Localization of Dirac Fermions in Finite-Temperature Gauge Theory

Abstract

It is by now well established that Dirac fermions coupled to non-Abelian gauge theories can undergo an Anderson-type localization transition. This transition affects eigenmodes in the lowest part of the Dirac spectrum, the ones most relevant to the low-energy physics of these models. Here we review several aspects of this phenomenon, mostly using the tools of lattice gauge theory. In particular, we discuss how the transition is related to the finite-temperature transitions leading to the deconfinement of fermions, as well as to the restoration of chiral symmetry that is spontaneously broken at low temperature. Other topics we touch upon are the universality of the transition, and its connection to topological excitations (instantons) of the gauge field and the associated fermionic zero modes. While the main focus is on Quantum Chromodynamics, we also discuss how the localization transition appears in other related models with different fermionic contents (including the quenched approximation), gauge groups, and in different space-time dimensions. Finally, we offer some speculations about the physical relevance of the localization transition in these models.

Localization at the quenched SU(3) phase transition

“…It is known that in the high temperature phase the lowest part of the Dirac spectrum consists of localized eigenmodes (for a recent review see [6]). Approaching the transition temperature from above, the mobility edge, separating the low localized modes from the bulk of the spectrum, moves toward zero, and when it reaches zero, all modes become delocalized, as expected in the low temperature phase.…”

Localization properties of Dirac modes at the Roberge-Weiss phase transition

“…A growing body of evidence indicates that center symmetry is intimately related also with another aspect of the physics of fermions that changes radically as the system transitions to the high temperature phase, namely the localization properties of the low-lying eigenmodes of the Dirac operator (see Ref. [51] for a recent review). It has been shown in a rather large variety of gauge theories, including QCD, that as the theory crosses over from the confined to the deconfined phase, the low Dirac modes turn from delocalized to localized, up to a "mobility edge" λ c in the spectrum, above which they are again delocalized [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68].…”

Localised Dirac eigenmodes and Goldstone's theorem at finite temperature

“…Ample evidence from lattice calculations shows that the lowest modes of the Euclidean Dirac operator / are localised in the high-temperature phase of QCD [1][2][3][4] and of other gauge theories [5][6][7][8][9][10][11][12][13] (see [14] for a recent review). Localised modes are supported essentially only in a finite spatial region whose size does not change as the system size grows.…”