Accurate densities of states for disordered systems from free probability: Live Free or Diagonalize
AAccurate densities of states for disordered systems from free probability: Live Free or Diagonalize
Matthew Welborn, ∗ Jiahao Chen 陳 家 豪 , † and Troy Van Voorhis ‡ Department of Chemistry, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
We investigate how free probability allows us to approximate the density of states in tight binding modelsof disordered electronic systems. Extending our previous studies of the Anderson model in one dimensionwith nearest-neighbor interactions [J. Chen et al. , Phys. Rev. Lett. , 036403 (2012)], we find that freeprobability continues to provide accurate approximations for systems with constant interactions on two- andthree-dimensional lattices or with next-nearest-neighbor interactions, with the results being visually indistin-guishable from the numerically exact solution. For systems with disordered interactions, we observe a small butvisible degradation of the approximation. To explain this behavior of the free approximation, we develop andapply an asymptotic error analysis scheme to show that the approximation is accurate to the eighth moment inthe density of states for systems with constant interactions, but is only accurate to sixth order for systems withdisordered interactions. The error analysis also allows us to calculate asymptotic corrections to the density ofstates, allowing for systematically improvable approximations as well as insight into the sources of error withoutrequiring a direct comparison to an exact solution.
I. INTRODUCTION
Disordered matter is ubiquitous in nature and in manmadematerials . Random media such as glasses , disordered al-loys , and disordered metals exhibit unusual properties result-ing from the unique physics produced by statistical fluctuations.For example, disordered materials often exhibit unusual elec-tronic properties, such as in the weakly bound electrons in metal–ammonia solutions , or in water . Paradoxically, disordercan also enhance transport properties of excitons in new pho-tovoltaic systems containing bulk heterojunction layers andquantum dots , producing anomalous diffusion effects which appear to contradict the expected effects of Anderson lo-calization . Accounting for the effects of disorder in electro-optic systems is therefore integral for accurately modeling and en-gineering second–generation photovoltaic devices .Disordered systems are challenging for conventional quantummethods, which were developed to calculate the electronic struc-ture of systems with perfectly known crystal structures. Determin-ing the electronic properties of a disordered material thus necessi-tates explicit sampling of relevant structures from thermodynam-ically accessible regions of the potential energy surface, followedby quantum chemical calculations for each sample. Furthermore,these materials lack long-range order and must therefore be mod-eled with large supercells to average over possible realizations ofshort-range order and to minimize finite-size effects. These twofactors conspire to amplify the cost of electronic structure calcu-lations on disordered materials enormously.To avoid such expensive computations, we consider instead cal-culations where the disorder is treated explicitly in the electronicHamiltonian. The simplest such Hamiltonian comes from the An-derson model , which is a tight binding lattice model of theelectronic structure of a disordered electronic medium. Despiteits simplicity, this model nonetheless captures the rich physicsof strong localization and can be used to model the conductiv-ity of disordered metals . However, the Anderson model can-not be solved exactly except in special cases , which compli-cates studies of its excitation and transport properties. Studyingmore complicated systems thus requires accurate, efficiently com-putable approximations for the experimental observables of inter-est.Random matrix theory offers new possibilities for developingaccurate approximate solutions to disordered systems . Inthis Article, we focus on using random matrix theory to con- struct efficient approximations for the density of states of a ran-dom medium. The density of states is one of the most importantquantities that characterize an electronic system, and a large num-ber of physical observables can be calculated from it . Further-more, it only depends on the eigenvalues of the Hamiltonian andis thus simpler to approximate, as information about the eigen-vectors is not needed. We have previously shown that highly ac-curate approximations can be constructed using free probabilitytheory for the simplest possible Anderson model, i.e. on a one-dimensional lattice with constant nearest-neighbor interactions .However, it remains to be seen if similar approximations are suf-ficient to describe more complicated systems, and in particular ifthe richer physics produced by more complicated lattices and byoff-diagonal disorder can be captured using such free probabilisticmethods.In this Article, we present a brief, self–contained introductionto free probability theory in Section II. We then develop approxi-mations from free probability theory in Section III that generalizeour earlier study in three ways. First, we develop analogousapproximations for systems with long range interactions, special-izing to the simplest such extension of a one-dimensional latticeswith next-nearest-neighbor interactions. Second, we study latticesin two and three dimensions. We consider square and hexagonaltwo-dimensional lattices to investigate the effect of coordinationon the approximations. Third, we also make the interactions ran-dom and develop approximations for these systems as well. Thesecases are summarized graphically in Figure 1 and are represen-tative of the diversity of disorder systems described above. Fi-nally, we introduce an asymptotic error analysis which allows usto quantify and analyze the errors in the free probability approxi-mations in Section IV. II. FREE PROBABILITYA. Free independence
In this section, we briefly introduce free probability by high-lighting its parallels with (classical) probability theory. One ofthe core ideas in probability theory is how to characterize therelationship between two (scalar-valued) random variables x and y . They may be correlated, so that the joint moment (cid:104) xy (cid:105) is notsimply the product of the individual expectations (cid:104) x (cid:105) (cid:104) y (cid:105) , or theymay be correlated in a higher order moment, i.e. there are some a r X i v : . [ c ond - m a t . d i s - nn ] M a y Free independence and the R -transform III NUMERICAL RESULTS Figure 1: The lattices considered in this work: (a) one-dimensionalchain with nearest neighbor interactions, (b) one-dimensional chain withmany neighbors, (c) two-dimensional square lattice, (d) two-dimensionalhexagonal (honeycomb) lattice, (e) three-dimensional cubic lattice, and(f) one dimensional-chain with disordered interactions. smallest positive integers m and n for which (cid:104) x m y n (cid:105) (cid:54) = (cid:104) x m (cid:105) (cid:104) y n (cid:105) . Ifneither case holds, then they are said to be independent, i.e. that alltheir joint moments of the form (cid:104) x m y n (cid:105) factorize into products ofthe form (cid:104) x m y n (cid:105) = (cid:104) x m (cid:105) (cid:104) y n (cid:105) . For random matrices, similar state-ments can be written down if the expectation (cid:104)·(cid:105) is interpreted asthe normalized expectation of the trace, i.e. (cid:104)·(cid:105) = N E Tr · , where N is the size of the matrix. However, matrices in general do notcommute, and therefore this notion of independence is no longerunique: for noncommuting random variables, one cannot simplytake a joint moment of the form (cid:104) A m B n · · · A m k B n k (cid:105) and assert itto be equal in general to (cid:104) A m + ··· + m k B n + ··· + n k (cid:105) . The complica-tions introduced by noncommutativity give rise to a different the-ory, known as free probability theory, for noncommuting randomvariables . This theory introduces the notion of free indepen-dence, which is the noncommutative analogue of (classical) inde-pendence. Specifically, two noncommutative random variables A and B are said to be freely independent if for all positive integers m ,..., m k , n ,..., n k , the centered joint moment vanishes, i.e. (cid:10) A m B n · · · A m k B n k (cid:11) = , (1)where we have introduced the centering notation A = A − (cid:104) A (cid:105) .This naturally generalizes the notion of classical independence tononcommuting variables, as the former is equivalent to requiringthat all the centered joint moments of the form (cid:10) x m y n (cid:11) vanish.If the expectation (cid:104) A (cid:105) is reinterpreted as the normalized expecta-tion of the trace of a random matrix A , then the machinery of freeindependence can be applied directly to random matrices . B. Free independence and the R -transform One of the central results of classical probability theory is that if x and y are independent random variables with distributions p X ( x ) and p Y ( y ) respectively, then the probability distribution of theirsum x + y is given by the convolution of the distributions, i.e. p X + Y ( y ) = ˆ ∞ − ∞ p X ( x ) p Y ( x − y ) dx . (2)An analogous result holds for freely independent noncommutingrandom variables and is known as the (additive) free convolution; this is most conveniently defined using the R -transform .For a probability density p ( x ) supported on [ a , b ] , its R -transform R ( w ) is defined implicitly via G ( z ) = lim ε → + ˆ ba p ( x ) z − ( x + i ε ) dx (3a) R ( w ) = G − ( w ) − w . (3b)These quantities have natural analogues in Green function the-ory: p ( x ) is the density of states, i.e. the distribution of eigen-values of the underlying random matrix; G ( z ) is the Cauchytransform of p ( x ) , which is the retarded Green function; and G − ( w ) = R ( w ) + / w is the self-energy. The R -transform al-lows us to define the free convolution of A and B , denoted A (cid:1) B,by adding the individual R -transforms R A (cid:1) B ( w ) : = R A ( w ) + R B ( w ) . (4)This finally allows to state that if A and B are freely independent,then the sum A + B must satisfy R A + B ( w ) = R A (cid:1) B ( w ) . (5)In general, random matrices A and B are neither classically in-dependent nor freely independent. However, we can always con-struct combinations of them that are always freely independent.One such combination is A + Q † BQ , where Q is a random orthog-onal (unitary) matrix of uniform Haar measure, as applied to realsymmetric (Hermitian) A and B . The similarity transform ef-fected by Q randomly rotates the basis of B , so that the eigen-vectors of A and B are always in generic position, i.e. that anyeigenvector of A is uncorrelated with any eigenvector of B . Thisis the main result that we wish to exploit. While in general A and B are not freely independent, and hence (5) fails to hold exactly,we can nonetheless make the approximation that (5) holds approx-imately , and use this as a way to calculate the density of states of arandom matrix H using only its decomposition into a matrix sum H = A + B . Our application of this idea to the Anderson model isdescribed below. III. NUMERICAL RESULTSA. Computation of the Density of States and its Free Approximant
We now wish to apply the framework of free probability theoryto study Anderson models beyond the one-dimensional nearest-neighbor model which was the focus of our initial study . Itis well-known that more complicated Anderson models exhibitrich physics that are absent in the simplest case. First, the one-dimensional Anderson Hamiltonian with long range interactionshas delocalized eigenstates at low energies and an asymmetricdensity of states, features that are absent in the simplest Ander-son model . These long range interactions give rise to slowlydecaying interactions in many systems, such as spin glasses and ionic liquids . Second, two-dimensional lattices can exhibitweak localization , which is responsible for the unusual con-ductivities of low temperature metal thin films . The hexag-onal (honeycomb) lattice is of particular interest as a tight bind-ing model for nanostructured carbon allotropes such as carbonnanotubes and graphene , which exhibit novel electronic2 II NUMERICAL RESULTS A Computation of the Density of States and its Free Approximant phases with chirally tunable band gaps and topological insula-tion . Third, the Anderson model in three dimensions exhibitsnontrivial localization phases that are connected by the metal–insulator transition . Fourth, systems with off-diagonal dis-order, such as substitutional alloys and Frenkel excitons in molec-ular aggregates , exhibit rich physics such as localization tran-sitions in lattices of any dimension , localization dependence onlattice geometry , Van Hove singularities , and asymmetries inthe density of states . Despite intense interest in the effects ofoff-diagonal disorder, such systems have resisted accurate model-ing . We are therefore interested to find out if our approxima-tions as developed in our initial study can be applied also to allthese disordered systems.The Anderson model can be represented in the site basis by thematrix with elements H i j = g i δ i j + J i j (6)where g i is the energy of site i , δ i j is the usual Kronecker delta,and J i j is the matrix of interactions with J ii =
0. Unless otherwisespecified, we further specialize to the case of constant interac-tions between connected neighbors, so that J i j = JM i j where J isa scalar constant representing the interaction strength, and M isthe adjacency matrix of the underlying lattice. Unless specifiedotherwise, we also apply vanishing (Dirichlet) boundary condi-tions, as this reduces finite-size fluctuations in the density of statesrelative to periodic boundary conditions. For concrete numericalcalculations, we also choose the site energies g i to be iid Gaussianrandom variables of variance σ and mean 0. With these assump-tions, the strength of disorder in the system can be quantified by asingle dimensionless parameter σ / J .The particular quantity we are interested in approximating is thedensity of states, which is one of the most important descriptorsof electronic band structure in condensed matter systems . It isdefined as the distribution ρ H ( x ) = (cid:42) ∑ j δ ( x − ε j ) (cid:43) (7)where ε j is the j th eigenvalue of a sample of H and the expectation (cid:104)·(cid:105) is the ensemble average.To apply the approximations from free probability theory,we partition our Hamiltonian matrix into its diagonal and off–diagonal components A and B . The density of states of A is sim-ply a Gaussian of mean 0 and variance σ , and for many of ourcases studied below, the density of states of B is proportional tothe adjacency matrix of well–known graphs and hence is knownanalytically. We then construct the free approximant H (cid:48) = A + Q T BQ (8)where Q is a random orthogonal matrix of uniform Haar mea-sure as discussed in Section II B, and find its density of states ρ H (cid:48) .Specific samples of Q can be generated by taking the orthogonalpart of the QR decomposition of matrix from the Gaussian or-thogonal ensemble (GOE) . We then average the approximatedensity of states over many realizations of the Hamiltonian and Q and compare it to the ensemble averaged density of states gener-ated from exact diagonalization of the Hamiltonian. We choosethe number of samples to be sufficient to converge the density ofstates with respect to the disorder in the Hamiltonian. While thisis not the most efficient way of computing free convolutions, itprovides a general and robust test for the quality of the free ap-proximation. The free approximant can be computed efficientlyusing numerical free convolution techniques . B. One-dimensional chain
We now proceed to apply the theory of the previous sectionto specific examples of the Anderson model on various lattices.Previously, we had studied the Anderson model on the one-dimensional chain : H i j = g i δ i j + J (cid:0) δ i , j + + δ i , j − (cid:1) , (9)which is arguably the simplest model of a disordered system. De-spite its simple tridiagonal form, this Hamiltonian does not havean exact solution for its density of states, and many approxima-tions for it have been developed . However, unlike the originalHamiltonian, the diagonal and off-diagonal components each havea known density of states when considered separately. To calcu-late the density of states of the Hamiltonian, we diagonalized 1000samples of 1000 × σ / J . C. One-dimensional lattice with non-neighbor interactions
Going beyond tridiagonal Hamiltonians, we next study the An-derson model on a one-dimensional chain with constant interac-tions to n neighbors. The Hamiltonian then takes the form: H Di j = g i δ i j + J (cid:34) n ∑ k = δ i , j + k + δ i + k , j (cid:35) . (10)where we use the superscript to distinguish the one-dimensionalmany-neighbor Hamiltonian from its higher dimensional analogs.Unlike the nearest-neighbor interaction case above, the density ofstates is known to exhibit Van Hove singularities at all but thestrongest disorder .We average over 1000 samples of 1000 × n = , ..., n = . The reproduction of singulari-ties by the free approximant parallels similar observations foundin other applications of free probability to quantum informationtheory . D. Square, hexagonal and cubic lattices
We now investigate the effect of dimensionality on the accuracyof the free approximant in three lattices. First, we consider theAnderson model on the square lattice, with Hamiltonian: H D = B D ⊗ I + I ⊗ B D + A (11)3 Square, hexagonal and cubic lattices III NUMERICAL RESULTS
Figure 2: Comparison of exact density of states (lines) with free probability approximant (circles) for the lattices in Figure 1, with (b) showing the caseof n = σ / J , the ratio of the noisiness of diagonal elements to the strength of off-diagonal interaction. In (f), theaxis is chosen to be the relative strength of off-diagonal disorder to diagonal disorder, σ ∗ / σ , with σ / J = -10 0 10 0.1 1 10x σ /J -10 0 10 0.1 1 10x σ /J -10 0 10 0.1 1 10x σ /J -10 0 10 0.1 1 10x σ /J -10 0 10 0.1 1 10x σ /J -10 0 10 0.1 1 10x σ */ σ (a) (b) (c) (d) (e) (f) ρ(x) ρ(x) where B D is the off-diagonal part of the H D defined in equa-tion (10), I is the identity matrix with the same dimensions as B D , A is the diagonal matrix of independent random site ener-gies of appropriate dimension, and ⊗ is the Kronecker (direct)product. We have found that a square lattice of 50 × = × n = , ..., × σ / J ∼ .
1) are also reproduced by the free approximation.Third, we consider the Anderson model on a cubic lattice,whose Hamiltonian is: H D = (cid:0) B D ⊗ I ⊗ I (cid:1) + (cid:0) I ⊗ B D ⊗ I (cid:1) + (cid:0) I ⊗ I ⊗ B D (cid:1) + A (12)Figure 2(e) shows the approximate and exact density of states cal-culated from 1000 samples of 1000 × × ×
10 cubic lattice which is significantly smallerin linear dimension than the previously considered lattices. Wetherefore observed oscillatory features in the density of states aris-ing from finite-size effects. Despite this, the free approximant is still able to reproduce the exact density of states quantitatively. Infact, if the histogram in Figure 2(e) is recomputed with finer his-togram bins to emphasize the finite-size induced oscillations, westill observe that the free approximant reproduces these features.
E. Off-diagonal disorder
Up to this point, all of the models we have considered haveonly site disorder, with no off-diagonal disorder. Free probabilityhas thus far provided a qualitatively correct approximation for allthese lattices. To test the robustness of this approximation, wenow investigate systems with random interactions. The simplestsuch system is the one-dimensional chain, with a Hamiltonian ofthe form: H i j = g i δ i j + h i (cid:0) δ i , j + + δ i , j − (cid:1) . (13)Unlike in the previous systems, the interactions are no longer con-stant, but are instead new random variables h i . We choose themto be Gaussians of mean J and variance ( σ ∗ ) . There are nowtwo order parameters to consider: σ ∗ / J , the relative disorder inthe interaction strengths, and σ ∗ / σ , the strength of off-diagonaldisorder relative to site disorder. As in the prior one-dimensionalcase, we average over 1000 realizations of 1000 × σ ∗ / σ , but not σ ∗ / J . In Figure 2(f), we demonstratethe results of varying σ ∗ / σ with σ / J =
1. In the limits σ ∗ / σ (cid:29) σ ∗ / σ (cid:28)
1, the free approximation matches the exact resultwell; however, there is a small but noticeable discrepancy betweenthe exact and approximate density of states for moderate relativeoff-diagonal disorder, though the quality of the approximation is4
V ERROR ANALYSIS mostly unaffected by the centering of the off-diagonal disorder. Inthe next section, we will investigate the nontrivial behavior of theapproximation with the σ ∗ / σ order parameter. IV. ERROR ANALYSIS
In our numerical experiments, we have found that the accuracyof the free approximation remains excellent for systems with onlysite disorder, regardless of the underlying lattice topology or thenumber of interactions that each site has. Details such as finite-size oscillations and Van Hove singularities are also capturedwhen present. However, when off-diagonal disorder is present,the quality of the approximation does vary qualitatively with theratio of off-diagonal disorder to site disorder σ ∗ / σ as illustratedin Section III E, and the error is greatest when σ ∗ ≈ σ . To un-derstand the reliability of the free approximant (8) in all these sit-uations, we apply an asymptotic moment expansion to calculatethe leading order error terms for the various systems. In general,a probability density ρ can be expanded with respect to anotherprobability density ˜ ρ in an asymptotic moment expansion knownas the Edgeworth series : ρ ( x ) = exp (cid:32) ∞ ∑ m = κ ( m ) − ˜ κ ( m ) m ! (cid:18) − ddx (cid:19) m (cid:33) ˜ ρ ( x ) (14)where κ ( m ) is the m th cumulant of ρ and ˜ κ ( m ) is the m th cumulantof ˜ ρ . When all the cumulants exist and are finite, this is an exactrelation that allows for the distribution ˜ ρ to be systematically cor-rected to become ρ by substituting in the correct cumulants. If thefirst ( n − ) cumulants of ρ and ˜ ρ match, but not the n th, then wecan calculate the leading-order asymptotic correction to ˜ ρ as: ρ ( x ) = exp (cid:32) κ ( n ) − ˜ κ ( n ) n ! (cid:18) − ddx (cid:19) n + . . . (cid:33) ˜ ρ ( x ) (15a) = (cid:32) + κ ( n ) − ˜ κ ( n ) n ! (cid:18) − ddx (cid:19) n + . . . (cid:33) ˜ ρ ( x ) (15b) = ˜ ρ ( x ) + ( − ) n n ! (cid:16) κ ( n ) − ˜ κ ( n ) (cid:17) d n ˜ ρ dx n ( x ) + O (cid:18) d n + ˜ ρ dx n + (cid:19) (15c) = ˜ ρ ( x ) + ( − ) n n ! (cid:16) µ ( n ) − ˜ µ ( n ) (cid:17) d n ˜ ρ dx n ( x ) + O (cid:18) d n + ˜ ρ dx n + (cid:19) (15d)where on the second line we expanded the exponential e X = + X + . . . , and on the fourth line we used the well-known re- lationship between cumulants κ and moments µ and the fact thatthe first n − ρ and ˜ ρ were identical by assumption.We can now use this expansion to calculate the leading-orderdifference between the exact density of states ρ H = ρ A + B , andits free approximant ρ H (cid:48) = ρ A (cid:1) B by setting ˜ ρ = ρ H (cid:48) and ρ = ρ H in (15d). The only additional data required are the moments µ ( n ) H = (cid:104) H n (cid:105) and µ ( n ) H (cid:48) = (cid:10) ( H (cid:48) ) n (cid:11) , which can be computed fromthe sampled data or recursively from the joint moments of A and B as detailed elsewhere . This then gives us a way to detect dis-crepancies, which is to calculate successively higher moments of H and H (cid:48) to determine whether the difference in moments is sta-tistically significant, and then for the smallest order moment thatdiffers, calculate the correction using (15d).The error analysis also yields detailed information about thesource of error in the free approximation. The n th moment of H is given by µ ( n ) H = (cid:104) H n (cid:105) = (cid:104) ( A + B ) n (cid:105) = ∑ m , n ,..., m k , n k ∑ kj = m j + n j = n (cid:104) A m B n · · · A m k B n k (cid:105) , (16)where the last equality arises from expanding ( A + B ) n in a non-commutative binomial series. If A and B are freely independent,then each of these terms must satisfy recurrence relations that canbe derived from the definition (1) . Exhaustively enumeratingand examining each of the terms in the final sum to see if theysatisfy (1) thus provides detailed information about the accuracyof the free approximation.We now apply this general error analysis for the specific sys-tems we have studied. It turns out that the results for systems withand without off-diagonal disorder exhibit different errors, and soare presented separately below. A. Systems with constant interactions
We have previously shown that for the one-dimensional chainwith nearest-neighbor interactions, the free approximant is exactin the first seven moments, and that the only term in the eighthmoment that differs between the free approximant and the exact H is (cid:68) ( AB ) (cid:69) . The value of this joint moment can be understoodin terms of discretized hopping paths on the lattice . Writing outthe term (cid:68) ( AB ) (cid:69) explicitly in terms of matrix elements and withEinstein’s implicit summation convention gives: (cid:68) ( AB ) (cid:69) = N E (cid:0) A i i B i i A i i B i i A i i B i i A i i B i i (cid:1) (17a) = N E (cid:0) ( g i δ i i ) (cid:0) JM i i (cid:1) (cid:0) g i δ i i (cid:1) (cid:0) JM i i (cid:1) (cid:0) g i δ i i (cid:1) (cid:0) JM i i (cid:1) (cid:0) g i δ i i (cid:1) (cid:0) JM i i (cid:1)(cid:1) (17b) = N E (cid:0) g i g i g i g i J M i i M i i M i i M i i (cid:1) . (17c)From this calculation, we can see that each multiplication by A weights each path by the site energy of a given site, g i , and each multiplication by B weights the path by J and causes the path to5 Systems with constant interactions IV ERROR ANALYSIS hop to a coupled site. The sum therefore reduces to a weightedsum over returning paths on the lattice that must traverse exactlythree intermediate sites. The only paths on the lattice with nearest-neighbors that satisfy these constraints are shown in Figure 3(a),namely ( i , i , i , i ) = ( k , k + , k , k + ) , ( k , k + , k + , k + ) , ( k , k − , k , k − ) , and ( k , k − , k − , k − ) for some startingsite k . The first path contributes weight E (cid:0) g k g k + (cid:1) J = E (cid:0) g k (cid:1) E (cid:0) g k + (cid:1) J = σ J while the second term has weight E (cid:0) g k g k + g k + (cid:1) J = E ( g k ) E (cid:0) g k + (cid:1) E ( g k + ) J =
0. Similarly,the third and fourth paths also have weight σ J and 0 respec-tively. Finally averaging over all possible starting sites, we ar-rive at the final result that (cid:68) ( AB ) (cid:69) = σ J with periodic bound-ary conditions and (cid:68) ( AB ) (cid:69) = ( − / N ) σ J with vanishingboundary conditions. We therefore see when N is sufficientlylarge, the boundary conditions contribute a term of O ( / N ) whichcan be discarded, thus showing the universality of this result re-gardless of the boundary conditions. Figure 3: (a) Diagrammatic representation of the four paths that con-tribute to the leading order error for the case of a two-dimensional squarelattice with constant interactions and nearest neighbors. Dots contributea factor of g i for site i . Solid arrows represent a factor of J . Each pathcontributes J (cid:10) g a (cid:11)(cid:10) g b (cid:11) = σ J to the error. (b) Build up of the diagram-matic representation the leading order error in the case of a 1D chainwith off-diagonal disorder. The two dashed arrows contribute a factor of µ − µ . Because of the disorder in the interactions, multiplication by B allows loops back to the same site. The first of these loops, (cid:68) AB (cid:69) , haszero expectation value because it contains an independent random vari-able of mean zero as a factor. Once two loops are present, the expectationvalue instead contains this random variable squared, which has nonzeroexpectation value. Applying the preceding error analysis, we observe that the re-sult from the one-dimensional chain generalizes all the other sys-tems with constant interactions that we have studied; the only dif-ference being that the coefficient 2 is simply replaced by n , thenumber of sites accessible in a single hop from a given lattice site.In order to keep the effective interaction felt by a site constant aswe scale n , we can choose J to scale as √ n . In this case, the freeapproximation converges to the exact result as n . We can generalize the argument presented above to explain why (cid:68)(cid:0) AB (cid:1) (cid:69) is the first nonzero joint centered moment, and thus whythe approximation does not break down before the eighth moment.Consider centered joint moments of the form: (cid:68) A a B b A a B b . . . A a n B b n (cid:69) (18)for positive integers { a i , b i } such that ∑ i ( a i + b i ) ≤
8. Since A is diagonal with iid elements, all powers of A n are also diagonalwith iid elements, and so A n =
0. Centered higher powers of B , B n , couple each site to other sites with interaction strengths J n ,but after centering, the diagonal elements of B n are zero and mul-tiplication by B n still represents a hop from one site to a differentcoupled site. Therefore, the lowest order nonzero joint centeredmoment requires at least four hops, so n ≥ a i = b i =
1, i.e. the term (cid:68) ( AB ) (cid:69) . B. Random interactions
When the off-diagonal interactions are allowed to fluctuate, thefree approximation breaks down in the sixth moment, where thejoint centered moment (cid:28)(cid:16) AB (cid:17) (cid:29) fails to vanish. We can un-derstand this using a generalization of the hopping explanationfrom before. In this case, B contains nonzero diagonal elements,which corresponds to a nonzero weight for paths that stay at thesame site. Thus, (cid:16) AB (cid:17) contains a path of nonzero weight thatstarts at a site and loops back to that site twice (shown in Fig-ure 3(b)). The overall difference in the moment of the exact dis-tribution from that in the free distribution is 2 σ (cid:0) µ − µ (cid:1) , where µ and µ are the fourth and second moments of the off-diagonaldisorder. As above, the σ component of this difference can beunderstood as the contribution of the two A s in the joint centeredmoment. The other factor, 2 (cid:0) µ − µ (cid:1) , is the weight of the pathof two consecutive self-loops. The sixth moment is the first tobreak down because, as before, we must hop to each node on ourpath twice in order to avoid multiplying by the expectation valueof mean zero, and (cid:0) AB (cid:1) is the lowest order term that allows sucha path.We summarize the the leading order corrections and errors inTable I. At this point, we introduce the quantity ˜ J = √ nJ , whichis an aggregate measure of the interactions of any site with all its2 n neighbors. As can be seen, the discrepancy occurs to eighthorder for all the studied systems with constant interactions, with anumerical prefactor indicative of the coordination number of thelattice, and the factor of 1/8! strongly suppresses the contributionof the error terms. Furthermore, for any given value of the total in-teraction ˜ J , the error decreases quickly with coordination number2 n , suggesting that the free probability approximation is exact inthe mean field limit of 2 n → ∞ neighbors. This is consistent withprevious studies of the Anderson model employing the coherentpotential approximation. In contrast, the system with off-diagonal disorder has a discrepancy in the sixth moment, whichhas a larger coefficient in the Edgeworth expansion (15d). Thisexplains the correspondingly poorer performance of our free ap-proximation for systems with off-diagonal disorder. Furthermore,the preceding analysis shows that only the first and second mo-ments of the diagonal disorder σ contribute to the correction co-6 CONCLUSION B Random interactions efficient, thus showing that this behavior is universal for disorderwith finite mean and standard deviation.
Table I: Coefficients of the leading-order error in the free probability ap-proximation in the Edgeworth expansion (15d).Order Term Coefficient1D 8 ( AB ) ˜ J σ / ( · )
2D square 8 ( AB ) ˜ J σ / ( · )
2D honeycomb 8 ( AB ) ˜ J σ / ( · )
3D cube 8 ( AB ) ˜ J σ / ( · )
1D with n nearest-neighbors 8 ( AB ) ˜ J σ / ( n · )
1D with off-diagonal disorder 6 (cid:0) AB (cid:1) σ (cid:0) µ − µ (cid:1) / V. CONCLUSION
Free probability provides accurate approximations to the den-sity of states of a disordered system, which can be constructedby partitioning the Hamiltonian into two easily-diagonalizable en-sembles and then free convolving their densities of states. Previ-ous work showed that this approximation worked well for theone-dimensional Anderson model partitioned into its diagonal andoff-diagonal components. Our numerical and theoretical study de-scribed above demonstrates that the same approximation schemeis widely applicable to a diverse range of systems, encompassingmore complex lattices and more interactions beyond the nearest-neighbor. The quality of the approximation remains unchangedregardless of the lattice as long as the interactions are constant, with the free approximation being in error only in the eighth mo-ment of the density of states. When the interactions fluctuate, thequality of the approximation worsens, but remains exact in thefirst five moments of the density of states.Our results strongly suggest that free probability has the po-tential to produce high-quality approximations for the propertiesof disordered systems. In particular, our theoretical analysis ofthe errors reveals universal features of the quality of the approx-imation, with the error being characterized entirely by the mo-ments of the relevant fluctuations and the local topology of thelattice. This gives us confidence that approximations constructedusing free probability will give us high-quality results with rigor-ous error quantification. This also paves the way for future inves-tigations for constructing fast free convolutions using numericalmethods for R -transforms, which would yield much faster meth-ods for constructing free approximations. Additionally, furtherstudies will be required to approximate other observables of inter-est such as conductivities and phase transition points. These willrequire further theoretical investigation into how free probabilitycan help predict properties of eigenvectors, which may involvegeneralizing some promising initial studies linking the statisticsof eigenvectors such as their inverse participation ratios to eigen-value statistics such as the spectral compressibility . Acknowledgements
This work was funded by NSF SOLAR Grant No. 1035400.M.W. acknowledges support from the NSF GRFP. We thank AlanEdelman, Eric Hontz, Jeremy Moix, and Wanqin Xie for insightfuldiscussions. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] J. M. Ziman,
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