Universality in Chiral Random Matrix Theory atβ=1andβ=4

Abstract

In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real (β = 1) and quaternion real (β = 4) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles (β = 2). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Brézin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for β = 2 as shown by Akemann, Damgaard, Magnea and Nishigaki.PACS numbers: 64.60. Cn, 05.45.+b, 11.30.Rd, 12.38.Aw Since its introduction in nuclear physics [1], Random Matrix Theory (RMT) has been applied successfully to many different branches of physics ranging from atomic physics to quantum gravity (for a recent comprehensive review we refer to [2]). One important common ingredient is that eigenvalue correlations appear to be insensitive to the details of the underlying Hamiltonian. The success of RMT is based on this type of universality, and it is no surprise that it has received a great deal of attention in recent literature [3-10] [11-20] [21-27]. What has been shown is that spectral correlators on the scale of the average eigenvalue spacing are insensitive to the details of the probability distribution of the matrix elements. Because of its mathematical simplicity most studies were performed for complex (β = 2) Hermitean RMT's. However, in the case of the classical RMT's it was shown that universality extends to real (β = 1) and quaternion real (β = 4) matrix ensembles [4,6,7]. This suggests that relations between correlation functions for different values of β which can be derived for a Gaussian probability distribution [28,29] might be valid for a wide class of probability distributions. The main goal of this paper is to establish such general relations. As a consequence, universality for the much simpler complex ensembles implies universality for the real and quaternion real ensembles.In this letter we address the question of microscopic universality for the chiral ensembles. These ensembles are relevant for the description of spectral correlations of the QCD Dirac operator. They also appear in theory of universal conductance fluctuations in mesoscopic systems [30,31]. In particular, they can be applied to spectral correlations near λ = 0. According to the Banks-Casher formula [32], this part of the spectrum is directly related to the order parameter Σ of the chiral phase transition (Σ = lim πρ(0)/V , where V is the volume of space time and ρ(λ) = k δ(λ−λ k )). It is therefore natural to introduce the microscopic limit where the variable u = λV Σ is kept fixed for V → ∞. For example, the microscopic spectral density is defined by [33] ρ S (u) = limwhere the average is over the distribution of the matrix elements of the D...

Power law deformation of Wishart–Laguerre ensembles of random matrices

“…The corresponding results for β = 1, 4 are given in terms of a Pfaffian of a matrix kernel [18], and for a discussion of a relation between the three universal kernels we refer to [20]. A feature we observe for all three β is that for α ≤ O(1) the oscillations of the Bessel density are completely smoothed out, apart from the first peak.…”

“…It is governed by the Bessel-law [15,4,16,17,18] labelled by β and M − N being finite. The robustness or universality under polynomial deformations of the Gaussian weight function has also been proven [19,20]. This perturbation destroys the independence of the uncorrelated degrees of freedom of X, without changing the microscopic correlation functions.…”

“…The corresponding microscopic densities of the WL ensembles are again universal [20]. Computed initially in [17,18] the obtained expressions can be simplified.…”

Spectral universality of real chiral random matrix ensembles

“…Recent progress was made by relating the kernel for the correlation functions of the chOE to the universal kernel of the chUE [10]. This method was based on a generalization of an operator construction of skew-orthogonal polynomials for the Wigner-Dyson ensembles [7] to the chiral ensembles.…”

“…This program has been carried out most completely for the Hermitian random matrix ensembles (denoted by the Dyson index β = 2; for recent reviews see [4,5]) which are mathematically much simpler than real or quaternion-real random matrix ensembles (with Dyson index β = 1 and β = 4, respectively). Nevertheless, several universality proofs are available for these ensembles as well [6,7,8,9,10,11,12,13,14].…”