A random matrix approach to the boson peak and Ioffe-Regel criterion in amorphous solids
AA random matrix approach to the boson peakand Ioffe-Regel criterion in amorphous solids
D. A. Conyuh ∗ and Y. M. Beltukov Ioffe Institute, 194021 Saint-Petersburg, Russia (Dated: July 27, 2020)We show that the correlated Wishart ensemble can be used to study general vibrational propertiesof stable amorphous solids with translational invariance. Using the random matrix theory, we foundthe vibrational density of states and the dynamical structure factor. We demonstrate the presenceof the Ioffe-Regel crossover between low-frequency propagating phonons and diffusons at higherfrequencies. The reduced vibrational density of states shows the boson peak, which frequency isclose to the Ioffe-Regel crossover.
Establishing the general vibrational properties inamorphous dielectrics (glasses) is one of the key prob-lems in the physics of disordered systems. The dominantpart of the vibrational spectrum above the Ioffe-Regelcrossover and below the mobility edge is occupied by dif-fusons [1, 2]. These delocalized vibrations are spread bymeans of diffusive energy transfer from atom to atom.The diffusons are responsible for the heat transfer inglasses in a wide range of temperatures. However, themechanism of these vibrations is still poorly understood.Another universal vibrational property of almost allglasses is an excess vibrational density of states (VDOS)well-known as the boson peak . The boson peak was ob-served using different experimental techniques: the Ra-man scattering [3, 4], the X-ray scattering [5], the inelas-tic neutron scattering [6], the far-infrared spectroscopy[7–9] and the temperature dependence of the heat ca-pacity [10–13]. Also, the boson peak was observed intwo-dimensional structures [14–16].There are several theoretical explanations of theseanomalies such as an effective medium theory of elas-ticity [17–21], soft-potential model [22–26], attributionto the transverse-acoustic van Hove singularity [27–29],the mode coupling theory [30]. However, despite a bignumber of articles about the boson peak, the nature ofthis phenomenon also remains under discussion [31–33].It was observed that the boson peak frequency ω b isclose to the frequency ω ir of the Ioffe-Regel crossover be-tween well-defined phonons with a long mean free pathand disordered vibrations, diffusons [34–36]. Therefore, ageneral theory of the boson peak and Ioffe-Regel criterioncan shed light on the nature of vibrations in amorphoussolids.To study these fundamental vibrational features ofamorphous solids, we use an approach based on the ran-dom matrix theory (RMT). This theory has importantapplications in many diverse areas of science and engi-neering [37–46]. Vibrations of amorphous solids are char-acterized by eigenvalues and eigenvectors of the dynam-ical matrix ˆ M . The presence of disorder in amorphoussystems leads to the random nature of the matrix ele-ments M ij . Therefore, the RMT can be applied to study vibrational properties of amorphous solids [47–50]. It isalso applicable to jammed solids [51], which is widelystudied nowadays [52, 53]. However, not every randommatrix ensemble takes into account special correlationsbetween matrix elements M ij in amorphous solids. Inthis work we consider a correlated ensemble, which takesinto account only two the most important properties ofamorphous solids: (i) the system is near the stable equi-librium position and (ii) the potential energy is invariantunder the translation of the system.In this paper we demonstrate that the general proper-ties (i) and (ii) determine a correlated random-matrix en-semble, which represents the most important propertiesof amorphous solids like the boson peak and the Ioffe-Regel crossover. Correlated Wishart ensemble. —The mechanical stabil-ity of amorphous solids is equivalent to the positive def-initeness of the dynamical matrix ˆ M . Any positive def-inite matrix ˆ M can be written as ˆ M = ˆ A ˆ A T and viceversa, ˆ A ˆ A T is positive definite for any (not necessarilysquare) matrix ˆ A [54]. Therefore, we can consider a N × K random matrix ˆ A to obtain a mechanically stablesystem with the dynamical matrix in the form ˆ M = ˆ A ˆ A T ,which is known as the Wishart ensemble [55, 56]. Eachcolumn of the matrix ˆ A represents a “bond” with a pos-itive potential energy [57] U k = 12 (cid:16) (cid:88) i A ik u i (cid:17) , (1)where u i is a displacement of i -th atom from the equilib-rium position. Each row of the matrix ˆ A corresponds tosome degree of freedom. The difference between the num-ber of bonds K and the number of degrees of freedom N plays a crucial role in vibrational and mechanical proper-ties. In a stable system with a finite rigidity, the numberof bonds should be larger than the number of degrees offreedom, which is known as the Maxwell counting rule.For the jammed solids, it was shown that many proper-ties (like the shear modulus and the Ioffe-Regel crossoverfrequency) are scaled with κ = ( K − N ) /N [51, 53].The bond energy U k should not depend on the shift u i → u i + const . Therefore, the matrix ˆ A obeys the sum a r X i v : . [ c ond - m a t . d i s - nn ] J u l rule (cid:80) i A ik = 0. It means that the matrix elements A ij are correlated . In the minimal model, we can assumethat an amorphous solid consists of statistically equiva-lent random bonds. In this case the pairwise correlationsbetween matrix elements A ij can be written as (cid:104) A ik A jl (cid:105) = 1 N C ij δ kl , (2)where ˆ C is some correlation matrix. One can see thatthe correlation matrix ˆ C is proportional to the averagedynamical matrix: ˆ C = NK (cid:104) ˆ M (cid:105) . For simplicity, we con-sider a scalar model of an amorphous solid on a simplecubic lattice with random bonds and unit lattice con-stant a = 1. In this case the average dynamical matrix (cid:104) ˆ M (cid:105) is a crystalline matrix. It is natural to assume thatthe crystalline matrix has simple bonds between near-est neighbors with a certain rigidity. In this case thematrix ˆ C has the following structure. The non-diagonalelements C ij = − Ω if atoms with indices i and j arenearest neighbors in the lattice and C ij = 0 otherwise.Diagonal elements are C ii = 6Ω . The constant Ω definesa typical frequency in the system.The correlation matrix ˆ C is a regular matrix, whichdescribes a simple cubic lattice with nearest neighbor in-teraction. Eigenvalues of the matrix ˆ C depend on thewavevector q which can be expressed as a dispersion law ω ( q ) = 4Ω (cid:16) sin q x q y q z (cid:17) . (3)Using the random matrix approach, it can be shown thatstatistical properties of the random matrix ˆ M are relatedto the known correlation matrix ˆ C . To find these prop-erties, we consider the corresponding resolvents:ˆ G ( z ) = (cid:28) z − ˆ M (cid:29) , ˆ G ( Z ) = 1 Z − ˆ C , (4)where z and Z are complex parameters. The averaging isperformed over different realizations of the random ma-trix ˆ M . In the thermodynamic limit N → ∞ there is afundamental duality relation between spectral propertiesof ˆ M and ˆ C [58]: Z ˆ G ( Z ) = z ˆ G ( z ) , (5)where complex parameters z and Z are related by a con-formal map Z ( z ) defined by the equation κ Z + Z N Tr ˆ G ( Z ) = z, (6)where parameter κ = ( K − N ) /N defines the relativeexcess of the number of bonds over the number of degreesof freedom, which controls a proportion between rigidityand disorder in the system. The duality relation makes itpossible to find the vibrational density of states (VDOS)and dynamical structure factor (DSF) of the dynamicalmatrix ˆ M . ω / Ω R = 1 R = 2theory a b ω / Ω FIG. 1. (Color online) The VDOS of the correlated Wishartensemble for κ = 0 and κ = 1. Color lines show the numericalsolution for the interaction between nearest neighbors ( R = 1)and next-nearest neighbors ( R = 2). Black lines show thetheoretical result (9). Vibrational density of states. —To analyze the VDOS g ( ω ), we consider the normalized trace of ˆ G ( z ), which isthe Stieltjes transform of g ( ω ): F ( z ) = 1 N Tr ˆ G ( z ) = (cid:90) g ( ω ) z − ω dω. (7)For regular correlation matrix ˆ C , we can calculate a sim-ilar quantity F ( Z ) = N Tr ˆ G ( Z ). Using the dispersionlaw for the cubic lattice (3), we find F ( Z ) = 12Ω W s (cid:18) Z − (cid:19) , (8)where W s is the third Watson integral [59]. On theone hand, from Eq. (5) we know the relation ZF ( Z ) = zF ( z ). On the other hand, we can express the VDOS as g ( ω ) = ωπ Im F ( ω − i g ( ω ) = 2 ωπ Im 1 Z ( ω ) , (9)where the complex parameter Z depends on the real pa-rameter ω through the following complex equation κ Z + Z F ( Z ) = ω . (10)This equation defines a contour on a complex plane,which is known as a critical horizon [60]. For a givenparameter ω , Eq. (10) has multiple solutions. We choosea physical one with Im Z ( ω ) < g ( ω ) > g ( ω ) in an im-plicit form, which can be solved numerically. The resultis presented in Fig. 1. For κ = 1 one can see a low fre-quency region with the Debye law g ( ω ) ∼ ω . However,for κ = 0 the VDOS has a constant low-frequency limit.Such behavior of the VDOS was observed in the ran-dom matrix model and the jamming transition [49, 53].The animated plot of the transition between crystallineVDOS ( κ = ∞ ) and a soft amorphous one ( κ = 0) ispresented in Supplemental Materials [61].Figure 1 demonstrates a good agreement between thetheory and the numerical VDOS calculated for finite in-teraction radius R for a system with 400 atoms using theKernel Polynomial Method [62, 63]. The nearest neigh-bor case R = 1 was considered before [49, 64]. The gen-eralization of the numerical model for arbitrary R is pre-sented in Supplemental Materials [61]. As the interactionradius increases, the difference between the theory andthe numerical calculation becomes negligible. Therefore,the theory is applicable for a finite interaction radius,which is important to describe amorphous solids. Dynamical structure factor. —To analyze the spatialstructure of the vibration modes, we calculate the DSF,which specifies the relation between the frequency ω andthe wavevector q [29]. The DSF can be calculated as S ( q , ω ) = 2 ωπ Im (cid:104) q | ˆ G ( ω ) | q (cid:105) . (11)Using the duality relation (5) and the dispersion law (cid:104) q | ˆ G ( Z ) | q (cid:105) = 1 / ( Z − ω ( q )), the resulting dynami-cal structure factor can be presented in the form of thedamped harmonic oscillator (DHO): S ( q , ω ) = 1 π ω Γ( q , ω )( ω − q E ( q , ω )) + ω Γ ( q , ω ) , (12)where the Young modulus is E ( q , ω ) = ω ( q ) q Re ω Z ( ω ) , (13)and the damping isΓ( q , ω ) = ω ( q ) Im ωZ ( ω ) = π ω ( q ) g ( ω ) . (14)Figure 2 shows the normalized DSF for different κ . For κ = 0 there is no exact relation between the frequency ω and the wavevector q . Such a broad DSF was attributedto diffusons [2, 49]. For κ = 1 in the low-frequency rangethere is a linear dispersion ω ∼ q with a small broaden-ing due to a small scattering of plane waves. Such low-frequency vibrations are propagating phonons . However,in the dominant frequency range there is a broad behav-ior of the DSF. Thus, for nonzero κ there is a crossoverbetween phonons and diffusons which is known as theIoffe-Regel crossover. In this paper we do not considerthe Anderson localization which affects only a small partof high-frequency vibrations [2, 49]. Ioffe-Regel criterion, phonons and diffusons. —To an-alyze the Ioffe-Regel crossover we consider the low-frequency region ω (cid:28) Ω. In this case we can use a small-argument expansion of F ( Z ). For any three-dimensionalsystem with a linear dispersion ω ( q ) = Ω q for q → q = 0 q = 1 di ff usons d i ff u s o n s p h o n o n s a b ω / Ω FIG. 2. (Color online) The dynamical structure factor for κ = 0 and κ = 1. For better visual performance the colorrepresents the normalized DSF S ( q , ω ) / max ω S ( q , ω ). this expansion has a form F ( Z ) = − a + √− Z π Ω + O ( Z ) . (15)For the cubic lattice under consideration a = Ω − (cid:112) w s / w s ≈ . ω (cid:28) Ω using an iterative solution of Eq. (10):1 Z ( ω ) = κ ω + 1 ω (cid:118)(cid:117)(cid:117)(cid:116) f ( ω ) + iω/ π Ω (cid:113) κ / ω (cid:112) f ( ω ) , (16)where f ( ω ) = κ ω − a . The sign of f ( ω ) signifi-cantly changes the behavior of Z ( ω ). The correspondingcrossover frequency ω c = κ / a separates the frequencydomain onto two regions.For κ (cid:28) ω < ω c and ω > ω c separately. For the VDOS we obtain g ( ω ) = ω π Ω a / (cid:115) ω c − (cid:112) ω c − ω ω c − ω , ω < ω c , (17) g ( ω ) = 2 aπω (cid:112) ω − ω c , ω > ω c . (18)There is a narrow smooth transition region between (17)and (18). However, this transition region is much smallerthan ω c for κ (cid:28)
1. In the low-frequency region ω (cid:28) ω c ,the VDOS has the Debye behavior g ( ω ) ∝ ω : g D ( ω ) = ω π Ω κ / , (19)which corresponds to a static Young modulus E = Ω κ .For κ = 0 the Young modulus becomes zero, whichmeans a soft system without propagation of phonons. FIG. 3. (Color online) The reduced VDOS g ( ω ) /g D ( ω ) fordifferent values of the parameter κ . Solid line marks thecrossover frequency ω c , dashed line marks the boson peakfrequency ω b = (cid:112) / ω c . Figure 3 demonstrates the boson peak in the reducedVDOS g ( ω ) /g D ( ω ) for different values of the parame-ter κ . The boson peak frequency ω b is close to thecrossover frequency ω c . For κ (cid:28) ω b = (cid:112) / ω c . As a result, the Young modulus E is proportional to the boson peak frequency ω b . Thisrelation was observed by other experimental and theo-retical groups [13, 65]. The height of the boson peak isproportional to κ − / , which diverges for κ →
0. Theboson peak was also observed in two-dimensional systemswith logarithmic scaling of the boson peak height [66].The obtained DSF (12) is defined by the Young mod-ulus E ( q , ω ) and the damping Γ( q , ω ). For κ (cid:28) q (cid:28)
1, the Young modulus has a separate form, whichdepends on the frequency only: E ( ω ) = Ω κ (cid:32) (cid:115) − ω ω c (cid:33) , ω < ω c , (20) E ( ω ) = Ω κ π Ω (cid:16) ω a (cid:17) / , ω > ω c . (21)For ω < ω c we can find a dispersion of phonons usingthe relation ω /q = E ( ω ): ω ( q ) = Ω aq (cid:112) q c − q , (22)where the crossover wavenumber q c = (cid:112) κ / a corre-sponds to the crossover frequency ω c . For low-frequencymodes with q (cid:28) q c , there is a linear dispersion ω ( q ) = √ E q .The damping Γ follows the vibrational density of states g ( ω ) (see Eq. (14)). For ω < ω c it can be written usingthe dispersion relation:Γ = q πa (cid:112) q c − q q c − q . (23) q = 10 = 10 = 10 = 10 = 0.01= 0.1 = 1 q q FIG. 4. (Color online) The damping Γ as a function of thewavevector q for different values of the parameter κ . Dashedlines mark the Rayleigh scattering Γ ∝ q and diffusion lawΓ ∝ q . Vertical solid lines mark the crossover wavenumber q c for the corresponding value of κ . For low-frequency modes with q (cid:28) q c , the dampingΓ( q ) ∼ q , which corresponds to the Rayleigh scatteringfrom disorder (Fig. 4). In amorphous bodies, additionalresonant scattering of phonons by quasilocal vibrationscan occur [26]. However, the number of quasilocal vibra-tions decreases with increasing relaxation time [67], andthis phenomenon goes beyond the general assumptions(i) and (ii) given in the introduction.The mean free path is defined by the group velocity v g = d ω ( q ) / d q and damping Γ as l = v g Γ = 16 π Ω a q ( q c − q ) q c − q . (24)The mean free path l becomes of the order of the wave-length λ = 2 π/q in the transition region near ω c (Fig. 5).It means, that the frequency ω c defines the Ioffe-Regelcrossover, which is usually written as l/λ ≈ / ω > ω c we considerthe dominant part of this frequency region: ω c (cid:28) ω (cid:28) Ω.In this case, the DSF (12) takes the diffusion form S ( q , ω ) = 1 π q , ω ) ω + Γ ( q , ω ) , (25)which verifies the notion of diffusons introduced in [1, 2].In the same frequency range g ( ω ) ≈ a/π and Γ = Dq ,where D = Ω a is a diffusivity. Previously, this form ofthe DSF was obtained numerically [49].Figure 4 shows a crossover between the low-frequencyRayleigh scattering Γ ∝ q and the diffusion dampingΓ ∝ q . Such a quadratic dependence above Ioffe-Regelcrossover was observed experimentally [68–70]. Isostatic state. —If κ = 0 the number of degrees offreedom N is equal to the number of bonds K . In the FIG. 5. (Color online) The ratio of the mean free path l tothe wavelength λ as a function of the reduced frequency ω/ω c for different values of the parameter κ . Horizontal dashedline marks the Ioffe-Regel criterion l ≈ λ/ jamming transition this state is known as the isostaticstate. In this case the macroscopic rigidity becomes zeroand the low-frequency VDOS does not follow the Debyelaw ( ω c = 0). Instead, there is a nonzero low-frequencyVDOS. Using the second-order approximation in the ran-dom matrix model, we obtain g is ( ω ) (cid:39) aπ − π Ω (cid:114) ω a . (26)This dependence has a linear form as function of √ ω (seeinset in Fig. 1a). Such a low-frequency cusp-like singu-larity of the isostatic VDOS was observed numerically inthe random matrix model [51] and the jamming transi-tion [53, 71]. In our model, this behavior is related tothe diffusion nature of vibrations in this frequency rangewith the DSF governed by Eq. (25).In summary, we have demonstrated that the randommatrix theory can be applied to study general vibra-tional properties of amorphous solids. Taking into ac-count only the most important correlations of randommatrices, which ensure the mechanical stability (i) andthe translation invariance (ii), we find the vibrationaldensity of states and the dynamical structure factor. Wedemonstrate the presence of the Ioffe-Regel crossover be-tween low-frequency propagating phonons and diffusonsat higher frequencies. The boson peak essentially appearsnear the Ioffe-Regel crossover. The obtained scaling re-lations correspond to transverse vibrational properties ofthe jammed solids if we put κ ∼ z − z c ∼ ∆ φ / andΩ ∼ ∆ φ ( α − / [53, 65].We wish to acknowledge D. A. Parshin, V. I. Kozub,Y. M. Galperin, and V. L. Gurevich for valuable discus-sions. The work was supported by the Russian Federa-tion President Grant no. MK-3052.2019.2. ∗ [email protected][1] P. B. Allen, and J. L. Feldman, Phys. Rev. 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