Anderson Localization in Quark-Gluon Plasma

Chiral random matrix model for critical statistics

We apply the Lefschetz thimble formulation of field theories to a couple of different problems. We first address the solution of a complex 0-dimensional phi^4 theory. Although very simple, this toy-model makes us appreciate a few key issues of the method. In particular, we will solve the model by a correct accounting of all the thimbles giving a contribution to the partition function and we will discuss a number of algorithmic solutions to simulate this (simple) model. We will then move to a chiral random matrix (CRM) theory. This is a somehow more realistic setting, giving us once again the chance to tackle the same couple of fundamental questions: how many thimbles contribute to the solution? how can we make sure that we correctly sample configurations on the thimble? Since the exact result is known for the observable we study (a condensate), we can verify that, in the region of parameters we studied, only one thimble contributes and that the algorithmic solution that we set up works well, despite its very crude nature. The deviation of results from phase quenched results highlights that in a certain region of parameter space there is a quite important sign problem. In view of this, the success of our thimble approach is quite a significant one.

Randomness on the lattice

The low-lying eigenvalue spectrum of the QCD Dirac operator in the epsilon-regime is expected to match with that of chiral Random Matrix Theory (ChRMT). We study this correspondence for the case including sea quarks by performing two-flavor QCD simulations on the lattice. Using the overlap fermion formulation, which preserves exact chiral symmetry at finite lattice spacings, we push the sea quark mass down to \sim 3 MeV on a 16^3\times 32 lattice at a lattice spacing a \simeq 0.11 fm. We compare the low-lying eigenvalue distributions and find a good agreement with the analytical predictions of ChRMT. By matching the lowest-lying eigenvalue we extract the chiral condensate, \Sigma(2 GeV)[MSbar] = [251(7)(11) MeV]^3, where errors represent statistical and higher order effects in the epsilon expansion. We also calculate the eigenvalue distributions on the lattices with heavier sea quarks at two lattice spacings. Although the epsilon expansion is not applied for those sea quarks, we find a reasonable agreement of the Dirac operator spectrum with ChRMT. The value of Sigma, after extrapolating to the chiral limit, is consistent with the estimate in the epsilon-regime.

Quark spectra in QCD are linked to fundamental properties of the theory including the identification of pions as the Goldstone bosons of spontaneously broken chiral symmetry. The lattice Overlap-Dirac operator provides a nonperturbative, ultraviolet-regularized description of quarks with the correct chiral symmetry. Properties of the spectrum of this operator and their relation to random matrix theory are studied here. In particular, the predictions from chiral random matrix theory in topologically non-trivial gauge field sectors are tested for the first time.

In the first section the dynamics of QCD at finite temperature and density are summarized. Color superconductivity that takes place at sufficiently high density and low temperature is reviewed and unsolved problems on the QCD phase diagram are presented. In the second section readers are introduced to QCD with the gauge group SU(2) (called two-color QCD), and the similarities to and differences between this theory and the standard three-color QCD are illustrated. An overview of chiral perturbation theory (ChPT) for two-color QCD is presented and used to discuss the phase diagram at finite density. In the third section the connection between small eigenvalues of the Dirac operator and spontaneous chiral symmetry breaking is delineated by reviewing the Banks-Casher relation, the Smilga-Stern relation and the Leutwyler-Smilga spectral sum rules. Chiral random matrix theory (ChRMT) is then introduced, and the universal correspondence of ChRMT to the e-regime of partially quenched ChPT is explained. Finally existing approaches to applying ChRMT to QCD with nonzero chemical potential are assessed.

In recent decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only. Up to now it was thought that determinants occur for ensembles with Dyson index β = 2, whereas Pfaffians only for ensembles with β = 1, 4. We derive a non-trivial Pfaffian determinant for β = 2 random matrix ensembles which is similar to the one for β = 1, 4. Thus, it unveils a hidden universality of this structure. We also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew-orthogonal polynomials corresponding to the Pfaffian. As a particular example, we consider the chiral unitary ensembles in great detail.

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to infinity simultaneously with the beta (inverse temperature) parameter going to zero. Our approach is based on the analysis of the (symmetric) Dunkl transform in this regime. As an application we obtain the LLN for the sums of random matrices as the inverse temperature goes to 0. This results in a one-parameter family of binary operations which interpolates between classical and free convolutions of the probability measures. We also introduce and study a family of deformed cumulants, which linearize this operation.

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel–Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.

Chiral random matrix theory and effective theories of QCD

Random Matrix Theory (RMT) has elaborated successful predictions for Dirac spectra in field theoretical models. However, a generic assumption by RMT has been a non-vanishing chiral condensate Σ in the chiral limit. Here we consider the 2-flavour Schwinger model, where this assumption does not hold. We simulated this model with dynamical overlap hypercube fermions, and entered terra incognita by analysing this Dirac spectrum. The usual RMT prediction for the unfolded level spacing distribution in a unitary ensemble is confirmed to a high precision. The microscopic spectrum does not perform a Banks-Casher plateau. Instead the obvious expectation is a density of the lowest eigenvalue λ1 which increases ∝ λ 1/3 1 . That would correspond to a scale-invariant parameter ∝ λV 3/4, which is, however, incompatible with our data. Instead we observe to high precision a scale-invariant parameter z ∝ λV 5/8. This surprising result implies a microscopic spectral density ∝ λ 3/5 1 , which still remains to be understood in the light of RMT.

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For a class of radially symmetric potentials with soft edges, we derive the asymptotic expansions of the log-partition functions up to and including the O(1)-terms as the number N of particles increases. Notably, our findings stress that the formulas of the $$O(\log N)$$ - and O(1)-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to an additive constant depending on N but not on the background potential. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the O(N)-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.

There is considerable interest in understanding the relation between random-matrix ensembles and quantum chaotic systems in the context of the universality of energy-level correlations. In this connection, while Gaussian ensembles of random matrices have been studied extensively, not much is known about ensembles with non-Gaussian weight functions. Dyson has shown that the n-level correlation functions can be expressed in terms of a kernel function involving orthogonal and skew-orthogonal polynomials--orthogonal for matrix ensembles with unitary invariance and skew orthogonal for ensembles with orthogonal and symplectic invariances. We have obtained the following results. (1) Skew-orthogonal polynomials of both types are derived for the Jacobi class of weight functions including the limiting cases of associated Laguerre and Hermite (or Gaussian). (2) Matrix-integral representations are given for the general weight functions. (3) Asymptotic forms of the polynomials are obtained rigorously for the Jacobi class and in the form of an ansatz for the general case. (4) For the three types of ensembles, the (asymptotic) n-level correlation functions with appropriate scaling are shown to be universal, being independent of the weight function and location in the spectrum, and identical with the well-known Gaussian results. This provides a rigorous justification for the universality of the Gaussian ensemble results observed in quantum chaotic systems. As expected, the level density is not universal.

We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter $\ensuremath{\lambda}$ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.