representations of quantum chaotic eigenstates and random polynomials

ar X iv :c ha o- dy n/ 96 12 00 6v 1 3 D ec 1 99 6 February 5, 2008 Parametric statistics of zeros of Husimi representations of quantum chaotic eigenstates and random polynomials Tomazˇ Prosen 1 Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia Abstract Local parametric statistics of zeros of Husimi representations of quantum eigen- states are introduced. It is conjectured that for a classically fully chaotic systems one should use the model of parametric statistics of complex roots of Gaussian random poly- nomials which is exactly solvable as demonstrated below. For example, the velocities (derivatives of zeros of Husimi function with respect to an external parameter) are pre- dicted to obey a universal (non-Maxwellian) distribution dP(v) dv2 = 2 piσ2 (1 + |v|2/σ2)−3, where σ2 is the mean square velocity. The conjecture is demonstrated numerically in a generic chaotic system with two degrees of freedom. Dynamical formulation of the “zero–flow” in terms of an integrable many-body dynamical system is given as well. 1 Introduction The intense research in the so-called quantum chaology has produced many different sig- natures of classical chaos in the corresponding quantum Hamiltonian systems. It has been found that energy spectra and eigenstates of classically fully chaotic quantum sys- tems have universal statistical properties which can be described by stochastic models with no free parameters, like random matrix theory (RMT). In the Bargmann (or Husimi) representation, eigenstates of quantum systems can be uniquely represented in terms of complex analytic functions in phase space variables. Quite recently, it has been proposed [7] that in case of 1-dim systems (or Poincare´ surface 1e-mail: prosen@fiz.uni-lj.si 1 of section reductions of 2-dim systems [10]), where phase space is two dimensional, and one has only one complex variable z = q+ ip, any eigenstate can be uniquely represented by the collection of (complex) zeros of its Bargmann or Husimi representation. This has been called the stellar representation. It has been found [10] that the structure of zeros of Husimi representation of an eigenstate is reminiscent of the structure of classical phase space: (i) in classically fully chaotic systems the zeros tend to spread uniformly over the whole classically allowed region of phase space [7], (ii) in classically integrable systems the zeros lie on 1-dim. anti-Stokes curves [7], (iii) while in generic mixed systems a part of zeros uniformly cover chaotic components of classical phase space, zeros in regular islands lie along 1-dim. classically invariant curves - tori, while substantial part of zeros lie on 1-dim. classically noninvariant (generalized anti-Stokes) curves. In the following we shall be interested only in the case of classically fully chaotic systems. Let us assume that the Hamiltonian of our chaotic system can be statistically described by Gaussian orthogonal/unitary (in the presence/absence of anti-unitary sym- metry) random matrix in a generic basis. If we choose the harmonic basis then the Bargmann representation of an arbitrary eigenstate is given by the entire function f(z) = ∞∑ n=0 cn√ n! zn, (1) and the Husimi representation in appropriately scaled units is |f(z)|2e−|z|2, where z = q + ip. cn are the coefficients of the expansion of that eigenstate in the harmonic basis, which are, by assumption and hence according to RMT, real/complex Gaussian (pseudo) random variables. Ideally, as RMT predicts, there should be no correlations among them 〈cnc∗m〉 = δnm, although we believe that for nonzero shortrange correlations among the coefficients cn our general conclusions are still valid. Since Taylor expansion of f(z) around arbitrary point z0 is convergent one may define and study Gaussian random polynomials of order N and the statistical properties of their roots and in the end, if neccesary, take the ‘thermodynamic’ limitN →∞ (in a sense of increasing number of zeros — quasi-particles) of Gaussian random holomorphic functions (1). The limit N → ∞ is in fact compatible with a semiclassical limit h¯ → 0 since the order N of a Taylor polynomial aproximating a Bargmann function (and its zeros) should be at least equal to the number of basis states covering the classically accessible area A of phase space (or surface of section), N > A/(2pih¯), while higher coefficients aN+n, n > 0, vanish rapidly with increasing n. One can also consider cases of different geometry, for example of kicked spin j systems, where quantum (eigen)states can be exactly represented in terms of the so-called SU(2) polynomials of finite order 2j. It has been shown recently by Hannay [5] for the case of complex coefficients and supplemented by Prosen [12] for the case of real coefficients that the statistics of zeros of Gaussian random polynomials are exactly solvable and all k-point correlation functions can be given in terms of simple analytical formulae. It has been demonstrated [6, 11] that the results obtained by random polynomials indeed reproduce the statistics of zeros of Husimi representations of chaotic eigenstates. It is the aim of this paper to introduce the local parametric statistics of zeros of Husimi representations of the quantum eigenstates. Let us take a family of Hamiltonian systems which smoothly depend upon an external parameter λ. Then the Bargmann represen- tation of a given eigenstate f(z, λ) and its zeros zn(λ) are also smooth functions of the 2 parameter λ. Therefore we can introduce the velocities vn as derivatives of zeros zn with respect to an external parameter λ vn(λ) = d dλ zn(λ) = −∂λf(zn, λ) ∂zf(zn, λ) (2) Then we define a parametric k-point correlation function of k zeros z = (z1, . . . , zk) and k velocities v = (v1, . . . , vk) as ρ˜k(z,v) = 〈 δ2k(z − z′)δ2k(v − v′) 〉 z′,v′ (3) where 〈〉z′,v′ represents an (ensemble) average over all k-tuples of zeros z′ and correspond- ing velocities v′ of a given state (or ensemble of states). In other words, ρ˜k(z,v)d 2kz d2kv is a probability to find a k-tuple of zeros and corresponding velocities in a small 4k-cube of volume d2kz d2kv around point (z,v). Integrating out the velocities one should obtain the usual k-point correlation function ρk(z) of zeros only [5, 12] ρ(z) = ∫ d2kvρ˜(z,v). (4) We show below, in section 2, that this parametric correlation functions can be explic- itly calculated for Gaussian ensembles of parameter-dependent random polynomials with either complex or real coefficients. In section 3, we shall verify our conjecture that the obtained result on parametric statistics of roots of statistical ensembles of random poly- nomials may be applied to quantum chaotic systems by presenting some numerical results obtained in a generic chaotic system, namely 2-dim semi-separable oscillator [9, 10]. In section 4, we write a closed system of ‘equations of motion’ for the zero-flow zn(λ) for the simplest, linear parametric dependence of the coefficients an, and stress the integrability of the underlying dynamical system. 2 Parametric statistics of roots of Gaussian random polynomials 2.1 Complex coefficients In this section we study parametric statistics of statistical ensembles of Gaussian random polynomials. In the first subsection we are dealing with the case of complex coefficients while slightly more complicated case of real coefficients will be dealt with in the next subsection. Let us write a random polynomial of order N in a form f(z, λ) = N∑ n=0 an(λ)z n (5) where the coefficients an(λ) (for a fixed realization) may depend smoothly on an external parameter λ. We will need only first derivatives with respect to λ so the parametric statistical Gaussian ensemble of random polynomials is completely specified by fixing λ and saying that an and ∂λan are complex Gaussian random variables with prescribed covariances 〈ana∗m〉 , 〈an∂λa∗m〉 , 〈∂λan∂λa∗m〉 3 which need not be further specified for the statement of the general result. Fixing the two k-tuples of complex numbers z and v, we define the 3k linear combinations of 2(N + 1) Gaussian random variables an, ∂λan, n = 0, . . . , N fj = f(zj, λ) = N∑ n=0 anz n j , (6) f ′j = ∂zf(zj, λ) = N−1∑ n=0 (n + 1)an+1z n j , (7) f˜j = d dλ f(zj , λ) = f¯j + vjf ′ j, f¯j = ∂λf(zj , λ) = N∑ n=0 ∂λanz n j (8) which are again Gaussian random variables. The joint probability distribution of 3k random variables ξ = (f ,f ′, f˜) can be therefore written as P (f ,f ′, f˜ ) = 1 pi3k det M˜ exp ( −ξ∗ · M˜−1ξ ) (9) where M˜ = M˜(z,v) is 3k × 3k Hermitian covariance matrix which is written in a block form as M˜ = 〈ξ ⊗ ξ∗〉 =   A B D˜B† C E˜ D˜ † E˜ † F˜   (10) Ajl = 〈fjf ∗l 〉 , Bjl = 〈 fjf ′ l ∗ 〉 , Cjl = 〈 f ′jf ′ l ∗ 〉 , (11) D˜jl = 〈 fj f˜ ∗ l 〉 , E˜jl = 〈 f ′j f˜ ∗ l 〉 , F˜jl = 〈 f˜j f˜ ∗ l 〉 . (12) Note that tilded symbols are used to denote matrices (or vectors or scalars) which explic- itly depend on the parametric velocities v. k-tuple z are the zeros if f = 0, and k-tuple v are the velocities if in addition f˜ ≡ (d/dλ)f = 0. So the parametric k-point correlation function (3) can be written as a linear transformation of a joint distribution P (ξ) ρ˜(z,v) = ∫ d2kf ′ ∂(f , f˜) ∂(z,v) P (0,f ′, 0) (13) = 1 pi3k detM ∫ k∏ j=1 d2f ′j |f ′j|4 exp ( −f ′∗ · L˜−1f ′ ) (14) where ∂(f , f˜)/∂(z,v) = ∏k j=1 |f ′j|4 is the Jacobian of the mapping (z,v)→ (f , f˜), and L˜ = C− B†A−1B− (E˜− B†A−1D˜)(F˜− D˜†A−1D˜)−1(E˜† − D˜†A−1B) (15) is the central k × k block of the inverse of covariance matrix, M˜−1. The dependence on positions of zeros z and velocities v is digged in the definitions of the matrices (10,11,12) In general, the dependence on velocities can be made explicit in the following way. Writing a diagonal velocity matrix as V = diag {vj, j = 1 . . . k} and using a definition (8) one can observe that the covariance matrices D˜, E˜, F˜ have a simple velocity dependence which can 4 be written in terms of their velocity-independent counterparts D,E,F D˜ = D+ BV†, Djl = 〈 fj f¯ ∗ l 〉 , E˜ = E+ CV†, Ejl = 〈 f ′j f¯ ∗ l 〉 , (16) F˜ = F+ E†V† +VE+VCV†, Fjl = 〈 f¯j f¯ ∗ l 〉 This relations can be used to prove that the determinant of the covariance matrix does not depend on velocities det M˜ = detM (17) where 3k × 3k matrix M is obtained from M˜ by replacing the blocks D˜, E˜, F˜ by D,E,F. Using some elementary algebra one can rewrite the matrix L˜ in the form which makes velocity dependence explicit L˜ −1 = G−1 + (V† +G−1K)(H−K†G−1K)−1(V+K†G−1) (18) where we have introduced the matrices G = C− B†A−1B = G†, H = F−D†A−1D = H†, (19) K = E− B†A−1D. The representation of parametric correlations ρ˜k in terms of moments of a Gaussian (14) is very convenient since it may be explicitly evaluated as the sum of all possible pairwise contractions of integration variables f ′j (Wick theorem) and expressed in a compact form following an approach of Hannay [5] ρ˜k(z,v) = det L˜ pi2k detM per ( L˜ L˜ L˜ L˜ ) (20) where permanent of a square matrix perS = ∑ p ∏ j Sjpj is a symmetric analog of a de- terminant detS = ∑ p(−)p ∏ j Sjpj where p are permutations with signatures (−)p. Inte- grating out the velocities, which can be done by putting expression (18) into eq. (14) and evaluating the inner Gaussian integrals in terms of new variables uj = vjf ′ j, one obtains the k−point correlation function of Hannay [5] ρk(z) = perG pik detA . (21) The formula (20) is a general result on parametric statistics of Gaussian random poly- nomials with complex coefficients. Its important feature is purely algebraic dependence on velocities in contradistinction with e.g. parametric energy level statistics (see e.g. [4], chapter 6) where velocities have Maxwellian distribution since the Hamiltonian of the energy level flow can be clearly written as the sum of the usual kinetic and potential part. In the important special case where (the coefficients of) the random polynomial and its parametric derivative are statistically uncorrelated 〈an∂λam〉 = 0 (22) 5 we obtain parametric correlation functions which are invariant under the change of sign of velocities ρ˜k(z,−v) = ρ˜k(z,v) since E = D = K = 0 and therefore L˜ −1 = G−1 +V†F−1V. In other words, the average velocity (and all its odd moments) is zero 〈v〉 = 0. In this case, the determinant of 3k×3k covariance matrix has also a simple factorization in terms of k × k matrices detM = detA detG detF. Since the 2−point parametric correlation function depends on 4-complex arguments it may be useful to define also the 2-point velocity-moments of the parametric correlation functions 〈 vk1v l 2v ∗ 1 mv∗2 n 〉 = 1 ρ2(z1, z2) ∫ d2v1d 2v2 v k 1v l 2v ∗ 1 mv∗2 nρ˜2(z1, z2, v1, v2) (23) which are different from zero only if k + l = m+ n and finite if k + l +m + n ≤ 4. The nontrivial velocity-moments, which still depend on the positions of two zeros z1 and z2, can be calculated using the Wick theorem from the representation (14). Let us quote the results for the driftless case (22) 〈v1v∗2〉 = G21F12/(G11G22 +G12G21), (24)〈 |v1|2|v2|2 〉 = (F11F22 + F12F21)/(G11G22 +G12G21), (25)〈 v21v ∗ 2 2 〉 = 2F 212/(G11G22 +G12G21). (26) So far the correlations between the coefficients of random polynomials an, ∂λan have been completely arbitrary! Now we specialize to the case where coefficients of random polynomials are δ−correlated 〈ana∗m〉 = δmnbn, 〈∂λan∂λa∗m〉 = δmnb¯n (27) where the variances bn, b¯n are still arbitrary. Introducing two polynomials g(s) = N∑ n=0 bns n, g¯(s) = N∑ n=0 b¯ns n (28) the relevant matrices can be expressed as Ajl(z) = g(zjz ∗ l ), (29) Bjl(z) = zjg ′(zjz ∗ l ), (30) Cjl(z) = g ′(zjz ∗ l ) + zjz ∗ l g ′′(zjz ∗ l ), (31) Fjl(z) = g¯(zjz ∗ l ). (32) Putting k = 1, 1−point parametric statistics can be explicitly written in an elegant factorized form ρ˜1(z, v) = ρ1(z) ν˜(v/σ(z)) σ2(z) (33) 6 where ρ1(z) = 1 pi d ds s d ds log g(s) ∣∣∣ s=|z|2 (34) is a general density of zeros as can be deduced from [5] and ν˜(v) = 2 pi (1 + |v|2)−3 (35) is an universal velocity distribution normalized to a unit mean square and σ2(z) = 〈|v|2〉 is a mean square velocity which is inversely proportional to the density of zeros σ2(z) = g¯(|z|2) g(|z|2) 1 piρ1(z) . (36) So, the theory of random polynomials predicts a universal form of a velocity distribution (35) when it is locally rescaled to a unit mean square local velocity. In case of eigenstates of RMT in the Bargmann representation one has bn = 1/n!, b¯n = σ2/n! and N →∞, so g(s) = exp(s), g¯(s) = σ2 exp(s) (37) and therefore the density distribution and the local mean square velocity are constant, ρ1(z) = 1/pi, 〈|v|2〉 = σ2, so one has ρ˜1(z, v) = 2 pi2σ2 (1 + |v|2/σ2)−3 (38) In this probably the most important particular case we are also able to give some details of 2-point parametric correlation function ρ˜2(z1, z2, v1, v2) which is only a function of the 4 real quantities instead of 8: distance between roots |z2−z1|, magnitudes of the velocities u1 = |v1|, u2 = |v2| and the angle between velocities φ = arccos(Rev1v∗2/|v1v2|). Writing s = |z2 − z1|2 and fixing the velocity scale by putting σ = 1 we may express 2−point parametric correlations as ρ˜2 = 4(es − 1)5α5(β2 + γ2 + 4βγ) pi2(β − γ)5 , (39) α = e2s − (s2 + 2)es + 1 β = es((es − 1)(es − 1− s) + αu21)((es − 1)(es − 1− s) + αu22) γ = e2s(es − 1)2(e−s − 1 + s)2 + α2u21u22 − 2αes(es − 1)(e−s − 1 + t)u1u2 cos(φ) Since this expression is quite complicated it is worthwhile to study its asymptotics for large and small distances √ s between the roots. The first two nonzero terms of the small s expansion are ρ˜2 = 48 pi4(2 + |v1 + v2|2)5 s 2 + 8(|v21 − v22 |2 − 8|v1 − v2|2) pi4(2 + |v1 + v2|2)6 s 3 +O(s4). (40) For large s asymptotics in the leading term, as we expect, the two velocities are uncorre- lated, while we give also the next term of an expansion in powers of e−s ρ˜2 = 4 pi4 [ 1 + 6e−s (1 + |v1|2)3(1 + |v2|2)3 (41) + s2(|v1|2 − 2)(|v2|2 − 2) + 12s(|v1|2 + |v2|2 − 1)− 9s|v1 + v2|2 − 9|v1 − v2|2 (1 + |v1|2)4(1 + |v2|2)4 e −s + O ( e−2s )] 7 Note that the small s expansion (40) should be understood strictly pointwise, while it is not termwise integrable with respect to velocities v1 and v2 as should be the case for the entire 2-point parametric correlation function ρ˜2 (3). It is useful to give also the velocity-moments (23) which in this case depend only on the distance between zeros √ s and probably still contain a lot of information about parametric 2-point statistics 〈v1v∗2〉 = −σ2es(es − 1)(e−s − 1 + s)/ω, (42)〈 |v1|2|v2|2 〉 = σ4(es + 1)(es − 1)2/ω, (43)〈 v21v ∗ 2 2 〉 = 2σ4(es − 1)3/ω, (44) ω = es(es − 1− s)2 + e2s(e−s − 1 + s)2. (45) 2.2 Real coefficients In this subsection we discuss the case of parametric statistics of the Gaussian random polynomials with real coefficients, i.e. an and ∂λan are real Gaussian random variables with prescribed covariances 〈anam〉 , 〈an∂λam〉 , 〈∂λan∂λam〉 . Fixing the two k-tuples of complex numbers z and v we define 6k real random vari- ables Refj , Imfj ,Ref ′ j , Imf ′ j,Ref˜j, Imf˜j , or equivalently, a vector of 3× 2k variables ξ = (f ,f ′, f˜) where f = (f1, f ∗ 1 , . . . , fk, f ∗ k ),f ′ = (f ′1, f ′ 1 ∗, . . . , f ′k, f ′ k ∗), f˜ = (f˜1, f˜ ∗ 1 , . . . , f˜k, f˜ ∗ k ) The joint distribution of ξ is now again a (real) Gaussian with various blocks and their derivations of covariance matrices which are now 2k × 2k matrices and are defined by the same formulae (10,11,12,16,18, 19). Using straightforward approach which follows the previous subsection and the derivation of nonparametric statistics for real coefficients [12] one derives the general formula for the parametric k−point correlation function of zeros of Gaussian random polynomials with real coefficients ρ˜k(z,v) = 1 (2pi)2k √ det L˜ detM sper   L˜1 1 L˜1 1 . . . L˜1 2k L˜1 2k L˜1 1 L˜1 1 . . . L˜1 2k L˜1 2k ... ... . . . ... ... L˜2k 1 L˜2k 1 . . . L˜2k 2k L˜2k 2k L˜2k 1 L˜2k 1 . . . L˜2k 2k L˜2k 2k   (46) which reduces to the corresponding nonparametric k−point correlation function [12] ρk(z) = sperG (2pi)k √ detA (47) after velocities are integrated out. Semi-permanent of a square 2m × 2m matrix (intro- duced in [12]) is a homogeneous polynomial of order m of the matrix elements sper S = jn 6=ln′∑ j1<...