Smallest Dirac eigenvalue distribution from random matrix theory

Abstract

We derive the hole probability and the distribution of the smallest eigenvalue of chiral hermitian random matrices corresponding to Dirac operators coupled to massive quarks in QCD. They are expressed in terms of the QCD partition function in the mesoscopic regime. Their universality is explicitly related to that of the microscopic massive Bessel kernel. PACS number(s): 05.45.+b, 12.38.Aw, 12.38.Lg There has long been an attractive idea that the lowenergy physics of a complex system can be described by a simple effective theory which respects the global symmetries of the original system. As an example, the quantum spectral statistics of a classically chaotic system is believed to be described by a random matrix theory belonging to the same universality class as the former [1]. One new manifestation of essentially the same idea is the recent observation that QCD Dirac operator spectra on the scale λ = O(1/V 4 ) (where V 4 is the space-time volume) measured in lattice Monte Carlo simulations [2] are in excellent agreement with the predictions from those large-N random matrix theories [3,4] that share the same global symmetries as QCD. The suitably rescaled (microscopic) spectral correlation functions thus seem to provide exact finite-size scaling functions for QCD in a finite volume. Very recently, the microscopic spectral correlators have been calculated from random matrix theories that include the effect of fermion determinants with masses m ≃ O(1/V 4 ) [5][6][7] (see also [8]). When λ and m are measured in units of the mean level spacing at zero virtuality, all the random matrix predictions turn out to be universal, i.e., insensitive to the details of the random matrix potential [4][5][6]. Although the question of whether or not QCD is included in the same universality class cannot be answered by demonstrating the existence of the wide range of universality within random matrix theories, it provides strong support for the former.From the field-theoretic point of view [9] it would be most surprising if these observables would not also be computable solely within the framework of finite-volume generating functionals (partition functions) for the order parameter ψ ψ . If not, large-N random matrix theory, which in principle is foreign to the pertinent field theory language, would seem to be a new ingredient required to describe the observed spectral correlators. It has recently been shown that a description entirely in terms of finite-volume partition functions is indeed also possible [10].In order to confirm by numerical simulations that the low-lying spectra of QCD Dirac operators can be described alternatively by large-N random matrix theories, it is in practice most convenient to measure the distribution of the smallest eigenvalue [2] and compare that to the random matrix prediction [11,7]. Since the smallest eigenvalue distribution largely consists of the first peak of the microscopic spectral density (see Fig. 1, ζ ≡ N λ), we expect it to be universal. In fact, the proven universality of the massive kerne...

Power law deformation of Wishart–Laguerre ensembles of random matrices

“…In WL the gap probability E(s) and the first eigenvalue distribution p(s) are explicitly known and universal in the microscopic large-N limit for all ν at β = 2, for odd values of ν and 0 at β = 1, and for ν = 0 at β = 4. This has been shown by various authors independently [16,38,39,37]. In some cases only finite-N results are know in terms of a hypergeometric function of a matrix valued argument [40,41], from which limits are difficult to extract.…”

“…Next we give the first eigenvalue distribution for general ν. Here we directly use the most compact universal expression [37] for ℘ (2) ν (y) in WL, without making the detour over E (2) (y) [39],…”

Distributions of Dirac operator eigenvalues

“…Clearly the approximation cannot be trusted in the region where, to this order, p (ν) 1 (ζ) goes negative. However, the comparison to the full result as it is conjectured from Random Matrix Theory [20] shows that for all practical purposes even this approximation that keeps only the two leading terms in the expansion is sufficient when comparing with lattice data.…”

“…[15,16]) and overlap fermions [17,18,19]. So far it is has only been possible to compare with the analytical predictions based on the conjectured Random Matrix Theory results [20,21], and an outstanding question has been whether these individual eigenvalue distributions also can be derived directly from the effective field theory. We will here rely only on what to date has been derived from field theory, namely the one-and two-point functions for the symmetry breaking class of QCD-like gauge theories (gauge theories with fermions transforming according to complex representations of the gauge group) [5,9].…”

Deconfinement, chiral transition and localisation in a QCD-like model

“…More precisely, chRMT makes definite predictions for the statistical properties of the microscopic spectrum, z, defined by rescaling the eigenvalues with the spectral density at the origin, z = λπρ(0). In particular, the distribution of the smallest eigenvalue in a given topological sector is known analytically: for example, the smallest rescaled eigenvalue z 1 = λ 1 πρ(0) in the trivial topological sector and in the quenched theory is expected to be distributed according to the following probability distribution function [47][48][49]:…”