Localisation in 2+1 dimensional SU(3) pure gauge theory at finite temperature

Abstract

I study the localisation properties of low Dirac eigenmodes in 2+1 dimensional SU(3) pure gauge theory, both in the low-temperature, confined and chirally-broken phase and in the high-temperature, deconfined and chirally-restored phase, by means of numerical lattice simulations. While these modes are delocalised at low temperature, they become localised at high temperature, up to a critical point in the Dirac spectrum where a BKT-type Anderson transition takes place. All results point to localisation appearing at the deconfinement temperature, and support previous expectations about the close relation between deconfinement, chiral symmetry breaking, and localisation. Contents1 Introduction 1 2 Localisation in lattice gauge theories 4 3 Lattice SU(3) pure gauge theory in 2+1 dimensions 7 4 Numerical results 9 4.1 Participation ratio 10 4.2 Spectral statistics 16 5 Conclusions and outlook 28with N f = 2 flavours of adjoint fermions [9] on the lattice: 2 here two phase transitions are present, a deconfining one at T dec and a chiral-symmetry-restoring one at T χ with T dec < T χ . Nonetheless, at T dec the chiral condensate jumps downwards, and a partial restoration of chiral symmetry happens via a first-order phase transition. It is also well known that the fate of chiral symmetry is determined by the spectrum of the Dirac operator near the origin. In fact, the celebrated Banks-Casher relation [11] establishes that the chiral condensate in the chiral limit is proportional to the spectral density of the Dirac operator near the origin. For finite but small quark masses, an accumulation of eigenmodes near the origin is still expected at low temperatures, leading to light pions and all the other phenomenological consequences for QCD due to its being close to a theory with spontaneously broken symmetry. At high temperatures, instead, the spectral density is expected to vanish near the origin, reflecting the restoration of chiral symmetry in the massless case. Given the close relation between confining and chiral properties of the theory, it is natural to wonder if confinement is somehow responsible for the accumulation of modes near the origin, and analogously if deconfinement causes the depletion of this spectral region. A similar question of course can be asked also for other gauge theories.Adding to the mistery, or possibly helping to solve it, a third phenomenon has been observed to take place in QCD around the critical temperature, namely the localisation of the lowest modes of the Dirac operator [12][13][14][15][16][17][18]. Numerical studies on the lattice have shown that while in the low-temperature phase all the Dirac modes are extended throughout the whole system, above T c the lowest modes get localised [12][13][14][15][16][17][18] on the scale of the inverse temperature [16]. More precisely, modes are localised up to a temperature-dependent critical point in the spectrum, λ c = λ c (T ). 3 At λ c , a second-order phase transition takes place in the spectrum [19], and modes become delocalised. This type of tr...

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

“…Localization in the presence of a sharp transition has been mostly investigated in pure gauge theories, selecting the "physical" center sector (i.e., real positive expectation value of the Polyakov loop) in the spontaneously broken phase. In these cases localization and deconfinement have been shown to coincide within numerical uncertainties [64][65][66][67][68]. The only study with dynamical fermions and a genuine phase transition is that of Ref.…”

“…Figure 2 shows that our procedure For finite statistics and volume one has necessarily to use sufficiently large finite bins in order to collect sufficiently many eigenvalues, and in regions where ρ(λ) is small the bin size may be comparable or even exceed the scale over which ρ(λ) varies appreciably. This leads to ρ(λ) ∆λ λ = 1 and in turn to s λ = 1 in that region [65]. This happens in the lowest part of the staggered spectrum at high temperature where the spectral density is small.…”

“…[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].…”