in particular certain universal properties of anharmonic damping, on superconductivity. Our focus regards the combination of microscopic-agnostic effective theories for bosonic mediators with the well-established BCS theory and Migdal–Eliashberg theory for superconductivity. We discuss in detail the theoretical frameworks, their possible implementation within first-principles methods, and the experimental probes for anharmonic decoherence. Finally, we present several concrete applications to emerging quantum materials, including hydrides, ferroelectrics and systems with charge density wave instabilities. Keywords: superconductivity, anharmonicity, quantum materials ∗ Author to whom any correspondence should be addressed. Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1 © 2024 The Author(s). Published by IOP Publishing Ltd https://doi.org/10.1088/1361-648X/ad2159 https://orcid.org/0000-0003-4829-1508 https://orcid.org/0000-0001-9392-7507 https://orcid.org/0000-0002-6673-7043 mailto:settychandan@gmail.com http://crossmark.crossref.org/dialog/?doi=10.1088/1361-648X/ad2159&domain=pdf&date_stamp=2024-2-1 https://creativecommons.org/licenses/by/4.0/ J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review 1. Introduction: anharmonicity and superconductivity 1.1. Historical perspective: anharmonicity and superconductivity In classic theories of phonon-mediated superconductivity, such as BCS theory and Migdal–Eliashberg theory, phonons were described as harmonic oscillators. In the second half of the 1970s, the discovery of the ‘high-Tc’ (for that time) super- conductivity in Niobium-based alloys which superconduct at temperatures T > 10K prompted a change in paradigm. Those materials were rich in structural instabilities often linked to quasi-localized lattice (ionic) vibrations, or coupled lattice-electronic instabilities of the Jahn–Teller type. Plakida and co-workers developed early models addressing the influ- ence, and enhancement, of the critical temperature Tc due to highly anharmonic quasi-local vibrations (QLVs), in a first series of papers [1, 2]. In these papers, the QLVsweremodeled as two-level anharmonic wells by means of a pseudospin formalism. Within the Eliashberg formalism, pairing proper- ties containing contributions from these QLVs, distinct from those of standard phonons, are shown to produce a significant enhancement of the superconducting Tc. Plakida et al [3] modified these models further to under- stand the high temperature superconductivity in the copper based materials such as La(Y)BaCuO [4]. In this model, the structural instability, again described in the form of anhar- monic wells with two levels, occurs due to the rotational motions of the apical oxygen that are located within the layered perovskite structure. Within this theory, highly anhar- monic motions by the apical oxygens lead to the enhance- ment of Tc. These motions are associated with an amplitude of displacement d that is much greater than the mean-squared displacement ⟨u2⟩ of the ions in the lattice. Given that λ, the electron–phonon coupling, scales as the ionic motions squared, along with the hierarchy of scales d2/⟨u2⟩ ≫ 1, one can justify a significant rise of Tc caused by the soft, local- ized unstable vibrations. Such an enhancement is also reflec- ted in an effectively stronger electron–phonon coupling with respect to that of the harmonic limit λharm, that is: λ/λharm ∼ d2/⟨u2⟩ ≫ 1. Furthermore, the enhancement may also occur via polaron formation, leading to bi-polaronic theories of high- Tc superconductivity [5]. While this enhanced electron–phonon coupling caused by localized soft vibrational modes of the oxygen atoms is irre- futable (and was shown early on in Raman experiments by Müller, Liarokapis, Kaldis and co-workers [6–8]), it does not fully explain the interesting phenomenology of cuprate superconductivity in its entirety. These include d-wave sym- metry of the paired wavefunction, a non-Fermi liquid normal phase and the effects of magnetic correlations. Additionally, these early models do not provide a systematic relationship between Tc and lattice anharmonicity, since they focus on two-level type excitations modeled as pseudospin excitations, thus leaving out all the usual descriptors of lattice anhar- monicity (i.e. phonon linewidth, Grüneisen parameter, etc). Finally, these models, while they predict an enhancement of Tc with anharmonic motions, they are unable to predict other regimes where, instead, the anharmonicity is detrimental for the superconductivity and thus causes a reduction of Tc. More recently, anharmonic extensions of conventional superconductivity theory have been proposed in the context of rattling modes in caged thermoelectric-type materials, such as filled skutterudites, β-pyrochlore oxides and clathrates [9]. A common feature of this material group is the existence of nano-size cages of light atoms. The ion enclosed in the cage, frequently called the guest ion, experiences a highly anhar- monic potential and it can perform large amplitude oscilla- tions, referred to as ‘rattling’. In the context of superconductivity in this class of materi- als (a known example is superconductivity in the β-pyrochlore oxides which appears to be enhanced by anharmonicity [10]), Oshiba and Hotta [9] developed a theory of superconductivity using the Holstein ‘small polaron’ model as a the starting point to treat electron–phonon contributions to the Hamiltonian where large screening effects lead to small polaron radius and large electron–phonon coupling constant. The next step was to apply Migdal–Eliashberg theory to evaluate the Tc as a func- tion of the quartic and sixth-order anharmonic coefficients in the lattice Hamiltonian. Themodel predicts an enhancement of Tc with increasing anharmonicity followed by a peak or max- imum and then a declining regime—a superconducting dome. The question of how anharmonicity affects superconduct- ivity at a more fundamental level has however remained unex- plored until recently. In [11] (see also [12]) the effect of anhar- monicity of phonons (both acoustic and optical) has been described at the level of BCS theory. For optical phonons, a non-monotonic dependence of Tc on the parameter which characterizes the lattice anharmonicity, i.e. the phonon damp- ing or linewidth, was predicted, with a dome in Tc. The enhancement can be explained thanks to dissipation acting to connect bosons at different energy scales that combine coherently to increase the effective electron–phonon coupling and Tc. Mathematically, the wave-vector dependence in the propagator is integrated out in the gap equation, and the integ- ration combines high and low energy phonons coherently to enhance the effective electron–phonon coupling and hence the Tc. Such a mechanism was previously discussed in the context of proton irradiated samples of La2−xBaxCuO4 [13]. Some experimental evidence of the enhancement regime came in the study of filled skutterudite LaRu4As12, by using electron irradiation to tune the phonon anharmonicity [14]. In practice, electron irradiation was used to suppress certain anharmonic phonon modes by creation of suitable defects. Upon suppressing the anharmonic phonon modes, the Tc was observed to decreasemonotonically in the investigated regime. The aim of this review is to further explore, on the basis of the available literature, the effects of lattice anharmon- icity, especially anharmonic damping, on superconductivity. The emerging picture is that phonon anharmonicity [15] can be viewed as a powerful means to determine significant vari- ations in Tc, including both enhancement and suppression. This becomes an issue of vital important in all the high-Tc 2 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review superconducting materials, from the cuprates (for the reasons explained above) to the high-pressure hydrides [16], where light hydrogen atoms perform huge anharmonic zero-point motions, and where anharmonicity is key to both determine phase stability and superconductivity. In these materials, and more in general in anharmonic crystals [17], clear guidelines or understanding about these effects are still lacking. It is hoped that theoretical concepts as embodied in an ‘anharmonic theory of superconductivity’ will provide a deeper under- standing of electron–phonon superconductivity in materials with non-trivial lattice effects, such as strong anharmonicity, phonon softening and lattice instabilities. 1.2. Boson damping mechanisms Different microscopic origins of damping for the bosons involved in the Cooper pairing of electrons can play a role in the superconductivity mechanisms. These include glassy damping due to disorder, damping due to electron–phonon interaction, damping due to phonon–phonon interactions, just to name the most important ones. In real materials with com- plex chemistry, the interatomic potential is very far from being harmonic, such that the intrinsically large anharmonicity of the lattice dynamics leads to strong damping of phonons due to phonon–phonon processes. The effect of anharmonicity on the phonon dispersion curves is twofold. On one hand, the bare phonon frequency gets strongly renormalized (typically, lowered), while on the other hand the lifetime of the phonon also gets reduced (damping). Anharmonicity arises from the non-harmonic character of the interatomic interactions, although it can also arise from the electron–phonon interaction itself. Here, we shall focus on the lattice anharmonicity (neglecting the contributions from electron–phonon interaction), and on the damping effect (i.e. we take the phonon frequencies in the effective theories as already renormalized). 1.2.1. Akhiezer damping of acoustic phonons. At finite tem- perature, the main mechanism for damping of acoustic phon- ons with long wavelengths is provided by the Akhiezer mech- anism, whose form can be derived directly from hydrodynam- ics (i.e. from viscoelasticity) [18]. From the point of view of elastodynamics, the viscous contribution to the stress, σ ′ ij (which is dissipative, and therefore odd under time reversal) is added to the elastic component of the stress σelij to make the overall total stress σij. In the context of linear viscoelasti- city theory [19], this is tantamount to assume the so-called Kelvin–Voigt model, which is the most suitable to describe viscoelastic solids (in contrast to the Maxwell model, which better applies to viscoelastic fluids). The corresponding elast- odynamical equations reduce then to [18, 20, 21]: ρ üi =∇jσij+ f exti (r) , σij = σel ij +σ ′ ij , (1) where Latin indices are used to denote spatial components. The above equation expresses that the internal stress force ∇kσik plus the external force density f exti (r) is equal to the acceleration of the elastic displacement field ui times the mass density ρ of the medium (Newton’s law). At a linearized level, i.e. for small deformations, the elastic contribution is given as usual by σel ij = CijklΥkl, where Cijkl is the elastic tensor and Υkl ≡ 1/2(∇i uj+∇jui ) the linear strain tensor. In all real solids (crystals with or without defects, glasses), there is a dissipative component σ ′ ij to the stress tensor due to the vis- cous component of the material response, which is propor- tional to the deformation rate. This contribution is ultimately due to anharmonicity of the lattice, and to finite temperature effects. Its structure is given by σ ′ ij = ηijkl∂tΥkl [18], and can be derived by symmetry arguments (hydrodynamics), or using the Rayleigh dissipation function [20, 21]. Here, ηijkl repres- ents the viscosity tensor. For isotropic systems, the elastic and viscosity tensors can be written solely in terms of four para- meters: the shear modulus G, the bulk modulus K, the shear viscosity η and the bulk viscosity ζ (see [18] for details). In the following, wewill restrict ourselves to this situation andwe will also neglect possible odd responses [22]. After standard manipulations, the elastodynamic equations with the viscous dissipative contribution can be written in the form of a forced damped harmonic oscillator. The transverse Green’s function in the mixed (k, t) representation, GT(k, t− t ′), is readily found by replacing the external force with a δ-function source:[ ∂2t +(η/ρ) k2 ∂t+(G/ρ) k2 ] GT (k, t− t ′) = δ (t− t ′) (2) and upon Fourier-transforming in time we get: GT (k,ω) = 1 −ω2 +(G/ρ) k2 − iω (η/ρ) k2 (3) and,mutatis mutandis, an analogous expression for the longit- udinal Green’s function. In general, we thus have the Green’s functions for longit- udinal (L) and transverse (T) modes in the following generic form: GL,T (k,ω) = 1 Ω2 L,T (k)−ω2 − iωΓL,T (k) , (4) with the poles providing the following set of dispersion rela- tions for transverse and longitudinal phonons, respectively: ωL,T = vL,T k− iDL,T k 2 , (5) v2T = G ρ , v2L = K + 2(d−1) d G ρ , (6) DT = η 2ρ , DL = 1 2ρ [ ζ + 2(d− 1) d η ] . (7) These expressions are valid only for low frequency/ wavevector, but the higher order terms can be systematically derived using a perturbative scheme known as the gradi- ent expansion. In general, vL > vT since K,G> 0 for all materials. Using equations (4) and (5) we therefore identify ΩL,T(k) = vL,T k and ΓL,T (k) = 2DL,T k 2, (8) i.e. a diffusive viscous damping, known as Akhiezer sound damping [23]. The root cause of Akhiezer damping is 3 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review anharmonicity, as will be discussed in the following of this section. Importantly, this framework recovers the ubiquitously observed finite temperature Γ∼ k2 scaling of the acoustic phonon linewidth, which does not depend on the microscopic details of the system. As a matter of fact, Akhiezer damping perfectly reproduces the experimental data at low wavevector (see for example [24] for a demonstration in a-Si3N4 and a- SiO2 using the data of [25–29]). The above derivation follows a hydrodynamic approach [20] which is agnostic of the short-scale physics; by compar- ing with the result of a microscopic approach based on the Boltzmann transport equation for phonons, it has been shown that [30] DL = CvTτU 2ρ ( 4 3 ⟨γ2xy⟩− ⟨γxy⟩2 ) ≈ CvTτU 2ρ ⟨γ2xy⟩ , (9) where the last approximation for acoustic modes can bemotiv- ated with the typical wild fluctuations of γ for low frequency vibrational excitations in both crystals [31] and metal alloys [32], hence ⟨γxy⟩ ≈ 0. Here, we neglected the contribution from the bulk vis- cosity ζ, since normally η≫ ζ. Furthermore, ⟨. . .⟩ indicates averaging with respect to the Bose–Einstein distribution as a weight, while γxy is the xy component of the tensor of Grüneisen constants. Also, Cv is the specific heat at con- stant volume, while τU is the average time interval between two Umklapp scattering events. Since τU ∼ T−1 (which is an experimental observation for most solids [30, 33]), the dif- fusive constant DL, and also the sound damping, are weakly dependent on temperature, i.e. a well-known experimental fact [33], in the Akhiezer regime. A substantially equivalent expression for the damping of longitudinal phonons, in terms of an average Grüneisen con- stant of the material γav, was proposed by Bömmel and Dransfeld [33], DL ≈ CvTτU 2ρ γ2av, (10) and provides a good description of the Akhiezer damping measured experimentally in quartz at T > 60K [33]. In turn, the Grüneisen constant γ, or at least the leading term [34] of γav or γxy above, can be directly related to the anharmonicity of the interatomic potential. For perfect crys- tals with pairwise nearest-neighbor interactions, the following relation holds [34] γ =−1 6 V ′ ′ ′ (a)a2 + 2 [V ′ ′ (a)a−V ′ (a)] V ′ ′ (a)a+ 2V ′ (a) , (11) where a is the equilibrium lattice spacing between nearest- neighbors, and V ′ ′ ′(a) denotes the third derivative of the interatomic potential V(r) evaluated at r= a. Hence, the phonon damping coefficient DL can be directly related to the anharmonicity of the interatomic potential via the Grüneisen coefficient and equation (11). 1.2.2. Klemens damping of optical phonons. The discussion in this subsection closely follows [35]. On general grounds, the lifetime of a optical phonon can be rationalized by looking at its microscopic decay processes which are ultimately related to anharmonic (phonon–phonon) interactions. As proved in the pioneeringwork byKlemens [36], the leading decay channel is the three-phonon scattering between two acoustic modes and an optical one, which can be described using Boltzmann form- alism and many-body perturbation theory. Above room tem- perature, higher order processes become relevant as well and cannot be neglected anymore [37]. Despite the various approx- imations, Klemens result is in good agreement with controlled experimental observations [38]. The damping of an optical phonon can be described via the analytical model derived by Klemens. The starting point was a master kinetic equation of the Boltzmann type for the phonon population, combined with many-body theory up to third order. This assumption means that only three-phonon processes are accounted for in describing the phonon decay. At high temperatures clearly also higher order terms may play a role [37] but it has been shown experimentally that the Klemens result is a reasonable description of the optical phonon damping in comparison with experimental data in many situations. The mean lifetime of the optical phonon is deduced within Klemens’ theory as follows: 1 τ = ω J 24π γ2G h̄ω Mv2 a3ω3 v3 C(α,β) [ 1+ 2 ex− 1 ] (12) where C(α,β) = 2√ 3 α−β α+β ; x= h̄ω 2kBT . (13) In the above formulae, ω refers to the phonon frequency in the undamped limit, and this applies to either longitudinal (LO) or transverse (TO) optical phonons. Furthermore, a3 is the volume per atom, M is the ion mass, and v is the acoustic phonon velocity within Debye approximation (since the decay process of the optical phonon may involve acoustic phon- ons). Also, J is the label of the allowed channels by which the optical mode decays into acoustic phonons, and γG is the Grüneisen parameter of the solid related to anharmonicity of the interatomic potential. Finally,C is a coefficient on the order of 0.1, which in the case of ionic crystals (e.g. alkali halides) depends on the spring constants of the two different ion spe- cies. This is because Klemens’ original derivation focused on ionic crystals. All these prefactors which appear in the Klemens formula equation (12) can be put into a single coefficient ξ, 1 τ = ω5ξ [ 1+ 2 ex− 1 ] , (14) where ξ ≡ J 24π γ2 h̄ Mv2 a3 v3 C(α,β) , (15) 4 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review and ω the frequency of the optical phonon. According to Klemens [36], the above expression equation (14) can be fur- ther simplified. Using the Debye model, and approximating the optical phonon frequency with the Debye frequency ωD, one gets 1 τ ∝ ω2 D. (16) Here, in good approximation, τ can be regarded as independ- ent of the wavevector k. In the next sections, we will simply assume that the optical mode is not too dispersing and its frequency ωopt can be approximated, at least in the limit of small wave-vector, by a constant. All in all, we will indicate as Klemens damping the assumption that τ−1 ∼ ω5 opt, where ωopt ≡ Reωopt(k= 0), and ωopt(k) is the dispersion relation of the optical mode. The main difference with the Akhiezer damping for acoustic phonons is that the Klemens damping is, at least in first approximation, independent of the wave-vector k, under the approximation detailed above. 1.3. The nature of the bosonic mediator The starting point of the phenomenological theoretical frame- work is the definition of the mediator φ responsible for the pairing and for the superconducting instability. For simpli- city, we will focus on phononic mediators. Despite most of the treatmentwill identify the latter with acoustic/optical phonons, we will keep the derivation as general as possible to account for alternative bosonic mediators such as spin waves/magnons. The fundamental object under scrutiny is the retarded Green’s function for the mediator φ which in Fourier space takes the general form Gφ (ω,k) = 1 −ω2 +Ω2 (k)− iωΓ(k) (17) where ω,k are respectively frequency and wave-vector. Under few assumptions, this is the most general expression for the Green’s function and Ω,Γ are, at this point, undetermined quantities which can be expanded in the low-energy limit, sometimes referred to as the hydrodynamic limit or gradient expansion, in a systematic power-series expansion in terms of k. Examples of this sort can be found in [39] for relativistic flu- ids, and in [40] for phases of matter with broken translations. Importantly, this expansion is in general not convergent [41]. To continue, isotropy is assumed, where k≡ |⃗k|. The exten- sion to anisotropic systems does not present any conceptual difficulties but it is rather cumbersome and therefore not expli- citly shown. The poles of the retarded Green’s function in equation (17) define the dispersion relation ω(k) of the corres- ponding excitations which is given by solving the following equation: −ω2 +Ω2 (k)− iωΓ(k) = 0 . (18) Here, we take the frequency ω to be complex and the wave- vector k to be real. The Green’s function presented in equation (17) corres- ponds to the following spectral function s(ω,k) s(ω,k)≡− 1 π ImGφ (ω+ iδ,k) = ωΓ(k) π [ (ω2 −Ω2 (k))2 +ω2Γ2 (k) ] , (19) which is directly accessible via scattering experiments and shows the typical Lorentzian shape. In the following, we will consider two fundamentally dif- ferent types of excitation. First, we discuss the scenario in which the mediator corresponds to a gapless mode whose dis- persion relation at low-energy is given by Ω(k) = vk+ . . . , Γ(k) = Dk2 + . . . (20) where the . . . indicate higher-order corrections in the wave- vector k. Here, v defines the propagation speed while Γ = Dk2 the diffusive sound attenuation. By abusing the lan- guage, we will refer to D as the diffusion constant. Intuitively, equation (20) can be identified as the low-energy solution of a dynamical equation of the type: ∂2ϕ ∂t2 + v2 ∂ϕ2 ∂x2 +D ∂ ∂t ∂ϕ2 ∂x2 = 0 (21) with ϕ(t,x) = e−iωt+i kxϕ0, where using isotropy we have assumed the spatial dependence to be only along the x direc- tion. Equation (21) must be taken with a grain of salt since low-energy sound modes usually appear in the context of hydrodynamics from a more complicated dynamics in terms of coupled fluctuations, e.g. particle number, momentum, energy, etc and not from the dynamics of a single low-energy variable. The collective variable ϕ is the dynamical field correspond- ing to the mediator φ. The most notable example obeying a dispersion as in equation (20) is that of acoustic phonons. In this concrete case, equation (21) corresponds to the dynamical equation obtained from viscoelasticity theory where ϕ is iden- tified with the infinitesimal displacement field [18]. For trans- verse (T) and longitudinal (L) acoustic phonons, one obtains (cfr section 1.2.1 above for the derivation): v2T = G ρ , v2L = K + 2(d−1) d G ρ , (22) DT = η ρ , DL = 1 ρ [ ζ + 2(d− 1) d η ] , (23) where G,K,η,ζ,ρ are respectively the static shear modulus, the static bulk modulus, the shear viscosity, the bulk viscosity and the mass density of the system [18]. We have defined with d the number of spatial dimensions and neglected the effects from thermal expansion. For the moment, we will be agnostic about the microscopic origin of the damping term Γ and we will take v,D as pure phenomenological parameters. Because of stability requirements, we have v2,D> 0. 5 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 1. The spectral function for an underdamped acoustic mode using equation (19) and the parameters defined in equation (20). The dashed line shows the real part of the dispersion relation ω = vk up to the Ioffe–Regel scale k⋆. The speed of sound is taken to unity v= 1 while the diffusion constant D= 0.03,1 (left, right). In this first situation, a fundamental scale in the problem is given by the so-called Ioffe–Regel (IR) wave-vector k⋆ [42], defined as the root of: Ω(k⋆) = πΓ(k⋆) . (24) The IR scale qualitatively indicates the energy at which the acoustic mediator loses its well-defined propagating nature and turns into a diffusive quasi-localized mode. Physically, the larger the attenuation constant∼D, the lower the energy at which the coherent nature of the mediator is lost. This is evid- ent in figure 1 in which the spectral function of the bosonic mediator is shown for two very different values of D. A second relevant scenario is defined by the following alternative choice Ω(k) = ω0 +αk2 + . . . , Γ(k) = Γ0 + . . . (25) where ω0 represents the energy gap (the ‘mass’ in particle physics jargon) and Γ0 the wave-vector independent scatter- ing rate. The parameter α takes into account the eventual mild k dependence in the dispersion relation of the mode. Finite values for ω0,Γ0 are prohibited for acoustic phonons because of their Goldstone mode nature but they can naturally appear once one considers optical modes which are not protected by any fundamental symmetry breaking pattern. While the sign of Γ0 is fixed by stability arguments to be positive, the one of α is a priori undetermined and strongly dependent on the microscopics of the system. A simplified possibility is to neglect the k dependence in equation (25), and consider a simpler dispersion relation: ω2 = ω2 0 − i ωΓ0 . (26) Once more, depending whether Re(ω)> Im(ω) or vice versa, the dynamics will result underdamped or overdamped. The transition roughly happens when ω0 ∼ Γ0. Therefore, we find convenient to define a dimensionless parameter: Γ̃≡ Γ0 ω0 , (27) such that in the regime Γ̃≪ 1 we have a coherent well-defined bosonic quasiparticle mediating the pairing, while for Γ̃≫ 1 the mediator becomes incoherent and does not correspond anymore to a well-defined quasiparticle. Notice that in general, effective parameters such as ω0,Γ0 are implicit functions of thermodynamic variables (temperat- ure, doping, etc). In order to reveal their explicit dependence a microscopic theory is needed. Nevertheless, we will see that in some scenarios (for example the soft mode instability mech- anism described in section 4.5) one can simply introduce a convenient parameterization and obtain interesting physical results. Later in the paper, we will use the results of this section to examine the manner in which superconducting proper- ties, like critical temperature, are affected by the damping and other parameters appearing in the dispersion relations, equations (20)–(25). 2. Damped bosons: experimental probes In this section we review various experimental probes used to quantify anharmonic damping in the pairing mediators. This can be achieved either by fitting the spectral line shape of the boson or by directly measuring their correlation functions. Typically, these observables are measured as a function of an external control parameter like pressure, carrier concentration, impurities, temperature etc. We broadly classify the probes according to their coupling to the charge, as in lattice based mediators like phonons, or to the spin, as in spin based medi- ators such as spin fluctuations. The list of experiments dis- cussed in each category below is by no means comprehensive but rather a collection of representative examples that allows the interested reader to further explore each topic. We begin with probes of anharmonic damping in phonons. 2.1. Raman scattering Raman scattering is the most widely used technique to meas- ure phonon properties (frequency shifts and linewidths) in quantum materials [43, 44]. The method involves measuring the frequency shift of visible light scattered from a sample at zero momentum. The anharmonic damping is then extracted by fitting the line shape with well known anharmonic models that contribute to the linewidth [17, 43, 45, 46]. Additionally, the polarization of light can be varied to access different symmetry channels of the crystal point group of the specific materials. In principle, several scattering processes contrib- ute to the phonon linewidth. These include electron–phonon, multi-phonon, impurity scattering, lattice dislocations etc and care must be taken to extract the purely anharmonic com- ponent. The classic BCS superconductor MgB2 serves as an illustrative example of extracting phonon anharmonicity using Raman scattering (see figure 2). Here, the E2g phonon mode, centered around ∼620 cm−1, has a large anomalous broad- ening of ~200–280 cm−1 [47–51]. The experiments were performed on clean samples and the electron–phonon coup- ling was calculated to account for only about 50 cm−1. The 6 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 2. Raman scattering spectra showing the broad E2g mode at ~620 cm−1 in MgB2 attributed to anharmonic phonon decay. Reprinted (figure) with permission from [47], Copyright (2002) by the American Physical Society. remaining scattering was attributed to multi-phonon decay from anharmonic effects (see [48] and references therein). More recently, similar broadening effects and anharmonic frequency shifts were also noticed in high pressure super- conductors such as hydrides [52–54] and TlInTe2 [55], and non-stoichiometric Fe based chalcogenide superconductors KyFe2−x(Se,S)2 [56, 57]. We will further discuss the relation- ship between superconductivity and anharmonicity for the spe- cific case of TlInTe2 later in this review. 2.2. Inelastic x-ray scattering Another widely used tool to extract anharmonic damping effects in phonons involves inelastic x-ray scattering (IXS) (see [58–60] for a review on phonon spectroscopy using IXS). The IXS technique is also a photon-in/photon-out process like Raman scattering, but is performed at higher (x-ray) energies and is momentum resolved. Here an incoming photon with a given energy and momentum scatters off a phonon in the sample to create an outgoing photon with a different energy and momentum. The frequency shift and linewidths at differ- ent momentum transfers are then fit to theoretical models to extract anharmonic damping effects. Like Raman scattering, polarization of the incoming photons can be manipulated to access different symmetry channels of the solid. In this regard, IXS has become an invaluable probe for mapping out phonon dispersion relations and extracting momentum and symmetry dependent spectral lineshapes [58–60]. IXS spectra for vari- ous metals and superconductors have been summarized in [58–60]. Returning to the prototypical example of MgB2, IXS spectra and phonon dispersions and linewidths were studied in [61–63]. Momentum resolved data suggested that the broad linewidth of the E2g phonon mode was dominated by the Γ-A direction of the Brillouin zone (see figure 3). The contribution to the linewidth from phonon anharmonicity was also found to Figure 3. IXS spectra in MgB2. (Top) Symmetry decomposed data with the E2g mode centered around 60 meV taken at a momentum point 0.6 Γ−A. (Bottom) Momentum resolved width of the E2g mode enhanced in the Γ−A direction. Reprinted (figure) with permission from [61], Copyright (2003) by the American Physical Society. be smaller than the electron–phonon broadening in apparent contradiction of earlier Raman scattering studies [47]. This disagreement between IXS and Raman scattering was later addressed in [63]. More recently, it was demonstrated that the E2g mode is strongly coupled to electrons and higher-order electron–phonon scatterings become relevant leading to large effective phonon–phonon dampings at zero momentum [64, 65]. As a result, anharmonic linewidths are sometimes dif- ficult to separate from the usual electron–phonon contribu- tion, since phonon–phonon scatterings mediated by higher- order electron–phonon scatterings have similar temperature dependence [64, 65]. Finally, phonon linewidths from IXS with momentum resolution have also been measured in other superconducting systems such as cuprates [66–69], iron super- conductors [70], soft phonon systems like CaAlSi [71] as well as the newly discovered kagome superconductors [72]. 2.3. Electron energy loss spectroscopy (EELS) Our focus so far was on purely photonic probes. We now turn to electronic scattering methods to probe phonon damp- ing on material surfaces and thin films. EELS is a popu- lar technique in this category where electrons with particu- lar energy (and momentum, as is the case in the momentum resolved counterpart M-EELS [73]) are shot into the sample to determine the nature of surface phonons [74]. The energy (and momentum) transfer to the sample is then determined 7 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review from the kinematics of the scattered electron. In superconduct- ors, EELS has been used predominantly to extract medium to high energy (~30 meV–1 eV) electronic properties and response functions. In principle, information of phononic lineshapes can also be obtained depending on the energy res- olution of the device; although, to our knowledge, a system- atic study that isolates the effects of mode specific anhar- monic phonon damping on the lineshape is currently missing. This is due to the fact that coupling of electronic probes to specific phonon symmetry modes is not straightforward with high resolution unlike photon based probes. Early EELS data in cuprates [75] studied and modeled surface optical phonon line shapes. Recently, M-EELS measurements in the normal state of the Cuprates [76] were analyzed [77, 78] to isol- ate the phononic and electronic components, and determine their independent and combined effects on correlation prop- erties of the strange metal. In MgB2 [79], several phonon excitation energies were associated to features obtained in the EELS spectra. In strontium ruthenate [80], the bulk and surface phonon lineshapes, and their coupling to quasi-1D electronic bands was explored. However, neither phonon broadening due to anharmonic damping nor its relationship to superconduct- ivity were systematically studied in these works. A comprehensive study of phonon broadening due to anhar- monic decay in single unit cell FeSe films on strontium titan- ate was examined by the authors of [81]. Properties of spe- cific phonon frequency branches (α and β modes) such as the energy and full width at half maximum (FWHM) as a func- tion of temperature was modeled. The anharmonic contribu- tion to the FWHM from multi-phonon decay processes was obtained by subtracting the T = 0 (electron–phonon) contribu- tion. Figure 4 shows the plots of the total FWHM (top panel) and the extracted anharmonic component (bottom panel). 2.4. Probes of Grüneisen constant The importance of the Grüneisen constant or Grüneisen para- meter as a quantitative estimate of the extent of anharmon- icity in superconductors has been recently emphasized. Since the Grüneisen constant essentially describes how the acoustic phonon frequencies change with volume, it can be measured by mechanical ways by linking to the nonlinear elastic beha- vior of the solid. Gilvarry [82–84] was able to connect the Murnaghan equation of state of nonlinear elasticity (linking changes in pressure to changes in volume) to the traditional Grüneisen assumption that the normal mode frequencies ωi of the lattice model for particles exhibiting anharmonic interactions should have the volume dependence, γi = ∂ lnωi ∂V (28) where V is the material volume so that γi describes how the normal mode frequencies change with material volume, regardless of the detailed molecular origin of the volume change. For reference, the Grüneisen exponent γi for an ideal harmonic (Debye) lattice material equals 1/3, i.e. for all of the Figure 4. (Top) Fits of the full width at half maximum (FWHM) for different anharmonic models with multi-phonon decay channels in single unit cell FeSe on strontium titanate substrate. (Bottom) Extracted phonon–phonon contribution to the FWHM by subtracting the T = 0 electron–phonon component in the two optical phonon anharmonic model. Reprinted (figure) with permission from [81], Copyright (2016) by the American Physical Society. normal modes. More generally, the Grüneisen parameter γG, represents an average over the normal modes of the material, so that γG normally differs from 1/3 in materials having more realistic intermolecular interactions. Gilvarry’s anharmonic extension of the Debye model assuming equation (28) and a constancy of the Poisson ratio, leads exactly to Murnaghan’s equation of state where the scaling exponent γM equals, γM = 2γG + 1 3 , (29) which provides a link between the microscopic atomic dynam- ics and macroscopic elasticity. A power-law scaling of the normal mode frequencies with V was originally motivated by schematic choices of anharmonic interparticle potentials (Mie or Lennard–Jones) where the repulsive and attractive contributions to this potential have variable power exponents. This was already considered by Grüneisen [85] and many others [82–84]. In general, the Grüneisen parameter γG can 8 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review be specialized to particular normal modes [31] or mode types (longitudinal or transverse) [86]. Often, however, an appro- priately defined average over all the modes of the system is assumed. This approximation seems to be particularly suit- able for glass-forming liquids and amorphous solids where the existence of well-defined normal modes of the type found in crystals is not so well defined. We then have a semi-empirical equation of state generaliz- ing the Debye theory in which there is an explicit link between the microscopic measure of anharmonicity γG and the macro- scopic measure derived empirically from nonlinear elasticity. The importance of the Murnaghan equation in understanding the temperature dependence of relaxation in condensed mater- ials has become appreciated [87]. From a thermodynamic per- spective, γG describes the rate of change of the pressure as the internal energy varies at a constant volume and this interpret- ation leads to an explicit expression in terms of the specific heat CV, the thermal expansion coefficient and the isothermal compressibility [82–84, 88], There are extensive tabulations of γG measured experi- mentally and the many properties to which it is interrelated [89–92]. The application of γG in materials science has been discussed recently in [93, 94]. Recently, there have been significant advances in the first principle computation of the average γG and the Grüneisen exponent for particular modes [31, 32]. 2.5. Inelastic neutron scattering (INS) Neutrons scattering off lattice vibrations forms another com- plementary probe of phonon dispersions and linewidths (see [95, 96] for early conceptual work). Unlike the previous probes, neutrons do not couple through the charge due to their charge neutrality. Rather the coupling to the lattice occurs through atomic displacements via interactions with the nuc- lei. These interactions are typically modeled with a short range ‘hard core’ isotropic potentials [97]. Like IXS and EELS, neut- ron scattering is capable of extracting mode and momentum resolved phonon dispersions and linewidths. Earlier INS work in Nb3Sn by Axe and Shirane [98] found abrupt changes in the lifetimes of certain transverse acoustic phonon modes near the superconducting transition temperature. They further dis- cussed certain empirical relationships between superconduct- ivity and damping induced by anharmonicity and electron– phonon coupling. In liquid helium, [99] used INS to extract phonon linewidths at various temperatures and wave vectors, and classified the total INS structure factor and dampings according to one-phonon and multi-phonon contributions. In the cuprates, early INS studies [100, 101] laid out the role of anharmonic phonon damping and electron–phonon coupling to the linewidths of various phonon modes as well as their relationship to superconductivity. In the iron based supercon- ductor CaFe2As2, large phonon linewidths were found [102] from INS opening up a possible role for anharmonicity in the pnictides. Yamaura et al [103] assigned excitation of two distinct phonon modes to different types of extremely anhar- monic phonons arising from ‘quantum rattling’ in deuterium doped LaFeAsO. In the YNi2B2C superconductor, momentum resolved INS phonon linewidths were studied in [104, 105]. These authors concluded that the scattering rate was dom- inated by electron–phonon coupling rather than anharmon- icity. Finally, in MgB2, first-principles calculations of lattice dynamics were performed and found to be in agreement with INS data [106]. In this work, a giant anharmonicity of the E2g in-plane boron phonons and nonlinear electron–phonon coup- ling was found to be important for understanding supercon- ductivity. Anomalous behavior of the phonon density of states (DOS) due to multi-phonon processes in MgB2 was further explored in [107]. 2.6. Atomic scattering Scattering of inert gas atoms such as Helium over crystal surfaces is another tool to probe properties of surface phon- ons [108–110]. The technique involves inert gas atoms with an initial momentum and energy incident on a crystal surface, and interacting with lattice vibrations via a generic two-body atomic potential. The energy andmomentum transferred to the phonons is measured from the inelastically scattered atoms; the kinematics and decay of the vibrations can then be mapped out using this information. Phonon dispersions obtained from surface scattering using He atoms for most part agree with EELS measurements, and the two techniques complement each other in covering much of the surface phonon vibrational spectra [110]. In superconductors, Helium atom scattering has been used over the last several years to extract the electron– phonon coupling constant [111–114]. Sklyadneva et al [111], for example, used inelastic Helium atom scattering to meas- ure electron–phonon coupling strengths for each phononmode in superconducting Pb films. The mode/momentum specificity of the couplings has yet to be properly exploited to study other superconducting families. Obtaining anharmonic effects including anharmonic phonon damping in superconductors using atomic scattering has been less explored. One rare example [115] is the case of metallic Aluminumwhere Helium atom scattering was used to obtain surface-phonon anharmon- icity and linewidths on the Al(100) and Al(111) surfaces. The results were shown to be in good agreement with molecular dynamics simulations over a range of wave vectors and tem- peratures. Over the last decade, due to the utility of atomic scattering to study properties surface phonons, the coupling between Dirac fermions and phonons on the surface of the strong topological insulators has also been studied in a mode- specific manner [116, 117]. We anticipate applications of this technique to measure low-lying ‘topological phonon’ [118, 119] surface modes and their potential relationship to super- conductivity in the near future [120]. 2.7. Point contact spectroscopy (PCS) Over the last several decades, the capability of designing nano- meter size orifices at the junction of metals and superconduct- ors has enabled a new spectroscopic tool to probe electronic properties. Termed as PCS, such an experimental geometry has been successful in quantifying various electronic relaxa- tion mechanisms in metals, superconductors, heavy fermion 9 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review systems (see [121–123] for detailed reviews) and non-fermi liquids [124]. Relevant to our discussion is the role played by PCS in extracting energy resolved electron–phonon inter- actions in metals and superconductors. For normal metals, the basic geometry consists of a nanometer(s)-thick dielec- tric layer that separates two metallic films. The dielectric layer contains a small constriction with a diameter of the order of the scattering length of the electron injected into it. The resistance of the ‘point’ contact is given approximately by the interpola- tion formula R≃ Rsh ( 1+ 3πd 16vfτe−ph ) , (30) τ−1 e−ph = 2π h̄ ˆ eV 0 dω α2 (ω)F(ω) . (31) Here Rsh = 16ρrl 3π d2 is the Sharvin resistance, ρr is the resistivity, l (τ−1 e−ph) is the electron–phonon scattering length (rate), vf the Fermi velocity, d the diameter of the constriction and eV the bias voltage. The information of electron–phonon coupling is contained in α(ω) and the phonon DOS is given by F(ω). The equation (31) is an interpolation formula for the contact res- istance between the clean (l> d; dominated by Rsh) and dirty (l< d; dominated by Maxwell resistance RM ≡ ρr/d) limits. The derivative of the bias dependent contact resistance is pro- portional to the second derivative of the voltage with respect to the current and is given by dR dV ∝ d2V dI2 = Rsh 3π2ed 8h̄vf α2F(eV) . (32) The key quantity on the right hand side is the Eliashberg func- tion α2F(eV)which is the convolution of the phonon DOS and the electron phonon coupling function. Thus from the deriv- ative of the contact resistance, information about the phonon linewidths can be obtained, and the method has been applied to a wide variety of metals to extract the Eliashberg func- tion [121]. A formal theory justifying equation (32) appears in [125, 126]. Returning to our prototypical case of MgB2, a damped maximumwas found above 60 meV (width ~15meV) consistent with the E2g phonon modes [127–129] observed in other probes. To our knowledge, there is currently no sys- tematic PCS study that separates the linewidth contributions originating from anharmonic and electron–phonon interaction effects. An effort in this direction could greatly complement existing Raman and neutron scattering analyses discussed in previous subsections. 2.8. Spin based techniques Pairing in a superconductor can also occur through a ‘spin- fluctuation’ based bosonic mediator as opposed to phonons [130]. Detecting anharmonic damping in such bosons requires spin-based experimental probes where the coupling of the probe to the boson occurs through the spin quantum number. Broadly speaking, there are two categories of techniques that can access damping effects in spin based mediators: resonance Figure 5. La139 NMR linewidth at low temperatures in La1.88Sr0.12CuO4. Inset shows the line shape at two different temperatures. Reprinted (figure) with permission from [138], Copyright (2008) by the American Physical Society. and magnetization based probes. In resonance based probes— examples include nuclear magnetic resonance (NMR), nuc- lear quadrupole resonance (NQR) and muon spin resonance (µSR)—a nucleus or incident muon spin in the sample pre- cesses at its Larmor frequency (ωl) determined by a combin- ation of an applied and internal magnetic fields. The spin can then decay due to its coupling to the environment, in this case, the fluctuating spins that mediate superconductivity, through hyperfine interactions (for conceptional foundations of the technique, see [131]). Under certain circumstances, either an enhanced decay rate or broadening of the precession linewidth at low temperature can imply a freezing of the spins to due to damping or ‘glassiness’ in the spin fluctuations. This occurs when spin correlation time becomes long enough to be com- parable to ω−1 l and can be indicative of a spin-glass phase with no long-rangemagnetic order. Existing evidence of glassy spin mediators in the cuprate and iron based superconductors has been established through NMR/NQR [132–138], µSR [139] and neutron scattering [139, 140]. Figure 5 shows the broad- ening of the La139 NMR linewidth in La1.88Sr0.12CuO4 as seen in [138]. Magnetization and magnetic susceptibility are other indicators of freezing of spins and spin-glass physics. Conventional signatures include shift of the ac susceptibility cusp with frequency [141, 142], irreversible dc magnetiza- tion in the field-cooled and zero field-cooled states [141–146] and a direct measurement of the Edwards–Anderson spin- glass order parameter [143, 144]. These techniques have been extensively applied in the cuprate [143, 144] and iron based [141, 142, 145, 146] superconductors where spin fluc- tuations are thought to play an important role in the pairing mechanism. A more thorough exposition of the aforemen- tioned topics can be found in [147]. 3. Phonon damping from first principles While the focus of this review is to phenomenologically under- stand the role of anharmonic damping of the bosonicmediators 10 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review on superconductivity, in this section we briefly review exist- ing literature that calculate boson damping/linewidth from first-principles. Our objective here is to open up the pos- sibility of integrating bosonic damping into first principles calculations of superconducting properties. Such a scheme could involve incorporating ab-initio data for both the bosonic damping and dispersion relations of real materials into well established routines that evaluate quantities such as Eliashberg functions, coupling constants and critical temperature. Most of the focus so far has been on the calculation of anhar- monic damping in phonons. The original theories of phonon damping due to anharmonicity [17, 43, 46, 148, 149] con- sidered anharmonic interactions up to fourth order contribu- tions to the Hamiltonian. Each term in the expansion is asso- ciated with harmonic (second-order) and anharmonic (higher order) force constants. The shift of the phonon frequencies and linewidths were evaluated by a diagrammatic perturbation expansion of the self-energy. The broadening of the phonon line was specifically determined using second order perturb- ation theory of the third order anharmonic term in the expan- sion of the total energy. This approach was applied to Raman linewidths of Si, Ge and α-Sn [43] and it was argued that a combination of optical and acoustical phonons were the key decay channels that contributed to the linewidths. More recently, Green’s function based methods have been advanced to study the role of anharmonic damping and other quantum effects on the phonon DOS [150] and electron–phonon couplings [151]. The simplest incorporation of theoretically determined linewidths into first principles is to evaluate the harmonic and anharmonic force constants ab initio. For example, the matrix elements of the anharmonic tensor that contribute to the linewidths are third-order differentials of the total free energy for a single unit cell with respect to the phonon dis- placement amplitudes. These can then be obtained via dens- ity functional perturbation theory [152–156]. This approach was applied to optical phonons at the zone center in Ge, Si, and C where the temperature and pressure dependencies of the linewidth were calculated [153, 154] and shown to be in good agreement with experiments. Similarly, longitudinal and transverse linewidths of zinc-blende semiconductors such as AlAs, GaAs, InP and GaP were determined [155] and the temperature dependence of the damping was shown to be consistent with Raman data. More recently, density func- tional second order perturbation theory was applied to noble metals [157] Cu, Ag and Au, and has been used to understand thermal conductivity and phonon linewidths in the dichalco- genide MoS2 [158]. A similar approach was used to show that dynamical phonon anomalies (beyond the Born–Oppenheimer approximation) can considerably modify the electron–phonon coupling strength λ and transition temperature Tc in conven- tional superconductors [159]. First principles calculations to evaluate accurate interatomic forces was also applied to study the thermodynamics of crystals at finite temperature taking into account anharmonic effects. Termed as ‘self-consistent ab initio lattice dynamics (SCAILD)’ [160], the method was employed to understand the stability of several body-centered cubic metals whose lattice structure was unstable at low tem- perature but stabilized at higher temperatures. An alternate approach toward first principle computation of linewidths is a simplified version of the Car–Parinello scheme [161, 162]. The Car–Parinello method unifies density func- tion theory and molecular dynamics simulations to provide an accurate description of the inter-atomic forces, ground state and finite temperature properties (like energy shifts and linewidths) of quantum mechanical systems. However, the method is computationally intensive when applied to real materials. Wang et al [163–165] simplified the scheme by calculating free energies and inter-atomic forces by com- bining molecular dynamics with an empirical tight-binding method rather than density functional theory. This enabled an approximate but efficient way of analyzing lattice and electronic properties. The method was used to study tem- perature dependence of anharmonic frequency shifts and linewidths in Si and diamond [163–165], and was shown to be in good agreement with data. A molecular dynamics based approach was also implemented to understand the stability of body-centered cubic lattice phases of Li and Zr at high temperatures [166]. First-principles evaluation of anharmonic phonon proper- ties hasmade rapid progress in recent years since the discovery of hydride superconductors. In the hydrides, anharmonicity is known to be substantial and standard perturbative approaches fail [167]. Typically, variational approaches such as the self- consistent harmonic approximation (SCHA) [168–170] are employed in non-perturbative settings. In the last couple of years, the method has also been generalized to include time dependent effects to simulate nuclear dynamics [171, 172]. However, this approach is computationally intensive and a stochastic version of the SCHA (called the SSCHA) has been explored [167, 173–176] to determine anharmonic free energy, thermal transport and superconducting properties. In the case of palladium and platinum hydrides, it was shown that phononic spectra are strongly renormalized by anharmon- icity and harmonic approximations overestimate the super- conducting transition temperature [174, 175]. The SSCHA allows computation of anharmonic phonon linewidths arising from phonon–phonon interactions. Figure 6 shows the full width half maximum of palladium hydride phonon spectrum obtained from SSCHA [173]. Alternatively, one can adopt the self-consistent phonon theory (SCP) with anharmonic force constants, see e.g. [177, 178] for recent developments and the current state of the art. These techniques directly stem from the original work of Born and Hooton [168, 179]. A brief review of various first principle methods for treating anharmonicity and phonon lifetimes can be found in [177]. Despite these attempts, currently there are no systematic studies that take into account anharmonic damping effects to examine super- conducting properties from first principles, and efforts in this direction are much needed. 11 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 6. Phononic dispersions and linewidths for palladium hydride at 295 K. Red (blue) dashed line is the spectrum from harmonic approximation (SSCHA). The shaded yellow region is the calculated linewidth. Reprinted (figure) with permission from [173], Copyright (2015) by the American Physical Society. 4. Damped bosons: minimal theory of superconductivity In this section, we would like to understand how the super- conducting properties, and specially the critical temperature Tc, are affected by the low-energy parameters appearing in the dispersion relations equations (20)–(25). In order to make the analysis more concrete, in the following we will expli- citly identify the bosonic mediator with acoustic and optical phononic modes. 4.1. BCS theory: acoustic phonons The discussion in the next two subsections follows [11]. We first consider the situation in which the bosonic medi- ators are acoustic phonons that obey the simple dispersion relation in equation (20). Moreover, in this section, we will limit ourselves to standard BCS superconductors described BCS theory (see [180] for a comprehensive review). SWe start our discussion from the Green’s function expressed in equation (17), which allows us to re-write the phonon propag- ator as: Π(Ωn,k) = G (iΩn,k) = 1 v2k2 +Ω2 n+Dk2|Ωn| . (33) Here, Ωn = 2πnT correspond to the bosonic Matsubara fre- quencies where T is the temperature of the system and n an integer index which serves as label. The phonon damping, or linewidth, which appears in the last term in the denominator of the above expression must be positive, due to the analytical properties of the response functions involved in the quantum theory of dissipative systems, see [181, 182]. Figure 7. Superconductivity mediated by acoustic phonons. The critical temperature Tc as a function of the damping constant D for several values of the velocity of acoustic phonons v ∈ [0.8,1.8] (from black to yellow). Reprinted (figure) with permission from [12], Copyright (2022) by the American Physical Society. Common algebraic manipulations [180, 183] yield to the superconducting gap equation, ∆(iωn,k) = g2 β V ∑ q,ωm ∆(iωm,k+ q) Π(k, iωn− iωm) ω2 m+ ξ2k+q+∆(iωm,k+ q)2 , (34) where g is the coupling that quantifies the attractive pairing interaction, V the volume and β ≡ 1/T. Moreover, ξk ≡ k2 −µ is the free electron dispersion in presence of a chemical poten- tial µ. For simplicity, we set the electron mass to 2me = 1 and work in reduced units. Finally, we assume the gap∆(iωn,k) = ∆ to be independent of the frequency and the wave-vector. Then, we replace the sum over the wave-vector k with a con- tinuous integration over the energy ξ, 1 V ∑ q → 1 (2π)d ˆ ddq→ ˆ N(ξ)dξ , (35) where N(ξ) is the DOS at energy ξ. We then assume a constant density of state N(ξ)≈ N(0). All in all, the superconducting critical temperature Tc can be readily obtained by imposing that the superconducting gap vanishes, ∆= 0. The behavior of the critical temperature Tc is plotted in figure 7 as a func- tion of the anharmonic damping parameterD. The figure illus- trates that Tc always decreases monotonically. Physically, this implies that anharmonicity, ∝D, is always detrimental for the onset of superconductivity under these assumptions (see [12] for more details). 4.2. BCS theory: optical phonons Until now, we have mainly considered the case in which the electronic pairing is mediated by the interaction with acous- tic phonons. In this case, because of their acoustic nature the damping is a quadratic function of the wave-vector k, i.e. Γ∼ k2 (Akhiezer mechanism). Here, we consider the altern- ative scenario in which the ‘glue’ is provided by optical phon- ons. In this case, the damping coefficient, or phonon life- time, is independent of the wave vector, as derived using 12 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review perturbation theory by Klemens [36]. Moreover, the life- time of the optical phonons is controlled by their decay into acoustic phonons and ultimately by the Grüneisen constant squared. For the optical phononmodes, we assume the following dis- persion relation Ωopt (k) = ω0 + αk2. (36) where ω0 is the optical phonon mass and α is curvature of the dispersion. We further choose a damping independent of wave vector, which we denote by Γ. With these assumptions, the bosonic propagator takes the following form Π(iΩn,k) = 1[ ω2 0 + 2ω0αk2 +O (k4) ] +Ω2 n+Γ |Ωn| . (37) We now implement the propagator Π(iΩn,k) above into the gap equation. The theoretical predictions for Tc as a function of anharmonic damping constant Γ that we obtain are shown in in figure 8. Our predictions demonstrate that low and moderate anhar- monic phonon damping can lead to an enhancement of the critical temperature. The rise of Tc is followed by a peak for optimal damping and then subsequently a decrease for very large values of anharmonic damping. Therefore, it is evident that anharmonic damping can lead to a substantial increase in Tc for a range of mass values. This behavior must be contrasted with the case of acoustic phonons where Tc is monotonically suppressed. Furthermore, as shown in the lower panel of figure 8, our model predicts that the damping-induced enhancement, and the peak, become larger upon decreasing the optical phonon energy gap ω0. Finally, one can also exam- ine the role of the curvature coefficient,α, in the optical disper- sion relation on the transition temperature. It affects the per- cent of enhancement as well as the peak value—both become larger as the coefficient α becomes smaller. Hence approach- ing a flat optical dispersion, typically seen in DFT computa- tions of optical phonons in hydride materials [167], is favor- able to the absolute value of Tc; however, the enhancement effect is reduced in the process. To understand the increase in Tc, we observe that phonon dispersion and anharmonic damp- ing behave in such a way as to superpose bosons at differ- ent energy scales. The superposition acts to combine vari- ous phonons with different energies coherently to increase the superconducting transition temperature. This occurs because, mathematically, the k-dependence in the propagator is integ- rated out in the gap equation. Such an integration combines low and high energy phonons coherently and, thereby, increas- ing the electron–phonon coupling effectively. In the oppos- ite limit when the linewidth dominates the dispersion spec- trum, the phonons are no longer able to provide a sufficiently strong pairing for the electrons. Thus there is eventually a suppression of Tc for sufficiently large Γ as demonstrated in figure 8. Figure 8. Optical phonon mediated superconductivity. Plots of critical temperature as a function of damping for various vo (top) and optical gap/mass (bottom). (Top) v2o ∈ [2,8] (from black to yellow), where v2o = 2ω0α, and the optical mass is fixed to ω0 = 0.3. (Bottom) ω0 ∈ [0.1,0.7] (from black to yellow) with a fixed value v2o = 4. Reprinted (figure) with permission from [12], Copyright (2022) by the American Physical Society. 4.3. Eliashberg theory for damped phonons In this section, we consider a different theoretical approach in which the superconducting transition is treated by means of Eliashberg theory [184, 185]. The discussion in this subsec- tion closely follows [184]. More concretely, as mediators, we 13 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review consider acoustic phonons with quadratic attenuation constant whose Green’s function is parameterized as usual: G(ω,k) = 1 ω2 − Ω2 (k) + iωΓ(k) , (38) with propagating term given byΩ2 = v2 k2 and the attenuation constant by Γ = Dk2. From equation (38), we can derive the corresponding spectral function which is given by [184]: B (ω,k) = ωΓ(k) π [ (ω2 − Ω2 (k))2 + ω2Γ2 (k) ] . (39) We can then express Eliashberg spectral function in the fol- lowing form [185]: α2F (⃗ k, k⃗ ′,ω ) ≡ N (µ) |g⃗k,⃗k ′ | 2B (⃗ k− k⃗ ′,ω ) . (40) In the formula above, N (µ) represents the electronic DOS computed at µ (chemical potential). Additionally, g⃗k,⃗k ′ is the electron–phonon matrix element. Following the steps in [185], spectral function that is averaged over the Fermi sur- face reads: α2F(ω) = 1 N (µ)2 ∑ k⃗,⃗k ′ α2F (⃗ k, k⃗ ′,ω ) δ ( ϵ⃗k − µ ) δ ( ϵ⃗k ′ − µ ) . (41) For simplicity, we take the matrix elements to be constant in wave-vector, i.e. g⃗k,⃗k ′ ≡ g. In this way, the previous equation takes the simplified form: α2F(ω) = g2 N (µ) ∑ k⃗,⃗k ′ B (⃗ k− k⃗ ′,ω ) δ (⃗ k2 − µ ) δ ( k⃗ ′ 2 − µ ) . (42) In the expression above, we have assumed the typical elec- tronic band of the form ϵ⃗k = k⃗2. To make further progress, we convert the previous sum into a 2D integral using the rela- tion ∑ k⃗ = V2 (2π)2 ´ kdkdϕk with the wave-vector amplitude k ∈ [0,∞] andϕk ∈ [0,2π]. Spatial isotropy dictates thatB(⃗k− k⃗ ′,ω) depends only on the difference (⃗k − k⃗ ′)2, which can be expressed in polar coordinates as(⃗ k − k⃗ ′ )2 = k2 + k ′ 2 − 2kk ′ cos(ϕk − ϕk ′). (43) All in all, we can perform the above integral and obtain the final result [184]: α2F(ω) = g2 4 (2π)4 N ˆ B ( X2,ω ) dϕk dϕk ′ (44) B ( X2,ω ) = ωDX2 π (ω2 − v2X2) 2 + ω2D2X4 (45) X2 ≡ 2µ (1 − cos(ϕk − ϕk ′)) (46) where the electronic DOS is assumed to be constant,N (µ) = N. This final integral in equation (44) can be performed numerically. At this point, we can use the standard definition for the electron–phonon mass enhancement parameter, λ(v,D) = 2 ˆ ∞ 0 α2F(ω) ω dω , (47) determining the effective (dimensionless) strength of the electron–phonon interactions. In order to estimate the crit- ical temperature Tc, we use the Allen–Dynes formula [186] given by: Tc = f1 f2ωlog 1.2 exp ( − 1.04 (1+λ) λ− u⋆ − 0.62λu⋆ ) (48) where ωlog = exp ( 2 λ ˆ ∞ 0 dω α2F(ω) ω lnω ) (49) represents the characteristic energy scale of phonons for pairing in the strong-coupling limit, while f 1, f 2 are semi- empirical correction factors, as defined in [186]. The para- meter u⋆ encodes the strength of the Coulomb interactions and it is determined experimentally and tabulated in the literature for various materials; we will take it as an external input from tabulated literature data. That said, all the SC properties are determined by the shape of the spectral function α2F(ω). As a concrete application of this framework, let us consider as a bosonic mediator an acoustic phonon described by the following choice, Ω(k) = vk− v 2kVH k2 , Γ(k) = Dk2 , (50) where v is the speed of sound, D the attenuation con- stant and kVH the location of the Van-Hove singularity. In figure 9, we show the results for this choice of mediators. The Eliashberg function α2F(ω) shows a clear peak which broaden upon increasing the attenuation constant D. Both the effective electron–phonon coupling λ and the critical temperature Tc decrease monotonically upon increasing the attenuation constant of the acoustic phonon which mediates the pairing. In other words, one finds that, in the case of acoustic phonons, the anharmonic damping is detrimental to superconductivity. 4.4. BCS theory: glassy spins Randomness, when taken into account collectively through electron correlation effects yields interesting phases of mat- ter [188]. The spin glass (SG) phase forms one such example which has been thoroughly studied as well as observed in the phase diagrams of correlated electronic systems [132–146, 189, 190]. The phenomenology of the SG is remarkable [147]. Typically, the SG phase is characterized by aging, linear in temperature of the AC susceptibility peak, hysteretic effects in 14 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 9. The Eliashberg function α2F(ω) corresponding to the mediator dispersion relation in equation (50). The insets show the coupling constant λ and Tc as a function of the damping parameter D. λ0 and Tc,0 are the values at the smallest damping D= D0 = 100. In this figure, the other parameters are fixed to v= 5000, kVH = 1, kF = 1/2, g2k = C(vk)2, C= 0.03 and µ∗ = 0.1. Reproduced from [187]. © IOP Publishing Ltd. All rights reserved. the DC magnetization and a cusp in the thermodynamic spe- cific heat, to name a few. On the theoretical side, SG order phase occurs when the spin average on each lattice site is non-zero, but vanishes when spatially averaged over the entire lattice [191]. For the purposes of this discussion, we note a key property of the temporal dependence τ of spin correlation function at the SG critical point: it follows a power law of the form [191–194] Π(τ)≡ [⟨Siµ (τ)Siµ (0)⟩]∼ 1 τ 2 . (51) We define Siµ as the spin at site i with its µth compon- ent, and the square and angular brackets denote the site and thermal averages respectively. In frequency space, the correl- ation function behaves as Π(ω)∼ |ω|, i.e. it is linear in fre- quency indicating that dissipation is a necessary (but not suf- ficient) condition for a SG. Spin fluctuations have been studied extensively as potential mediators of superconductivity [195, 196]. In the proximity of a SG phase, correlators of the form appearing in equation (51) modify the spin fluctuation propagator [13]. Since the dissip- ative component arises from randomness that is exclusively a property of the spin sector, it constitutes an anharmonicity of the spin mediator. To see how this occurs, we write the model for the total bosonic propagator by additing the dis- sipative (anharmonic) contribution from equation (51). As a result, the total action consists of a free term S0[Ψ,Ψ∗] and a dissipative term Sdiss[Ψ,Ψ∗]. These are given by S [Ψ,Ψ∗] = S0 [Ψ,Ψ ∗] + Sdis [Ψ,Ψ ∗] S0[Ψ,Ψ ∗] = ˆ ddrdτ [ κ|∇Ψ(r, τ) |2 + |∂τΨ(r, τ) |2 +M2|Ψ(r, τ) |2 ] , Sdiss [Ψ,Ψ ∗] = ∑ k,ωn (2η |ωn|) |Ψ(k,ωn) |2. Here Ψ is the bosonic field, ωn is the Matsubara frequency, η is the dissipation (anharmonicity) parameter, κ is the energy scale of the bosonic velocity (or spatial stiffness), and the para- meterM2 is the square mass that is proportional to the inverse correlation length. The bosonic propagator Π(k, iωn− iωm) for the action S[Ψ,Ψ∗] takes the form Π(k, iωn) = α κk2 +ω2 n + 2η|ωn|+M2 . Here k= |k| and α is a constant with dimensions of energy (see for example [195]). We can now substitute Π(k, iωn− iωm) into equation (34). We assume a frequency independent and isotropic s-wave gap (henceforth denoted by ∆) and perform the momentum and Matsubara summations. The equation determining Tc (setting ∆= 0) then becomes [13] 1=−λ [ ψ ( 1 2 + η ′−iκ 2πTc ) 2(η ′ − iκ)2 + ψ ( 1 2 + η ′+iκ 2πTc ) 2(η ′ + iκ)2 + κ2 − η ′2 (κ2 + η ′2) 2ψ ( 1 2 ) − π2η ′ 4πTc (η ′2 +κ2) ] , (52) where η ′ ≡ 2η and ψ(x) is the digamma function. The equation (52) can be solved numerically to study the effect of η on Tc. As will be further elucidated below, Tc follows a non-monotonic behavior with η where the optimal value is set by the stiffness κ. A potentially interesting limiting case is that of κ→ 0, in which case the gap equation reduces to 1= λ(η− i M̄) −1 2i M̄ [ ψ ( 1 2 + η 2πTc − i M̄ 2πTc ) −ψ ( 1 2 )] + c.c, (53) where M̄≡ √ M2 − η2. As we will see below, in this limit, Tc monotonically decreases with η. 4.5. Optical soft mode instabilities and structural transitions Soft phonon modes appear near structural transitions in which a higher-symmetry crystal structure transforms into a lower-symmetry one [197]. Typical examples of this sort are ferroelectric and ferroelastic transitions [198, 199]. Here, we address the question of how the appearance of soft mode 15 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 10. The dynamics of the critical modes from equation (56) as a function of n− nc. Here, Γ = 1. From white color to black color we change the control parameter from n= 0 to n= nc as the red arrows indicate. instabilities in a metallic state might affect the critical temper- ature of a near superconducting transition. For simplicity, we will focus on the usage of BCS theory and on the situation in which the soft mode is a optical excitation whose dynamics is described by equation (26). Solving equation (26), we obtain the simple dispersion relation: ω =± √ ω2 0 − Γ 4 − i 2 Γ (54) from which we can identify the real part as the renormalized energy and the imaginary part as the inverse lifetime τ−1. In the limit of Γ→ 0, the relation above coincides with the Einstein approximation ω = ω0. In order to continue, and as we will see later to make contact with realistic materials, we assume a soft mode instability in which the energy of the soft mode is well described (at least close enough to the critical point) by the mean-field Curie–Weiss law: ω2 0 ∼ |n− nc|, (55) with n an external parameter driving the instability [200]. By doing so, the dynamics of the low-energy critical modes is defined by the following dispersion relations: ω = 1 2 ( ± √ 4|n− nc| −Γ2 − iΓ ) . (56) As evident from equation (56), and as shown in figure 10, at the critical point one of the two modes becomes strongly overdamped, ω =−iΓ, while the other approaches the origin of the complex plane,ω= 0, moving along the imaginary axes. This second mode is the mode responsible for the instability at n= nc. Figure 11. The critical temperature as a function of n− nc for v2 = µ= 1 and increasing the linewidth Γ from black to light blue. Reprinted (figure) with permission from [201], Copyright (2022) by the American Physical Society. Using the framework described in the previous section, we can then compute the critical temperature Tc as a function of the distance from the critical point, n− nc (see [201] for details). The critical temperature displays a dome shaped beha- vior centered at the critical point which is shown in figure 11 for different values of the damping parameterΓ. A larger value of Γ corresponds to a more pronounced maximum at n= nc. In summary, this simple model predicts the appearance of a dome shaped critical temperature Tc which is maximized at the location of the critical point [201]. Such a result provides a viable explanation (see [202] for a different explanation based on the concept of quantum criticality) for the superconducting dome experimentally observed in various ferroelectric mater- ials such as SrTiO3 [203] (see section 5.4). 4.6. Kohn-like soft phonon instabilities Phonon softening is a general phenomenon in condensed mat- ter physics which is not restricted to structural phase trans- itions. A particularly interesting scenario is that associated with the formation of charge order in metallic states, in which softening emerges in the form of ‘Kohn anomaly’ [204]. Charge correlations soften the energy of the acoustic phon- ons whose dispersion presents a pronounced dip at a finite value of the wave-vector. The frequency might be even go to zero at a specific critical point, which signals the onset of charge density waves (CDWs) formation, as rationalized in 1D by the so-called Peierls instability [205]. This type of softening is profoundly different from the one described in the previous section as it is localized in a small region of finite wave-vector and does not appear at k= 0. More in general, Kohn-like instabilities, defined as a localized decrease of the energy of acoustic phonons in a finite and small interval of wave-vectors, appear not only in association to CDW form- ation. They appear more generally whenever important nest- ing is exhibited by the Fermi surface, such as in NbC1−xNx andNbN rocksalt structures [206]. Intriguingly, similar soften- ing mechanisms have been also reported in the acoustic 16 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review dispersion relations of specific amorphous systems known as ‘strain-glasses’ [207]. Here, wewant to assess the effects of Kohn-like instabilities in the metallic state of a superconducting material. The discus- sion in this subsection follows [187]. Despite the main interest of this analysis is the question of coexistence and/or competi- tion between CDW and superconductivity, we will leave the model as generic as possible in order to account for other possibilities not related to CDW. Moreover, for brevity, we will consider only the case of acoustic phonons, although the same qualitative behaviors hold for optical phonons as well. For more details and for the case of optical modes we refer to [187]. As a phenomenological description of phonon softening, we consider the standard dispersion relation for acoustic phon- ons extracted from the denominator of the mediator Green’s function in equation (17) with Ω(k) = p(k) ( vk− v 2kVH k2 ) , Γ(k) = Dk2 . (57) In the equation above, kVH is the wave-vector corresponding to the end of the Brillouin zone, the Van-Hove wave-vector. Most importantly, the softening of the dispersion is parameterized by the function p(k) which is chosen as p(k) = 1− ζ exp [ − ( k/kVH −α β )2 ] . (58) The softening dip is assumed to be of Gaussian shape. The parameter ζ determines the depth of the softening, α controls the wave-vector at which the dip appears and β its width. After assuming this dispersion, we can use the Eliashberg theory for damped bosonic mediators outlined in section 4.3 to compute the various superconducting properties. In figure 12, we show the results as a function of the depth of the softening dip ζ. Upon increasing the depth of the soften- ing region, the value of the Eliashberg electron–phonon coup- ling λ grows monotonically and it can be strongly enhanced. Nevertheless, this enhancement of the electron–phonon coup- ling is not always reflected in an increase of the supercon- ducting temperature. On the contrary, a non-monotonic beha- vior is observed in the critical temperature, which first grows with softening roughly linearly, but then decreases quickly after a critical value ζc. Intuitively, this non-monotonic trend is explained by the competition of the different factors appearing in the Allen–Dynes formula, equation (48). More precisely, despite the fact that the electron–phonon coupling λ grows with softening, the logarithmic average frequency ωlog (49) decreases with it. As a consequence, a maximum value of Tc appears at ζ = ζc. From a different perspective, the appearance of a maximum is compatible with the concept of ‘optimal fre- quency’ developed by Bergmann and collaborators [208]. In a nutshell, the idea is that a weight transfer in the Eliashberg function α2F(ω), which in our case is induced by soften- ing, is beneficial to superconductivity only when it is near the ‘optimal frequency’, defined as the location in which the functional derivative δTc δα2F(ω) is maximal. The optimal condi- tion coincides with the maximum in Tc at ζc. The value of ζc and the maximum increase in the critical temperature, Tc(ζc)/Tc(0), are strongly sensitive to the para- meters of the model. Nevertheless, some general conclusions can be reached. In particular, in a weakly-coupled supercon- ductor, the increase of Tc due to softening can, in general, be very large (over one order of magnitude in Tc) but at the same time it requires a substantial degree of softening, i.e. a relatively large value of ζ (see example in the bottom panel of figure 12). On the contrary, for strongly-coupled systems, the increase of Tc is more limited but the degree of soften- ing needed to reach it is also smaller (see the central panel of figure 12). A more complete case study about the effects of Kohn-like softening on Tc, including the case of optical modes, can be found in [187]. 5. Applications to emerging quantum materials 5.1. Cuprates It may not have been a sheer coincidence that the major break- through in high-Tc superconductivity, i.e. the Nobel-prize win- ning discovery of the cuprate rare-earth oxides by Bednorz and Müller in 1986 [4], came in that same Zurich IBM lab after more than 15 years of studying the dielectric properties and soft-mode transitions in strontium titanate. Bednorz and Müller’s original intuition was that certain oxides could host Jahn–Teller type composites made of an electron plus a local lattice distortion that could travel as whole through the lat- tice, thus leading to a very strong electron–phonon coupling. While lattice distortion and strong electron–phonon coupling have certainly been recognized to be important factors for the high Tc of the cuprates, other non-trivial (e.g. magnetic) phe- nomena have since also been observed, which also appear to strongly affect the Tc. Importantly, in a series of papers by Liarokapis, Kaldis and co-workers, Raman spectroscopy studies of the Cu- bonded oxygen atoms and associated Raman-active modes, highlighted a number of striking phonon-softening instabil- ities. The in-plane (Ag) oxygen vibrations in YBa2Cu3Ox were shown in [6] to suffer a major softening right at the optimal doping x≈ 6.92 that corresponds to the highest Tc. Concomitantly, a displacive structural phase transition involving the Cu2O planes (basically a dimpling of the planes) was demonstrated in [7] to also occur at a value of oxygen dop- ing very close to the optimal one for Tc. Recent experiments where the cuprate superconductor La2−xBaxCuO4 (LBCO) at 1 8 doping was irradiated with pro- tons [209] observed a (radiation) disorder induced enhance- ment of Tc despite the proximate CDW ordering temper- ature being unaffected by irradiation. The measurements found up to a 50% increase in Tc with the dosage of radiation. Above a critical value of the dosage, Tc was gradually suppressed until superconductivity was destroyed. To understand these observations, scenarios involving the competition between CDW and superconductivity seem 17 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 12. The effects of Kohn-like softening in the acoustic dispersion on the superconducting properties. (Top) the dispersion relation of the bosonic mediator for different values of the softening depth ζ and the corresponding behavior of the Eliashberg coupling λ. λ0 stands for λ(ζ = 0). (Center and Bottom) the critical temperature Tc as a function of smaller and large softening depth ζ respectively. The vertical dashed line indicates the location of the optimal condition ζ = ζc. Reproduced from [187]. © IOP Publishing Ltd. All rights reserved. promising at first sight, especially given their proximity in the phase diagram. However, given that the CDW transition tem- perature seems unaffected by irradiation, it is unlikely that a mechanism involving the competition between two mean field phases [210] is at play. It is also unclear how scalar dis- order affects two mean field phases in an asymmetric fashion without any parameter dependence, except under special cir- cumstances [210, 211] which may not hold for LBCO. Hence, to explain the non-monotonic Tc dependence in LBCO as a function of irradiation, a mechanism that does not involve any competition between CDW and superconductivity is a poten- tial candidate. Setty [13] made the case for enhanced Tc due to glassy dissipation in spin fluctuation mediator. The SG phase has been observed in proximity to superconductivity in the cuprate [133–139, 143, 144, 189, 190] and iron superconduct- ors [132, 140–142, 145, 146]. It is thus reasonable to include the effects of the SG phase on the superconducting pairing. Further there is plenty of direct experimental data [132, 134, 136, 137, 139, 140, 143, 145, 190] supporting a dissipative nature of the spin fluctuations mediating Cooper pairing. See [13] for a brief review of these experiments and their relev- ance to superconductivity in LBCO. From these discussions, the premise of a SG induced dissipative pairing mediator in LBCO has firm experimental support. We now follow [13] which argues that a non-local dissipative mediator can explain the proton irradiation experiments in [209] despite the fact that the proximate CDW transition is unaffected by disorder. According to this proposal, disorder acts as an external tun- ing knob of the parameter η; hence, increased irradiation leads to larger dissipation in the pairing mediator. Then, for weak dissipation, Tc rises and above a critical value of η it gradu- ally falls. To see this, the local (κ= 0) and non-local (κ ̸= 0) gap equations (53) and (52) can be solved for Tc as a func- tion of the dissipation parameter η. Figure 13 shows a plot of the solutions. For the case of a local mediator, the Tc falls monotonically with dissipation. On the other hand, when the mediator is non-local, Tc is non-monotonic with dissipation parameter and reaches an optimal value at an η value set by the stiffness. This can be understood from the energy integral leading to equation (52) above. For a non-local mediator, this integral forces the gap equation to acquire dissipative contri- butions that both increase and decrease the effective coupling constant. Note that weakly dissipative bosonic modes at differ- ent energy scales act coherently to enhance Tc but eventually destroy superconductivity when dissipation dominates all the other energy scales. Consequently, a non-monotonic behavior in Tc follows. Thus the SG dissipative mechanism described above is a way to raise Tc that does not rely on ‘tug-of-war’ - like scenarios between two competing phases. Further analysis of the superconducting gap and specific heat as a function of the dissipation parameter can be found in [13]. 5.2. Hydrides Many recent studies have pointed to the possibility of achieving room-temperature superconductivity in the hydride 18 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 13. (Top) Superconducting critical temperature Tc (normalized to the case of zero dissipation and bosonic mass) as a function of the dissipation parameter η for different masses M for a local mediator (κ= 0). (Bottom) For a non-local mediator (κ ̸= 0): plot of the dimensionless T̄c = Tc/κ as a function of η̄ ′ = η ′/κ for different dimensionless coupling strengths λ̄= λ/κ2 and M= 0. The Tc peak is determined by κ, the energy scale arising from the bosonic spatial stiffness/velocity. compounds at high pressure [16], following on Ashcroft’s early intuition for metallic hydrogen [212]. Recent exper- imental evidence points at superconductive behavior in nitrogen-doped lutetium hydride thin films at pressures as low as 10 kbar [213]. While it is clear that phonon dispersion curves are strongly renormalized (i.e. lowered in energy) by anharmonicity in the hydrides [174, 175, 214], a clear picture about the effect of the ubiquitous large anharmonicity on the superconductivity of these systems is missing. In particular, systematic studies of the anharmonic phonon linewidths and the effect thereof on the Cooper pairing are currently lacking. Since these systems exhibit high-T superconductivity at high pressures, the inter- play between lattice dynamics under pressure, and anharmon- icity, which leads to the resulting electron–phonon coupling, is expected to be non-trivial. In particular, the effects of pres- sure are twofold, on one hand there exists a critical pressure to stabilize the superconducting lattice structure [215–217], while on the other hand there are (hitherto much less explored) effects of pressure-mediated phonon dynamics on the pairing mechanism [55, 218]. Recent progress [219] has identified the phononEumode as the one mainly responsible for the pairing in atomic hydrogen Figure 14. Superconducting critical temperature Tc and electron– phonon coupling constant λ as a function of pressure computed with and without the anharmonic corrections for the P63/mmc phase of ScH6. Reprinted (figure) with permission from [220], Copyright (2021) by the American Physical Society. at high pressure. The effect of phonon anharmonicity on the superconducting critical temperature Tc has, instead, remained poorly understood. The anharmonic extension of BCS the- ory to include the effect of anharmonic damping on the pair- ing mechanism [11], has shown that anharmonicity can either enhance the Tc or lower it, depending on the extent of phonon damping (moderate or very large, respectively), for the case of optical phonons, whereas for acoustic phonons the effect is always to cause a depression of the Tc. This theory [11] thus might explain why the Tc is much lowered by huge anharmonicity of the low-lying optical phon- ons in aluminum [220, 221], palladium [174, 175], and plat- inum hydrides [175]. Conversely, an enhancement of the superconducting Tc, in a regime of moderate anharmonicity, may be responsible for the observed enhancement of Tc due to anharmonicity in the high-pressure P63/mmc phase of ScH6 as reported in [220], see figure 14. Importantly, in the current literature, e.g. [220], the effects of anharmonicity are mostly considered at the level of the renormalization of the bare energy. So far, the effects of the anharmonic linewidth has been largely over- looked (see nevertheless [222] for a recent discussion about it). In future studies, it could be useful to carry more system- atic studies of the effect of phonon anharmonicity on the Tc by more closely combining theoretical concepts [11], and atom- istic computations [223]. 5.3. The case of TlInTe2 The discussion in this subsection closely follows [218]. TlInTe2 undergoes a superconducting transition at a pres- sure of 5.7 GPa with a Tc ≃ 4K [55]. The Tc behaves non-monotonically with further increasing the pressure—it decreases initially and climbs again with the minimum value 19 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review of Tc occurring at 10 GPa. Concurrently, ab initio electronic structure calculations found a Lifshitz transition induced change in Fermi surface topology between 6.5 and 9GPa and the formation of enlarged electron pockets at the Fermi level. Additionally, x-ray diffraction and Raman scattering measure- ments performed at high pressure found that the Ag phonon mode begins to soften [55]. Naively, the V-shaped Tc beha- vior may be attributable to a combination of softening of the Ag phonon mode and variations in the electronic DOS with pressure. However, the Tc appears to get reduced exactly in the regime where there is an increase in DOS from the elec- tron pocket due to the Lifshitz transition. Moreover, the sup- pression of Tc with increasing phonon frequency involves the Bergmann–Rainer criterion which would, in turn, require a second dip in Tc. This feature is, however, absent in the exper- imental observations. Hence, we can rule out a dominant role of electronic DOS or phonon frequency shifts in understand- ing the observed Tc dependence. Here, we consider the role of both phonon frequency and linewidth in determining Tc as a function of pressure in TlInTe2. Raman scattering linewidths extracted as a function of pressure indicate that anharmonicity in this material is in an optimal range—weak enough so that the phonons remain coherent, but strong enough so as to have significant effects on the superconducting properties. In particular, as we will see below, Tc correlates positively with the ratio of the linewidth to the peak frequency, Γ/ω0. The possibility of excluding other electronic DOS and phonon frequency-shift effects on the pair- ing renders TlInTe2 an ideal playground to test the role of anharmonic boson damping. At this juncture, the properties of the normal state, strength of specific electron–phonon coup- lings, pairing symmetry etc are not completely determined in TlInTe2. But the formalism and conclusions presented below are general enough so that the above uncertainties can be accommodated as more experimental data becomes available. We begin by studying how the optical phonons of a crys- tal lattice are affected by external pressure. Predominantly, pressure acts to induce a volume contraction (negative volume change) in the material. We can relate change in volume to a change in phonon frequency through the Grüneisen parameter, γ =−d lnω ′/ d lnV, via the relation [224]: ω ′ (V) ω ′ P=0 = ( V V0 )−γ . (59) Here the optical phonon energy at zero ambient pressure is denoted by ω ′ P=0 and the relations above apply to individual phonon modes with frequency ω ′. The change in pressure can be written in terms of the volume change, as described by the Birch–Murnaghan equation of state [225] (see also section 2.4 above). The equation provides an expression for the pressure P(V) and is derived based on nonlinear elasticity theory. We then replace V with ω ′ in (59) to obtain a relationship between applied pressure and optical phonon frequency ω ′ [224] given by P(X) = 3 2 b0 ( X7 −X5 ) [ 1+ η ( 1−X2 )] , (60) where we have defined X≡ (ω ′/ω ′ P=0) 1/3γ . Next, we invert the above equation (60) and obtain ω ′ as a function of P. We see that ω ′ is a function that monotonically increases with P in the regime of interest, and is modulated by anharmon- icity through γ. We have also defined b0 = B0/γ0 where B0 the bulk modulus and η = (3/4)(4−B ′ 0) with B ′ 0 = dB0/dP. The frequency ω ′ refers to the real part of the phonon dis- persion (including the renormalization shift from anharmon- icity [226]). The imaginary part of the dispersion relation is related to the phonon damping coefficient Γ (the inverse phonon lifetime). These quantities are given by the following relations (see for example equations (23)–(27) in [226]) ω2 = ω2 0 − iωΓ + O ( q2 ) , (61) ω ′ ≡ Re(ω) = 1 2 √ 4ω2 0 − Γ2 +O ( q2 ) , (62) Γ 2 ≡ Im(ω) +O ( q2 ) . (63) The phonon linewidth Γ can, in principle, be evaluated from quantitative microscopics using the Self-Consistent Phonon (SCP) methodology [177, 226] for specific systems [178]. Here we rather focus on generic qualitative trends in terms of the effect of Γ on the pairing and on Tc. As a concrete application of this model, we consider the case of TlInTe2 using the data reported in [55]. We fit the bare frequency ω0 and the linewidth Γ of the optical mode Ag, which is the dominant one in the electron pairing, as a function of the pressure P. The results for the linewidth are shown in the top panel of figure 16. We then use these func- tions as an input into the theoretical gap-equation framework and predict the corresponding Tc. First, we use the fitting for the energy ω0(P) together with the Klemens expression for the linewidth Γ = αω5 0 , with α a phenomenological parameter. The results for different values ofα are shown in figure 15. The critical temperature Tc decreases monotonically with the pres- sure. An approximately linearly decaying trend of Tc with P has been recently observed in the strongly anharmonic AlH3 high-pressure hydride [220] as well as in the SC-I phase of CeH10 in [227]. In more standard systems, a linear decay of Tc with increasing P has been reported in the literature for simple (e.g. elemental) superconductors [228–230]. In order to improve the results, we also used the fitted linewidth from the experimental data, available in [55] for TlInTe2. The pre- dicted critical temperature is compared with the experimental data in the bottom panel of figure 16. The agreement, at least at qualitative level, is good. In particular, the theoretical predic- tion shows a minimum in the critical temperature at a pressure which roughly corresponds to the position of the minimum in Γ/ω0. This is rationalized, within the theoretical framework, by noticing that in the so-called coherent (moderate-damping) regime, where Γ/ω0 ≪ 1, the behavior of Tc positively correl- ates with the ratio Γ/ω0. Similar considerations also appear to hold for the Osmium based pyrochlore superconductors where Tc shows an optimum value as a function of anharmonicity parameters [231, 232]. 20 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review Figure 15. The normalized superconducting transition temperature as a function of pressure for various Klemens damping parameters Γ = αω5 0 where ω0 is the energy of the optical mode at zero wavevector. Here, α increases from 0.1 to 1.0 for red to purple curves. Reprinted (figure) with permission from [233], Copyright (2022) by the American Physical Society. 5.4. SrTiO3 and BaTiO3 The discussions in this subsection follow from [201]. Superconductivity in the quantum paraelectric semiconductor SrTiO3 has recently attracted much attention in view of dif- ferent experimental protocols that have been discovered in order to promote superconductivity, often via a supercon- ducting ‘dome’, in the vicinity of the ferroelectric instabil- ity. These methods include doping in terms of carrier concen- tration, isotopic doping, mechanical strain, and dislocations. Since the pioneering experimental work of Müller and co- workers [234], this material has been classified as a quantum paraelectric, in the sense that although a ferroelectric instabil- ity appears to be approached upon lowering the temperature below about 30K, eventually the TO mode energy remains finite and real, and no condensation of the TO mode into the ferroelectric phase occurs (hence no real soft mode instabil- ity occurs). The common explanation for this phenomenon is that large quantum fluctuations of the lattice (in partic- ular, zero-point motions of the oxygen atoms) prevent the mode condensation and the corresponding ‘freezing’ of atomic position into polar order. Hence the ferroelectric transition is de facto suppressed and the expected Curie–Weiss beha- vior of the dielectric constant is instead replaced by a low- T plateau [234], hence the quantum paraelectric phase at T < 4K.Mechanical strain has been shown to be amost effect- ive way of re-establishing ferroelectricity in SrTiO3, to the point that even ferroelectricity at room temperature and above has been demonstrated for SrTiO3 material under strain. Also, electron doping and oxygen doping have proved to be effective ways of inducing the ferroelectric instability and thus stabilize the ferroelectric phase. Themechanism bywhich Figure 16. (Top) Phonon linewidth (black symbols) extracted from experimental data [55] versus optimal phenomenological fit (red solid line). (Bottom) Comparison between the theoretically obtained calculations for the critical temperature (red solid line) and experimental data (black symbols). The dashed black line is an interpolation of the experimental data as a visual guide. For details about the parameters and the numerical procedure see [233]. Reprinted (figure) with permission from [233], Copyright (2022) by the American Physical Society. superconductivity occurs in this material has been thought for a long time to be puzzling because SrTiO3 behaves as a superconductor even at very low carrier doping levels. Early evidence however has been collected pointing to the fact that it is indeed the soft transverse optical (TO) phonons which mediate the electron pairing [235] in doped SrTiO3, a mech- anism recently confirmed [236]. More recently DFT calcula- tions have demonstrated that indeed the maximum of the dome observed in the superconducting Tc in SrTiO3 does coincide with the TO mode energy crossing zero, i.e. with the ferro- electric criticality [202]. This has become widely known as the ‘quantum criticality’ paradigm for superconductivity in SrTiO3, in view of the fact that large lattice fluctuations upon approaching the ferroelectric transition are thought to promote the Cooper pairing, and these fluctuations at such low temper- atures are quantum, since the oxygens motion is of zero-point type. In reality, however, these are just large atomic fluctuations about the equilibrium positions in the lattice, and quantum or not, they are always associated with large anharmonicity, simply because atoms displaced far away from the harmonic bonding minimum locally experience a large anharmonic 21 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review potential, thus leading to huge values of the cubic derivative of the potential V ′ ′ ′(a) in (11), and hence to large values of the Grüneisen parameter. This is indeed what happens, and giant values of Grüneisen parameter γ have been indeed observed both numerically and experimentally [237]. This picture of giant anharmonicity assisting superconductivity even at low carrier concentrations in SrTiO3 is further corroborated by recent anharmonic phonon calculations [238]. Finally, a similar dome in the superconducting Tc with a maximum coinciding with the ferroelectric transition at which the TO mode goes to zero has been observed in the stand- ard ferroelectric compound BaTiO3. Also in this case very large anharmonicity of the TOphononwhich explodes towards the ferroelectric transition, accompanies the superconducting dome. The fact that a very similar phenomenology is shared by quantum paraelectric SrTiO3 and standard ferroelectric BaTiO3 thus strongly suggests that the so-called quantum crit- icality may not be the peculiarity behind the superconducting properties of SrTiO3, and points rather a unifying common mechanism. A recent proposal is that anharmonicity in SrTiO3 leads to an even stronger pairing tendency, according to the anhar- monic damping enhancement of Tc mechanism proposed in [11, 201] (see section 4.5 for details), which therefore explains the surprising fact that superconductivity in SrTiO3 occurs at very low electron doping. Cfr figure 11 in section 4.5 where the experimentally observed dome in Tc is reproduced in model calculations that fully take into account the increase in the phonon anharmonic linewidth upon approaching the ferroelec- tric transition on both sides [201]. But the general mechanism, by which superconductivity is enhanced by phonon damping is the same which is operative in standard (non-quantum) fer- roelectrics such as BaTiO3. 6. Outlook and conclusions 6.1. Open issues Several questions regarding the broader implications of bosonic damping effects on superconductivity remain pertin- ent. These questions range from material specific aspects that require a generic design principle of superconducting materi- als where boson anharmonic decoherence is a key player, to more fundamental aspects that may require new theoretical frameworks to deal with the role of dissipation on the super- conducting ground state. In regards to the former, there is an immediate need to integ- rate first principles evaluation of phonon/bosonic linewidths into routines (such as Electron–PhononWannier (EPW) [223]) that evaluate material-specific superconducting properties. This could help make predictions about the role of bosonic anharmonic decay on Eliashberg functions, superconduct- ing coupling constants, and critical temperatures. Moreover, studying the role of multiple phononic branches and symmetry allowed dissimilarities in their anharmonicities can help make a clearer connection to real material systems. It is our hope that this review puts together various relatively disconnected theoretical, numerical and experimental works on the topic of anharmonic decoherence and superconductivity, and motiv- ates future numerical work in this direction. On the formalism side, there are several open questions that remain unanswered. The simplest extension to the results presented would incorporate the full electronic and bosonic self-energies, including anharmonic decoherence effects, into the Eliashberg formalism [239]. This would give a better understanding on the robustness of the conventional BCS results when retardation effects are taken into consideration. Moreover, in the presence of a lattice, discrete spatial sym- metries constrain the momentum dependence of higher order anharmonic terms in the Hamiltonian. The role of these sym- metries and its interplay with decoherence and supercon- ductivity is an interesting question worth exploring. Further, whether anharmonic damping can be a primary driver of super- conductivity instead of playing a catalyst is also unclear. Going beyond infinitesimal weak coupling effects is another promising avenue. This includes role of anharmonic damping on BCS-BEC (Bose–Einstein Condensation) crossover phys- ics [240] and strong coupling superconductivity. Additionally, at intermediate coupling scales and anisotropic interactions, the superconducting ground state is modulated [241, 242]; the role of anharmonicity on such ground states could have con- sequences on their stability. A more ambitious goal would be to include electron–electron correlation effects and under- stand their co-action with anharmonicity and superconductiv- ity. While this might seem like a difficult endeavor, recent pro- gress has been made [243–245] in obtaining exact solutions to the pairing problem in the presence of long range interactions. Therefore, gaining an intuitive understanding of the cooper- ation between anharmonic damping, strong correlations and superconductivity is a realistic possibility. 6.2. Summary In this review, we have addressed the role of anharmonic deco- herence/damping effects on superconducting pairing proper- ties using minimal theoretical models. We have adopted a mechanistic approach to describe the physics of how super- conducting properties such as Eliashberg functions, coup- ling strengths and transition temperatures are affected, and applied these mechanisms to phenomenologically describe experiments on a variety of emerging quantum materials. The central theme of the review emphasizes the qualitative role played by damping, and the use of simplifying assumptions, rather than an elaborate implementation of atomistic first prin- ciple simulations. However, our objective is to highlight these simplemechanistic effects tomotivate further works that could combine first principles evaluation of bosonic lineshapes with superconductivity routines such as EPW for material specific results. To this end, this review provided a basic introduc- tion to phonon anharmonicity and various bosonic damping mechanisms that may be relevant to superconductivity. We then reviewed several experimental probes that can be used to measure anharmonic damping effects in pairing mediators fol- lowed by a brief interlude into existing literature that directly calculates phonon linewidths from first principles. The bulk of the remainder of the paper focused on minimal theories of 22 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review superconductivity driven by damped bosons including phon- ons and glassy spin fluctuations models, and how they can be relevant to emerging quantum materials. As an outlook, we presented several outstanding problems and natural exten- sions of the current work that remain currently unaddressed. While a complete picture that delineates the role of anharmon- icity and damping effects on superconductivity remains elu- sive, we believe that its fuller understanding holds the potential for interesting fundamental physics, novel numerical imple- mentations, and innovative design of quantum materials and experimental realizations. In particular, we anticipate broad implications of the anharmonic physics of superconductiv- ity also in nanostructured devices [246, 247] and disordered materials [248]. Data availability statement All data that support the findings of this study are included within the article (and any supplementary files). Acknowledgments M B acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01) and the sponsorship from the Yangyang Development Fund. A Z gratefully acknowledges funding from the European Union through Horizon Europe ERC Grant Number: 101043968 ‘Multimech’, and from US Army Research Office through Contract No. W911NF-22-2-0256. ORCID iDs Chandan Setty https://orcid.org/0000-0003-4829-1508 Matteo Baggioli https://orcid.org/0000-0001-9392-7507 Alessio Zaccone https://orcid.org/0000-0002-6673-7043 References [1] Vujicic G, Aksenov V, Plakida N and Stamenkovic S 1979 Phys. Lett. A 73 439 [2] Vujicic G M, Aksenov V L, Plakida N M and Stamenkovic S 1981 J. Phys. C: Solid State Phys. 14 2377 [3] Plakida N M, Aksenov V L and Drechsler S L 1987 Europhys. Lett. 4 1309 [4] Bednorz J G and Müller K A 1986 Z. Phys. B 64 189 [5] Alexandrov A S and Mott N F 1996 Polarons and Bipolarons (World Scientific) [6] Poulakis N, Palles D, Liarokapis E, Conder K, Kaldis E and Müller K A 1996 Phys. Rev. B 53 R534 [7] Kaldis E, Röhler J, Liarokapis E, Poulakis N, Conder K and Loeffen P W 1997 Phys. Rev. Lett. 79 4894 [8] Liarokapis E 2019 Condens. Matter 4 87 [9] Oshiba K and Hotta T 2011 J. Phys. Soc. Japan 80 094712 [10] Isono T, Iguchi D, Matsubara T, Machida Y, Salce B, Flouquet J, Ogusu H, Yamaura J-i, Hiroi Z and Izawa K 2013 J. Phys. Soc. Japan 82 114708 [11] Setty C, Baggioli M and Zaccone A 2020 Phys. Rev. B 102 174506 [12] Setty C, Baggioli M and Zaccone A 2022 Phys. Rev. B 106 139901 [13] Setty C 2019 Phys. Rev. B 99 144523 [14] Mizukami Y, Kończykowski M, Tanaka O, Juraszek J, Henkie Z, Cichorek T and Shibauchi T 2020 Phys. Rev. Res. 2 043428 [15] Wei B, Sun Q, Li C and Hong J 2021 Sci. China Phys. Mech. Astron. 64 117001 [16] Quan Y, Ghosh S S and Pickett W E 2019 Phys. Rev. B 100 184505 [17] Cowley R A 1968 Rep. Prog. Phys. 31 123 [18] Chaikin P M and Lubensky T C 1995 Principles of Condensed Matter Physics (Cambridge University Press) [19] Christensen R 2003 Theory of Viscoelasticity (Civil, Mechanical and Other Engineering Series) (Dover Publications) [20] Landau L and Lifshitz E 2013 Fluid Mechanics vol 6 (Elsevier Science) [21] Landau L, Lifshitz E, Kosevich A, Sykes J, Pitaevskii L and Reid W 1986 Theory of Elasticity: Volume 7 (Course of Theoretical Physics) (Elsevier Science) [22] Fruchart M, Scheibner C and Vitelli V 2022 arXiv:2207.00071 [23] Akhiezer A I 1939 J. Phys. (Moscow) 1 277 [24] Liao Y and Shiomi J 2021 J. Appl. Phys. 130 035101 [25] Moon J, Hermann R P, Manley M E, Alatas A, Said A H and Minnich A J 2019 Phys. Rev. Mater. 3 065601 [26] Baldi G, Giordano V M, Monaco G and Ruta B 2010 Phys. Rev. Lett. 104 195501 [27] Devos A, Foret M, Ayrinhac S, Emery P and Rufflé B 2008 Phys. Rev. B 77 100201 [28] Benassi P, Caponi S, Eramo R, Fontana A, Giugni A, Nardone M, Sampoli M and Viliani G 2005 Phys. Rev. B 71 172201 [29] Vacher R and Pelous J 1976 Phys. Rev. B 14 823 [30] Maris H J 1971 Physical Acoustics vol VIII, ed W Mason and R Thurston (Academic) [31] Cuffari D and Bongiorno A 2020 Phys. Rev. Lett. 124 215501 [32] Yang Z-Y, Wang Y-J and Zaccone A 2022 Phys. Rev. B 105 014204 [33] Bömmel H E and Dransfeld K 1960 Phys. Rev. 117 1245 [34] Krivtsov A M and Kuz’kin V A 2011 Mech. Solids 46 387 [35] Casella L and Zaccone A 2021 J. Phys.: Condens. Matter 33 165401 [36] Klemens P G 1966 Phys. Rev. 148 845 [37] Balkanski M, Wallis R F and Haro E 1983 Phys. Rev. B 28 1928 [38] Hase M, Ushida K and Kitajima M 2015 J. Phys. Soc. Japan 84 024708 [39] Kovtun P 2012 J. Phys. A: Math. Theor. 45 473001 [40] Baggioli M and Goutéraux B 2023 Rev. Mod. Phys. 95 011001 [41] Grozdanov S, Kovtun P K, Starinets A O and Tadić P 2019 Phys. Rev. Lett. 122 251601 [42] Ioffe A and Regel A 1960 Prog. Semicond. 4 237 [43] Menéndez J and Cardona M 1984 Phys. Rev. B 29 2051 [44] Devereaux T P and Hackl R 2007 Rev. Mod. Phys. 79 175 [45] Cowley R 1964 Proc. Phys. Soc. 84 281 [46] Cowley E and Cowley R A 1965 Proc. R. Soc. A 287 259 [47] Quilty J, Lee S, Yamamoto A and Tajima S 2002 Phys. Rev. Lett. 88 087001 [48] Blumberg G, Mialitsin A, Dennis B, Zhigadlo N and Karpinski J 2007 Physica C 456 75 [49] Goncharov A F, Struzhkin V V, Gregoryanz E, Hu J, Hemley R J, Mao H-k, Lapertot G, Budko S and Canfield P 2001 Phys. Rev. B 64 100509 [50] Mialitsin A, Dennis B, Zhigadlo N, Karpinski J and Blumberg G 2007 Phys. Rev. B 75 020509 [51] Renker B, Schober H, Adelmann P, Bohnen P, Ernst D, Heid R, Schweiss P and Wolf T 2003 J. Low Temp. Phys. 131 411 23 https://orcid.org/0000-0003-4829-1508 https://orcid.org/0000-0003-4829-1508 https://orcid.org/0000-0001-9392-7507 https://orcid.org/0000-0001-9392-7507 https://orcid.org/0000-0002-6673-7043 https://orcid.org/0000-0002-6673-7043 https://doi.org/10.1016/0375-9601(79)90111-7 https://doi.org/10.1016/0375-9601(79)90111-7 https://doi.org/10.1088/0022-3719/14/17/010 https://doi.org/10.1088/0022-3719/14/17/010 https://doi.org/10.1209/0295-5075/4/11/016 https://doi.org/10.1209/0295-5075/4/11/016 https://doi.org/10.1007/BF01303701 https://doi.org/10.1007/BF01303701 https://doi.org/10.1103/PhysRevB.53.R534 https://doi.org/10.1103/PhysRevB.53.R534 https://doi.org/10.1103/PhysRevLett.79.4894 https://doi.org/10.1103/PhysRevLett.79.4894 https://doi.org/10.3390/condmat4040087 https://doi.org/10.3390/condmat4040087 https://doi.org/10.1143/JPSJ.80.094712 https://doi.org/10.1143/JPSJ.80.094712 https://doi.org/10.7566/JPSJ.82.114708 https://doi.org/10.7566/JPSJ.82.114708 https://doi.org/10.1103/PhysRevB.102.174506 https://doi.org/10.1103/PhysRevB.102.174506 https://doi.org/10.1103/PhysRevB.106.139901 https://doi.org/10.1103/PhysRevB.106.139901 https://doi.org/10.1103/PhysRevB.99.144523 https://doi.org/10.1103/PhysRevB.99.144523 https://doi.org/10.1103/PhysRevResearch.2.043428 https://doi.org/10.1103/PhysRevResearch.2.043428 https://doi.org/10.1007/s11433-021-1748-7 https://doi.org/10.1007/s11433-021-1748-7 https://doi.org/10.1103/PhysRevB.100.184505 https://doi.org/10.1103/PhysRevB.100.184505 https://doi.org/10.1088/0034-4885/31/1/303 https://doi.org/10.1088/0034-4885/31/1/303 https://arxiv.org/abs/2207.00071 https://doi.org/10.1063/5.0050159 https://doi.org/10.1063/5.0050159 https://doi.org/10.1103/PhysRevMaterials.3.065601 https://doi.org/10.1103/PhysRevMaterials.3.065601 https://doi.org/10.1103/PhysRevLett.104.195501 https://doi.org/10.1103/PhysRevLett.104.195501 https://doi.org/10.1103/PhysRevB.77.100201 https://doi.org/10.1103/PhysRevB.77.100201 https://doi.org/10.1103/PhysRevB.71.172201 https://doi.org/10.1103/PhysRevB.71.172201 https://doi.org/10.1103/PhysRevB.14.823 https://doi.org/10.1103/PhysRevB.14.823 https://doi.org/10.1103/PhysRevLett.124.215501 https://doi.org/10.1103/PhysRevLett.124.215501 https://doi.org/10.1103/PhysRevB.105.014204 https://doi.org/10.1103/PhysRevB.105.014204 https://doi.org/10.1103/PhysRev.117.1245 https://doi.org/10.1103/PhysRev.117.1245 https://doi.org/10.3103/S002565441103006X https://doi.org/10.3103/S002565441103006X https://doi.org/10.1088/1361-648x/abdb68 https://doi.org/10.1088/1361-648x/abdb68 https://doi.org/10.1103/PhysRev.148.845 https://doi.org/10.1103/PhysRev.148.845 https://doi.org/10.1103/PhysRevB.28.1928 https://doi.org/10.1103/PhysRevB.28.1928 https://doi.org/10.7566/JPSJ.84.024708 https://doi.org/10.7566/JPSJ.84.024708 https://doi.org/10.1088/1751-8113/45/47/473001 https://doi.org/10.1088/1751-8113/45/47/473001 https://doi.org/10.1103/RevModPhys.95.011001 https://doi.org/10.1103/RevModPhys.95.011001 https://doi.org/10.1103/PhysRevLett.122.251601 https://doi.org/10.1103/PhysRevLett.122.251601 https://doi.org/10.1103/PhysRevB.29.2051 https://doi.org/10.1103/PhysRevB.29.2051 https://doi.org/10.1103/RevModPhys.79.175 https://doi.org/10.1103/RevModPhys.79.175 https://doi.org/10.1088/0370-1328/84/2/311 https://doi.org/10.1088/0370-1328/84/2/311 https://doi.org/10.1098/rspa.1965.0179 https://doi.org/10.1098/rspa.1965.0179 https://doi.org/10.1103/PhysRevLett.88.087001 https://doi.org/10.1103/PhysRevLett.88.087001 https://doi.org/10.1016/j.physc.2007.02.011 https://doi.org/10.1016/j.physc.2007.02.011 https://doi.org/10.1103/PhysRevB.64.100509 https://doi.org/10.1103/PhysRevB.64.100509 https://doi.org/10.1103/PhysRevB.75.020509 https://doi.org/10.1103/PhysRevB.75.020509 https://doi.org/10.1023/A:1022926530542 https://doi.org/10.1023/A:1022926530542 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review [52] Pickard C J, Errea I and Eremets M I 2020 Annu. Rev. Condens. Matter Phys. 11 57 [53] Drozdov A P, Eremets M I, Troyan I A, Ksenofontov V and Shylin S I 2015 Nature 525 73 [54] Snider E, Dasenbrock-Gammon N, McBride R, Wang X, Meyers N, Lawler K V, Zurek E, Salamat A and Dias R P 2021 Phys. Rev. Lett. 126 117003 [55] Yesudhas S et al 2021 Inorg. Chem. 60 9320 [56] Ignatov A, Kumar A, Lubik P, Yuan R, Guo W, Wang N, Rabe K and Blumberg G 2012 Phys. Rev. B 86 134107 [57] Lazarević N, Lei H, Petrovic C and Popović Z V 2011 Phys. Rev. B 84 214305 [58] Burkel E 2000 Rep. Prog. Phys. 63 171 [59] Krisch M and Sette F 2006 Light Scattering in Solid IX (Springer) pp 317–70 [60] Baron A Q 2009 arXiv:0910.5764 [61] Shukla A et al 2003 Phys. Rev. Lett. 90 095506 [62] Baron A, Uchiyama H, Tanaka Y, Tsutsui S, Ishikawa D, Lee S, Heid R, Bohnen K-P, Tajima S and Ishikawa T 2004 Phys. Rev. Lett. 92 197004 [63] d’Astuto M, Calandra M, Reich S, Shukla A, Lazzeri M, Mauri F, Karpinski J, Zhigadlo N, Bossak A and Krisch M 2007 Phys. Rev. B 75 174508 [64] Novko D 2018 Phys. Rev. B 98 041112 [65] Novko D, Caruso F, Draxl C and Cappelluti E 2020 Phys. Rev. Lett. 124 077001 [66] Uchiyama H, Baron A, Tsutsui S, Tanaka Y, Hu W-Z, Yamamoto A, Tajima S and Endoh Y 2004 Phys. Rev. Lett. 92 197005 [67] Fukuda T, Mizuki J, Ikeuchi K, Yamada K, Baron A and Tsutsui S 2005 Phys. Rev. B 71 060501 [68] Graf J, d’Astuto M, Jozwiak C, Garcia D, Saini N L, Krisch M, Ikeuchi K, Baron A, Eisaki H and Lanzara A 2008 Phys. Rev. Lett. 100 227002 [69] Le Tacon M, Bosak A, Souliou S, Dellea G, Loew T, Heid R, Bohnen K, Ghiringhelli G, Krisch M and Keimer B 2014 Nat. Phys. 10 52 [70] Fukuda T et al 2008 J. Phys. Soc. Japan 77 103715 [71] Kuroiwa S, Baron A, Muranaka T, Heid R, Bohnen K-P and Akimitsu J 2008 Phys. Rev. B 77 140503 [72] Li H et al 2021 Phys. Rev. X 11 031050 [73] Vig S et al 2017 SciPost Phys. 3 026 [74] Ibach H and Mills D L 2013 Electron Energy Loss Spectroscopy and Surface Vibrations (Academic) [75] Phelps R, Akavoor P, Kesmodel L, Barr A, Markert J, Ma J, Kelley R and Onellion M 1994 Phys. Rev. B 50 6526 [76] Mitrano M et al 2018 Proc. Natl Acad. Sci. 115 5392 [77] Huang E W, Limtragool K, Setty C, Husain A A, Mitrano M, Abbamonte P and Phillips P W 2021 Phys. Rev. B 103 035121 [78] Setty C, Padhi B, Limtragool K, Abbamonte P, Husain A A, Mitrano M and Phillips P W 2018 arXiv:1803.05439 [79] Sahadev N, Biswas D N, Srinivas K, Manfrinetti P, Palenzona A and Maiti K 2012 AIP Conf. Proc. 1447 841–2 [80] Wang Z et al 2017 Nat. Phys. 13 799 [81] Zhang S et al 2016 Phys. Rev. B 94 081116 [82] Gilvarry J J 1957 J. Appl. Phys. 28 1253 [83] Gilvarry J J 1956 Phys. Rev. 102 331 [84] Gilvarry J J 1956 Phys. Rev. 102 325 [85] Grüneisen E 1912 Ann. Phys., Lpz. 344 257 [86] Wang R J, Wang W H, Li F Y, Wang L M, Zhang Y, Wen P and Wang J F 2003 J. Phys.: Condens. Matter 15 603 [87] Douglas J F and Xu W-S 2021 Macromolecules 54 3247 [88] Gilvarry J J 1955 J. Chem. Phys. 23 1925 [89] Macdonald J R 1966 Rev. Mod. Phys. 38 669 [90] Anderson O L 1966 J. Phys. Chem. Solids 27 547 [91] Barker Jr R 1967 J. Appl. Phys. 38 4234 [92] Sharma B 1983 Polymer 24 314 [93] Anderson O L 2000 Geophys. J. Int. 143 279 [94] de la Roza A O, Luaña V and Flórez M 2016 An Introduction to High-Pressure Science and Technology (CRC Press) pp 25–72 [95] Waller I and Froman P 1952 Ark. Fys. 4 [96] Elliott R and Thorpe M 1967 Proc. Phys. Soc. 91 903 [97] Hudson B S 2006 Vib. Spectrosc. 42 25 [98] Axe J and Shirane G 1973 Phys. Rev. B 8 1965 [99] Cowley R and Woods A 1971 Can. J. Phys. 49 177 [100] Reichardt W et al 1994 J. Supercond. 7 399 [101] Chou H, Yamada K, Axe J, Shapiro S, Shirane G, Tanaka I, Yamane K and Kojima H 1990 Phys. Rev. B 42 4272 [102] Mittal R et al 2009 Phys. Rev. Lett. 102 217001 [103] Yamaura J-I et al 2019 Phys. Rev. B 99 220505 [104] Pintschovius L, Weber F, Reichardt W, Kreyssig A, Heid R, Reznik D, Stockert O and Hradil K 2008 Pramana 71 687 [105] Weber F, Pintschovius L, Reichardt W, Heid R, Bohnen K-P, Kreyssig A, Reznik D and Hradil K 2014 Phys. Rev. B 89 104503 [106] Yildirim T et al 2001 Phys. Rev. Lett. 87 037001 [107] Muranaka T, Yokoo T, Arai M, Margiolaki E, Brigatti K, Prassides K, Petrenko O and Akimitsu J 2002 J. Phys. Soc. Japan 71 338 [108] Cabrera N, Celli V and Manson R 1969 Phys. Rev. Lett. 22 346 [109] Manson R and Celli V 1971 Surf. Sci. 24 495 [110] Benedek G and Toennies J P 1994 Surf. Sci. 299 587 [111] Sklyadneva I Y, Benedek G, Chulkov E V, Echenique P M, Heid R, Bohnen K-P and Toennies J 2011 Phys. Rev. Lett. 107 095502 [112] Benedek G, Bernasconi M, Bohnen K-P, Campi D, Chulkov E V, Echenique P M, Heid R, Sklyadneva I Y and Toennies J P 2014 Phys. Chem. Chem. Phys. 16 7159 [113] Benedek G, Manson J R, Miret-Artés S, Ruckhofer A, Ernst W E, Tamtögl A and Toennies J P 2020 Condens. Matter 5 79 [114] Anemone G et al 2021 npj 2D Mater. Appl. 5 25 [115] Gester M, Kleinhesselink D, Ruggerone P and Toennies J 1994 Phys. Rev. B 49 5777 [116] Zhu X, Santos L, Sankar R, Chikara S, Howard C, Chou F, Chamon C and El-Batanouny M 2011 Phys. Rev. Lett. 107 186102 [117] Zhu X, Santos L, Howard C, Sankar R, Chou F, Chamon C and El-Batanouny M 2012 Phys. Rev. Lett. 108 185501 [118] Stenull O, Kane C and Lubensky T 2016 Phys. Rev. Lett. 117 068001 [119] Liu Y, Chen X and Xu Y 2020 Adv. Funct. Mater. 30 1904784 [120] Di Miceli D, Setty C and Zaccone A 2022 arXiv:2203.03499 [121] Naidyuk Y G and Yanson I K 2005 Point-Contact Spectroscopy vol 145 (Springer Science & Business Media) [122] Jansen A G M, Van Gelder A and Wyder P 1980 J. Phys. C: Solid State Phys. 13 6073 [123] Naidyuk Y G and Yanson I K 2003 arXiv:physics/0312016 [physics.pop-ph] [124] Lee W-C, Park W K, Arham H Z, Greene L H and Phillips P 2015 Proc. Natl Acad. Sci. 112 651 [125] Yanson I 1983 Sov. J. Low Temp. Phys. 9 343 [126] Kulik I 1992 Sov. J. Low Temp. 18 450 [127] Naidyuk Y G, Yanson I, Kvitnitskaya O, Lee S and Tajima S 2003 Phys. Rev. Lett. 90 197001 [128] Yanson I and Naidyuk Y G 2004 Low Temp. Phys. 30 261 [129] Samuely P, Szabo P, Kacmarcik J, Klein T and Jansen A 2003 Physica C 385 244 [130] Dahm T, Hinkov V, Borisenko S V, Kordyuk A A, Zabolotnyy V B, Fink J, Büchner B, Scalapino D J, Hanke W and Keimer B 2009 Nat. Phys. 5 217 [131] Bloembergen N, Purcell E M and Pound R V 1948 Phys. Rev. 73 679 24 https://doi.org/10.1146/annurev-conmatphys-031218-013413 https://doi.org/10.1146/annurev-conmatphys-031218-013413 https://doi.org/10.1038/nature14964 https://doi.org/10.1038/nature14964 https://doi.org/10.1103/PhysRevLett.126.117003 https://doi.org/10.1103/PhysRevLett.126.117003 https://doi.org/10.1021/acs.inorgchem.0c03795 https://doi.org/10.1021/acs.inorgchem.0c03795 https://doi.org/10.1103/PhysRevB.86.134107 https://doi.org/10.1103/PhysRevB.86.134107 https://doi.org/10.1103/PhysRevB.84.214305 https://doi.org/10.1103/PhysRevB.84.214305 https://doi.org/10.1088/0034-4885/63/2/203 https://doi.org/10.1088/0034-4885/63/2/203 https://arxiv.org/abs/0910.5764 https://doi.org/10.1103/PhysRevLett.90.095506 https://doi.org/10.1103/PhysRevLett.90.095506 https://doi.org/10.1103/PhysRevLett.92.197004 https://doi.org/10.1103/PhysRevLett.92.197004 https://doi.org/10.1103/PhysRevB.75.174508 https://doi.org/10.1103/PhysRevB.75.174508 https://doi.org/10.1103/PhysRevB.98.041112 https://doi.org/10.1103/PhysRevB.98.041112 https://doi.org/10.1103/PhysRevLett.124.077001 https://doi.org/10.1103/PhysRevLett.124.077001 https://doi.org/10.1103/PhysRevLett.92.197005 https://doi.org/10.1103/PhysRevLett.92.197005 https://doi.org/10.1103/PhysRevB.71.060501 https://doi.org/10.1103/PhysRevB.71.060501 https://doi.org/10.1103/PhysRevLett.100.227002 https://doi.org/10.1103/PhysRevLett.100.227002 https://doi.org/10.1038/nphys2805 https://doi.org/10.1038/nphys2805 https://doi.org/10.1143/JPSJ.77.103715 https://doi.org/10.1143/JPSJ.77.103715 https://doi.org/10.1103/PhysRevB.77.140503 https://doi.org/10.1103/PhysRevB.77.140503 https://doi.org/10.1103/PhysRevX.11.031050 https://doi.org/10.1103/PhysRevX.11.031050 https://doi.org/10.21468/SciPostPhys.3.4.026 https://doi.org/10.21468/SciPostPhys.3.4.026 https://doi.org/10.1103/PhysRevB.50.6526 https://doi.org/10.1103/PhysRevB.50.6526 https://doi.org/10.1073/pnas.1721495115 https://doi.org/10.1073/pnas.1721495115 https://doi.org/10.1103/PhysRevB.103.035121 https://doi.org/10.1103/PhysRevB.103.035121 https://arxiv.org/abs/1803.05439 https://doi.org/10.1063/1.4710265 https://doi.org/10.1063/1.4710265 https://doi.org/10.1038/nphys4107 https://doi.org/10.1038/nphys4107 https://doi.org/10.1103/PhysRevB.94.081116 https://doi.org/10.1103/PhysRevB.94.081116 https://doi.org/10.1063/1.1722628 https://doi.org/10.1063/1.1722628 https://doi.org/10.1103/PhysRev.102.331 https://doi.org/10.1103/PhysRev.102.331 https://doi.org/10.1103/PhysRev.102.325 https://doi.org/10.1103/PhysRev.102.325 https://doi.org/10.1002/andp.19123441202 https://doi.org/10.1002/andp.19123441202 https://doi.org/10.1088/0953-8984/15/3/324 https://doi.org/10.1088/0953-8984/15/3/324 https://doi.org/10.1021/acs.macromol.1c00075 https://doi.org/10.1021/acs.macromol.1c00075 https://doi.org/10.1063/1.1740606 https://doi.org/10.1063/1.1740606 https://doi.org/10.1103/RevModPhys.38.669 https://doi.org/10.1103/RevModPhys.38.669 https://doi.org/10.1016/0022-3697(66)90199-5 https://doi.org/10.1016/0022-3697(66)90199-5 https://doi.org/10.1063/1.1709110 https://doi.org/10.1063/1.1709110 https://doi.org/10.1016/0032-3861(83)90269-0 https://doi.org/10.1016/0032-3861(83)90269-0 https://doi.org/10.1046/j.1365-246X.2000.01266.x https://doi.org/10.1046/j.1365-246X.2000.01266.x https://doi.org/10.1088/0370-1328/91/4/318 https://doi.org/10.1088/0370-1328/91/4/318 https://doi.org/10.1016/j.vibspec.2006.04.014 https://doi.org/10.1016/j.vibspec.2006.04.014 https://doi.org/10.1103/PhysRevB.8.1965 https://doi.org/10.1103/PhysRevB.8.1965 https://doi.org/10.1139/p71-021 https://doi.org/10.1139/p71-021 https://doi.org/10.1007/BF00724577 https://doi.org/10.1007/BF00724577 https://doi.org/10.1103/PhysRevB.42.4272 https://doi.org/10.1103/PhysRevB.42.4272 https://doi.org/10.1103/PhysRevLett.102.217001 https://doi.org/10.1103/PhysRevLett.102.217001 https://doi.org/10.1103/PhysRevB.99.220505 https://doi.org/10.1103/PhysRevB.99.220505 https://doi.org/10.1007/s12043-008-0257-z https://doi.org/10.1007/s12043-008-0257-z https://doi.org/10.1103/PhysRevB.89.104503 https://doi.org/10.1103/PhysRevB.89.104503 https://doi.org/10.1103/PhysRevLett.87.037001 https://doi.org/10.1103/PhysRevLett.87.037001 https://doi.org/10.1143/JPSJS.71S.338 https://doi.org/10.1143/JPSJS.71S.338 https://doi.org/10.1103/PhysRevLett.22.346 https://doi.org/10.1103/PhysRevLett.22.346 https://doi.org/10.1016/0039-6028(71)90277-9 https://doi.org/10.1016/0039-6028(71)90277-9 https://doi.org/10.1016/0039-6028(94)90683-1 https://doi.org/10.1016/0039-6028(94)90683-1 https://doi.org/10.1103/PhysRevLett.107.095502 https://doi.org/10.1103/PhysRevLett.107.095502 https://doi.org/10.1039/c3cp54834a https://doi.org/10.1039/c3cp54834a https://doi.org/10.3390/condmat5040079 https://doi.org/10.3390/condmat5040079 https://doi.org/10.1038/s41699-021-00204-5 https://doi.org/10.1038/s41699-021-00204-5 https://doi.org/10.1103/PhysRevB.49.5777 https://doi.org/10.1103/PhysRevB.49.5777 https://doi.org/10.1103/PhysRevLett.107.186102 https://doi.org/10.1103/PhysRevLett.107.186102 https://doi.org/10.1103/PhysRevLett.108.185501 https://doi.org/10.1103/PhysRevLett.108.185501 https://doi.org/10.1103/PhysRevLett.117.068001 https://doi.org/10.1103/PhysRevLett.117.068001 https://doi.org/10.1002/adfm.201904784 https://doi.org/10.1002/adfm.201904784 https://arxiv.org/abs/2203.03499 https://doi.org/10.1088/0022-3719/13/33/009 https://doi.org/10.1088/0022-3719/13/33/009 http://arxiv.org/abs/physics/0312016 https://doi.org/10.1073/pnas.1422509112 https://doi.org/10.1073/pnas.1422509112 https://doi.org/10.1103/PhysRevLett.90.197001 https://doi.org/10.1103/PhysRevLett.90.197001 https://doi.org/10.1063/1.1704612 https://doi.org/10.1063/1.1704612 https://doi.org/10.1016/S0921-4534(02)02344-4 https://doi.org/10.1016/S0921-4534(02)02344-4 https://doi.org/10.1038/nphys1180 https://doi.org/10.1038/nphys1180 https://doi.org/10.1103/PhysRev.73.679 https://doi.org/10.1103/PhysRev.73.679 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review [132] Dioguardi A et al 2013 Phys. Rev. Lett. 111 207201 [133] Imai T and Hirota K 2018 J. Phys. Soc. Japan 87 025004 [134] Hunt A, Singer P, Cederström A and Imai T 2001 Phys. Rev. B 64 134525 [135] Baek S-H, Loew T, Hinkov V, Lin C, Keimer B, Büchner B and Grafe H-J 2012 Phys. Rev. B 86 220504 [136] Wu T et al 2013 Phys. Rev. B 88 014511 [137] Julien M-H, Borsa F, Carretta P, Horvatić M, Berthier C and Lin C 1999 Phys. Rev. Lett. 83 604 [138] Mitrović V, Julien M-H, De Vaulx C, Horvatić M, Berthier C, Suzuki T and Yamada K 2008 Phys. Rev. B 78 014504 [139] Sternlieb B et al 1990 Phys. Rev. B 41 8866 [140] Rømer A T et al 2013 Phys. Rev. B 87 144513 [141] Ryu H, Wang K, Opacic M, Lazarevic N, Warren J B, Popovic Z V, Bozin E S and Petrovic C 2015 Phys. Rev. B 92 174522 [142] Ryu H et al 2015 Phys. Rev. B 91 184503 [143] Chou F, Belk N, Kastner M, Birgeneau R and Aharony A 1995 Phys. Rev. Lett. 75 2204 [144] Wakimoto S, Ueki S, Endoh Y and Yamada K 2000 Phys. Rev. B 62 3547 [145] Grinenko V et al 2012 arXiv:1203.1585 [146] Yadav C and Paulose P 2010 J. Appl. Phys. 107 083908 [147] Mydosh J 2015 Rep. Prog. Phys. 78 052501 [148] Maradudin A and Fein A 1962 Phys. Rev. 128 2589 [149] Cowley R 1965 J. Physique 26 659 [150] Singh A and Singh N P 2023 Mater. Today Proc. 79 39 [151] Bianco R and Errea I 2023 arXiv:2303.02621 [152] Gonze X and Vigneron J-P 1989 Phys. Rev. B 39 13120 [153] Debernardi A and Baroni S 1994 Solid State Commun. 91 813 [154] Debernardi A, Baroni S and Molinari E 1995 Phys. Rev. Lett. 75 1819 [155] Debernardi A 1998 Phys. Rev. B 57 12847 [156] Debernardi A, Ulrich C, Syassen K and Cardona M 1999 Phys. Rev. B 59 6774 [157] Tang X and Fultz B 2011 Phys. Rev. B 84 054303 [158] Li W, Carrete J and Mingo N 2013 Appl. Phys. Lett. 103 253103 [159] Girotto N and Novko D 2023 Phys. Rev. B 107 064310 [160] Souvatzis P, Eriksson O, Katsnelson M I and Rudin S P 2008 Phys. Rev. Lett. 100 095901 [161] Car R and Parrinello M 1985 Phys. Rev. Lett. 55 2471 [162] Car R and Parrinello M 1988 Phys. Rev. Lett. 60 204 [163] Wang C, Chan C T and Ho K 1989 Phys. Rev. B 39 8586 [164] Wang C, Chan C T and Ho K 1989 Phys. Rev. B 40 3390 [165] Wang C, Chan C T and Ho K 1990 Phys. Rev. B 42 11276 [166] Hellman O, Abrikosov I and Simak S 2011 Phys. Rev. B 84 180301 [167] Errea I, Calandra M, Pickard C J, Nelson J, Needs R J, Li Y, Liu H, Zhang Y, Ma Y and Mauri F 2015 Phys. Rev. Lett. 114 157004 [168] Born M and Hooton D J 1955 Z. Phys. 142 201 [169] Hooton D 1955 London, Edinburgh Dublin Phil. Mag. J. Sci. 46 422 [170] Koehler T R 1966 Phys. Rev. Lett. 17 89 [171] Siciliano A, Monacelli L, Caldarelli G and Mauri F 2023 Phys. Rev. B 107 174307 [172] Monacelli L and Mauri F 2021 Phys. Rev. B 103 104305 [173] Paulatto L, Errea I, Calandra M and Mauri F 2015 Phys. Rev. B 91 054304 [174] Errea I, Calandra M and Mauri F 2013 Phys. Rev. Lett. 111 177002 [175] Errea I, Calandra M and Mauri F 2014 Phys. Rev. B 89 064302 [176] Errea I, Calandra M, Pickard C J, Nelson J R, Needs R J, Li Y, Liu H, Zhang Y, Ma Y and Mauri F 2016 Nature 532 81 [177] Tadano T and Tsuneyuki S 2018 J. Phys. Soc. Japan 87 041015 [178] Tadano T and Tsuneyuki S 2015 Phys. Rev. B 92 054301 [179] Klein M L and Horton G K 1972 J. Low Temp. Phys. 9 151 [180] Marsiglio F and Carbotte J 2008 Superconductivity (Springer) pp 73–162 [181] Caldeira A and Leggett A J 1983 Ann. Phys., NY 149 374 [182] Weiss U 2012 Quantum Dissipative Systems 4th edn (World Scientific) [183] Kleinert H 2018 Collective Classical and Quantum Fields (World Scientific) [184] Baggioli M, Setty C and Zaccone A 2020 Phys. Rev. B 101 214502 [185] Marsiglio F and Carbotte J 2008 Superconductivity (Springer) pp 73–162 [186] Allen P B and Dynes R C 1975 Phys. Rev. B 12 905 [187] Jiang C, Beneduce E, Baggioli M, Setty C and Zaccone A 2023 J. Phys.: Condens. Matter 35 164003 [188] Lee P A and Ramakrishnan T 1985 Rev. Mod. Phys. 57 287 [189] Cordero F, Paolone A, Cantelli R and Ferretti M 2001 Phys. Rev. B 64 132501 [190] Julien M-H 2003 Physica B 329 693 [191] Read N, Sachdev S and Ye J 1995 Phys. Rev. B 52 384 [192] Bray A and Moore M 1985 Phys. Rev. B 31 631 [193] Miller J and Huse D A 1993 Phys. Rev. Lett. 70 3147 [194] Dalidovich D and Phillips P 1999 Phys. Rev. B 59 11925 [195] Bennemann K-H and Ketterson J B 2008 Superconductivity: Volume 1: Conventional and Unconventional Superconductors Volume 2: Novel Superconductors (Springer Science & Business Media) [196] Hirschfeld P, Korshunov M and Mazin I 2011 Rep. Prog. Phys. 74 124508 [197] Venkataraman G 1979 Bull. Mater. Sci. 1 129 [198] Xu Y 2013 Ferroelectric Materials and Their Applications (Elsevier) [199] Gonzalo J A 2006 Effective Field Approach to Phase Transitions and Some Applications to Ferroelectrics vol 76 (World Scientific) [200] Cowley R 2012 Integr. Ferroelectr. 133 109 [201] Setty C, Baggioli M and Zaccone A 2022 Phys. Rev. B 105 L020506 [202] Edge J M, Kedem Y, Aschauer U, Spaldin N A and Balatsky A V 2015 Phys. Rev. Lett. 115 247002 [203] Koonce C S, Cohen M L, Schooley J F, Hosler W R and Pfeiffer E R 1967 Phys. Rev. 163 380 [204] Kohn W 1959 Phys. Rev. Lett. 2 393 [205] Peierls R, Peierls R, Peierls R and Press O U 1955 Quantum Theory of Solids (International Series of Monographs on Physics) (Clarendon) [206] Blackburn S, Coté M, Louie S G and Cohen M L 2011 Phys. Rev. B 84 104506 [207] Ren S, Zong H-X, Tao X-F, Sun Y-H, Sun B-A, Xue D-Z, Ding X-D and Wang W-H 2021 Nat. Commun. 12 1 [208] Bergmann G and Rainer D 1973 Z. Phys. 263 59 [209] Leroux M et al 2018 arXiv:1808.05984 [210] Fernandes R, Vavilov M and Chubukov A 2012 Phys. Rev. B 85 140512 [211] Mishra V and Hirschfeld P 2016 New J. Phys. 18 103001 [212] Ashcroft N W 1968 Phys. Rev. Lett. 21 1748 [213] Dasenbrock-Gammon N et al 2023 Nature 615 244 [214] Dogan M, Oh S and Cohen M L 2022 J. Phys.: Condens. Matter 34 15LT01 [215] Pickard C J and Needs R J 2007 Nat. Phys. 3 473 [216] Pickett W E 2022 arXiv:2204.05930 [cond-mat.supr-con] [217] Zurek E and Bi T 2019 J. Chem. Phys. 150 050901 [218] Setty C, Baggioli M and Zaccone A 2021 Phys. Rev. B 103 094519 [219] Verma A K, Modak P, Schrodi F, Aperis A and Oppeneer P M 2021 Phys. Rev. B 103 094505 [220] Hou P, Belli F, Bianco R and Errea I 2021 J. Appl. Phys. 130 175902 25 https://doi.org/10.1103/PhysRevLett.111.207201 https://doi.org/10.1103/PhysRevLett.111.207201 https://doi.org/10.7566/JPSJ.87.025004 https://doi.org/10.7566/JPSJ.87.025004 https://doi.org/10.1103/PhysRevB.64.134525 https://doi.org/10.1103/PhysRevB.64.134525 https://doi.org/10.1103/PhysRevB.86.220504 https://doi.org/10.1103/PhysRevB.86.220504 https://doi.org/10.1103/PhysRevB.88.014511 https://doi.org/10.1103/PhysRevB.88.014511 https://doi.org/10.1103/PhysRevLett.83.604 https://doi.org/10.1103/PhysRevLett.83.604 https://doi.org/10.1103/PhysRevB.78.014504 https://doi.org/10.1103/PhysRevB.78.014504 https://doi.org/10.1103/PhysRevB.41.8866 https://doi.org/10.1103/PhysRevB.41.8866 https://doi.org/10.1103/PhysRevB.87.144513 https://doi.org/10.1103/PhysRevB.87.144513 https://doi.org/10.1103/PhysRevB.92.174522 https://doi.org/10.1103/PhysRevB.92.174522 https://doi.org/10.1103/PhysRevB.91.184503 https://doi.org/10.1103/PhysRevB.91.184503 https://doi.org/10.1103/PhysRevLett.75.2204 https://doi.org/10.1103/PhysRevLett.75.2204 https://doi.org/10.1103/PhysRevB.62.3547 https://doi.org/10.1103/PhysRevB.62.3547 https://arxiv.org/abs/1203.1585 https://doi.org/10.1063/1.3392797 https://doi.org/10.1063/1.3392797 https://doi.org/10.1088/0034-4885/78/5/052501 https://doi.org/10.1088/0034-4885/78/5/052501 https://doi.org/10.1103/PhysRev.128.2589 https://doi.org/10.1103/PhysRev.128.2589 https://doi.org/10.1051/jphys:019650026011065900 https://doi.org/10.1051/jphys:019650026011065900 https://doi.org/10.1016/j.matpr.2022.08.208 https://doi.org/10.1016/j.matpr.2022.08.208 https://arxiv.org/abs/2303.02621 https://doi.org/10.1103/PhysRevB.39.13120 https://doi.org/10.1103/PhysRevB.39.13120 https://doi.org/10.1016/0038-1098(94)90654-8 https://doi.org/10.1016/0038-1098(94)90654-8 https://doi.org/10.1103/PhysRevLett.75.1819 https://doi.org/10.1103/PhysRevLett.75.1819 https://doi.org/10.1103/PhysRevB.57.12847 https://doi.org/10.1103/PhysRevB.57.12847 https://doi.org/10.1103/PhysRevB.59.6774 https://doi.org/10.1103/PhysRevB.59.6774 https://doi.org/10.1103/PhysRevB.84.054303 https://doi.org/10.1103/PhysRevB.84.054303 https://doi.org/10.1063/1.4850995 https://doi.org/10.1063/1.4850995 https://doi.org/10.1103/PhysRevB.107.064310 https://doi.org/10.1103/PhysRevB.107.064310 https://doi.org/10.1103/PhysRevLett.100.095901 https://doi.org/10.1103/PhysRevLett.100.095901 https://doi.org/10.1103/PhysRevLett.55.2471 https://doi.org/10.1103/PhysRevLett.55.2471 https://doi.org/10.1103/PhysRevLett.60.204 https://doi.org/10.1103/PhysRevLett.60.204 https://doi.org/10.1103/PhysRevB.39.8586 https://doi.org/10.1103/PhysRevB.39.8586 https://doi.org/10.1103/PhysRevB.40.3390 https://doi.org/10.1103/PhysRevB.40.3390 https://doi.org/10.1103/PhysRevB.42.11276 https://doi.org/10.1103/PhysRevB.42.11276 https://doi.org/10.1103/PhysRevB.84.180301 https://doi.org/10.1103/PhysRevB.84.180301 https://doi.org/10.1103/PhysRevLett.114.157004 https://doi.org/10.1103/PhysRevLett.114.157004 https://doi.org/10.1007/BF01329422 https://doi.org/10.1007/BF01329422 https://doi.org/10.1080/14786440408520575 https://doi.org/10.1080/14786440408520575 https://doi.org/10.1103/PhysRevLett.17.89 https://doi.org/10.1103/PhysRevLett.17.89 https://doi.org/10.1103/PhysRevB.107.174307 https://doi.org/10.1103/PhysRevB.107.174307 https://doi.org/10.1103/PhysRevB.103.104305 https://doi.org/10.1103/PhysRevB.103.104305 https://doi.org/10.1103/PhysRevB.91.054304 https://doi.org/10.1103/PhysRevB.91.054304 https://doi.org/10.1103/PhysRevLett.111.177002 https://doi.org/10.1103/PhysRevLett.111.177002 https://doi.org/10.1103/PhysRevB.89.064302 https://doi.org/10.1103/PhysRevB.89.064302 https://doi.org/10.1038/nature17175 https://doi.org/10.1038/nature17175 https://doi.org/10.7566/JPSJ.87.041015 https://doi.org/10.7566/JPSJ.87.041015 https://doi.org/10.1103/PhysRevB.92.054301 https://doi.org/10.1103/PhysRevB.92.054301 https://doi.org/10.1007/BF00654839 https://doi.org/10.1007/BF00654839 https://doi.org/10.1016/0003-4916(83)90202-6 https://doi.org/10.1016/0003-4916(83)90202-6 https://doi.org/10.1103/PhysRevB.101.214502 https://doi.org/10.1103/PhysRevB.101.214502 https://doi.org/10.1103/PhysRevB.12.905 https://doi.org/10.1103/PhysRevB.12.905 https://doi.org/10.1088/1361-648X/acbd0a https://doi.org/10.1088/1361-648X/acbd0a https://doi.org/10.1103/RevModPhys.57.287 https://doi.org/10.1103/RevModPhys.57.287 https://doi.org/10.1103/PhysRevB.64.132501 https://doi.org/10.1103/PhysRevB.64.132501 https://doi.org/10.1016/S0921-4526(02)01997-X https://doi.org/10.1016/S0921-4526(02)01997-X https://doi.org/10.1103/PhysRevB.52.384 https://doi.org/10.1103/PhysRevB.52.384 https://doi.org/10.1103/PhysRevB.31.631 https://doi.org/10.1103/PhysRevB.31.631 https://doi.org/10.1103/PhysRevLett.70.3147 https://doi.org/10.1103/PhysRevLett.70.3147 https://doi.org/10.1103/PhysRevB.59.11925 https://doi.org/10.1103/PhysRevB.59.11925 https://doi.org/10.1088/0034-4885/74/12/124508 https://doi.org/10.1088/0034-4885/74/12/124508 https://doi.org/10.1007/BF02743964 https://doi.org/10.1007/BF02743964 https://doi.org/10.1080/10584587.2012.663634 https://doi.org/10.1080/10584587.2012.663634 https://doi.org/10.1103/PhysRevB.105.L020506 https://doi.org/10.1103/PhysRevB.105.L020506 https://doi.org/10.1103/PhysRevLett.115.247002 https://doi.org/10.1103/PhysRevLett.115.247002 https://doi.org/10.1103/PhysRev.163.380 https://doi.org/10.1103/PhysRev.163.380 https://doi.org/10.1103/PhysRevLett.2.393 https://doi.org/10.1103/PhysRevLett.2.393 https://doi.org/10.1103/PhysRevB.84.104506 https://doi.org/10.1103/PhysRevB.84.104506 https://doi.org/10.1038/s41467-021-26029-w https://doi.org/10.1038/s41467-021-26029-w https://doi.org/10.1007/BF02351862 https://doi.org/10.1007/BF02351862 https://arxiv.org/abs/1808.05984 https://doi.org/10.1103/PhysRevB.85.140512 https://doi.org/10.1103/PhysRevB.85.140512 https://doi.org/10.1088/1367-2630/18/10/103001 https://doi.org/10.1088/1367-2630/18/10/103001 https://doi.org/10.1103/PhysRevLett.21.1748 https://doi.org/10.1103/PhysRevLett.21.1748 https://doi.org/10.1038/s41586-023-05742-0 https://doi.org/10.1038/s41586-023-05742-0 https://doi.org/10.1088/1361-648X/ac4c62 https://doi.org/10.1088/1361-648X/ac4c62 https://doi.org/10.1038/nphys625 https://doi.org/10.1038/nphys625 http://arxiv.org/abs/2204.05930 https://doi.org/10.1063/1.5079225 https://doi.org/10.1063/1.5079225 https://doi.org/10.1103/PhysRevB.103.094519 https://doi.org/10.1103/PhysRevB.103.094519 https://doi.org/10.1103/PhysRevB.103.094505 https://doi.org/10.1103/PhysRevB.103.094505 https://doi.org/10.1063/5.0063968 https://doi.org/10.1063/5.0063968 J. Phys.: Condens. Matter 36 (2024) 173002 Topical Review [221] Rousseau B and Bergara A 2010 Phys. Rev. B 82 104504 [222] Dangic D, Monacelli L, Bianco R, Mauri F and Errea I 2023 arXiv:2303.07962 [cond-mat.supr-con] [223] Poncé S, Margine E R, Verdi C and Giustino F 2016 Comput. Phys. Commun. 209 116 [224] Kunc K, Loa I and Syassen K 2003 Phys. Rev. B 68 094107 [225] Birch F 1947 Phys. Rev. 71 809 [226] Xu Y, Wang J-S, Duan W, Gu B-L and Li B 2008 Phys. Rev. B 78 224303 [227] Chen W, Semenok D V, Huang X, Shu H, Li X, Duan D, Cui T and Oganov A R 2021 arXiv:2101.01315 [cond-mat.supr-con] [228] Brandt N B and Ginzburg N I 1969 Sov. Phys. - Usp. 12 344 [229] Boughton R, Olsen J and Palmy C 1970 Chapter 4 Pressure effects in superconductors Progress in Low Temperature Physics (Elsevier) pp 163–203 [230] Lorenz B and Chu C 2005 High pressure effects on superconductivity Frontiers in Superconducting Materials ed A V Narlikar (Springer) pp 459–97 [231] Hiroi Z, Yamaura J-i and Hattori K 2011 J. Phys. Soc. Japan 81 011012 [232] Hattori K and Tsunetsugu H 2010 Phys. Rev. B 81 134503 [233] Setty C, Baggioli M and Zaccone A 2022 Phys. Rev. B 106 139902 [234] Müller K A and Burkard H 1979 Phys. Rev. B 19 3593 [235] Appel J 1969 Phys. Rev. 180 508 [236] Enderlein C, de Oliveira J F, Tompsett D A, Saitovitch E B, Saxena S S, Lonzarich G G and Rowley S E 2020 Nat. Commun. 11 4852 [237] Franklin J, Xu B, Davino D, Mahabir A, Balatsky A V, Aschauer U and Sochnikov I 2021 Phys. Rev. B 103 214511 [238] He X, Bansal D, Winn B, Chi S, Boatner L and Delaire O 2020 Phys. Rev. Lett. 124 145901 [239] Marsiglio F 2020 Ann. Phys., NY 417 168102 [240] Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80 885 [241] Setty C, Zhao J, Fanfarillo L, Huang E W, Hirschfeld P J, Phillips P W and Yang K 2022 arXiv:2209.10568 [242] Setty C, Fanfarillo L and Hirschfeld P 2021 arXiv:2110. 13138 [243] Setty C 2020 Phys. Rev. B 101 184506 [244] Setty C 2021 Phys. Rev. B 103 014501 [245] Phillips P W, Yeo L and Huang E W 2020 Nat. Phys. 16 1175 [246] Fomin V M 2021 Self-Rolled Micro- and Nanoarchitectures (De Gruyter) [247] Travaglino R and Zaccone A 2023 J. Appl. Phys. 133 033901 [248] Zaccone A 2023 Theory of Disordered Solids (Springer) 26 https://doi.org/10.1103/PhysRevB.82.104504 https://doi.org/10.1103/PhysRevB.82.104504 http://arxiv.org/abs/2303.07962 https://doi.org/10.1016/j.cpc.2016.07.028 https://doi.org/10.1016/j.cpc.2016.07.028 https://doi.org/10.1103/PhysRevB.68.094107 https://doi.org/10.1103/PhysRevB.68.094107 https://doi.org/10.1103/PhysRev.71.809 https://doi.org/10.1103/PhysRev.71.809 https://doi.org/10.1103/PhysRevB.78.224303 https://doi.org/10.1103/PhysRevB.78.224303 http://arxiv.org/abs/2101.01315 https://doi.org/10.1070/PU1969v012n03ABEH003900 https://doi.org/10.1070/PU1969v012n03ABEH003900 https://doi.org/10.1143/JPSJ.81.011012 https://doi.org/10.1143/JPSJ.81.011012 https://doi.org/10.1103/PhysRevB.81.134503 https://doi.org/10.1103/PhysRevB.81.134503 https://doi.org/10.1103/PhysRevB.106.139902 https://doi.org/10.1103/PhysRevB.106.139902 https://doi.org/10.1103/PhysRevB.19.3593 https://doi.org/10.1103/PhysRevB.19.3593 https://doi.org/10.1103/PhysRev.180.508 https://doi.org/10.1103/PhysRev.180.508 https://doi.org/10.1038/s41467-020-18438-0 https://doi.org/10.1038/s41467-020-18438-0 https://doi.org/10.1103/PhysRevB.103.214511 https://doi.org/10.1103/PhysRevB.103.214511 https://doi.org/10.1103/PhysRevLett.124.145901 https://doi.org/10.1103/PhysRevLett.124.145901 https://doi.org/10.1016/j.aop.2020.168102 https://doi.org/10.1016/j.aop.2020.168102 https://doi.org/10.1103/RevModPhys.80.885 https://doi.org/10.1103/RevModPhys.80.885 https://arxiv.org/abs/2209.10568 https://arxiv.org/abs/2110.13138 https://arxiv.org/abs/2110.13138 https://doi.org/10.1103/PhysRevB.101.184506 https://doi.org/10.1103/PhysRevB.101.184506 https://doi.org/10.1103/PhysRevB.103.014501 https://doi.org/10.1103/PhysRevB.103.014501 https://doi.org/10.1038/s41567-020-0988-4 https://doi.org/10.1038/s41567-020-0988-4 https://doi.org/10.1063/5.0132820 https://doi.org/10.1063/5.0132820 Anharmonic theory of superconductivity and its applications to emerging quantum materials 1. Introduction: anharmonicity and superconductivity 1.1. Historical perspective: anharmonicity and superconductivity 1.2. Boson damping mechanisms 1.2.1. Akhiezer damping of acoustic phonons. 1.2.2. Klemens damping of optical phonons. 1.3. The nature of the bosonic mediator 2. Damped bosons: experimental probes 2.1. Raman scattering 2.2. Inelastic x-ray scattering 2.3. Electron energy loss spectroscopy (EELS) 2.4. Probes of Grüneisen constant 2.5. Inelastic neutron scattering (INS) 2.6. Atomic scattering 2.7. Point contact spectroscopy (PCS) 2.8. Spin based techniques 3. Phonon damping from first principles 4. Damped bosons: minimal theory of superconductivity 4.1. BCS theory: acoustic phonons 4.2. BCS theory: optical phonons 4.3. Eliashberg theory for damped phonons 4.4. BCS theory: glassy spins 4.5. Optical soft mode instabilities and structural transitions 4.6. Kohn-like soft phonon instabilities 5. Applications to emerging quantum materials 5.1. Cuprates 5.2. Hydrides 5.3. The case of TlInTe2 5.4. SrTiO3 and BaTiO3 6. Outlook and conclusions 6.1. Open issues 6.2. Summary References