Localization of Dirac modes in finite-temperature Z2 gauge theory on the lattice

Abstract

We study the localization properties of the eigenmodes of the staggered Dirac operator across the deconfinement transition in finite-temperature Z3 pure gauge theory on the lattice in 2+1 dimensions. This allows for nontrivial tests of the sea-islands picture of localization, according to which low modes should localize on favorable Polyakov-loop fluctuations in the deconfined phase of a gauge theory. We observe localized low modes in the deconfined phase of the theory, both in the real Polyakov-loop sector, where they are expected, and in the complex Polyakov-loop sectors, where they are not. Our findings expose the limitations of the standard sea-islands picture, and call for its refinement. An improved picture, where spatial hopping terms play a more prominent role, is proposed and found to be in excellent agreement with numerical results.

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

“…This picture relates localized modes with local fluctuations of the Polyakov loop away from its ordered value (i.e., 1) in the high-temperature phase, which provide "energetically" favorable locations for the fermions. Numerical support for this picture has been obtained in various cases [62,63,68,69].…”

“…[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]. The strong connection observed between localization and deconfinement is clarified by the mechanism provided by the "sea/islands picture" of localization proposed in Refs.…”

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.…”

“…The situation is instead not so clear for gauge theories, where the physical meaning of the localisation of Dirac modes has proved to be more elusive. There is, however, growing evidence of an intimate connection between localisation and deconfinement: in a variety of systems with a genuine deconfinement transition, localisation of the low Dirac modes appears in fact precisely at the critical point [7][8][9][10][11][12][13]. This is true even for the simplest model displaying a deconfinement transition, namely 2+1 dimensional Z 2 gauge theory [12].…”

“…There is, however, growing evidence of an intimate connection between localisation and deconfinement: in a variety of systems with a genuine deconfinement transition, localisation of the low Dirac modes appears in fact precisely at the critical point [7][8][9][10][11][12][13]. This is true even for the simplest model displaying a deconfinement transition, namely 2+1 dimensional Z 2 gauge theory [12]. Theoretical arguments for this behaviour have also been discussed in the literature [20][21][22].…”