Critical statistics at the mobility edge of QCD Dirac spectra

Abstract

We examine statistical fluctuation of eigenvalues from the near-edge bulk of QCD Dirac spectra above the critical temperature. For completeness we start by reviewing on the spectral property of Anderson tight-binding Hamiltonians as described by nonlinear σ models and random matrices, and on the scale-invariant intermediate spectral statistics at the mobility edge. By fitting the level spacing distributions, deformed random matrix ensembles which model multifractality of the wave functions typical of the Anderson localization transition, are shown to provide an excellent effective description for such a critical statistics. Next we carry over the above strategy for the Anderson Hamiltonians to the Dirac spectra. For the staggered Dirac operators of QCD with 2+1 flavors of dynamical quarks at the physical point and of SU(2) quenched gauge theory, we identify the precise location of the mobility edge as the scaleinvariant fixed point of the level spacing distribution. The eigenvalues around the mobility edge are shown to obey critical statistics described by the aforementioned deformed random matrix ensembles of unitary and symplectic classes. The best-fitting deformation parameter for QCD at the physical point turns out to be consistent with the Anderson Hamiltonian in the unitary class. Finally, we propose a method of locating the mobility edge at the origin of QCD Dirac spectrum around the critical temperature, by the use of individual eigenvalue distributions of deformed chiral random matrices.

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

“…In figure 5 we plot the second moment of the unfolded spacing distribution, s 2 , against I s 0 , with each data point corresponding to a specific point in the spectrum, and to a given system size and value of the gauge coupling. The data points indeed arrange themselves rather precisely on a single curve, which furthermore compares well with the one obtained in QCD [34,51].…”

Critical behaviour in the QCD Anderson transition

“…Comparing with analogous results for the Anderson model, it turns out that the spectral statistics at the critical point in the two models are compatible [18].…”

“…The transition from Poisson to Wigner-Dyson behaviour in finite volume is therefore expected to be described by a universal one-parameter family of random matrix models [18]. Comparing with analogous results for the Anderson model, it turns out that the spectral statistics at the critical point in the two models are compatible [18].…”

Chiral transition as Anderson transition

“…For the QCD Dirac spectrum we showed that this transition is a second order phase transition. Most likely this is the case for our toy model too, since around the critical point in the spectrum (the mobility edge), where I 0.5 is volume independent, this quantity takes the same value as in QCD, I 0.5 (λ c ) = 0.187 [14], both for β = 5.120 and 5.130. We also see from Fig.…”