Heterogeneous Elasticity: The tale of the boson peak
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Heterogeneous Elasticity: The tale of the boson peak
Walter Schirmacher and Giancarlo Ruocco , Institut f ¨ u r Physik, Universit ¨ a t Mainz, Staudinger Weg 7, D-55099 Mainz, Germany Fondazione Istituto Italiano di Tecnologia (IIT), Center for Life Nano Science,Viale Regina Elena 291, I00161 Roma, Italy and Department of Physics, University of Rome ‘La Sapienza’, Piazzale Aldo Moro, 5, I00185, Rome, Italy
The vibrational anomalies of glasses, in particular the boson peak, are addressed from the stand-point of heterogeneous elasticity, namely the spatial fluctuations of elastic constants caused by thestructural disorder of the amorphous materials.In the first part of this review article a mathematical analogy between diffusive motion in adisordered environment and a scalar simplification of vibrational motion under the same conditionis emploited. We demonstrate that the disorder-induced long-time tails of diffusion correspondto the Rayleigh scattering law in the vibrational system and that the cross-over from normal toanomalous diffusion corresponds to the boson peak. The anomalous motion arises as soon as thedisorder-induced self-energy exceeds the frequency-independent diffusivity/elasticity. For this modela variational scheme is emploited for deriving two mean-field theories of disorder, the self-consistentBorn approximation (SCBA) and coherent-potential approximation (CPA). The former applies if thefluctuations are weak and Gaussian, the latter applies for stronger and non-Gaussian fluctuations.In the second part the vectorial theory of heterogenous elasticity is presented and solved in SCBAand CPA, introduced for the scalar model. Both approaches predict and explain the boson-peak andthe associated anomalies, namely a dip in the acoustic phase velocity and a characteristic strongincrease of the acoustic attenuation below the boson peak. Explicit expressions for the density ofstates and the inelastic Raman, neutron and X-ray scattering laws are given. Recent conflictingways of explaining the boson-peak anomalies are discussed.
I. INTRODUCTION
The vibrational properties of disordered solids arequite different from those of crystals [1, 2]. While theharmonic vibrational spectra of crystals are given by thedispersion relations obtained from the crystalline sym-metry groups, those of amorphous solids, in particularglasses exhibit anomalous, continuous spectra, which area matter of debate since 60 years [2–7].The first evidence of something unusual in the vibra-tional spectra of glasses came from Raman scattering[8–13]. In the THz or 100 wavenumber regime one ob-served a broad maximum, where usually either no inten-sity or very sharp peaks due to low-lying optical modeswere observed. Because this maximum obeyed the fre-quency/temperature dependence of the Bose function n ( ω )+1 = [1 − exp {− ¯ hω/k B T } ] − ( ω/ π is the frequencyand k B T is the Boltzmann constant times the tempera-ture) one called this maximum “boson peak” [14]. Herewe note that because the Raman scattering intensity isproportional to n ( ω ) + 1 times the Raman spectral func-tion χ ′′ R ( ω ) [13, 15, 16], the latter must be temperatureindependent if the entire temperature dependence comesfrom the Bose function. This points to a harmonic ori-gin of the boson peak and questions all interpretations interms of anharmonic interactions.A maximum in the frequency range ∼ ∼ g ( ω ) with respect to Debye’s g ( ω ) ∝ ω law,which appears as a peak in the reduced DOS g ( ω ) /ω . This was confirmed experimentally by means of inelasticincoherent neutron [18] and nuclear scattering [19], whichboth measures directly the DOS. Further, the boson peakturned out to be related to a maximum in the “reduced”specific heat C ( T ) /T in the 10K regime [18, 20], which isalso called “boson peak”. Right at the boson-peak tem-perature the thermal conductivity of the same glasses ex-hibits a characteristic shoulder or dip in its temperaturedependence [21], which turned out to be an “upside-downboson peak” [22].These are not the only low-temperature thermalanomalies of glasses. Below the boson peak, in the ∼ T law,but varies approximately linearly with temperature. Thethermal conductivity varies quadratically in this regime.These findings have been attributed to the existence ofbistable structural arrangements, which allow for tunnel-ing between the two positions, giving rise to a tunnelsplitting (two-level systems (TLS), tunneling systems)[23–27]. If the energy separations of the TLS are as-sumed to have a broad distribution, the 1K anomaliescan be explained. Independent evidence for the existenceof the TLS comes from ultrasonic and nuclear magneticresonance data [28, 29].Inelastic X-ray scattering contributed more anomalousfeatures in the boson-peak regime [30]. In particular thegroup velocity of longitudinal sound (dispersion of theBrillouin line position) was found to exhibit a minimumat the wavenumber corresponding to the boson peak, andthe sound attenuation (width of the Brillouin line) wasfound to increase strongly with frequency in the boson-peak regime (Rayleigh scattering) [31–36]. These anoma-lies were confirmed in computer simulations [37, 38]. Wecall all these features the boson-peak related anomalies.The theoretical interpretation of the boson peak has avery diverse history and gave rise to several controversies[3]. In one line of argumentation for explaining the boson-peak related anomalies the authors tried to find a clas-sical analogon of the tunneling model, the soft-potentialmodel [39–41]. The bistable configurations, which can bedescribed by an anharmonic double-well potential with arather shallow barrier between the wells, were assumed tohave a distribution of their characteristic parameters, inparticular the second-order coefficient. It was found thatin general the statistics of the vibrational excitations ofsuch defect potentials would give rise to a constant den-sity of eigenvalues g ( ω ), which leads to a DOS g ( ω ) ∝ ω .On the other hand, so the argument, at lower frequencies(below a cross-over frequency ω c ) the balance betweenunstable configurations with negative second-order coef-ficients and the anharmonic fourth-order ones give rise toa defect DOS ∝ ω . Therefore the reduced DOS g ( ω ) /ω increases ∝ ω below ω c and decreases ∝ ω − above ω c .In between, i.e. near ω c is then the boson peak [39, 42].Arguments that the boson peak, by its temperaturecharacteristics, is not an anharmonic phenomenon weremet by the authors of the soft-potential model by notingthat the anharmonic interaction only acts in producingthe soft configurations in the quenching process [43]. Inthe quenched state they are supposed to act like localharmonic oscillators, similar to heavy-mass atoms, cou-pled to the acoustic harmonic degrees of freedom [44–47].Another attempt to explain the boson-peak anomalywas the phonon-fracton model [48–52]. It was postulatedthat disordered solids exhibit a certain degree of fractalstructure. A fractal is a self-similar structure [53, 54],which has a non-integer dimensionality D , which issmaller than the embedding dimensionality d . Real frac-tals like sponges or trees have a smallest and largest scale,in which the self-similarity holds. The smallest scale ise.g. the smallest pore diameter of a sponge, the largestis the correlation length ξ . For scales larger than ξ theobject looks like an ordinary material in which the massscales as L d , where L is the size. For scales smaller than ξ the mass scales as L D . Alexander and Orbach [48]have shown that the vibrational degrees of freedom scalewith a fractal dimensionality d s < D (spectral dimen-sionality) and that the DOS of such an object obeyesa Debye law below ω ξ = 2 πv/ξ , where v is the soundvelocity. Above ω ξ the DOS behaves as g ( ω ) ∝ ω d s − .They found that in all fractal structures they investigated d s ≈ /
3. The specific model employed by the phonon-fracton supporters was a percolating lattice, i.e. a cu-bic lattice in which a certain percentage of bonds 1 − p (carrying neares-neighbor force constants) was missing.If the bond concentration p is larger than the criticalconcentration p c , which determines the connectedness ofthe structure a finite correlation length ξ ∝ ( p − p c ) ν exists ( ν is the order-parameter exponent [52]). Calcula-tions using the coherent-potential approximation (CPA) [55–57] showed that in between the Debye and the frac-ton regime an enhancement over the Debye g ( ω ) ∝ ω law was present [49, 50]. This was, for the time being,a satisfactory explanation of the boson peak. However,numerical simulations of the percolation-phonon-fractonmodel [52] showed that the phonon-fracton crossover inthe DOS of this model occurs very smooth without anyexcess over the Debye law. Obviously the excess in thecalculations [49, 50] had been an artifact of the CPA.Another argument against the phonon-fracton model ascandidate for explaining the boson-peak anomalies is that- apart from aerogels [58, 59] glasses do not show anyself-similary, which should show up (but does not) asan enhanced small-angle scattering in neutron or X-raydiffraction data.In other articles reflecting on the vibrational anomaliesof glasses the boson peak and the anomalous shoulder inthe temperature dependence of the thermal conductiv-ity were also considered to be related to Anderson lo-calization of sound [51, 60, 61]. It was observed thatnear the boson-peak frequency the mean-free path of theacoustic excitations is of the order of the sound wave-length. According to a rule, coined by Ioffe and Regelin their survey of electronic conduction in semiconduc-tors [62], the notion of a mean-free path, which impliesa wave, which is occasionally scattered by an inhomo-geneity, breaks down once the mean-free path becomesequal to the wavelength (Ioffe-Regel limit). Mott [63]conjectured that Anderson localization occurs for elec-trons near the Ioffe-Regel energy. For phonon it wasassumed [61, 64], that the crossover from extended toAnderson-localized states would take place near the Ioffe-Regel frequency and that the boson peak would mark theonset of localized states. In particular it was thought,that the presence of localized states would cause the dipin the thermal conductivity [61]. Later, more detailedtheoretical investigations showed, however, that the An-derson transition in realistic solids does not occur nearthe Ioffe-Regel crossover, but in a much higher frequencyrange near the Debye frequency [65–71].Later many researchers were intested in whether theboson-peak frequency coincides exactly with the Ioffe-Regel frequency, given by the implicit relation ω IR = π Γ( ω IR ), where Γ( ω ) is the sound attenuation coefficient[72, 73]. Γ( ω ) and hence ω IR is, of course, different forlongitudinal and transverse sound waves. It appearedthat in materials governed by hard-sphere-like potentialsthe boson-peak coincides with the transverse Ioffe-Regelfrequency [38, 74], whereas in network glasses with thelongitudinal one [75–78].In 1991 Schirmacher and Wagener [79] exploited themathematical analogy between a single-particle randomwalk and harmonic phonons [80] using an off-lattice ver-sion of the CPA. They demonstrated that the cross-overfrom a frequency-independent conductivity/diffusivity toa frequency-dependent one [81, 82] corresponds to theonset of the frequency dependence of the complex soundvelocity. The latter was found to lead to a boson peak inthe vibrational DOS.Because understanding this analogy is essential forgrasping the essence of the BP-related anomalies, we de-vote a whole section of the present review to this analogy.However, because the CPA in the case of the percolat-ing lattice predicted a boson peak [49, 50], which did notexist in the simulation of the same system [52] a check wasneeded, whether the resulting boson-peak enhancementof the DOS was not an artifact of the CPA like in thephonon-fracton model. Therefore in Ref. [68] the latticeversion of the CPA [83–85] was compared to a numericalcalculation for a cubic lattice with fluctuating nearest-neighbor force constants. Both simulation and CPA cal-culation showed a boson peak, and good agreement be-tween CPA and the numerical spectrum was found. Soit was demonstrated that effective-medium calculationsare reliable for investigating the influence of disorder onthe harmonic spectrum of a model solid. The breakdownof the CPA in the case of the phonon-fracton model wasobviously due to the critical fluctuations in this model,which are not generic for disordered solids.The boson peak in the model calculations of Ref. [68]was identified to be caused by very small positive andnegative force constants, and being a precursor of an in-stability, which happens for stronger disorder [69, 70].With the help of the numerical calculation in Ref.[68] it could also decided, whether the vibrational statesnear and above the boson peak were localized or ex-tended. This was achieved by means of the statistics ofthe distances between the eigenvalues. Near and abovethe boson peak they showed the so-called Gaussian-ortogonal-ensemble (GOE) statistics of random-matrixtheory [86, 87], which proves that the correspondingstates are delocalized. At high frequencies, near theupper band edghe a transition to Poissonian statisticswas observed, which is evidence for a delocalization-localization transition in this regime, in agreement withearlier estimates [65, 66]. On the other hand, the GOEstatistics is a sign for the so-called level repulsion, show-ing that each eigenvalue is non-degenerate due to theabsence of symmetries in the disordered system. As thegeneric spectrum of random matrices is not a Debye spec-trum (which is highly degenerate) the boson peak marksthe transition from a Debye to a random-matrix-typespectrum [6].The spectra calculated in Ref. [68] for a cubic latticewith very small disorder exhibited the usual van-Hovesingularities, which appear as a result of the leveling-off ofthe crystalline phonon dispersions ω ( k ) at the Brillouin-zone boundary [88]. With increasing disorder the datashowed that the sharp van-Hove peak became rounderand is shifted downwards and gradually transformed tothe low-frequency boson peak. This led Taraskin etal. [89] to the conclusion that the boson peak is justa crystal-like van-Hove peak, modified by disorder. Thisappeared as a rather unexpected conclusion, because avan-Hove singularity is a typical signature of a crystallinestructure with long-range order and was not known to exist in glasses. However, until now, the boson peak ex-planation as a glassy version of a van-Hove singularity isstill considered to be an alternative to the disorder ex-planation [90–92]. We further comment on this in sectionIV.Quite recently in an experimental study of a macro-scopic disordered model glass it was shown that thedisorder-induced maximum of the reduced DOS and thatinduced by the van-Hove mechanism are, in fact, two dif-ferent phenomena [93].The random-matrix aspect of the disorder-vibrationproblem was elaborated further in the literature [94–104],in particular by means of the euclidean random-matrixtheory [96–103].As in Ref. [68] a disorder-induced boson peak wasfound and shown to be the precursor of an instability(“phonon-saddle transition” [100]).A quite different and interesting approach to theboson-peak anomaly was worked out [105] in the frame-work of the mode-coupling theory of the glass transi-tion . This theory of glassy freezing in its original form[106, 107] describes the idealized glass transition as adynamical transition towards a non-ergodic state lead-ing to a frozen-in additional contribution to the staticlongitudinal susceptibility. This, in turn leads to a char-acteristic hump in the density fluctuation spectrum ofthe idealized glass (proportional to the neuton scatteringlaw S ( k, ω )), which was identified with the boson-peakanomaly found in neutron scattering experiments. Inter-estingly this theory already predicted the characteristicdip in the longitudinal group velocity, which was foundlater to be associated with the boson peak, as mentionedabove [31, 35, 37, 38].We feel that an important step in the understand-ing of the boson peak was achieved by working outheterogeneous-elasticity theory [22]. In this phenomeno-logical theory it is assumed that the shear modulus inordinary elasticity theory [108] is assumed to exhibit spa-tial fluctuations. The resulting stochastic equations weresolved by field-theoretical techniques [65, 109], resultingin a mean-field theory, called self-consistent Born approx-imation (SCBA). The SCBA is obtained from assuminga Gaussian distribution of the elasticity fluctuations andthat the relative width of the Gaussian is a small param-eter (“disorder parameter” γ ). Again, if the disorder pa-rameter becomes larger than a critical one, an instabilityoccurs, which is due to too many regions with negativeelastic constants.Shortly before heterogeneous-elasticity theory hadbeen worked out a series of papers appeared, in whichmolecular-dynamics simulations of glasses were investi-gated for their elastic and vibrational properties [110–113]. It turned out that, by applying external forces, in-deed heterogeneous shear deformations are present. Theauthors showed that in regions with strong deformationsthe shear response is highly non-affine, i.e. the displace-ments do not follow the direction of the applied stress. Inthe non-affine regions the local shear moduli were foundto be very small and even negative, a finding, which wasobserved also in other simulations [38, 114, 115]. Thisnicely confirmed the model assumption of heterogeneous-elasticith theory. In Ref. [38] a direct comparison be-tween the elasticity fluctuations in a simulated glass andthe theory of Ref. [22] was made and good agreementwas found. We shall comment on this article in moredetail below.In later simulations it was shown [116, 117] that the re-gions in the simulated disordered solids, which have pro-nounced non-affine response, are also regions with stronginversion-symmetry breaking. The authors introducedquantitative measures of this inversion-symmetry break-ing and found a unique correlation with the height of theboson peak. This was also found in the quoted investi-gation of a macroscopic model of a disordered solid, inwhich soft spots with strong inversion-symmetry break-ing were shown to contribute predominantly to the bosonpeak [93].An important aspect of the boson peak appearedwhen it was shown [118] with the help of heterogeneous-elasticity theory that the excess DOS ∆ g ( ω ) = g ( ω ) − g D ( ω ) with respect to the Debye DOS g D ( ω ) is pro-portional to the sound attenuation in the boson-peakfrequency range. Heterogeneous-elasticity provides anexpression for the disorder-induced sound attenuationas imaginary part of frequency-dependent elastic coef-ficients (see section ..). These enter into the spectralfunctions, and the DOS enhancement is just producedby the sound attenuation, which increases rapidly in theboson-peak frequency regime due to Rayleigh scattering.The rest of this contribution is organized as fol-lows: In section II. the mathematical analogy betweendiffusion and scalar elasticity is investigated in de-tail and two mean-field or effective-medium theories,the self-consistent Born approximation (SCBA) and thecoherent-potential approximation (CPA) for this modelare introduced and solved. A pedagogical derivation,which is more simple than the original field-theoreticalone, is presented. We show that the boson-peak re-lated anomalies are the analogon of the crossover froma frequency-independent diffusivity/conductivity to afrequency-dependent one.In section III. the full vectorial heterogeneous elastic-ity theory is presented and solved in SCBA and CPA.The salient features of the boson-peak related anomaliesof glasses are discussed with the help of these mean-fieldtheories. In section IV. we discuss recent conflicting the-ories of these anomalies. II. “SCALAR ELASTICITY” ANDDIFFUSION-VIBRATION ANALOGYA. Diffusion-vibration analogy
A simplified version of heterogeneous elasticity theory,which proved to be helpful in understanding the spectral properties of disordered solids [46, 79, 100, 119–121] isrepresented by a scalar wave equation with a spatiallyfluctuating elastic constant K ( r ) ≡ ρ e K ( r ) ( ρ is the massdensity) ∂ ∂t u ( r , t ) = ∇ e K ( r ) ∇ u ( r , t ) (1)The elastic coefficient e K ( r ), which is the square of thelocal sound velocity, v ( r ) may be sub-divided into anaverage elasticity e K = h e K i and deviations ∆ e K ( r ) e K ( r ) = v ( r ) = e K + ∆ e K ( r ) (2)If we replace the second time derivative in Eq. (1) bya first one, we arrive at a diffusion equation, which de-scribes the random walk of a particle, which encounters aspatially varying diffusivity, e.g. due to a spatially vary-ing activation energy ∂∂t n ( r , t ) = ∇ D ( r ) ∇ n ( r , t )= ∇ (cid:20) D + ∆ D ( r ) (cid:21) ∇ n ( r , t ) (3)with D , again, denoting the avarage diffusivity and∆ D ( r ) the fluctations. n ( r , t ) is the probability densityfor finding the particle within a volume element around r at time t . The Green’s function of Eq. (1) in frequencyspace obeys the equation (cid:18) − z − ∇ e K ( r ) ∇ (cid:19) G ( r , r ′ )= (cid:18) − z − ∇ (cid:2) e K + ∆ e K ( r ) (cid:3) ∇ (cid:19) G ( r , r ′ ) ≡ A [ z, r , e K ( r )] G ( r , r ′ ) = δ ( r − r ′ ) (4)with z = ω + iǫ, ǫ → +0. The operator A [ z, r , e K ( r )] is theoperator-inverse of the Green’s function. On the otherhand the Green’s function corresponding to the hetero-geneous diffusion equation (3) in frequency space is s − ∇ D ( r ) ∇ G ( r , r ′ ) = δ ( r − r ′ ) (5)with s = iω + ǫ . So all calculations done for the scalar vibration prob-lem (4) can be taken over for the diffusion problem (5)provided we identify D ↔ e K , s ↔ − z , or iω ↔ − ω . In the next three subsection we shall demonstrate bymeans of different approximation schemes (Born approx,Self-consistent Born approx. and coherent-potential ap-prox.) that the quenched glassy disorder induces a char-acteristic frequency dependence to the macroscopic dif-fusivity/elasticity. So these approximation schemes actas a coarse-graining scheme, which converts spatial fluc-tuations to frequency dependences.In this context we may distinguish between three char-acteristic scales: the microscopic scale, the mesoscopicscale and the macroscopic one. The microscopic scaleis the molecular one and may be described by micro-scopic quantum or classical equations of motions. Themesoscopic scale is a scale of 5 or 6 atomic or molec-ular diameter. This is the minimal scale at whichone may define local diffusivities or elastic constants[38, 114, 115, 121, 122], which exhibit spatial fluctua-tions in structurally disordered materials. The macro-scopic scale is the experimental one (mm or cm), inwhich the macroscopic diffusivities or elastic coefficientsare frequency-dependent.
B. Low frequency limit: Born approximation,Rayleigh scattering and long-time tails
The solution of Eq. (4) or (5) without fluctuations isgiven in k space by ( k is the wave vector correspondingto r − r ′ ≡ ˜ r ) G ( k, z ) = 1 − z + e K k (6)The disorder-averaged full Green’s function should alsoonly depend on | ˜ r | only and therefore may be representedas [120] h G ( z ) i k = G ( k , z ) = 1 − z + k ( e K − Σ( z )) ≡ − z + k Q ( z ) (7)where Σ( z ) = Σ ′ ( ω ) + i Σ ′′ ( ω ) is the self-energy func-tion, which describes the influence of the fluctuations∆ e K ( r ) or ∆ D ( r ). Here we have defined a frequency-dependent elasticity Q ( z ) = e K − Σ( z ), corresponding toa frequency-dependent diffusivity D ( s ) = D − Σ( s ). Theformer may be identified with the square of a frequency-dependent sound velocity v ( z ), i.e. Q ( z ) = v ( z ) .To lowest order in the fluctuations one obtains bystraightforward perturbation theory [120] the Born ap-proximation Σ( z ) = γ V X k k G ( k , z ) , (8)where P k ≡ V (2 π ) R d k , V is the sample volume, and γ = h ∆ e K i V c , (9)i.e. the variance of e K ( r ) times a coarse-graining volume V c , which serves to calculate the local elastic coefficient[38, 120, 122].If one imposes an upper cutoff k max in the wavenumberintegration, the integral in Eq. (8) can be done exactly.For small frequencies we obtain∆Σ( s ) = Σ( z ) − Σ(0) ∝ s / (10)For the vibrational problem s / → iω . We now show that this leads to Rayleigh’s ω scatter-ing law [123, 124]:We may be interested in the wave intensity given bythe modulus of Eq. (7) | G (˜ r , z ) | = (cid:18) π e K (cid:19) e − ˜ r/ℓ ( ω ) (11)with the mean-free path1 ℓ ( ω ) = 2Im (cid:8) ωv ( ω ) (cid:9) ≈ Σ ′′ ( ω ) ωv = γ π (cid:18) ωv (cid:19) (12)where v = q e K .Eq. (12) constitutes the Rayleigh scattering law [123,124]. It holds for harmonic excitations in the presenceof quenched disorder [103, 120], provided the disorderfluctuations do not exhibit long-range order [120, 125,126].Rayleigh scattering in Glasses is usually obscured byanharmonic sound attenuation, which prevails in the sub-THz frequency range. It has been observed experimen-tally in the THz regime in some glasses [31, 35] as wellin computer simulations [37, 38].In the diffusion problem D ( s ) is the frequency-dependent diffusivity, which can be shown [127] to be theLaplace transform of the velocity autocorrelation func-tion Z ( t ) of the moving particle D ( s ) = R ∞ dte − st Z ( t )We apply the Tauberian theorem [128]lim s → D ( s ) ∝ s − ρ ⇔ lim t →∞ D ( t ) ∝ t ρ − (13)from which we concludelim t →∞ Z ( t ) ∝ t − / , (14)a behaviour well known for particles performing a ran-dom walk in a quenched-disordered environment [129,130].On the other hand, by the Nernst-Einstein relation σ ( s ) = σ ′ ( ω ) + iσ ′′ ( ω ) = ne k B T (15)the frequency-dependent diffusivity is related to the dy-namic conductivity, the real part of which, σ ′ ( ω ) is thealternate-current (AC) conductivity. Therefore, one ex-pects a non-analytic low-frequency dependence of the ACconductivity increment σ ′ ( ω ) − σ (0) ∝ ω / . Indeed, sucha behavior has been observed in amorphous semiconduc-tors [131]. We come back to this in the subsection onstrong disorder. C. Weak disorder and the self-consistent Bornapproximation (SCBA)
A well-known characteristic of the AC conductivity indisordered materials is that beyond a characteristic fre-quency ω ∗ it starts to increase with frequency, in manycases with a characteristic power law σ ′ ( ω ) ∝ ω α where α is smaller than 1 and takes values around 0.8 [81, 82, 132].For the random walker in the disordered environmentthis means that the mean-square distance walked doesnot increase linearly with time but sublinearly with ex-ponent 1 − α . Such a behavior has been termed anoma-lous diffusion. So the cross-over at ω ∗ corresponds to atransition from anomalous diffusion for times t < /ω ∗ to normal diffusion for t > /ω ∗ .As pointed out in Refs. [79, 121] the cross-over at ω ∗ -if transformed from the diffusion to the scalar-vibrationalsystem - corresponds to the boson peak. In other words:it correspondss to the begin of the frequency dependenceof e K ( ω ) = v ( ω ). By the Kramers-Kronig correspon-dence this implies the onset of an imaginary part of e K ( ω )which becomes of the order of its real part.A minimal theory for the boson peak, in fact, can beobtained by the self-consistent version of the Born ap-proxmation. It is obtained by replacing the bare Green’sfunction in Eq. (8) by the full Green’s function:Σ( z ) = γ V X | k |≤ k max k G ( k , z ) (16)with G ( k, z ) given by Eq. (7). For the ultraviolet cut-off k max one should take the inverse of the length scalewhich is the diameter of the coarse-graining volume V c used to define the local elastic constant/local diffusiv-ity [120]. On the other hand this length scale shouldbe of the order of the correlation length ξ of the elastic-ity/diffusivity fluctuations [125]. In this work we treatthese two length scale as being the same. We thereforecall the cutoff k max = k ξ .While the “derivation” of Eq. (16) is, of course, justan ad-hoc replacement, a proper derivation is achieved byfield-theoretical techniques [46] in analogy to the deriva-tion of the nonlinear sigma model for electrons [109, 133–135], classical sound waves [65, 125] and electromagneticwaves [136, 137].A royal road for this derivation is to minimize the fol-lowing (dimensionless, frequency-dependent) mean-fieldfree energy or effective action with respect to Σ( z ) S eff [Σ( z )] = S med [Σ( z )] + S SCBA [Σ( z )] (17)with S med [Σ( z )] = S med [ Q ( z )] (18)= Tr ln (cid:18) A (cid:2) z, r , e K − Σ( z ) | {z } Q ( z ) (cid:3)(cid:19) = X k ln (cid:18) A (cid:2) z, k , Q ( z ) (cid:3)(cid:19) and S SCBA [Σ( z )] = 12 Vγ Σ( z ) (19)The first term, S med [Σ( z )], is the generalized free energyof the effective medium [138], which is a medium without disorder, in which the fluctuating force constants in theoperator A [ z, r , e K ( r )] are replaced by the homogeneous(but frequency-dependent) force constant Q ( z ) = K − Σ( z ). The trace can therefore be calculated in k space.The Fourier transform of A [ z, r , Q ( z )] is given by A [ z, k , Q ( z )] = − z + k Q ( z ) (20)This is no more the operator-inverse but just the ordinaryinverse of the mean-field Green’s function G ( k , z ) = 1 A [ z, k , Q ( z )]= 1 − z + k Q ( z ) (21)The second term, S SCBA [Σ( z )] arises [22, 65] from aGaussian configuration average of the full Green’s func-tion with distribution density P (cid:2) ∆ e K ( r ) (cid:3) = P exp {− γ Z d r [∆ e K ( r )] } (22)It is easily verified that the SCBA (16) is obtained byminimizing S eff of Eq. (18). This corresponds to thesaddle-point approximation of the effective field theoryderived for the appropriate stochastic Helmholtz equa-tion [22, 65, 109, 119]. The saddle-point approximationrelies on the large prefactor of S SCBA ∝ /γ , i.e. theSCBA has its validity range for( i ) small disorder h K i /K ≪ ii ) Gaussian disorder, Eq. (22).As in our calculation of the Rayleigh scattering and thenon-analyticity of D ( s ) the SCBA transforms the micro-scopic wave equation with fluctuating elastic coefficient (cid:2) − z − ∇ e K ( r ) ∇ (cid:3) u ( r , z ) (23)into a macroscopic mean-field wave equation (cid:2) − z − e K ( z ) ∇ (cid:3) u ( r , z ) (24)The fluctuations of e K ( r ), (represented by the variance)determine the frequency dependence of e K ( z ).It is useful to formulate the SCBA, Eq. (16) in dimen-sionless units (indicated by a hat). We measure velocitiesin units of q e K , lengths in units of k ξ , and angular fre-quencies in units of k ξ q e K . (In the diffusion problemdiffusivities are measured in units of D and angular fre-quencies in units of D k ξ .) In these units the SCBA, Eq.(16) takes the form b Σ(ˆ z ) = Σ( z ) e K = 3 b γ Z d ˆ k ˆ k ˆ k − ˆ z + ˆ k [1 − b Σ(ˆ z )] (25)with the dimensionless “disorder parameter” b γ = γ ν e K = V c ν h (∆ e K ) i e K (26)For ˆ z = 0 Eq. (25) takes the form b Σ(0) = b γ − b Σ(0 (27)This quadratic equation has the solution b Σ(0) = 12 (cid:20) − p − b γ (cid:21) (28)We observe that for b γ > b γ c = 1 / b γ > b γ c the SCBA predicts eigenvalues ω <
0. This can be ratio-nalized by the fact that the SCBA is obtained from as-suming a Gaussian distribution (22). If the width of thisdistribution exceed a critical value, there exist too manylocal regions with negative elastic coefficients, which thenleads to the instability. Within this model a strong bosonpeak is obtained if the disorder parameter b γ approachesthe critical value b γ c .If we assume that the Debye cutoff k D = p π N/V coincides with k ξ ( N is the number of atoms or molecularunits and V the sample volume), the density of states isgiven by g (ˆ ω ) = 2ˆ ωπ Z d ˆ k ˆ k Im ( − ˆ z + k (cid:2) − b Σ(ˆ z ) (cid:3) ) (29)In Fig. 1 we have plotted the frequency-dependentdiffusivity (panel a) together with the DOS, divided bythe Debye DOS g D ˆ ω = 3ˆ ω / ˆ ω D (reduced DOS), withˆ ω D = q − b Σ , for different values of b γ near b γ c . We seethat the boson peaks indeed coincide with the onset ofthe frequency dependence of D ( ω ). D. Strong disorder and the coherent-potentialapproximation (CPA)
It is clear from Fig. 1 that the frequency dependence ofthe diffusivity (and hence of the AC conductivity), pre-dicted by the SCBA is rather weak, as compared to thestrong frequency dependence of the conductivity in ionicconducting glasses or amorphous semiconductors [81, 82].This is so, because the SCBA is restricted to very weakdisorder. In materials with activated ionic or electronichopping conduction, on the other hand, the local diffusiv-ities fluctuate very strongly, because they depend expo-nentially on local activation energies and local tunnelingdistances [139, 140].A rather successful and reliable mean-field theory forstrong disorder is the coherent-potential approximation(CPA). It was widely used for electronic structure cal-culations of disordered crystals [141–143], and later to -4 -3 -2 -1 ω∧ D ( ω ) ∧ -4 -3 -2 -1 ω ∧ g ( ω ) / g D ( ω ) ∧∧ ab FIG. 1. Panel a: Frequency-dependent diffusivity, calculatedin SCBA, Eq. (25) with disorder parameters (ˆ γ − ˆ γ c ) / ˆ γ c =10 − (blue dashes), 10 − (orange dash-dots), 10 − (red line).Panel b: Reduced density of states g ( ω ) /g D ( ω ) for the sameparameters as in panel a and setting k ξ = k D . Vc FIG. 2. Visualization of the continuum version of the CPA:Inside a volume V c of the effective medium with homogeneouselasticity Q ( z ) the homogeneous one is replaced by the fluctu-ating one e K ( r ), which gives rise to a perturbation e K ( r ) − Q ( z ).The CPA postulate is to minimize the influence of this pertur-bation, i.e. forcing the averaged T matrix of the perturbationto be equal to zero. diffusion and vibrational properties of disordered crys-tals [68, 83–85, 89]. In this lattice version of the CPAone consider a certain lattice site of the ordered effectivemedium, in which the potentials [141–143], the force con-stants or the diffusivities [68, 83–85, 89] are homogeneous(“coherent”) but frequency-dependent. At this specialsite the effective medium is replaced by the real medium,causing a “perturbation” of the effective medium. En-forcing now the averaged T matrix of this perturbationto vanish gives the self-consistent CPA equation for thecoherent potential.A version suitable for non-crystalline materials hasbeen worked out by S. K¨ohler and the present authorsusing field-theoretical techniques. [121]. The resultingCPA equation may be pedagogically visualized in thefollowing way, see Fig. 2: in a certain region of the ef-fective medium the frequency-dependent elasticity Q ( z )is replaced by the fluctuating one e K ( r i ), where r i is themid-point of the region. The CPA postulate takes theform * e K ( r i ) − Q ( z )1 + Λ( z )[ e K ( r i ) − Q ( z )] + = 0 (30)The average is over the distribution of the elasticity val-ues e K ( r i ) ≡ e K i with distribution density P ( e K i ). Thelatter may be taylored to the statistics of the disorderedmaterial at hand.Λ( z ) generalizes the effective-medium propagator inthe lattice CPA. Both Λ( z ) and Q ( z ) may be obtainedby minimizing the following mean-field action S eff [ Q ( z ) , Λ( z )] = S med [ Q ( z )] + S CPA [ Q ( z ) , Λ( z )] (31)where S med [ Q ( z )] is given by Eq. (18) and the CPA ac-tion by S CP A [ Q ( z ) , Λ( z )] = VV c (cid:28) ln (cid:18) z )[ e K ( r i ) − Q ( z )] (cid:19)(cid:29) (32)The large parameter of the saddle-point approximation,which validates the mean-field approximation, is now not the inverse disorder parameter, as in the case of theSCBA, but the parameter V /V c . This is the reason, whythe CPA is not restricted to small disorder. Varying theaction with respect to Λ( z ) we obtain the CPA equation(30), which can be put into the equivalent forms *
11 + Λ( z )[ e K i − Q ( z )] + = 1 (33) Q ( z ) = * e K i z )[ e K i − Q ( z )] + . (34)Varying the action with respect to Q ( z ) gives a relationfor Λ( z ): X k k − z + k Q ( z ) = VV c Λ( z ) *
11 + Λ( z )[ e K i − Q ( z )] + = VV c Λ( z ) , (35)where the second line follows from Eq. (33).It is advantageous to normalize the k integration to 1Λ( z ) = V c V X k k − z + k Q ( z ) = pχ ξ ( z ) (36) ν / Hz -7 -6 -5 -4 -3 -2 -1 σ ( ω ) / Ω c m - ν / Hz g ( ω ) / g D ( ω ) ab FIG. 3. Panel a: Full lines: Frequency-dependent diffusivity,calculated in CPA, Eq. (74), with a flat distribution of ac-tivation energies, Eq. (41). Symbols: AC conductivity dataof the ionic-conducting glass Sodium Trisilicate, compiled byWong and Angell [144]. with disorder parameters e γ = 7.92( ↔ T = 1673 K); = 10.41 ( ↔ T = 1273 K); = 17.53 ( ↔ T =756 K); = 29.06 ( ↔ T = 456 K); (from top to bottom).Panel b: Reduced density of states g (ˆ ω ) /g D (ˆ ω ) for the equiv-alent distribution density (42) for the same parameters as inpanel a. with χ ξ ( z ) = 3 k ξ Z k ξ dkk k − z + k Q ( z ) (37)and p = V c k ξ π (38) p should be smaller than 1, and It has been argued in Ref.[121] that one may identify p with the continuum perco-lation concentration p c . One has χ ξ ( z = 0) = 1 /Q (0).The CPA equation (34) now takes the form Q ( z ) = * e K i pχ ξ ( z )[ e K i − Q ( z )] + . (39)As in the case of the SCBA the CPA turns the waveequation with fluctuating elasticity into a mean-fieldwave equation with frequency-dependent elastic coef-ficient Q ( z ) or a diffusion equation with frequency-dependent diffusivity D ( s ).Let us now identify e K i with a spatially fluctating dif-fusivity D i and assume that the fluctations are caused bya fluctuating activation energy D i = D e − E i /k B T (40)If we now impose a constant distribution density of acti-vation energies P ( E i ) 1 E c θ ( E c − E i ) (41)i.e. a flat distribution with cutoff E c . For the diffusivi-ties D i (or elasticities e K i ) this transforms to a truncatedinverse-power distribution [121] P ( e K i ) = 1 µ /µ e K i µ ≤ e K i ≤ µ (42)with µ = µ e − E c /k B , and we have1 h e K i h e K i − h e K i ) = 12 E c /k B T ≡ e γ (43)Low temperatures in the diffusion problem obviouslymean strong disorder.In panel a of Fig. 3 we have plotted AC conductivitydata for the glassy ionic conductor Sodium Trisilicate[144]. Together with these data we plot the CPA resultwith a distribution of activation energies as given in Eq.(41). In panel b of this figure we have plotted the reducedDOS of the equivalent vibrational problem with inverse-power distribution (42). The DOS for the scalar phononproblem is given by g ( ω ) = 2 ωπ Im (cid:26) N X | k |≤ k D − z + k Q ( z ) (cid:27) (44)Apart from the perfect agreement between the conduc-tivity data and the CPA curves we note that the (ratherstrong) boson peak precisely mark the crossover fromfrequency-independent to frequency-dependent conduc-tivity of the diffusion problem. In the vibrational prob-lem this crossover means a transition from a regimewith Debye waves and Rayleigh scattering to a non-Debye regime, in which the vibrational excitations arenot waves. These excitations have been called “diffusons”[145], because their intensity obeys a diffusion equationlike light in turbid media [146]. On the other hand, thevibrational excitations in this regime show the statisticalproperties of random matrices [68, 94, 104]. Thereforethe frequency regime above the boson peak may also becalled “random-matrix regime” [6, 7].Returning to the diffusion problem, we pointed outthat in the frequency range below the crossover (thefrequency-independent regime) the Rayleigh scatteringcorresponds to a contribution to the frequency depen-dent conductivity with ∆ σ ′ ( ω ) ∝ ω / . This may beexperimentally verified by considering the dielectric lossfunction ǫ ′′ ( ω ) ∝ ω (cid:2) σ ′ ( ω ) − σ (0) (cid:3) (45) log ω/ω -3-2.5 l og { [ σ ( ω ) − σ ( )] / ω } ω -1 0 1 FIG. 4. Connected symbols: AC loss function [ σ ( ω ) − σ (0)] /ω ,measured in sputtered amorphous silicon [131] for three dif-ferent temperatures ( T = 51 K, 77 K, 154 K). The scalingfrequency ω ( T ) is proportional to σ (0 , T ). Full thick line:CPA calculation for a constant distribution of activation en-ergies. In the frequency-independent regime this function in-creases with frequency ǫ ′′ ∝ ω / , whereas in thefrequency-dependent regime it decreases due to the sub-linear behavior of σ ′ ( ω ). The maximum of the loss func-tion corresponds to the maximum of the reduced DOS inthe vibrational problem (boson peak).It has been noted in the literature [132] that the ACconductivity data taken at different temperatures showuniversal behavior, if the data are divided by the DCconductivity σ (0 , T ) and the freqency by the crossoverfrequency ω ( T ) ∝ σ (0 , T ). For the loss function thismeans that ǫ ′′ ( ω/ω ) should also be the same for differenttemperatures.In Fig. 4 we show the loss function of sputtered amor-phous silicon [131] against ω/ω for three temperatures,together with the result of the CPA, which predict thescaling and the crossover from ǫ ′′ ∝ ω / to the decreasewith frequency. We discuss the corresponding scaling ofthe vibrational DOS in section III D. III. HETEROGENEOUS-ELASTICITY THEORYA. Model
We now formulate the full heterogeneous-elasticity the-ory for vector displacements u ( r , t ) The equation of mo-tion for an elastic medium with spatially fluctuatingshear modulus G ( r ) As locally the translational and rotational invariance is brokenone should in principle work with the full fourth-rank Hooketensor C ijkl ( r ) instead of equation (46). The latter is an ap-proximation to keep the model tractable. (cid:2) ∂ ∂t − ∇ · f M ( r ) ∇ · + ∇ × e G ( r ) ∇ × (cid:3) u ( r , ω ) = 0 (46)with the reduced shear modulus e G ( r ) = G ( r ) /ρ and thereduced longitudinal modulus f M ( r ) = M ( r ) /ρ = e K + 43 e G ( r ) (47)In this formulation the dilatational (bulk) modulus K = e K/ρ is assumed not to exhibit spatial fluctuations, i.e.the fluctuations of the shear modulus G ( r ) = e G ( r ) /ρ areassumed to affect the traceless stress and strain tensorsonly [7, 121].Eqs. (46) may be decoupled by introducing longitu-dinal and transverse displacements u L ( r , t ) and u T ( r , t )with ∇ × u L ( r , t ) = 0 and ∇ · u T ( r , t ) = 0 (48)In the frequency domain we then have0 = (cid:18) − z − ∇ · f M ( r ) ∇ · (cid:19) u L ( r , z ) ≡ A L [ z, r , e G ( r )] u L ( r , z ) (49)0 = (cid:18) − z + ∇ × e G ( r ) ∇ × (cid:19) u T ( r , z ) ≡ A T [ z, r , e G ( r )] u T ( r , z ) (50) B. Self-consistent Born approximation
As in the case of the scalar model, the SCBA and theCPA, (see next subsection) serve to calculate the fre-quency dependence of the reduced frequency-dependentshear modulus Q ( z ) = G ( z ) /ρ = Q ′ ( ω ) − iQ ′′ ( ω )= v T ( z ) = e G − Σ( z ) (51)and longitudinal modulus f M ( z ) = M ( z ) /ρ = e K + 43 Q ( z ) = f M ′ ( ω ) − i f M ′′ ( ω )= v L ( z ) = e K + 43 [ e G − Σ( z )] (52)which enter into macroscopic mean-field equations of mo-tion0 = − z − f M ( z ) ∇ u L ( r , z ) ≡ A L [ z, r , Q ( z )] u L ( r , z )(53)0 = − z − Q ( z ) ∇ u T ( r , z ) ≡ A T [ z, r , Q ( z )] u T ( r , z )(54) The effective action for deriving the SCBA is S eff [Σ( z )] = S med [Σ( z )] + S SCBA [Σ( z )] (55)Here S SCBA [Σ( z )] is given by Eq. (19) with γ = V c (cid:10) [ e G ( r ) − e G ] (cid:11) (56)The difference from the scalar model is that we now dealwith 3-dimensional vectors. Therefore the trace includesa sum over the 3 cartesian degrees of freedom. Thismeans that one has to sum the longitudinal contribu-tion once and that of the transverse twice. Explicitly S med [Σ( z )] takes the form S med [Σ( z )] = Tr ln (cid:18) A (cid:2) z, Q ( z ) (cid:3)(cid:19) = X k ln (cid:18) A L (cid:2) z, k , e G − Σ( z ) (cid:3)(cid:19) +2 X k ln (cid:18) A T (cid:2) z, k , e G − Σ( z ) (cid:3)(cid:19) (57)If we now vary S eff of Eq. (55) with respect to Σ( z ), i.e. ∂S eff ∂ Σ( z ) = 0 we get Σ( z ) = γ k ξ π χ ξ ( z ) (58)with χ ξ ( z ) = 43 χ ξL ( z ) + 2 χ ξT ( z ) (59)This weighted susceptibility is given in terms of the locallongitudinal and transverse susceptibilities χ ξL,T ( z ) = 3 k ξ Z k ξ dkk χ L,T ( k, z ) (60)with the k dependent susceptibilities χ L,T ( k, z ) = k G L,T ( k, z ) = k − z + k v L,T ( z ) (61) G L,T ( k, z ) are the longitudinal and transverse Green’sfunctions Eq. (58) together with Eqs. (51), (52),(59), (60), and (61) establish the self-consistent vectorSCBA equations for calculating the frequency-dependentreduced shear modulus Q ( z ), and from this the relevantmeasurable quantities: • Vibrational density of states g ( ω ) = 2 ω π k D Z k D dkk (cid:18) G ′′ L ( k, ω ) + 2 G ′′ T ( k, ω ) (cid:19) (62) We use sans serif for the Green’s functions G L,T ( k, z ) in order todistinguish them from the shear modulus G . • Specific heat C ( T ) ∝ Z ∞ dωg ( ω )( ω/T ) e ¯ hω/k B T [ e ¯ hω/k B T − (63) • Longitudinal and transverse sound attenuationΓ
L,T Γ L ( ω ) = ωM ′′ ( ω ) /M ′ ( ω )Γ T ( ω ) = ωG ′′ ( ω ) /G ′ ( ω ) (64) ⇔ v L,T ( z ) = Re (cid:8) v L,T (cid:9)(cid:2) − i Γ L,T /ω (cid:3) • Longitudinal and transverse mean-free paths ℓ L,T ℓ L,T ( ω ) = Γ L,T ( ω )2 v L,T (0) (65) • Thermal conductivity [22] κ ( T ) ∝ Z ∞ dωℓ T ( ω )( ω /T ) e ¯ hω/k B T [ e ¯ hω/k B T − (66) • Coherent inelastic neutron and X-ray scattering in-tensity S ( k, ω ) ∝ − e ¯ hω/k B T | {z } n ( ω ) + 1 Im (cid:8) χ L ( k, ω ) (cid:9) (67) • Depolaritzed Raman scattering intensity [7, 147] I V H ( ω ) ∝ [ n ( ω ) + 1]Im (cid:8) χ k p ( ω ) (cid:9) (68)It should be noted that the same susceptibility combina-tion, namely χ ξ ( ω ), which enters into the SCBA equation(58), appears also in the Raman intensity, Eq. (68), al-beit with a cutoff k P given by the fluctuations of thepockels constants and not the elastic constants [7, 147].It should further be noted that the Raman in-tensity is not given by the Shuker-Gammon formula I V H ( ω ) ∝ [ n ( ω ) + 1] g ( ω ) /ω [13]. If one divides the ex-pression of Eq. (68) for I V H ( ω ) by Shuker and Gam-mon’s expression one obtains the frequency-dependentcoupling constant C ( ω ), which had been inserted as ad-ditional factor into the Shuker-Gammon formula in orderto reconcile neutron and Raman-scattering results for thevibrational DOS [14, 16, 147–150]. C. General features of the disorder-inducedvibrational anomalies: Comparison of the SCBAversion of heterogeneous-elasticity theory with asimulation
As stated in the introduction, the disorder-induced vi-brational anomalies of glasses and other disordered solidsfeature three phenomena, which are related to each other g ( ω ) / ω a Q ’ ( ω ) ; / ( M ~ ’ ( ω )-- K ~ ) ω Q " ( ω ) ; / M ~ " ( ω ) bc FIG. 5. Comparison of results of a soft-spheremolecular-dynamics simulation (symbols) with the predictionof heterogeneous-elasticity theory in self-consistent Born ap-proximation (SCBA). We show the real parts (upper panel)and imaginary parts (lower panel) of the frequency-dependentshear modulus G (Ω T ) and the quantitity 3 / M (Ω L ) − ˜ K ) forthree temperatures (5 · − , 5 · − , 5 · − , in Lennard-Jonesunits) with K = 30 .
4. The SCBA parameters are γ − γ c = 0 . K/G = 3 . ( i ) The cross-over from Debye- to non-Debye behaviorof the vibrational DOS, leading to a maximum inthe reduced DOS g ( ω ) /ω , the boson peak;( ii ) a pronounced dip in the real part of the elasticcoefficients (and their square-root, the frequency-dependent velocities) near the boson-peak fre-quency;( iii ) a strong increase of the sound attenuation belowthe boson-peak frequency (Rayleigh scattering),which enters into the imaginary parts of the elasticcoefficients via Eq. (64).These three anomalies are displayed in Fig. 5, in whichthe results of a molecular-dynamic simulation is com-pared with the prediction of heterogeneous-elasticity the-ory, solved in SCBA [38]. In this simulation a binary soft-sphere potential (i.e. a Kob-Andersen-type [151] binaryLennard-Jones potential without the attractive part) wastaken for ten million particles. Such a potential mim-ics a metallic glass. The longitudinal and transverse2frequency-dependent moduli were obtained from deter-mining the longitudinal and transverse current correla-tion functions C L,T ( k, ω ) = K B T ωπm G ′′ ( k, ω ) (69)where m is the paticles’ mass. The simulation was runfor three very different temperatures deep in the glassystate (see the figure caption). The very large particlenumber made it possible to avoid finite-size effect in theboson-peak frequency range. The data for the complexlongitudinal modulus was converted to Q ( z ) = G ( z ) /ρ via the inverse of Eq. (52) Q ( z ) = 34 [ f M ( z ) − e K ] (70)in order to compare the data with the measured Q ( z ).It is seen in the figure that – as the longitudinal andtrasverse data ly on top of each other – the bulk modulus e K indeed does not appreciably depend on frequency andcorrespondingly has a negligible imaginary part.It is the strength of the present theory that it easilyallows to explain how the three anomalies are related toeach other. It was pointed out by Schirmacher et al. 2007[118] that one can deduce from Eq. (62) and using the3rd line of (64) a relation between the DOS and Γ( ω ): g ( ω ) − g D ( ω ) ∝ Γ( ω ) ∝ ω Σ ′′ ( ω ) (71)This means that the disorder-induced frequency depen-dence of the elastic coefficient, controlled by Σ( ω ) is re-sponsible for the boson peak. This can already under-stood using lowest-order perturbation theory, which leadsto Rayleigh scattering.On the other hand, the real parts and the imaginaryparts of the frequency-dependent moduli are related tothose of the response functions χ ( z ) = χ ′ ( ω ) − iχ ′′ ( ω ).These have, due to the causality requirement (the answermust occur at a time later than the question), a one-to-one correspondence by the Kramers-Kronig relation [152] χ ′ ( ω ) = 1 π P Z ∞ d ¯ ω χ ′′ (¯ ω )¯ ω − ω (72)This is most easily visualized by looking at the simulateddata of the real and imaginary parts of Q ( z ) = e G − Σ( z ),displayed in panels b and c. Usually the real part of ananalytic function, such as χ ξ ( z ), has a maximum near thebottom of the spectrum [ χ ξ ] ′′ ( ω ) and a minimum nearthe top. Now, because not Q ( z ) but Σ( z ) is proportionalto χ ξ ( z ) via the SCBA relation (58), Q ′ ( ω ) displays a minimum near the begin of the “disorder spectrum”, or“random-matrix spectrum” [ χ ξ ] ′′ ( ω ), obviously markedby the boson peak, displayed in panel a.In the boson-peak regime and above, the data for verydifferent temperatures collapse, which proves that theanomalies must be of harmonic origin. This is not so inthe very low frequency regime, where anharmonic effectsbecome visible [153, 154]. ω / ω D g ( ω ) / g D ( ω ) k D /k ξ = 1k D /k ξ = 2k D /k ξ = 3k D /k ξ = 4 FIG. 6. Reduced density of states g ( ω ) /g D ( ω ) vs. the rescaledfrequency ( ω/ω D )( k D /k x i ) = ω/v D k ξ for different values ofthe ratio k ξ /k D . The other parameters are γ/G = 1, G min =0 . D. Coherent-Potential Approximation (CPA)
The vector CPA may be derived in the same way asthe vector SCBA from the effective action S eff [ Q ( z )] = S med [ Q ( z )] + S CP A [ Q ( z )] (73)where S med [ Q ( z )] is given by Eq. (57) of the vector SCBAand S CP A [ Q ( z )] by Eq. (32) of the scalar CPA. Thevector CPA equation takes the same form as the scalarone: Q ( z ) = * e G i pχ ξ ( z )[ e G i − Q ( z )] + . (74)but now with fluctuating shear moduli e G i and the vectorversion of χ ξ ( z ), Eq. (59).As in the scalar model the CPA has several advantagescompared to the SCBA:– one can treat arbitrary distributions P ( G );– one is not restricted to weak disorder;– one needs not (but can) take negative values of G into account.For the calculations presented in Figs. 6 to 8 we used theCPA equation (74), together with (62) with a truncatedGaussian distribution of shear moduli of the form P ( G ) = P θ ( G − G min ) e − ( G − G ) / γ (75)where θ ( x ) is the Heaviside step function and G min is thelower cutoff. In these calculations we used the renormal-ized value of G (i.e. the self-consistently calculated one)for evaluating the Debye frequency ω D and Debye DOS3 ( ω / ω D ) g ( ω ) / g D ( ω ) G min /G = 0.0G min /G = - 0.05G min /G = - 0.1G min /G = - 0.015 FIG. 7. Reduced density of states g ( ω ) /g D ( ω ) vs. the rescaledfrequency ( ω/ω D ) for different values of the lower cutoff G min of the Gaussian distribution P ( G ). The other parameters are γ = /G = 1 . , k D /k ξ = 2. g D ( ω ) in terms of the longitudinal and transverse soundvelocities v L = K + Q (0), v T = Q (0) ω D = k D (cid:20) (cid:18) v L + 2 v T (cid:19) (cid:21) − / (76) g D ( ω ) = 3 ω /ω D (77)For the bulk modulus of the calculations we used thevalue K = 3 . G and for the cutoff parameter the value e ν = 1, which implies k D /k ξ = √ ξ/a , where a = p V /N is the mean intermolecular distance. The distribution ofshear moduli (75) involves three parameters G , γ and G min . Because G is used to fix the elastic-constant scalethere remain three adjustable parameters to fix the stateof elastic disorder of the material, namely k D /k ξ ∼ ξ/a , γ and G min . The latter (which we used with negativevalues or equals zero) specify the amount of regions withnegative shear modulus (soft regions) in the material. Ascan be seen from Figs. 6 to 8 increasing ξ and | G min | en-hances the BP and shifts its position to lower frequencies,whereas increasing γ just leads to an enhancement, whilekeeping the BP position constant. It has been pointedout in the literature [155–158] that the position of theboson peak in relation to the Debye frequency correlateswith the inverse correlation length of density and elastic-ity fluctuations.Let us discuss our findings further in terms of mea-sured vibrational spectra of materials, in which an exter-nal parameter (temperature, pressure or the amount ofpolymerization) is changed. If the Debye frequency (de-pending on the moduli K and G ) is changed, this leadsto a modification of the spectrum, which has been called elastic-medium transformation . This transformation istaken care of, if the DOS is represented in a normalized ( ω / ω D ) g ( ω ) / g D ( ω ) γ /G = 0.25 γ /G = 0.5 γ /G = 1.0 γ /G = 2.0 FIG. 8. Reduced density of states g ( ω ) /g D ( ω ) vs. the rescaledfrequency ( ω/ω D ) for different values of the width parameter γ . The other parameters are G min /G = − . k D /k ξ = 2. way, as is the case in Figs. 6 to 6. A number of boson-peak data, if normalized in this way, lead to a universalcurve, i.e. all data points fall onto the same curve if re-plotted, taking the elastic transformation into account[32, 34, 59, 90, 92, 159]. Other investigation reveal a deviation from this scaling [160–168]. In terms of ourmodel calculations this means, if the state of disorder isnot changed, but just the value of the mean elastic con-stants or the density, which go into the Debye frequency,this corresponds to elastic-transformation scaling. In theother cases obviously the state of disorder is changed bychanging the external conditions.A very interesting case in which the elastic transfor-mation scaling does not hold has been reported recently:the case of prehistoric amber. [168] measured the tem-perature dependence of the specific heat of the hyperagedand rejuvenated material. The height of the boson peak,taken from a C ( T ) /T curve is by 22 % lower in thehyperaged material, compared with the rejuvenated one.An elastic transformation using the change in the Debyefrequency determined by the authors would only lead toa difference by 7.4 %. IV. DISCUSSION AND CONCLUSIONS
Reading the text of the present article, one could beconvinced that the boson peak in glasses and the asso-ciate vibrational anomalies can be satisfactory explainedby the presence of the structural disorder, leading to spa-tial fluctuations of elastic coefficient, in particular theshear modulus. However, in the community working ex-perimentally and theoretically on the vibrational proper-ties of disordered and complex condensed matter there isno agreement about this. Whereas many authors agree,that the boson-peak-anomalies in glasses are due to thestructural disorder, others maintain that this is not so4and that the anomalies may be explained by conven-tional crystalline solid-state theory. Crystalline phonontheory is based on the phonon dispersions [169], reflect-ing the crystal symmetry group and anharmonic inter-actions, leading to renormalization and viscous damping[170–172].As mentioned in the introduction, Chumakov et al.[90, 92] argue that the boson peak is “identical” [90] toa washed-out van-Hove singularity, i.e. just the result ofthe bending-over of the lowest (transverse) phonon dis-persion near the first Brillouin zone. In fact, there is ex-perimental evidence for crystal-like features in the spec-trum of glasses and even liquids [173, 174]. These arecaused by the short-range order, which is documentedin the static structure factor S ( k ) of glasses. The peakposition k of S ( k ) corresponds to the diameter of thefirst Brillouin zone of a crystal. Correspondingly, k / α -Quartz and glassy SiO , densified to match the crys-talline density, accompanied with a numerical lattice-dynamics calculation of the crystalline phonon disper-sions, Baldi et al. [174] find a crossover of the sound at-tenuation of the glass from a Rayleigh Γ ∝ ω behaviourto a quadratic one at ¯ hω c ∼ ω c the two attenuation coefficients match.The interpretation of the authors is that below ω c theglassy attenuation is due to elastic heterogeneities, above ω c the spectrum is essentially the same of that of thepolycrystal. These findings are certainly at variance withthe claim that the boson-peak-type scenario leading to the Γ ∝ ω → ω crossover would have something to dowith the bending-over of the transverse dispersion of thecrystal. Obviously in this material the boson-peak fre-quency and the van-Hove-singularity are the same. Thismight also be the case for the other examples found bythe authors of refs. [90, 92].As indicated in the introduction, in the investigationof a two-dimensional macroscopic model glass the co-existence of crystal-like dispersions with the boson peakwas observed [93], but the frequencies of the boson peakand that of the transverse van-Hove singularity are quitedifferent. The boson peak was shown to show all salientfeatures which identify its origin arising from the disor-der. The conclusion is that the boson-peak, namely thedisorder-induced peak in the reduced DOS is not “iden-tical” with a washed-out van-Hove singularity. In someglasses their frequencies are just very near to each other.A recent interpretation of boson-peak related damp-ing phenomena claims that the boson peak would be a“universal phenomenon” both in crystals and in glasses[175].The “universality” of the boson peak in solids (bothperfectly ordered crystals and glasses), if there is one,suggested by these authors, may be identified with thefact that the boson peak is linked with a Ioffe-Regeltype of crossover from ballistic phonon propagation toa scattering-dominated regime where the exctiation isquasi-localized. This is where the ”universality” ends,because the scattering mechanism is clearly different inglasses and perfectly ordered crystals. In the formercase it is due to (harmonic) disorder, as predicted byheterogeneous-elasticity theory, whereas in the latter caseof ordered crystals it is due to anharmonicity.In conclusion, the boson-peak related vibrationalanomalies in glasses can consistently be explained bythe presence of spatial fluctuations of elastic coefficients(elastic heterogeneity) with the help of the SCBA andCPA. [1] S. R. Elliott. The Physics of Amorphous Materials .Longman, New York, 1984.[2] K. Binder and W. Kob.
Glassy Materials and DisorderedSolids: An Introduction . World Scientific, London, 2011.[3] See Refs. [2, 4–7] for recent reviews.[4] T. Nakayama.
Rep. Prog. Phys. , 65:1195, 2002.[5] M. Klinger.
Phys. Reports , 492:111, 2010.[6] W. Schirmacher. phys. stat. sol. (b) , 250:937, 2013.[7] W. Schirmacher, T. Scopigno, and G. Ruocco.
J. Non-cryst. Sol. , 407:133, 2014.[8] P. Flubacher, A. J. Leadbetter, and J. A. Morris.
J.Phys. Chem. Solids , 12:53, 1959.[9] A. J. Leadbetter and J. A. Morrison.
Phys. Chem.Glasses , 4:188, 1963.[10] A. J. Leadbetter.
Phys. Chem. Glasses , 9:1, 1968.[11] A. J. Leadbetter.
J. Chem. Phys. , 51:779, 1969.[12] R. Shuker and R. W. Gammon.
Phys. Rev. Lett. , 25:222,1970. [13] R. Shuker and R. W. Gammon. In M. Balkanski, editor,
Proceedings of the Second International Conference onLight Scattering in Solids , page 334. Flammarion, Paris,1971.[14] J. J¨ackle. In W. A. Phillips, editor,
Amor-phous Solids:Low-Temmperature Properties , page 135.Springer, Heidelberg, 1981.[15] A. J. Martin and W. Brenig. phys. status solidi (b) ,64:163, 1974.[16] B. Schmid and W. Schirmacher.
Phys. Rev. Lett. ,103:169702, 2009.[17] U. Buchenau, N. N¨ucker, and A. J. Dianoux.
Phys. Rev.Lett. , 53:2316, 1984.[18] J. Wuttke, W. Petry, G. Coddens, and F. Fujar.
Phys.Rev. E , 52:4026, 1995.[19] A. I. Chumakov et al.
Phys. Rev. Lett. , 92:245508, 2004.[20] R. C. Zeller and R. O. Pohl.
Phys. Rev. B , 4:2029, 1971.[21] J. J. Freeman and A. C. Anderson.
Phys. Rev. B , Europhys. Lett. , 73:892, 2006.[23] W. A. Phillips.
J. Low Temp. Phys. , 7:351, 1972.[24] P. W. Anderson, B. I. Halperin, and C. M. Varma.
Phi-los. Mag. , 25:1, 1972.[25] W.A. Phillips.
Rep. Prog. Phys. , 50:1557, 1987.[26] C. C. Yu and A. J. Leggett.
Comm. Cond. Matter.Phys. , 14:231, 1988.[27] A. W¨urger. From coherent tunneling to relaxation.In
Springer Tracts in Modern Physics , volume 135.Springer, Heidelberg, 1996.[28] S. Hunklinger and A. K. Raychaudhuri. In D. F. Brewer,editor,
Progress in Low-Temperature Physics , volume 9,page 265. North-Holland, Amsterdam, 1986.[29] S. Hunklinger. Tunneling states. In M. F. Thorpe andM. I. Mitkova, editors,
Amorphous Insulators and Semi-conductors .[30] F. Sette, M. H. Krisch, C. Masciovecchio, G. Ruocco,and G. Monaco.
Science , 280:1550, 1998.[31] G. Monaco and V. M. Giordano.
PNAS , 106:3659, 2009.[32] G. Baldi, A. Fontana, G. Monaco, L. Orsingher, S. Rols,F. Rossi, and B. Ruta.
Phys. Rev. Lett. , 102:195502,2009.[33] G. Baldi, V. M. Giordano, G. Monaco, and B. Ruta.
Phys. Rev. Lett. , 104:195501, 2010.[34] B. Ruta, G. Baldi, V. M. Giordano, L. Orsingher,S. Rols, F. Scarponi, and G. Monaco.
J. Chem. Phys. ,113:041101, 2010.[35] Giacomo Baldi, Valentina M. Giordano, and GiulioMonaco.
Phys. Rev. B , 83:174203, 2011.[36] B. Ruta, G. Baldi, F. Scarponi, D. Fioretto, M. Gior-dano, and G. Monaco.
J. Chem. Phys. , 137:214502,2012.[37] G. Monaco and S. Mossa.
PNAS , 106:16907, 2009.[38] A. Marruzzo, W. Schirmacher, A. Fratalocchi, andG. Ruocco.
Scientific Reports , 3:1407, 2013.[39] V. G. Karpov, M. I. Klinger, and F. N. Ignatiev.
Sov.Phys. JETP , 84:760, 1983.[40] U. Buchenau, Yu. M. Galperin, V. L. Gurevich, andH. R. Schober.
Phys. Rev. B , 43:5039, 1991.[41] See Refs. [5, 42] for the extended literature on soft po-tentials and quasi-local oscillators.[42] H. R. Schober, U. Buchenau, and V. L. Gurevich.
Phys.Rev. B , 89:014204, 2014.[43] V. L. Gurevich, D. A. Parshin, and H. R. Schober.
Phys.Rev. B , 67:094203, 2003.[44] E. N. Economou.
Green’s function in quantum physics .Springer-Verlag, Heidelberg, 1971.[45] A. L. Burin, L. A. Maksimov, and I. Ya. Polishchuk.
Physica B , 210:15, 1995.[46] E. Maurer and W. Schirmacher.
J. Low-TemperaturePhys. , 137:453, 2004.[47] W. Schirmacher.
J. Non-Cryst. Sol. , 357:518, 2011.[48] S. Alexander and R. Orbach.
J. Phys. (Paris) Lett. ,43:L625, 1982.[49] B. Derrida, R. Orbach, and Kin-Wah Yu.
Phys. Rev. B ,29:6645, 1984.[50] O. Entin-Wohlman, S. Alexander, R. Orbach, and Kin-Wah Yu.
Phys. Rev. B , 29:4588, 1984.[51] S. Alexander.
Physica , 140A:397, 1986.[52] T. Nakayama, K. Yakubo, and R. L. Orbach.
Rev. Mod.Phys. , 66:381, 1994.[53] B. B. Mandelbrot.
The Fractal Geometry of Nature .Freeman, New York, 1989. [54] W. Schirmacher. Theory of liquids and other disor-dered media. In
Lecture Notes in Physics , volume 887.Springer, Heidelberg, 2015.[55] I. Webman.
Phys. Rev. Lett. , 47:1496, 1981.[56] S. Summerfield.
Sol. State Comm. , 39:401, 1981.[57] T. Odagaki and M. Lax.
Phys. Rev. B , 24:5284, 1981.[58] E. Courtens, J. Pelous, J. Phalippou, R. Vacher, andT. Woignier.
Phys. Rev. Lett. , 58:128, 1987.[59] S. Caponi et al.
Phys. Rev. Lett. , 102:027402, 2009.[60] E. Akkermans and R. Maynard.
Phys. Rev. B , 32:7850,1985.[61] J. E. Graebner, B. Golding, and L. C. Allen.
Phys. Rev.B , 34:5696, 1986.[62] A. F. Ioffe and A. R. Regel.
Prog. Semicond. , 4:237,1960.[63] N. F. Mott.
Metal-insulator transitions, 2nd Edition .Taylor & Francis, London, 1990.[64] E. Akkermans and R. Maynard.
Phys. Rev. B , 32:7850,1985.[65] S. John, H. Sompolinky, and M. J. Stephen.
Phys. Rev.B , 28:5592, 1983.[66] P. B. Allen and J. L. Feldman.
Phys. Rev. Lett. , 62:645,1989.[67] W. Schirmacher and M. Wagener.
Sol. State Comm. ,86:597, 1993.[68] W. Schirmacher, G. Diezemann, and C. Ganter.
Phys.Rev. Lett. , 81:136, 1998.[69] W. Schirmacher S. Pinski and R. A. Rmer.
Europhys.Lett. , 97:16007, 2012.[70] W. Schirmacher S. Pinski, T. Whall, and R. A. Rmer.
J. Condens. Matt. , 24:405401, 2012.[71] C. Tomaras and W. Schirmacher.
J. Phys. Condens.Matter , 25:495402, 2013.[72] S. N. Taraskin and S. R. Elliott.
J. Phys.: Condens.Matter , 11:A219, 1999.[73] B. Ruffl´e, G. Guimbretire, E. Courtens, R. Vacher, andG. Monaco.
Phys. Rev. Lett. , 96:045502, 2006.[74] H. Shintani and H. Tanaka.
Nature Materials , 7:870,2008.[75] M. Foret, E. Courtens, R. Vacher, and J.-B. Suck.
Phys.Rev. Lett. , 77:3831, 1996.[76] M. Foret, E. Courtens, R. Vacher, and J.-B. Suck.
Phys.Rev. Lett. , 78:4669, 1997.[77] E. Rat, M. Foret, E. Courtens, R. Vacher, and M. Arai.
Phys. Rev. Lett. , 83:1355, 1999.[78] S. N. Taraskin and S. R. Elliott.
Phys. Rev. B , 61:12017,2000.[79] W. Schirmacher and M. Wagener.
Philos. Mag. B ,65:861, 1992.[80] S. Alexander, J. Bernasconi, W. R. Schneider, andR. Orbach.
Rev. Mod. Phys. , 53:175, 1981.[81] A. K. Jonscher.
Nature , 267:673, 1977.[82] A.R. Long.
Adv. Phys. , 31:553, 1982.[83] I. Webman.
Phys. Rev. Lett. , 47:1496, 1981.[84] T. Odagaki and M. Lax.
Phys. Rev. B , 24:5284, 1981.[85] S. Summerfield.
Sol. State Comm. , 39:401, 1981.[86] M. L. Mehta.
Random Matrices .[87] F. M. Izrailev.
Phys. Rep. , 196:300, 1990.[88] N. W. Ashcroft and N. D. Mermin.
Solid State Physics .Saunders College, Philadelphia, 1976.[89] S. N. Taraskin, S. Elliott, Y. H. Loh, and G. Nataranjan.
Phys. Ref. Lett. , 86:1255, 2001.[90] A. I. Chumakov et al.
Phys. Rev. Lett. , 106:8519, 2011.[91] R. Zorn.
Physics , 4:44, 2011. [92] A. I. Chumakov et al. Phys. Rev. Lett. , 112, 2014.[93] Yinqiao Wang, L. Hong, Yujie Wang, W. Schirmacher,and J. Zhang.
Phys. Rev. B , 98:174207, 2018.[94] S. K. Sarkar, G. S. Matharoo, and A. Pandey.
Phys.Rev. Lett. , 92:215502, 2004.[95] R. K¨uhn and U. Horstmann.
Phys. Rev. Lett. , 78:4067,1997.[96] M. M´ezard, G. Parisi, and A. Zee. Spectra of euclideanrandom matrices.
Nucl. Phys. B , 559:689, 1999.[97] V. Mart´ın-Mayor, G. Parisi, and P. Verroccio.
Phys.Rev. E , 62:2373, 2000.[98] T. S. Grigera, V. Mart´ın-Mayor, G. Parisi, and P. Ver-rocchio.
Phys. Rev. Lett. , 87:085502, 2001.[99] V. Mart´ın-Mayor, M. M´ezard, G. Parisi, and P. Verroc-chio.
J. Chem. Phys. , 114:8068, 2001.[100] T. S. Grigera, V. Mart´ın-Mayor, G. Parisi, and P. Ver-rocchio.
Nature , 422:289, 2003.[101] S. Ciliberti, T. S. Grigera, V. Mart´ın-Mayor, G. Parisi,and P. Verrocchio.
J. Chem. Phys. , 119:8577, 2003.[102] C. Ganter and W. Schirmacher.
Philos. Magazine ,91:1894, 2011.[103] T. S. Grigera, V. Mart´ın-Mayor, G. Parisi, P. Urbani,and P. Verrocchio.
J. Statist. Mech: Theory and Exper-iment , 2011:P02015, 2011.[104] Y. M. Beltukov, V. I. Kozub, and D. A. Parshin.
Phys.Rev. B , 87:134203, 2013.[105] W. G¨otze and M. R. Mayr.
Phys. Rev. E , 61:578, 1999.[106] U. Bengtzelius, W. G¨otze, and A. Sj¨olander. Dynamicsof supercooled liquids and the glass transition.
J. Phys.C , 17:5915, 1984.[107] W. G¨otze.
Complex Dynamics of Glass-Forming Liq-uids . Oxford Univ. Press, Oxford, 2009.[108] L. D. Landau and E. M. Lifshitz.
Theory of Elasticity .Pergamon Press, 1959.[109] A. J. McKane and M. Stone. Localization as an al-ternative to Goldstone’s theorem.
Ann. Phys. (N. Y.) ,131:36, 1981.[110] J. P. Wittmer, A. Tanguy, J.-L. Barrat, and L. Lewis.
Europhys. Lett. , 57:423, 2002.[111] F. L´eonforte, R. Boissi`ere, A. Tanguy, J.-P. Wittmer,and J.-L. Barrat.
Phys. Rev. B , 72:224206, 2005.[112] F. L´eonforte, A. Tanguy, J.-P. Wittmer, and J.-L. Bar-rat.
Phys. Rev. Lett. , 97:055501, 2006.[113] M. Tsamadosand A. Tanguy, C. Goldenberg, and J.-L.Barrat.
Phys. Rev. B , 80:026112, 2009.[114] S. G. Mayr.
Phys. Rev. B , 79:060201, 2009.[115] H. Mizuno, H. Mossa, and J.-L. Barrat.
Europhys. Lett. ,104:56001, 2013.[116] R. Milkus and A. Zaccone.
Phys. Rev. B .[117] Y. Yang, Y.-J. Wang, E. Ma, A. Zaccone, L. H. Dai,and M. Q. Jiang.
Phys. Rev. Lett. [118] W. Schirmacher, G. Ruocco, and T. Scopigno.
Phys.Rev. Lett. , 98:025501, 2007.[119] W. Schirmacher, E. Maurer, and M. P¨ohlmann. phys.stat. sol. (c) , 1:17, 2004.[120] C. Ganter and W. Schirmacher.
Phys. Rev. B ,82:094205, 2010.[121] S. K¨ohler, R. Ruocco, and W. Schirmacher.
Phys. Rev.B , 88:064203, 2013.[122] J. F. Lutsko.
J. Appl. Phys. , 65:2991, 1988.[123] J. W. Strutt Lord Rayleigh.
Philos. Mag. , 41:241, 1871.[124] J. W. Strutt Lord Rayleigh.
Philos. Mag. , 47:375, 1899.[125] S. John and M. J. Stephen.
Phys. Rev. B , 28:6358, 1983.[126] S. Gelin, H. Tanaka, and A. Lemaˆıtre.
Nature Materials , 15:1177, 2016.[127] J.P. Hansen and I.R. McDonald.
Theory of Simple Liq-uids . Elsevier Science, Amsterdam, 2006.[128] W. Feller.
An Introduction to probability theory and itsapplications , volume II.[129] M. H. Ernst, J. Machta, J. R. Dorfman, and H.
J.Statist. Phys. , 34:413, 1984.[130] J. Machta, M. H. Ernst, and H. van Beijeren and.
J.Statist. Phys. , 35:413, 1984.[131] A. R. Long, J. McMillan, N. Balkan, and S. Summer-field.
Philos. Mag. B , 58:153, 1988.[132] J. C. Dyre and T. B. Schrøder.
Rev. Mod. Phys. , 72:873,2000.[133] F. Wegner. The mobility edge problem: continuoussymmetry and a conjecture.
Z. Phys. B , 35:207, 1979.[134] K. B. Efetov, I. Larkin A, and D. E. Khmelnitsk.
SovietPhys. JETP , 52:568, 1980.[135] Lothar Sch¨afer and Franz Wegner. Disordered systemwith n orbitals per site: Lagrange formulation, hyper-bolic symmetry, and Goldstone modes. Zeitschrift f¨urPhysik B Condensed Matter , 38(2):113–126, 1980.[136] Sajeev John. Strong localization of photons in cer-tain disordered dielectric superlattices.
Phys. Rev. Lett. ,58:2486–2489, 1987.[137] W. Schirmacher, B. Abaie, A. Mafi, G. Ruocco, andM. Leonetti.
Phys. Rev. Lett. , 120:067401.[138] D. Vollhardt.
AIP Conf. Proc. , 1297:339, 2010.[139] H. B¨ottger and V. V. Bryksin.[140] A. I. Efros and B. I. Shklovski˘i.
Electronic propertiesof doped semiconductors . Springer-Verlag, Heidelberg,1984.[141] P. Soven.
Phys. Rev. , 156:809, 1967.[142] D. W. Taylor.
Phys. Rev. , 156:1017, 1967.[143] H. Ebert, D. K¨odderitzsch, and volume = 74 pages =096501 year = 2011 J. Min´ar, journal = Rep. Prog.Phys.[144] J. Wong and C. A. Angell.
Glass: Structure by Spec-troscopy . M. Dekker, New York, 1976.[145] P. B. Allen, J. L. Feldman, and J. Fabian.
Philos. Mag-azine B , 79:1715, 1999.[146] A. Ishimaru.
Wave propagation and scattering in ran-dom media, V . Academic Press, New York, 1978.[147] B. Schmid and W. Schirmacher.
Phys. Rev. Lett. ,100:137402, 2008.[148] V. K. Malinovsky and A. P. Sokolov.
Sol. St. Comm. ,57:757, 1986.[149] G. Viliani et al.
Phys. Rev. B , 52:3346, 1995.[150] V. N. Novikov and E. Duval.
Phys. Rev. Lett. ,103:169701, 2009.[151] W. Kob and H. C. Andersen.
Phys. Rev. Lett. , 73:1376,1994.[152] J. D. Jackson.
Classical Electrodynamics . Wiley, NewYork, 1962.[153] A. Marruzzo, S. K¨ohler, A. Fratalocchi, G. Ruocco, andW. Schirmacher.
Eur. Phys. J. Special Topics , 216:83,2013.[154] C. Ferrante, E. Pontecorvo, G. Cerullo, A. Chiasera,G. Ruocco, W. Schirmacher, and T. Scopigno.
NatureComm. , 4:1793, 2013.[155] E. Duval, A. Boukenter, and T. Achibat.
J. Phys. Con-dens. Matter , 2:10227, 1990.[156] S. R. Elliott.
Europhys. Lett. , 19:201, 1992.[157] A. P. Sokolov et al.
Phys. Rev. Lett. , 69:1540, 1992. [158] L. Hong, V. N. Novikov, and A. P. Sokolov. J. Noncryst.Sol. , 117:357, 2013.[159] A. Monaco et al.
Phys. Rev. Lett. , 97:135501, 2006.[160] M. Zanatta et al.
Phys. Rev. B , 81:212201, 2010.[161] M. Zanatta et al.
J. Chem. Phys. , 135:174506, 2011.[162] L. Hong et al.
Phys. Rev. B , 78:134201, 2008.[163] S. Caponi et al.
Phys. Rev. B , 76:092201, 2007.[164] B. Ruffl´e, S. Ayrinhac, E. Courtens, R. Vacher, M. Fore-tand A. Wischnewski, and U. Buchenau.
Phys. Rev.Lett. , 104:067402, 2010.[165] K. Niss et al.
Phys. Rev. Lett. , 99:055502, 2007.[166] L. Orsingher et al.
J. Chem. Phys. , 132:124508, 2010.[167] S. Corezzi et al.
J. Phys. Chem. B , 117:14477, 2013.[168] T. P´erez-Casta neda et al.
Phys. Rev. Lett. , 112:165901,2014.[169] A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova.
Theory of lattice dynamics in the harmonicapproximation, solid-state physics, Suppl.3 . AcademicPress, New York, 1971.[170] W. G¨otze and K. H. Michel. In G. K. Horton and A. A.Maradudin, editors,
Dynamical Properties of Solids , vol-ume I., chapter 9. North Holland, Amsterdam, 1974.[171] H. Horner. In G. K. Horton and A. A. Maradudin, ed-itors,
Dynamical Properties of Solids , volume I., chap-ter 9. North Holland, Amsterdam, 1974.[172] J. Scheipers and W. Schirmacher.
Z. Phys. B , 103:547,1996.[173] V. M. Giordano and G. Monaco.
PNAS , 107:21985,2010.[174] G. Baldi, M. Zanatta, E. Gilioli, V. Milman, K. Refson,B. Wehinger, B. Winkler, A. Fontana, and G. Monaco.
Phys. Rev. Lett. , 110:185503, 2013.[175] M. Baggioli and A. Zaccone., 110:185503, 2013.[175] M. Baggioli and A. Zaccone.