Universality and the QCD Anderson Transition

Abstract

We study the Anderson-type transition previously found in the spectrum of the QCD quark Dirac operator in the high temperature, quark-gluon plasma phase. Using finite size scaling for the unfolded level spacing distribution, we show that in the thermodynamic limit there is a genuine mobility edge, where the spectral statistics changes from Poisson to Wigner-Dyson statistics in a non-analytic way. We determine the correlation length critical exponent, ν, and find that it is compatible with that of the unitary Anderson model. PACS numbers: 12.38.Gc,72.15Rn,12.38.Mh,11.15.Ha The idea of Anderson localization is more than half a century old [1]. Anderson localization consists in the spatial localization of the states of a system due to quantum interference effects, caused by the presence of disorder. Its simplest realization is provided by the Anderson tightbinding model that aims at describing electronic states in a "dirty" conductor, by mimicking the effect of impurities through a random on-site potential. In three dimensions, as soon as the random potential is switched on, localized states appear at the band edge. However, states remain extended around the band center, beyond a critical energy called the "mobility edge". Increasing the amount of disorder, i.e., increasing the width of the distribution of the random potential, the mobility edge moves towards the band center, and above a certain critical disorder all the states become localized (see Refs. [2,3]).Originally proposed to explain the loss of zero temperature conductance as a result of disorder, localization was later found in a much wider range of physical systems. Anderson transitions have been demonstrated with electromagnetic and sound waves as well as cold atoms (see Ref.[4] and references therein) and recently in strongly interacting matter in its high temperature quark-gluon plasma phase [5]. The last item of the list is rather peculiar since in that case localization occurs on a vastly different length and energy scale from all previously known cases, namely on subnuclear rather than atomic scales.In the microscopic description of strongly interacting matter provided by quantum chromodynamics (QCD), a central role is played by the Dirac operator. Its spectrum encodes important properties of quarks and hadrons. At low temperature, the lowest lying quark eigenmodes of the Dirac operator have long been known to be extended, and the corresponding spectrum to obey WignerDyson statistics as predicted by random matrix theory (RMT) [6]. This has been successfully exploited to study the low-energy properties of QCD [6]. In contrast, in the high-temperature quark-gluon plasma phase no similar description of the low-lying quark modes was available until recently. It was first suggested by García-García and Osborn that the transition from the hadronic to the quark-gluon plasma phase might be an Andersontype transition [7]. Using lattice QCD they qualitatively demonstrated that heating the system through the critical temperature makes the quark states mor...

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

“…The curve λ c (T ) reaches zero at a temperature compatible with T c [7], as determined from thermodynamic observables [2,3]. The transition in the spectrum from localised to delocalised modes, taking place at the critical point λ c , was shown to be a genuine second-order phase transition [9], analogous to the metal-insulator transition in the Anderson model [38][39][40], which describes non-interacting electrons in a disordered crystal. Furthermore, in ref.…”

“…Furthermore, in ref. [9] the correlation-length critical exponent was found to be compatible with that of the 3D unitary Anderson model [41]. Here "unitary" refers to the symmetry class of the model in the Random Matrix Theory (RMT) classification of random matrix ensembles [42], which is shared by the staggered Dirac operator [35].…”

“…While at low temperatures the low-lying eigenmodes of the Dirac operator are delocalised on the entire lattice volume [35,36], above the crossover temperature, T c [2,3], they become spatially localised [4][5][6][7][8][9][10][11] on the scale of the inverse temperature [5,7,11]. From now on we focus on staggered fermions, both for simplicity and because of the larger amount of available evidence.…”

“…In ref. [9] this was determined to be I crit s 0 0.1966 (25) in QCD with 2 + 1 flavours of staggered fermions. Since I crit s 0 is determined by the critical eigenvalue statistics, which is believed to be universal as well as volume independent [45], this value should be attained at λ c in all those models displaying a localisation transition along the spectrum which belongs to the same universality class as that of the 3D unitary Anderson model.…”

“…In recent years, it has become more and more clear that the decrease in the spectral density of low modes is accompanied by a change in the localisation properties of the corresponding eigenvectors: while at low temperatures the low modes are extended throughout the whole spatial volume, above the crossover temperature the lowest modes are spatially localised on the scale of the inverse temperature. Evidence for this behaviour has been obtained by means of lattice simulations with different fermion discretisations, namely staggered [4][5][6][7][8][9], overlap [6,10] and domain wall [11] fermions.…”