Asymptotic level spacing distribution for a q-deformed random matrix ensemble

We obtain the asymptotic behaviour of the nearest-neighbour level spacing distribution for a q-deformed unitary random matrix ensemble. The q-deformed unitary random matrix ensemble introduced in [1] describes a transition in spectral statistics from the highly correlated Gaussian unitary ensemble [2] (GUE, q → 1) towards a completely uncorrelated Poisson ensemble (q → 0) as a function of q. Such transitions occur in a wide variety of physical systems including disorder and chaos [3]. In particular, the transition in the nearest-neighbour spacing distribution obtained from the model seems to show a remarkable similarity with that obtained numerically from a microscopic tight-binding Anderson model [4] describing the metal–insulator transition in disordered conductors. This similarity holds despite the fact that the problem of disordered conductors is known to have a small additional level-interaction [5, 6] which is not taken into account in [1]. (This term is known to change the numerical value of the variance of conductance by a very small amount in quasi-one dimension.) The similarity is all the more remarkable because it includes an apparent fixed point at s = s0 in the spacing distribution P(s) (s0 ' 1.8 for unitary ensembles and s0 ' 2 for orthogonal ensembles). The fixed point was used in [4] to propose the area A = ∫ ∞ s0 P(s) ds as a one-parameter characterization of the spacing distribution. The length and disorder dependences as well as the scaling behaviour of the normalized parameter γ = (A − AW)/(AP − AW), where the subscripts W and P refer to the Wigner and the Poisson limits, were studied in detail numerically and shown to contain information about the metal–insulator transition. However, the level spacing distribution of the q-ensemble requires the evaluation of a Fredholm determinant of the two-level kernel over a finite interval, and while the kernel was obtained analytically in [1], the determinant was evaluated numerically for various values of q. It is clearly of interest to evaluate analytically the spacing distribution as a function of q in order to understand the role of q-deformation in these systems by, e.g., relating q to the parameters A and/or γ . While it has not been possible to obtain the complete spacing distribution analytically for all s even in the GUE limit [2], the work of [4] shows that because of the (apparent) fixed point, P(s > s0) already contains valuable information. In the present work we obtain the asymptotic behaviour of the spacing distribution both near GUE and Poisson limits using the Szegö theorem on the asymptotic behaviour of a related Toeplitz † Also at the Department of Mathematics. 0305-4470/96/164853+05$19.50 c © 1996 IOP Publishing Ltd 4853 4854 K A Muttalib and J R Klauder determinant. Unfortunately, the method is not accurate enough to obtain information about the (apparent) fixed point itself, which requires evaluation of the terms independent of s as well. However, by exploiting the existence of at least an approximate fixed point, we are able to determine the leading s-independent (but q-dependent) terms and obtain a reasonably good analytical expression for the asymptotic spacing distribution for s > s0 near the GUE as well as the Poisson limits; this can then be used to make at least qualitative connection with the numerically accessible parameters A or γ . Moreover, a qualitative understanding of the q-deformation of the spacing distribution obtained from our results should provide a basis for a more careful analysis to understand the nature of the fixed point. We will assume that the q-ensemble is characterized, in the properly scaled variables where the average spacing between adjacent levels is unity, by the two-level kernel obtained in [1]: K(ζ − η; q 6 1) ≈ β 2π sin[(ζ − η)π ] sinh[(ζ − η)β/2] q = e −β. (1) Although the above kernel is valid only in restricted regimes [1], it is in these regimes that the kernel is translationally invariant and seems to be model independent [7]. We will obtain the asymptotic level spacing distribution given by this kernel. For comparison, the corresponding kernel for a GUE is given by K(ζ − η; q = 1) = sin[(ζ − η)π ] (ζ − η)π (2) which is universal, independent of any parameter. The asymptotic behaviour of the level spacing for this kernel has been obtained in a variety of ways [8, 9], although a complete expression valid for all spacing is not available. Given the kernel K , the probability E(t) that the interval (−t, t) does not contain any level can be obtained as a Fredholm determinant of the integral equation [2] ∫ t −t K(ζ − η)f (η) dη = λf (ζ ). (3) Changing variables such that the range of the integral is (−1, 1), one gets E(t) = det(1 − Kt) (4) where Kt(ζ − η; q) ≈ βt 2π sin[(ζ − η)πt] sinh[(ζ − η)βt/2] . (5) The nearest-neighbour spacing distribution is then given by P(s) = d 2 ds2 E(s) s = 2t. (6) We first note that the kernel Kt can be written as the Fourier transform of the function φ(k) = sinh[b] cosh[bk/πt] + cosh[b] b = 2π2 β . (7) To exploit the connection with the Toeplitz determinant, consider the function f (θ) = 1 − sinh[b] cosh[bθ/a] + cosh[b] . (8) Then in the limit N → ∞ and Na → 2πt 1, we can identify det[1 − Kt ] with the N × N Toeplitz determinant DN(f ) = det [ 1 2π ∫ π −π f (θ)ei(j−k)θ dθ ]