An Ising-Anderson model of localisation in high-temperature QCD

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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...

Section: Contentsmentioning

confidence: 99%

mentioning

confidence: 99%

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

Giordano

2019

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“…Moreover, the mechanism for localisation proposed in refs. [6,12,13] is based on the presence of "islands", where the Polyakov lines fluctuate away from the ordered (trivial) value, in the "sea" of ordered Polyakov lines. Such "islands" provide an "energetically" favourable location for the eigenmodes.…”

Section: Jhep02(2017)055mentioning

confidence: 99%

“…[6] by studying the correlation of the Dirac eigenfunctions with the fluctuations of the Polyakov loop on SU(2) gauge configurations, and in refs. [12,13] by designing toy models that should feature localisation precisely through the proposed mechanism. Further evidence has been recently obtained in ref.…”

Section: Jhep02(2017)055mentioning

confidence: 99%

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

Giordano¹,

Katz²,

Kovács³

et al. 2017

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Abstract:We study the problems of deconfinement, chiral symmetry restoration and localisation of the low Dirac eigenmodes in a toy model of QCD, namely unimproved staggered fermions on lattices of temporal extension N T = 4. This model displays a genuine deconfining and chirally-restoring first-order phase transition at some critical value of the gauge coupling. Our results indicate that the onset of localisation of the lowest Dirac eigenmodes takes place at the same critical coupling where the system undergoes the first-order phase transition. This provides further evidence of the close relation between deconfinement, chiral symmetry restoration and localisation of the low modes of the Dirac operator on the lattice.

An Anderson-like model of the QCD chiral transition

Giordano

Kovács

Pittler

2016

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Abstract:We study the problems of chiral symmetry breaking and eigenmode localisation in finite-temperature QCD by looking at the lattice Dirac operator as a random Hamiltonian. We recast the staggered Dirac operator into an unconventional three-dimensional Anderson Hamiltonian ("Dirac-Anderson Hamiltonian") carrying internal degrees of freedom, with disorder provided by the fluctuations of the gauge links. In this framework, we identify the features relevant to chiral symmetry restoration and localisation of the low-lying Dirac eigenmodes in the ordering of the local Polyakov lines, and in the related correlation between spatial links across time slices, thus tying the two phenomena to the deconfinement transition. We then build a toy model based on QCD and on the DiracAnderson approach, replacing the Polyakov lines with spin variables and simplifying the dynamics of the spatial gauge links, but preserving the above-mentioned relevant dynamical features. Our toy model successfully reproduces the main features of the QCD spectrum and of the Dirac eigenmodes concerning chiral symmetry breaking and localisation, both in the ordered (deconfined) and disordered (confined) phases. Moreover, it allows us to study separately the roles played in the two phenomena by the diagonal and the off-diagonal terms of the Dirac-Anderson Hamiltonian. Our results support our expectation that chiral symmetry restoration and localisation of the low modes are closely related, and that both are triggered by the deconfinement transition.