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At zero temperature the lowest part of the spectrum of the QCD Dirac operator is known to consist of delocalized modes that are described by random matrix statistics. In the present paper we show that the nature of these eigenmodes changes drastically when the system is driven through the finite temperature cross-over. The lowest Dirac modes that are delocalized at low temperature become localized on the scale of the inverse temperature. At the same time the spectral statistics changes from random matrix to Poisson statistics. We demonstrate this with lattice QCD simulations using 2 + 1 flavors of light dynamical quarks with physical masses. Drawing an analogy with Anderson transitions we also examine the mobility edge separating localized and delocalized modes in the spectrum. We show that it scales in the continuum limit and increases sharply with the temperature.

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...

At low temperature the low end of the QCD Dirac spectrum is well described by chiral random matrix theory. In contrast, at high temperature there is no similar statistical description of the spectrum. We show that at high temperature the lowest part of the spectrum consists of a band of statistically uncorrelated eigenvalues obeying essentially Poisson statistics and the corresponding eigenvectors are extremely localized. Going up in the spectrum the spectral density rapidly increases and the eigenvectors become more and more delocalized. At the same time the spectral statistics gradually crosses over to the bulk statistics expected from the corresponding random matrix ensemble. This phenomenon is reminiscent of Anderson localization in disordered conductors. Our findings are based on staggered Dirac spectra in quenched lattice simulations with the SU(2) gauge group.

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.

We investigate the QCD phase diagram for nonzero background magnetic fields using first-principles lattice simulations. At the physical point (in terms of quark masses), the thermodynamics of this system is controlled by two opposing effects: magnetic catalysis (enhancement of the quark condensate) at low temperature and inverse magnetic catalysis (reduction of the condensate) in the transition region. While the former is known to be robust and independent of the details of the interactions, inverse catalysis arises as a result of a delicate competition, effective only for light quarks. By performing simulations at different quark masses, we determine the pion mass above which inverse catalysis does not take place in the transition region anymore. Even for pions heavier than this limiting value -where the quark condensate undergoes magnetic catalysis -our results are consistent with the notion that the transition temperature is reduced by the magnetic field. These findings will be useful to guide low-energy models and effective theories of QCD.

We present first evidence for the Landau level structure of Dirac eigenmodes in full QCD for nonzero background magnetic fields, based on first principles lattice simulations using staggered quarks. Our approach involves the identification of the lowest Landau level modes in two dimensions, where topological arguments ensure a clear separation of these modes from energetically higher states, and an expansion of the full four-dimensional modes in the basis of these two-dimensional states. We evaluate various fermionic observables including the quark condensate and the spin polarization in this basis to find how much the lowest Landau level contributes to them. The results allow for a deeper insight into the dynamics of quarks and gluons in background magnetic fields and may be directly compared to low-energy models of QCD employing the lowest Landau level approximation.Comment: 21 pages, 19 figure

We investigate the Anderson transition found in the spectrum of the Dirac operator of Quantum Chromodynamics (QCD) at high temperature, studying the properties of the critical quark eigenfunctions. Applying multifractal finite-size scaling we determine the critical point and the critical exponent of the transition, finding agreement with previous results, and with available results for the unitary Anderson model. We estimate several multifractal exponents, finding also in this case agreement with a recent determination for the unitary Anderson model. Our results confirm the presence of a true Anderson localization-delocalization transition in the spectrum of the quark Dirac operator at high-temperature, and further support that it belongs to the 3D unitary Anderson model class.