Dyson's disordered linear chain from a random matrix theory viewpoint
aa r X i v : . [ m a t h - ph ] J a n DYSON’S DISORDERED LINEAR CHAIN FROM A RANDOM MATRIXTHEORY VIEWPOINT
PETER J. FORRESTER
Abstract.
The first work of Dyson relating to random matrix theory, "The dynamics of adisordered linear chain”, is reviewed. Contained in this work is an exact solution of a so-calledType I chain in the case of the disorder variables being given by a gamma distribution. Theexact solution exhibits a singularity in the density of states about the origin, which has since beenshown to be universal for one-dimensional tight binding models with off diagonal disorder. Wediscuss this context and also point out some universal features of the weak disorder expansionof the exact solution near the band edge. Further, a link between the exact solution, and atridiagonal formalism of anti-symmetric Gaussian β -ensembles with β proportional to /N , ismade. Introduction
In the early 1960’s Dyson, starting with the publication [19] and building on work of Wigner fromthe 1950’s, developed a theory of random matrices for applications to universal aspects of quantumspectra as determined by global symmetries. For reference, we remark that these early works areconveniently reprinted and reviewed in a book edited by Porter [39]. Whereas Wigner focussedon modelling the Hamiltonian using Hermitian random matrices, Dyson considered ensembles ofunitary matrices more fundamental due to there being a unique invariant measure; see Section Iand the beginning of Section II of [19], and also the review [15]. In addition to putting in place themathematical framework, an extensive theory was developed in relation to the statistical propertiesof the eigenvalues of the new ensembles.It is no exaggeration to say these contributions of Dyson to random matrix theory and itsapplications are celebrated achievements. Lesser known is the fact that these series of works werenot the first time Dyson had use for random matrices, nor the first time that he had the needto develop theory relating to random matrices in a pioneering fashion. The title to these claimsgoes instead to Dyson’s 1953 work “The dynamics of a disordered linear chain” [18]. From theviewpoint of foundational knowledge, revisiting [18] provides a valuable lesson in the methods andmotivations of random matrices. And with Dyson’s recent passing at age 96 on February 28th2020, drawing attention to [18] is also a contribution to paying tribute to his seminal contributionsto the field generally.There has been an earlier contextual review of [18], in the book of reprints with introductorytext on mathematical physics in one-dimension written in the mid 1960’s [32, Ch. 2] by Lieb andMattis. In addition to discussing follow up works from the original aims and objectives of [18], thisbook also contains reprints of the those papers. These follow ups are in relation to the propertiesof the disordered chain. A mathematical follow up written in 1960, with the aim of putting alllimiting procedures in [18] on a rigorous footing, can be found in the work [40] by a student of
V. Marčenko, cited in the 1973 survey of Pastur [38] on the spectra of random self adjoint operators.This latter reference also contains a discussion of [18].In Section 2 an account is given of the salient content of [18] from a random matrix theoryviewpoint. Some subsequent refinements to aspects of the working are covered in Section 3, andalso attention is drawn to universal features of the exact solvable case found by Dyson. These aretypically from the literature on localisation and the one-dimensional Anderson model. In Section4 a link between Dyson’s solvable case, and a tridiagonal formalism of anti-symmetric Gaussian β -ensembles, with β proportional to /N and for N → ∞ , is made.2. Overview of Dyson’s paper
Coupled harmonic oscillators and tridiagonal matrices.
Dyson’s Introduction in [18]makes it clear that his motivation was to present a mathematical model of a disordered system.The particular choice made was a system of N masses { m i } Ni =1 , confined to a line and each coupledto their nearest neighbour by (fictitious) springs with corresponding springs constants { K i } Ni =1 , andobeying Hooke’s law. With free boundary conditions, the displacements from equilibrium of thepositions { u i } Ni =1 of each mass obey the coupled set of Newton’s equations m j ¨ u j = K j ( u j +1 − u j ) + K j − ( u j − − u j ) . (2.1)Here K = K N = 0 in keeping with free boundary conditions. The disorder is introduced bychoosing the masses, or the spring constants, or possibly a combination of both from a probabilitydistribution function.With a = [ a i ] Ni =1 , introduce the notation diag a for a matrix with entries given by a along thediagonal and zero elsewhere. And with b = [ b i ] N − i =1 , introduce too the notation diag + b ( diag − b )for a matrix with non-zero entries only on the first diagonal above (below) the main diagonal, withthose entries given by b = [ b i ] N − i =1 . To make use of this notation, set u = [ u i ] Ni =1 , α = [ − K j /m j − K j − /m j ] Nj =1 α = [ K j /m j ] N − j =1 , α − = [ K j /m j +1 ] N − j =1 . We then have that the system (2.1) is equivalent to the second order matrix differential equation ¨ u = Au , A = diag α + diag + α + diag − α − . (2.2)Separating variables by writing u = e iωt U , where U is independent of t , shows the allowed valuesof − ω are given by the eigenvalues of the tridiagonal matrix A .Instead of considering this eigenvalue problem, Dyson chose to first transform the second ordersystem (2.2) into a first order system by changing variables y j = m / j u j ( j = 1 , . . . , N ), thendefining { z j } N − j =1 by ¨ z j = − λ / j y j + λ / j +1 y j +1 , (2.3)where λ j − = K j /m j , λ j = K j /m j +1 . (2.4)Introducing too y = [ y z y z · · · y n ] T and with Λ the (2 N − × (2 N − anti-symmetric tridiagonal matrix Λ = diag + [ λ / j ] N − j =1 − diag − [ λ / j ] N − j =1 , (2.5) YSON’S DISORDERED LINEAR CHAIN 3 the second order matrix differential equation (2.2) is seen to be equivalent to the first order matrixdifferential equation ˙ y = Λ y . (2.6)Separating variables by writing y = e iωt Y , where Y is independent of t , shows the allowed valuesof ω are given by the ( N − positive eigenvalues of the matrix i A , as well as the zero eigenvalue.The latter occurs due to the choice of free boundary conditions.Thus through either (2.2) or (2.6) Dyson was faced with the problem of quantifying the eigen-value distribution for a (random) tridiagonal matrix. It was immediately realised that simplifyingfeatures could be expected in the limit N → ∞ . Thus define M ( µ ) as the proportion of frequencies { ω j } with ω j ≤ µ . Dyson hypothesised that for N → ∞ [18, Eq. (10)] D ( µ ) := dMdµ (2.7)is well defined, with D ( µ ) corresponding to the density of states for the square eigenvalues. Underthis assumption the function Ω( x ) := lim N →∞ N − N − X j =1 log(1 + xω j ) , (2.8)referred to in [18] as the characteristic function of the chain, is also well defined and can expressedin terms of D ( µ ) according to [18, Eq. (11)] Ω( x ) = Z ∞ log(1 + xµ ) D ( µ ) dµ. (2.9)Moreover, it is noted that this can be inverted (differentiate and apply the Sokhotski–Plemelj —also associated with Stieltjes–Perron — formula) to deduce [18, Eq. (13)] D (1 /x ) = − x lim ǫ → + π Im Ω ′ ( − x + iǫ ) . (2.10)It is also noted that in the limit x tends to − z on the negative real axis from above, Im log(1+ xµ ) =0 ( iπ ) for zµ < ( zµ > and thus [18, Eq. (12)] Im 1 π lim ǫ → + Ω( − z + iǫ ) = Z ∞ /z D ( µ ) dµ = 1 − M (1 /z ) . (2.11)Note the consistency between (2.11) and (2.10).2.2. A continued fraction formula for Ω( x ) . Dyson expands the logarithm in (2.8) to deduce N − X j =1 log(1 + xω j ) = ∞ X n =1 ( − n n x n Tr Λ n . (2.12)After some intricate combinatorial analysis of Tr Λ n , it is shown that for large N (2.12) can beexpressed in terms of the continued fraction [18, Eq. (33)] ξ ( a ) = xλ a / (1 + xλ a +1 / (1 + xλ a +2 / (1 + · · · (2.13)Substituting in (2.8), this leads to the formula [18, Eq. (34)] Ω( x ) = lim N →∞ N N − X a =1 log(1 + ξ ( a )) . (2.14) PETER J. FORRESTER
In §3.1 below, subsequent simplified derivations of (2.14) will be given [2, 12] which make use ofalgebraic rather than combinatorial properties of Λ .From the structure of (2.13) and (2.14), it is observed in [18] that the simplest type of disorder toimpose is to choose { λ a } from a common probability distribution. The coupled spring and massessystem is then referred to as a Type I disordered chain. With the probability density function(PDF) of the continued fraction (2.13) denoted F ( ξ ) , (2.14) then reads [18, Eq. (46)] Ω( x ) = 2 Z ∞ F ( ξ ) log(1 + ξ ) dξ. (2.15)An alternative type of disorder introduced in [18], giving rise to what is termed a Type IIdisordered chain, is when each mass m j is an independent identically distributed random variablechosen with PDF G ( m ) , and with the spring constants all equal to the same value K . Then, from(2.4) [18, Eq. (47)], λ j − = λ j − = K/m j so the random variables { λ j } are constrained to be equal in pairs, while from (2.13) ξ (2 j ) and ξ (2 j − have different distributions. Defining [18, Eq. (48)] η j = 1 ξ (2 j ) , and with F ( η ) denote the corresponding PDF, manipulation of (2.14) shows [18, Eq. (53)] Ω( x ) = Z ∞ dη F ( η ) Z ∞ d ˜ m G ( ˜ m ) log (cid:16) /η ) + x ( K/ ˜ m ) (cid:17) . (2.16)As a distribution of masses of special interest, suppose [18, Eq. (54)] G ( ˜ m ) = pδ ( ˜ m − m ) + (1 − p ) δ ( ˜ m − M ) (2.17)so that the chain consists of two masses m, M with concentrations p and (1 − p ) respectively. Then(2.16) reads [18, Eq. (56)] Ω( x ) = Z ∞ dη F ( η ) (cid:18) p log (cid:16) /η ) + x ( K/m ) (cid:17) + (1 − p ) log (cid:16) /η ) + x ( K/M ) (cid:17)(cid:19) . (2.18)2.3. A functional equation in the case of a Type I chain and an exact solution.
Thecontinued fraction (2.13) obeys the functional equation [18, Eq. (43)] ξ ( a ) = xλ a / (1 + ξ ( a + 1)) . (2.19)For a Type I chain, the random variables λ a and ξ ( a + 1) are uncorrelated and moreover ξ ( a ) and ξ ( a + 1) have the same distribution, leading to the equality in law between random variables ξ ,and a combination of λ and ξ , ξ d = xλ/ (1 + ξ ) . (2.20)Recalling now that the PDF for ξ has been denoted F ( ξ ) above (2.15), and with the PDF for thedistribution of λ to be denoted G ( λ ) , we see that (2.20) implies [18, equivalent to Eq. (44)] F ( t ) = Z ∞ dλ G ( λ ) Z ∞ dξ F ( ξ ) δ (cid:16) t − xλ ξ (cid:17) . (2.21)With α, κ > , suppose now [18, equivalent to eq. (57)] G ( λ ) = κ α Γ( α ) λ α − e − κλ . (2.22) YSON’S DISORDERED LINEAR CHAIN 5
Then, by taking the Mellin transform of both sides of (2.21) it is straightforward to verify the factthat the solution of (2.21) is [18, Eq. (59)] F ( t ) = F α ( t ) = 1 K α ( x ) t α − (1 + t ) α e − κt/x , (2.23)where K α ( x ) is the normalisation given as an integral by K α ( x ) = Z ∞ t α − (1 + t ) α e − κt/x dt. (2.24)Substituting in (2.15) shows [18, Eq. (60)] Ω( x ) = 2 L α ( x ) K α ( x ) , (2.25)where K α ( x ) is given by (2.24) and L α ( x ) is given by L α ( x ) = Z ∞ t α − (1 + t ) α log(1 + t ) e − κt/x dt. (2.26) Remark . Γ[ α, p ] the gamma distribution with PDF proportional to x α − e − px supported on x > . Denote by K[ α, β, p ] the Kummer type II distribution with PDF proportional to x α − e − px / (1 + x ) α + β supported on x > . A result attributed to Letac in an unpublished manuscript (see [28,Remark 2.2]) gives that with X d = Γ[ α, p ] and Y d = K[ α + β, − β, p ] , X Y d = K[ α, β, p ] . (2.27)With β = 0 this reduces to Dyson’s result relating to (2.20).In view of (2.10), to compute the density of states, it is necessary to analytically continue both(2.24) and (2.26) for negative x . This is done in [18, Appendix III], and the result is substitutedin the integrated form of (2.7) [18, equivalent to final equality in eq. (12)] M ( x ) = 1 − Z ∞ x D ( µ ) dµ. (2.28)In the case that α = n ∈ Z + , the explicit evaluation of (2.28) was presented [18, Eq. (63)].Attention was drawn to the x → + singularity [18, consequence of eq. (72)] M ( x ) ∼ c (log x ) (2.29)for some (explicit) c , now referred to as the Dyson spectral singularity.Attention was also drawn to the α → ∞ behaviour. In this limit, after setting κ = α in (2.22),the PDF for { λ j } has the asymptotic form [18, Eq. (58)] G ( λ ) ∼ (cid:16) α π (cid:17) / e − α ( λ − / (2.30)of a Gaussian centred about λ = 1 . To leading order each λ j is equal to , and there is no disorder.It then follows from (2.20) that [18, Eq. (37) with λ = 1 ] ξ = 12 (cid:16) (1 + 4 x ) / − (cid:17) , (2.31) PETER J. FORRESTER which substituted in (2.14) gives [18, Eq. (38)] Ω( x ) = 2 log (cid:16)
12 ((1 + 4 x ) / + 1) (cid:17) . (2.32)Substituting this in (2.10) implies for the density of states [18, Eq. (41) with λ = 1 ] D ( µ ) = π √ µ − µ , µ < , µ > λ. (2.33)A corollary of (2.33), obtained by substituting in (2.28), is that the integrated density of states forthe chain with no disorder is [18, Eq. (74) with λ = 1 ] M ∞ ( x ) = π Arccos(1 − x/ , µ < , µ > . (2.34)In relation to corrections to this behaviour due to disorder, denote by M n ( x ) the integrated densityof states in the case { λ j } are distributed with PDF specialised to α = κ = n . From his exact result,Dyson showed that for n → ∞ [18, Eq. (75)] M n ( x ) ∼ π Arcos(1 − x/
2) + πn /x − / , < x < − γπ exp (cid:16) − γ − n (sinh γ − γ ) (cid:17) , x > − (cid:16) / (cid:17) (cid:16) n (cid:17) / , x = 4 , (2.35)where γ = Arcosh (( x/ − .3. Some subsequent refinements
Ratios of characteristic polynomials and Dyson’s continued fraction.
It was notedby Bellman [2] and Dean [12] that Dyson’s combinatorial derivation of (2.14) could be simplifiedby adopting an algebraic approach. For this purpose, in the notation of the paragraph including(2.2) introduce the general Hermitian tridiagonal matrix T n = diag [ a i ] ni =1 + diag + [ b i ] n − i =1 + diag − [¯ b i ] n − i =1 . (3.1)The corresponding (modified) characteristic polynomial is P n ( y ) = det( I n − y T n ) = n Y i =1 (1 − yλ ( n ) i ) , (3.2)where { λ ( n ) i } are the eigenvalues of T n . By expanding det( I n − y T n ) along the final row, { P n ( y ) } is seen to obey the three-term recurrence P n ( y ) = (1 − ya n ) P n − ( y ) − y | b n − | P n − ( y ) , P ( y ) := 1 . (3.3)In terms of r n ( y ) := P n ( y ) /P n − ( y ) , (3.3) reads r n ( y ) = (1 − ya n ) − y | b n − | r n − ( λ ) . (3.4)With a j = 0 ( j = 1 , . . . , n ) and upon the relabelling b n − j b j , iteration of (3.4) shows lim n →∞ r n +1 − j ( y ) = 1 − y | b j | / (1 − y | b j +1 | ) / (1 − y | b j +2 | ) / (1 − · · · . (3.5) YSON’S DISORDERED LINEAR CHAIN 7
Furthermore, in terms of { r n ( λ ) } P n ( y ) = n Y j =1 r n +1 − j ( y ) . (3.6)The significance of this setting is that in the case n = 2 N − , with diagonal entries given by b j = iλ j , we have that T n = Λ and thus P n ( y ) = N − Y j =1 (1 − y ω j ) . (3.7)Substituting (3.7) in (3.6) with − y replaced by x , taking the logarithm and dividing by (2 N − ,then making use of (3.5) we see that (2.14) is reclaimed.3.2. Type II chain and the work of Schmidt.
After separating variables as described below(2.2), the equations of motion (2.1) can be rearranged to read U j +1 = (cid:16) K j − /K j ) − ω m j /K j (cid:17) U j − ( K j − /K j ) U j − . (3.8)To do this requires K j = 0 for each j = 1 , . . . , N . This therefore excludes the free boundary condi-tions as used by Dyson, since then K N = 0 (recall the text below (2.1)). A compatible alternativeis to use fixed boundary conditions, specified by U = U N +1 = 0 . We remark that with (3.8)multiplied through by K j ( j = 1 , . . . , N ) free and fixed boundary conditions are indistinguishable.Note too, following Schmidt [41], that an equivalent way to write (3.8) is as the × matrixrecurrence " U j +1 U j = T j " U j U j − , T j = " K j − /K j ) − ω m j /K j − K j − /K j , (3.9)which implies the matrix product formula " U N +1 U N = T N T N − · · · T " U U . (3.10)Iterating (3.8) starting with U = 0 determines { U j } j =1 , ,... up to an overall scalar factor c say.We see that U j +1 is a polynomial of degree j in ω . Denoting the corresponding zeros by { µ ( j ) l } jl =1 ,this allows us to write U j +1 = c j Y l =1 ( − m l /K l )( ω − µ ( j ) l ) . (3.11)To obtain a more explicit characterisation of { µ ( j ) l } jl =1 , in (3.8) change variables by writing V j +1 = 1 ω j j Y l =1 (cid:16) − K l m l (cid:17) U j +1 , y = 1 /ω . (3.12)This gives V j +1 = 1 − y (cid:16) − K j m j − K j − m j (cid:19) V j − y (cid:16) K j − m j − m j (cid:17) V j − . (3.13)With A denoting the tridiagonal matrix specified in (2.2), and A n denoting its top n × n submatrix,comparison with (3.3) shows V n +1 = ˜ c det( I n + y A n ) , (3.14)where ˜ c is an arbitrary scalar. In particular, it follows that { µ ( j ) l } jl =1 are equal to the nonzeroeigenvalues of − A j +1 . Since A N = A it follows that for U N +1 = 0 as required by fixed boundary PETER J. FORRESTER conditions, we must have that − ω in (3.8) corresponds to the eigenvalues of A . This has beennoted below (3.9) for the case of free boundary conditions.The above theory implies that upon consideration of the ratios ˜ r n := U n /U n − , formulas equiv-alent to (3.6) and (3.7) hold. Thus we have U N +1 = c N Y l =1 ( − m l /K l )( ω − ω l ) = N Y j =0 ˜ r N +1 − j . (3.15)However { ˜ r n } relate to the matrix A , whereas { r n } relate to the anti-symmetric matrix Λ so theyhave different distributions. In fact for a Type II chain, characterised by all spring constants beingequal, there are simplifications which result by considering { ˜ r n } .First, in the setting of a Type II chain, it follows from (3.8) that ˜ r j +1 = (2 − ω m j /K ) − / ˜ r j . (3.16)For n , large let w ( z ) = w ( z ; ω ) denote the PDF for the distribution of ˜ r n . Proceeding as inthe derivation of (2.21), and specialising to the case of a diatomic chain as specified in (2.17) fordefiniteness, we see that w ( z ) satisfies the functional equation [41, equivalent to Eq. (II,16)] w ( z ) = p z w (cid:16) − mµ K − z (cid:17) + (1 − p ) 1 z w (cid:16) − M µ K − z (cid:17) . (3.17)Next, as a variant of Dyson’s characteristic function (2.8) define ˜Ω( y ) = lim N →∞ N N X j =1 log( ω j − y )= lim N →∞ N N X j =1 log U N +1 (cid:12)(cid:12)(cid:12) ω = y + (cid:16) p log m + (1 − p ) log M − log K (cid:17) . (3.18)Comparison with (2.8) shows ˜Ω( − /z ) = − log z + Ω( z ) . (3.19)In contrast to (2.8), ˜Ω( y ) is not real for positive real values of the argument y ( x in (2.8)), sincefor ω j < y log( ω j − y ) = log | ω j − y | + iπ. (3.20)A useful consequence is that analogous to (2.28), it follows Im 1 π ˜Ω( y ) = M ( y ) . (3.21)From the second equality in (3.18) and the definition of w ( z ) it also follows that analogous to(2.15) ˜Ω( y ) = Z ∞−∞ (log z ) w ( z ; y ) dz. (3.22)Taking imaginary parts using (3.21) then gives [41, Eq. (II,26)] M ( y ) = Z −∞ w ( z ; y ) dz = Pr { w ( z ; y ) ≤ } . (3.23)We remark that due to this development of Schmidt to Dyson’s pioneering work, (3.17), alongwith (2.21) in the case of the Type I chain, is nowadays typically referred to as an example of aDyson–Schmidt equation for the stationary distribution of the corresponding stochastic sequences. YSON’S DISORDERED LINEAR CHAIN 9
Also of note is that the final equality in (3.23) is well suited to numerical approximation, whereasformalisms based on (2.11) require analytic continuation.
Remark .
1. Consider the Type I disordered chain in Dyson’s anti-symmetric tridiagonal matrixformulation. The characteristic polynomial Q n ( y ) = det( u I n − Λ n ) , where Λ n is the top n × n block of Λ as specified by (2.5) satisfies the recurrence Q n +1 ( y ) = yQ n ( y ) − λ n Q n − ( y ) . Introducing the ratios s n = Q n ( y ) /Q n − ( y ) , then writing s n = y (1 + ˜ s n ) we see that the randomvariable ˜ s corresponding to the limiting distribution of ˜ s n must satisfy the equality in law ˜ s d = − ( λ/y )1 + ˜ s . (3.24)This is identical to (2.20) except that the positive parameter x is now equal to the negativeparameter − /y . However, in relation to Dyson’s exact solution in the case that the distributionof λ is specified by (2.22), having the parameter positive is an essential ingredient. In particular,no analogous exact solution is known in relation to (3.24).2. Related to the above point is the simplification — for example the existence of an exact solution— which results by the consideration of Ω( z ) (or equivalently from (3.19) the consideration of ˜Ω( − /z ) ) for z positive and real, and then analytically continuing as required by (2.11) and (2.10).For related uses of this strategy, applied to multichannel models, see [26, 48].Schmidt’s reformulation of the second order difference system (3.8) in the matrix form (3.9) issignificant as perhaps the first applied problem giving rise to a product of random matrices, as seenin (3.10). In the case of the Type II chain, each matrix in (3.10) is independent and identicallydistributed. Starting from the early 1960’s, products of random matrices with independent andidentically distributed elements attracted much attention in the mathematics literature, and manysignificant theoretical developments have followed [23, 24, 30, 37]. A key quantity in such studiesas they relate to (3.10) is γ := lim N →∞ N log q U N +1 + U N = lim N →∞ N log | U N +1 | , (3.25)referred to as the Lyapunov exponent. In this setting it is usual to refer to the limiting PDF of U N +1 /U N as specifying an invariant measure.For Type II chains, it follows from the first equality in (3.15) substituted in (3.25) that γ = Z ∞ log | ω − µ | D ( µ ) dµ + 1 K h log | m |i , (3.26)where D ( µ ) denotes the density of the squared singular eigenvalues; cf. (2.9). Nearly two decadesafter Dyson’s work, it was understood by Thouless [46] that through (3.26) there is a link betweenthe density of states and the localisation length in a one-dimensional disordered system — thelatter being an interpretation of /γ ; see the review [10]. Due to this conceptual advance, (3.26)is nowadays typically referred to as the Thouless formula, although some authors simultaneouslycite both Dyson and Thouless; see e.g. [25]. Type I chains near zero frequency.
The natural discretisation of the one-dimensionalSchrödinger equation (cid:16) − d dx + V ( x ) (cid:17) ψ ( x ) = Eψ ( x ) on the integer lattice is − ( ψ n +1 + ψ n − − ψ n ) + V n ψ n = Eψ n . (3.27)This is to be compared with the difference equation (3.8) in the case of a type II chain − ( U n +1 + U n − − U n ) − ω K m n U n = 0 . (3.28)While the discretisation of the Laplace operator − d /dx is evident, there is no direct analogybetween { E, V n } in (3.27) and { ω , m n } in (3.28).On the other hand, consider Dyson’s Type I chain in the anti-symmetric tridiagonal form (2.6).The corresponding equation for the eigenvalues and eigenvectors can be written iλ / n − φ n − − iλ / n +1 φ n +1 = ωφ n , (3.29)where { φ n } N − n =1 are the components of the eigenvector. This is recognised as an example of theSchrödinger equation for the tight binding Hamiltonian (one-dimensional Anderson model) withrandom off diagonal elements and constant diagonal the latter being absorbed into the energy E to give ω in (3.29).A number of works have given consideration to the limiting ω → + form of the density ofstates as implied by (3.29) for a general distribution of the non-negative random variable λ n ,assuming finite second moment; see for example [4, 14, 20, 45]. The conclusion of these works isthat the singularity (2.29) exhibited for the special distribution of λ n (2.22) actually holds in thegeneral case, and thus is a universal feature of both Dyson’s Type I chain, and the one-dimensionalAnderson model with off diagonal disorder. The same singularity is also seen in the density of statesfor the one-dimensional XY model with random coupling constants [42], and the one-dimensionalrandom mass Dirac Hamiltonian [44], both systems being related to the Dyson’s Type I chain.A closely related general behaviour can be seen from the solution of (3.29) with ω = 0 [8, 45, 51].After iteration, and setting φ = 1 for normalisation, we see that for n even φ n +1 = n/ Y l =1 λ l − λ l +1 . According to the central limit theorem log | φ n +1 | will, to leading order for large n , be proportionalto √ n , with the proportionality constant given in terms of the variance of λ n . In contrast, typicallyfor ω > the values of log | φ n +1 | obtained by iterating (3.29) will be proportional to n . Themodification of this conclusion in the case that the variance diverges has been the subject of therecent work [31]; see too the earlier work [3].3.4. Weak disorder limit.
We know from (2.30) that for α → ∞ the special PDF (2.22) forthe couplings { λ n } is to leading order a Gaussian centred at λ = 1 with variance /α . Thiscircumstance, which perturbs about the chain with no disorder, is referred to as weak disorder.Systematic weak disorder expansion methods have been devised (see e.g. [6] and references therein),typically specialised to the setting of the discrete Schrödinger equation (3.27) and so not directlyapplicable to disordered chains. Nonetheless, comparison of the results which follow from Dyson’sexactly solvable Type I chain for large α with results from the weak disorder expansion relating to YSON’S DISORDERED LINEAR CHAIN 11 (3.27) (the pertinent ones are conveniently summarised in [33, §2.2]), show a number of quantitativesimilarities.We consider first the Lyapunov exponent (3.26). A result of Thouless [47] gives that in theweak disorder limit of (3.27), with the variance of { V n } equal to /α , the leading large α form for | E | < is γ ∼ α (1 − ( E/ ) . (3.30)For Type I chains, we see from the definition (3.25) and (3.29) that in terms of D ( µ ) γ = 14 Z ∞ log | ω − µ | D ( µ ) dµ − h log | λ |i . (3.31)Making use of (2.33) it follows that with no disorder γ = 14 π Z µ − µ log | ω − µ | dµ = 0 , (3.32)where the final equality is valid for ω < ; see e.g. [21, §1.4.2]. This fact could also be deduceddirectly from the definition (3.25) since without disorder the components of the eigenvectors donot exponentially increase or decrease but rather oscillate, in keeping with the underlying coupledspring model having all masses equal.Let ( γ ) denote the term proportional to /α in the large α expansion of γ , and similarly themeaning of (Ω( x )) . We see from (3.31) and (2.9) that ( γ ) = 14 π (cid:16) lim ǫ → + Re Ω( − /ω + iǫ ) (cid:17) − . (3.33)Making use of (2.25) and workings in [18, Appendix IV] gives, for α ∈ Z + lim ǫ → + Re Ω( − /ω + iǫ ) = 2 L α ( − ω ) /K α ( − ω ) , (3.34)where, with f ( ξ, ω ) = log ξ − log(1 + ξ ) + ξω , g (0) α ( ξ ) = 1 ξ , g (1) α ( ξ ) = 1 ξ log(1 + ξ ) , (3.35)we have [18, Eqns. (A.17) and (A.19)] L α ( − ω ) = Z −∞ g (1) α ( ξ ) e αf ( ξ,ω ) dξ (3.36) K α ( − ω ) = Z −∞ g (0) α ( ξ ) e αf ( ξ,ω ) dξ (3.37)In both (3.36) and (3.37) the contours of integration are to run along the upper half plane side ofthe negative real axis.Moreover, it is noted in [18, Appendix IV] that for < ω < there is a single saddle point inthe upper half plane [18, Eq. (A.21)] η = 12 (cid:16) − i ((4 /ω ) − / (cid:17) . (3.38)It is noted too that by deforming the contours in (3.36), (3.37) to pass through this point at angle π/ , the large α asymptotic expansion follows by expanding the integrand about this point. Doing this, setting f ( ξ, ω ) = f ( ξ ) for notational convenience, and evaluating the corresponding Gaussianintegrals gives lim ǫ → + Re Ω( − /ω + iǫ ) = −
12 1 η (1 + η ) f ′′′ ( η ) | f ′′ ( η ) | + Re 12 i (1 + η ) | f ′′ ( η ) | = 1 + 12((4 /ω ) − . Now substituting in (3.33) gives ( γ ) = 18((4 /ω ) − . (3.39)Comparing with (3.30) shows the functional forms agree for E = ω → − .For the discrete Schrödinger equation (3.27) it is known [7, 13, 22, 27, 29] that in the limit E → the leading weak disorder expansion of the Lyapunov exponent obeys the scaling law interms of Airy functions γ ∼ (cid:16) α (cid:17) / F (cid:16) (2 α ) / ( | E | − (cid:17) , (3.40)where F ( x ) = Ai( x )Ai ′ ( x ) + Bi( x )Bi ′ ( x )(Ai( x )) + (Bi( x )) = Re e − πi/ Ai ′ ( e − πi/ x )Ai( e − πi/ x ) . (3.41)Precisely this scaling function was obtain by Smith [42, Eq. (4.14)] in the case of Dyson’s exactlysolvable Type I chain with α → ∞ . It was found by extending the asymptotic analysis of Dysonto uniformly account for there being two coalescing saddle points as ω → . Remark . It is noted in [7, §4.1] that Im e − πi/ Ai ′ ( e − πi/ x )Ai( e − πi/ x ) = 1 π ((Ai ( x )) + (Bi ( x )) ) . (3.42)The working in [42] then implies that (3.42) plays the role of the scaling function F in (3.40) forthe weak disorder expansion of the density of states near ω = 2 for Dyson’s type I chain. Also ofinterest is the fact that for large x the leading asymptotics of (3.42) is x / e − x / / , where theparticular functional form of the exponential is known in the theory of disordered systems as aLifshitz tail; see [9, §7.1].4. Inhomogeneous Type I chains related to Gaussian anti-symmetric matrices
Let X be a real N × N standard Gaussian matrix. The corresponding symmetric matrix ( X + X T ) is said to be a member of the Gaussian orthogonal ensemble; see e.g. [21, §1.1]. Thisensemble relates to the broader theme of disordered chains and tight binding Hamiltonians throughthe property that it permits a similarity transformation (using Householder reflection matrices) to atridiagonal matrix with entries on and above the diagonal again independent [50]. On the diagonalthey are unchanged, each being given by a standard Gaussian. On the sub-diagonal directly abovethe diagonal they are distributed by ( ˜ χ N − , ˜ χ N − , . . . . ˜ χ ) , where ˜ χ k denotes the square root of thegamma distribution Γ[ k/ , . Moreover, with the latter generalised to ( ˜ χ ( N − β , ˜ χ ( N − β , . . . . ˜ χ β ) ,where β > is a parameter, it was shown in [16] that the eigenvalue PDF can be explicitlycomputed, and is proportional to N Y l =1 e − x l / Y ≤ j This is the definition of the Gaussian β -ensemble.As discussed in [5], this class of random tridiagonal matrices is of interest from the viewpoint ofstochastic Schrödinger operators in one-dimension with a random potential decaying as | x | − α . Itis known that the exponent α equalling / separates localised and extended states, and it turnsout that the random tridiagonal matrices giving rise to the Gaussian β -ensemble corresponds tothis critical case.An anti-symmetric Hermitian matrix can be formed out of a real Gaussian matrix X by forming i times ( X − X T ) . It is known [17] that reduction of the latter to an anti-symmetric tridiagonalform using Householder transformations gives for the entries directly above the diagonal the samedistribution as in the symmetric case, ( ˜ χ N − , ˜ χ N − , . . . . ˜ χ ) . Furthermore, as a generalisation,if an anti-symmetric tridiagonal matrix is constructed with entries directly above the diagonaldistributed by ( ˜ χ ( N − β/ , ˜ χ β ( N − / , . . . , ˜ χ β ) , (4.1)it was shown in [17] that the eigenvalue PDF can be explicitly determined. The precise functionalform depends on the parity of N . 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