Density of Quasi-localized Modes in Glasses: where are the Two-Level Systems?
DDensity of Quasi-localized Modes in Glasses: where are the Two-Level Systems?
Avanish Kumar , Itamar Procaccia , and Murari Singh Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel. Center for OPTical IMagery Analysis and Learning,Northwestern Polytechnical University, Xi’an, 710072 China. Dept. of Physics, University of Texas, Austin Tx. 78712
The existence of a constant density of two-level systems (TLS) was proposed as the basis of someintriguing universal aspects of glasses at ultra-low temperatures. Here we ask whether their existenceis necessary for explaining the universal density of states quasi-localized modes (QLM) in glasses atultra-low temperatures. A careful examination of the QLM that exist in a generic atomistic modelof a glass former reveals at least two types of them, each exhibiting a different density of states,one depending on the frequency as ω and the other as ω . The properties of the glassy energylandscape that is responsible for the two types of modes is examined here, explaining the analyticfeature responsible for the creations of (at least) two families of QLM’s. Although adjacent wellscertainly exist in the complex energy landscape of glasses, doubt is cast on the relevance of TLS forthe universal density of QLM’s. Introduction : Low frequency vibrations in solids aredelocalized Debye modes. In contradistinction, athermalglasses exhibit an excess of low-frequency modes whichhave attracted considerable attention over the years dueto their important contributions to the Boson peak, toenhanced heat capacity and to plastic responses [1, 2].For many years the unusual universal features of glassesat ultra-low temperatures, were explained by a “tunnel-ing model” assuming the existence of a constant distribu-tion of two-level tunneling states (TLS) [3, 4]. In classicalmechanics such two-level systems are envisaged as vibra-tions over a ‘double well’ neighborhood on the energylandscape, with small barriers in between the two wells.While there are no doubts about the existence of suchclose-by wells on the very complex energy landscape ofgeneric glasses [5, 6], there are questions about their rele-vance. Indeed, doubts about the importance of TLS wererepeatedly voiced [7–10]. In this Letter we ask a very spe-cific question, i.e. whether the existence of double-wellstructures is important for explaining the universal den-sity of states of QLM’s in generic glasses. To this aimwe employ a generic atomistic model of a glass formerfor which the low-frequency vibrations and their analyticstructure can be computed explicitly. The model allowsa direct demonstration of the universality of the densityof states of quasi-localized modes, but the analysis showsexplicitly that TLS play no role in establishing this uni-versal behavior.On purely theoretical grounds it was predicted formore than thirty years now [11–17] that in athermalamorphous solids the access modes should exhibit a den-sity of states D ( ω ) with a universal dependence on thefrequency ω , i.e. D ( ω ) ∼ ω in all dimensions . (1)A successful model for the justification of this result isthe so-called “soft potential” model [13] which was pre-sented as an extension of the TLS model. To motivate the analysis presented below it is worthwhile to presentthe thinking behind this model that leads to Eq. (1).To start, we recall that “modes” in amorphous solidsemerge in the harmonic approximation. Consider aglassy system of N particles in a volume V at a temper-ature T = 0; The Hamiltonian of the system is denoted U ( r , r , · · · r N ) and the Hessian matrix is defined as H ij ≡ ∂ U ( r , r , · · · r N ) ∂ r i ∂ r j . (2)The “vibrational modes” at T = 0 are nothing but theeigenfunctions of the Hessian, denoted below as Ψ ( k ) ,each associated with an eigenvalue λ k . If the system isperturbed in direction of one pure eigenfunction it os-cillates indefinitely with frequency ω k , ω k = λ k . Sincethe Hessian is real and symmetric the eigenvalues andthe associated frequencies are all real. When the systemis stable the eigenvalues are all positive with the excep-tion of Goldstone modes associated with symmetries forwhich λ k = 0.The derivation of Eq. (1) goes beyond the harmonicapproximation. Probably the clearest derivation was pre-sented by Gurarie and Chalker [14]. It starts by denotingthe position of the particles as q ≡ { r i } Ni =1 . Next assumethat the energy landscape has minimum at a phase point q . Consider then a close-by point q A (cid:54) = q which forconvenience is chosen to be q A = 0; the potential energyis measured from this point, i.e. U ( q A ) = 0. Finally, as-sume that there exists a QLM whose vibrations are tak-ing place along a ‘reaction coordinate’ s with the energyexpanded around our reference point to fourth order: U ( s ) = (cid:88) n =1 a n n ! s n . (3)We recall that the configuration space is multi-dimensional, and it is therefore crucial to specify thedirection of this ‘reaction coordinate”. Different expan-sions are obtained in different directions. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Since such QLM’s are distributed in phase-space wecan have many such local expansions, and it is assumed that due to the glassy disorder the coefficients a n arequasi-random, with a probability distribution function(PDF) P ( a , a , a , a ) that is finite and smooth. To pro-ceed, consider next a similar expansion but now aroundthe minimum in the potential q . Since this is a minimumthe expansion to the same order reads U ( s ) = (cid:88) n =2 b n (cid:104) ( s − s ) n n ! − ( − s ) n n ! (cid:105) , (4)where s is the value of the assumed “reaction coordi-nate” at q . Finally with the same logic one assume thatthere exists a pdf P ( q , b , b , b ). Recall that by defini-tion b = λ = ω , the eigenvalue of the eigenfunction ofthe Hessian associated with this “reaction coordinate”.The next step in the argument is exact, being simplya consequence of change of variables. Demanding that P ( q , b , b , b ) = | J | P ( a , a , a , a ) ,J ≡ ∂ ( a , a , a , a ) ∂ ( q , b , b , b ) , (5)After little algebra [14] one obtains the result P ( q , b , b , b ) = | b | P ( a , a , a , a ) . (6)If one can now integrate out q , b and b without restric-tions (assuming smoothness and continuity of our PDF),one ends up with P ( b ) ∝ b , or equivalently , D ( ω ) ∝ ω . (7)To obtain Eq. (1) one needs to add another crucial as-sumption, i.e that the minimum in question is locally low-est , or, in other words, if there is another minimum in thequartic expansion (4), it is a higher minimum than theone around which we expand. This is where the pictureof double minimum sneaks in, and we will argue belowthat it is neither required nor supported by simulationdata. The condition for being at a lower minimum is | b | ≤ (cid:112) b b . (8)Integrating out b from Eq.(6) with this restriction inmind leads to P ( b ) ∝ b / , or equivalently , D ( ω ) ∝ ω . (9)In the rest of this letter we will make these considerationsconcrete in the context of a typical model of a glass for-mer, and present a critical assessment of the assumptionsmade. Model of glass former : We use here a standard poly-dispersed model of N = 4000 particles in two dimensions,with density ρ = 1 in a square box of area A = 4000. -1 0 100.030.06 -1 0 1-0.0300.030.06-1 0 100.030.06 -1 0 1-0.0300.030.06-1 0 100.020.04 -1 0 100.030.06 U ( s ) s s (a)(b)(c) (d)(e)(f) FIG. 1. Typical portraits of modes from group A (panels (a),(b) and (c)) and of group B (panel (d), (e) and (f).)
The units of length is σ min as defined below. The binaryinteractions are φ ( r ij ) = (cid:15) (cid:18) σ ij r ij (cid:19) + C + C (cid:18) r ij σ ij (cid:19) + C (cid:18) r ij σ ij (cid:19) (cid:15) = 1 , C = − . , C = 2 . , C = − . . The unit of energy will be (cid:15) and Boltzmann’s constantwill be unity. The interaction length was drawn from aprobability distribution P ( σ ) ∼ /σ in a range between σ min and σ max : σ ij = σ i + σ j (cid:104) − . (cid:12)(cid:12)(cid:12) σ i − σ j (cid:12)(cid:12)(cid:12)(cid:105) ,σ max = 1 . / . , σ min = σ max / . . (10)The parameters are chosen to avoid crystallization andto allow enough smooth derivatives of the Hamiltonian.The system is thermalized at T = 0 . T = 0 using conjugategradient methods. Results and analysis : We have created 6 × inde-pendent realizations of this glass, and after cooling eachto an inherent state at T = 0 we computed the Hessianthere and diagonalized it. Having 8000 modes (eigen-functions) for each realization (i.e. 4.8 × modes all inall), we first collected those with a small participation ra-tio. Here the participation ratio of the k th mode P R ( k ) is defined as P R ( k ) ≡ [ N ( Ψ ( k ) · Ψ ( k ) ) ] − . (11)Focusing on modes whose participation ratio is smallerthan 0.27 we found 403 QLM’s. Each one of these wasanalyzed to provide an expansion of the type of Eq. (4).To find the expansion in the direction of the eigenfunction Ψ ( k ) we define b ( k )2 ≡ Ψ ( k ) · H · Ψ ( k ) ,b ( k )3 ≡ Ψ ( k ) i [ ∂ U∂ r i ∂ r j ∂ r (cid:96) Ψ ( k ) (cid:96) ] Ψ ( k ) j b ( k )4 ≡ Ψ ( k ) i [ Ψ ( k ) (cid:96) ∂ U∂ r i ∂ r j ∂ r (cid:96) ∂ r m Ψ ( k ) m ] Ψ ( k ) j , (12)where repeated indices are summed upon. We shouldstress that in computing the expansion coefficients wehave selected the direction of the harmonic eigenfunctionas the relevant direction. It is possible that expanding inanother direction can lead to a different quartic structure,revealing other features of the energy landscape [6, 18].We choose the direction of Ψ ( k ) since we want b ( k )2 tobe the frequency of our QLM, the one that contributesto the density of states leading to Eq. (1). Excitationsin other directions will lead to a response with a host offrequencies.Having computed the coefficients, we separated our403 QLM’s into two group, 278 of them satisfying theinequality (8), and 125 for which | b ( k )3 | > (cid:113) b ( k )2 b ( k )4 . Itturns out that not even one of the modes that satisfythe constraint (8) has a double-well structure. Denotingthe group of modes that satisfy Eq. (8) as group A andthose that do not as group B, we show in Fig. 1 typicalportraits of the quartic expansion of both groups.According to the analytic discussion presented abovewe expect that the density of states of modes of group Awill obey Eq.(1) although they have no double well struc-ture. On the other hand, the modes of group B that dohave a double well structure are expected to have a den-sity of states proportional to ω [19]. These expectationsare born out by the data. In Fig. 2 we show the dou-ble logarithmic plots of the density of states of modes ofgroup B in panel (a) and of group A in panel (b).It is important to notice that the modes of group B arevery fragile, with the minimum that is found in the nu-merics at q being separated by a very shallow maximumfrom a deeper minimum. This is the case for all the 125modes belonging to group B. Scanning the magnitude ofthe barriers we find that all the 125 barriers span between3.4 × − and 4.0 × − . Due to these very shallow max-ima we expect that the physical significance of modesof group B were quite limited. Any temperature fluc-tuations (or any external strain) will potentially cause a ω ω (a)(b) D ( ω ) ω D ( ω ) FIG. 2. Density of states of modes from group B (panel (a)and of group A (panel (b). transition to the deeper minimum, adding them to modesof group A where the inequality (8) is obeyed. After sucha transition they would contribute to the ω density ofstates. In contradistinction, the modes of group A arevery robust, being very stable against small perturba-tions. The interesting point to observe is that they havenothing to do with two-level systems or double-well po-tentials.We should examine which of the coefficients in the ex-pansion (12) is mostly responsible for the split betweenthe two groups A and B of QLM’s. To this aim we definethe ratio β ( k ) ≡ b ( k )3 / (cid:113) b ( k )2 b ( k )4 and present in Fig. 3 thePDF’s P ( β ( k ) ), where k is taken from all the modes ofboth groups. Clearly, for group B β ( k ) is excluded fromsmall values, being by definition larger than unity . Thequestion remains however which of the three coefficients, b ( k )2 b ( k )3 or b ( k )4 is mostly responsible for the separationto the two groups. Close examination shows that b ( k )2 and b ( k )4 span the same range in groups A and B. It is -4 -2 0 2 400.20.40.60.81 -4 -2 0 2 400.10.20.30.40.5 P ( β ) β P ( β ) (a)(b) FIG. 3. The PDF P ( β ) for group A (panels (a)) and for groupB (panel (b). b ( k )3 which differs greatly. In Fig. 4 we show the PDF’s ofthe cubic coefficient. The PDF of group A peaks at verysmall values around zero, whereas for group B it has adip at small values, and the distribution includes muchlarger (an order of magnitude large) values than in groupA. We conclude that it is the cubic term that determineswhether a given QLM belongs to group A or B in thepresent case. Of course the existence of a large value of b ( k )3 leads to the existence of the deeper minimum awayfrom q in all the profiles of the modes belonging to groupB. We should stress at this point that in the present studyall the coefficients that are quoted are not taken from apresumed distribution or a random matrix theory. Theyare all computed explicitly for the chosen model of glassformer, using its actual Hamiltonian and the modes ofthe Hessian. They represent the actual microscopic en- -1 -0.5 0 0.5 1 1.500.511.522.53-10 -5 0 5 1000.10.20.30.4 (a) (b) P ( b ) b P ( b ) FIG. 4. The PDF P ( b ( k )3 ) for group A (panels (a)) and forgroup B (panel (b). vironment of the glass, being a true realization of thephysical nature of QLM’s computed from scratch. Thusthe lack of two-level systems should be taken as an in-dication of the generic physics of ultra-low temperatureglasses. We also mention in passing that the coefficientsof the expansion are not randomly distributed. What isthe precise source of this lack of randomness is an inter-esting question for future research, as it appears to beat the basis of glassy randomness. Although amorphous,glassy solids do not provide license for any random PDF.Correlations exist and need to be taken into account.As said in the introduction, in spite of the time-honored belief in two-level systems for the study oflow-temperature physics of glassy solids, not everybodyagreed [20, 21]. Yu and Legget [7] for example wrotequite explicitly: “what we are disputing is the claim thatthe TLS model...is the unique and universal explanationof the behavior, in particular...of amorphous solids below1K.” As a conclusion of the present study we concur. Atleast as far as the universal density of states of QLM’sEq. (1) is concerned, we have demonstrated by explicitenumeration of the modes, that those giving rise to thisdensity of states are robust, single-minimum states, thathave nothing to do with two-level systems. Moreover,those modes that do show two minima are all very fragile,and if they would remain relevant in any physical con-text, their density of states will go like ω rather than ω . Acknowledgments : We thank Edan Lerner for use-ful discussions and comments. This work has been sup-ported in part by the Minerva Foundation, Munich, Ger-many, and by the US-Israel Binational Science Founda-tion. [1] W. A. Phillips and A. Anderson,
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