Universal statistics of wave functions in chaotic and disor - dered systems

– We study a new statistics of wave functions in several chaotic and disordered systems: the random matrix model, band random matrix model, the Lipkin model, chaotic quantum billiard and the 1D tight-binding model. Both numerical and analytical results show that the distribution function of a generalized Riccati variable, defined as the ratio of components of eigenfunctions on basis states coupled by perturbation, is universal, and has the form of Lorentzian distribution. Statistics of eigenfunctions in disordered and chaotic systems are of great interest in many branches of physics such as condensed matter physics, nuclear physics, chemical physics, and in particular, in the new developing field of quantum chaos[1, 2, 3]. However, although eigenfunctions contain more information than eigenenergies, much less has been studied compared with the eigenenergies. The mostly studied property is the statistics of wave function amplitude |ψ(r)|2 and two point correlation function 〈ψ(r1)ψ(r2)〉. The former one has been found to be the PorterThomas distribution for time-reversal invariant systems whose classical counterpart is chaotic. This is predicted by the Random Matrix Theory [4, 5, 6], and confirmed numerically[8, 9] and experimentally[10]. The later one, which is proportional to J0(kd) (J0 is the Bessel function of the zeroth order, k the wave vector and d the distance between the two points), was proposed by Berry[5] for a chaotic billiard within an assumption that the wave function is a superposition of plane waves with random coefficients. The two point correlation function has been extended to large separation by Hortikar and Srednicki[7] recently. A beautiful review for the correlations of wave functions in disordered systems has been given recently by Mirlin [11]. (∗) To whom correspondence should be addressed. E-mail: [email protected] Typeset using EURO-TEX 2 EUROPHYSICS LETTERS In this Letter, we would like study another interesting statistics, namely the statistics of the Riccati variable of wave functions (see, e.g., [12]). The Riccati variable has been studied in one-dimensional disordered system, where it is related to the reflection coefficient of waves. As we shall see later, the Riccati variable can be easily related to the two-point correlation function, but it contains more information than the correlation function. Furthermore, the Riccati variable gives information about local fluctuation properties of wave functions, and the form of its distribution is related to properties of the perturbation of the physical models. To seek a general universality of this quantity among different quantum systems, our study will be extended to a wide range of random matrix model, disordered and chaotic systems such as the Random Matrix Model (RMM)[13], Band Random Matrix Model (BRMM)[14], the Lipkin model [15] with chaotic behavior in the classical limit, a chaotic quantum billiard model[16] and a 1D tight-binding model in the case of weak disorder[17]. In order to study the chaotic systems, the Riccati variable for the 1D tight-binding model is first extended to a series types of Riccati variable, namely, type I, type II and so on. Then, the statistics for type I Riccati variable is found analytically to be of a Lorentzian form for the RMM. Based on this fact, we conjecture that the distribution function of the Riccati variable of type I is universal for all chaotic and disordered systems. Our numerical results on several such models support the conjecture. Furthermore, the form of the distribution of the other types of Riccati variable gives information on properties of chaotic eigenfunctions which can not be supplied by, e.g., the statistics of intensity of wave functions. Let’s first consider a general case in which the Hamiltonians can be written in the form