Poisson statistics in the high temperature QCD Dirac spectrum
Tamás G. Kovács, Ferenc Pittler
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7 pages, 6 figures. Talk presented at the XXIX International Symposium on Lattice Field Theory - Lattice 2011, Lake Tahoe, California, USA Subjects: High Energy Physics - Lattice (hep-lat)
We analyze the eigenvalue statistics of the staggered Dirac operator above $T_{c}$ in QCD with 2+1 flavors of dynamical quarks. We use physical quark masses in our simulations. We compare the eigenvalue statistics from several parts of the Dirac spectrum with the predictions of Random Matrix Theory for this universality class and with Poisson statistics. We show that at the low end of the spectrum the eigenmodes are localized and obey Poisson statistics. Above a boundary region the eigenmodes become delocalized and obey Random Matrix statistics. Thus the QCD Dirac spectrum with physical dynamical quarks also has the Poisson to Random Matrix transition previously seen in the quenched SU(2) theory.
Several lattice calculations which probe the chiral and topological structure of QCD are discussed. The results focus attention on the low-lying eigenmodes of the Dirac operator in typical gauge field configurations.
We discuss the weak coupling expansion of lattice QCD with the overlap Dirac operator. The Feynman rules for lattice QCD with the overlap Dirac operator are derived and the quark self-energy and vacuum polarization are studied at the one-loop level. We confirm that their divergent parts agree with those in the continuum theory.
In the presence of a non-vanishing chemical potential the eigenvalues of the Dirac operator become complex. We use a random matrix model approach to calculate analytically all correlation functions at weak and strong non-Hermiticity for three-dimensional QCD with broken flavor symmetry and four-dimensional QCD in the bulk.
Random matrix theory and chiral Lagrangians offer a convenient tool for the exact calculation of microscopic spectral correlators of the Dirac operator in a well-defined finite-volume scaling regime.
We compute fermionic observables relevant to the study of chiral symmetry in quenched QCD using the Overlap-Dirac operator for a wide range of the fermion mass. We use analytical results to disentangle the contribution from exact zero modes and simplify our numerical computations. Details concerning the numerical implementation of the Overlap-Dirac operator are presented.
We compare eigenvalue correlations of the Dirac operator with a chemical potential obtained from lattice simulations of quenched QCD with analytic predictions obtained from chiral effective theories in the zero-momentum limit. By comparing the density and two-point correlation function we show that the analytic results agree with QCD at low energies. We also examine the scale (Thouless energy) up to which the zero-momentum approximation is valid.
We outline our implementation of the adaptive thick-restart Lanczos algorithm ($a$-TRLan) for the projection of the low-lying eigenmodes of the overlap Dirac operator in lattice QCD, and compare the performances of our code and the widely used package ARPACK.
We investigate the eigenvalue spectrum of the staggered Dirac matrix in full QCD with two colors and finite chemical potential. Along the strong-coupling axis up to the temperature phase transition, the low-lying Dirac spectrum is well described by random matrix theory (RMT) and exhibits universal behavior. The situation is discussed in the chirally symmetric phase and no universality is seen for the microscopic spectral density.
Measurements of the lowest-lying eigenvalues of the staggered fermion Dirac operator are made on ensembles of equilibrium gauge field configurations in quenched SU(3) lattice gauge theory. The results are compared with exact analytical predictions in the microscopic finite-volume scaling regime.
I define lattice fermions in five Euclidean dimensions and the corresponding effective theory in four dimensions. The main properties of these theories include the suppression of high momentum modes of the lattice Dirac operator and their ability to continuously interpolate between quenched and dynamical fermions. In particular, the standard formulation of lattice QCD can be viewed as a limiting case of the theory.
The distribution of the low-lying QCD Dirac spectrum is analyzed by means of partial quenched chiral perturbation theory. We identify an energy scale below which the valence quark mass dependence of the QCD partition function is given by chiral Random Matrix Theory. This is the the domain where the low-energy QCD partition function is dominated by the fluctuations of the mass term of the Goldstone bosons. In mesoscopic physics this domain is known the ergodic domain and the corresponding energy scale is the Thouless energy.
The microscopic spectral eigenvalue correlations of QCD Dirac operators in the presence of dynamical fermions are calculated within the framework of Random Matrix Theory (RMT). Our approach treats the low--energy correlation functions of all three chiral symmetry breaking patterns (labeled by the Dyson index $β=1,2$ and 4) on the same footing, offering a unifying description of massive QCD Dirac spectra. RMT universality is explicitly proven for all three symmetry classes and the results are compared to the available lattice data for $β=4$.
We compute the ratio $Λ_L/Λ_{\bar{MS}}$ between the scale parameter $Λ_L$, associated with a lattice formulation of QCD using the overlap-Dirac operator, and $Λ_{\bar{MS}}$ of the $\bar{\rm MS}$ renormalization scheme. To this end, the necessary one-loop relation between the lattice coupling $g_0$ and the coupling renormalized in the $\bar{\rm MS}$ scheme is calculated, using the lattice background field technique.
We investigate the eigenvalue spectrum of the staggered Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis.
All microscopic correlation functions of the spectrum of the Hermitian Wilson Dirac operator with any number of flavors with equal masses are computed. In particular, we give explicit results for the spectral density in the physical case with two light quark flavors. The results include the leading effect in the discretization error and are given for fixed index of the Wilson Dirac operator. They have been obtained starting from chiral Lagrangians for the generating function of the Dirac spectrum. Microscopic correlation functions of the real eigenvalues of the Wilson Dirac operator are comput…
It is well known that the block Krylov subspace solvers work efficiently for some cases of the solution of differential equations with multiple right-hand sides. In lattice QCD calculation of physical quantities on a given configuration demands us to solve the Dirac equation with multiple sources. We show that a new block Krylov subspace algorithm recently proposed by the authors reduces the computational cost significantly without loosing numerical accuracy for the solution of the O(a)-improved Wilson-Dirac equation.
We derive an analytical gauge-invariant formula between the Polyakov loop $L_P$ and the Dirac eigenvalues $λ_n$ in QCD, i.e., $L_P \propto \sum_n λ_n^{N_t -1} \langle n|\hat U_4|n \rangle$, in ordinary periodic square lattice QCD with odd-number temporal size $N_t$. Here, $|n\rangle$ denotes the Dirac eigenstate, and $\hat U_4$ temporal link-variable operator. This formula is a Dirac spectral representation of the Polyakov loop in terms of Dirac eigenmodes $|n\rangle$. Because of the factor $λ_n^{N_t -1}$ in the Dirac spectral sum, this formula indicates negligibly small contribution of low-ly…
For theories with a sign problem there is no analog of the Banks-Casher relation. This is true in particular for QCD at nonzero quark chemical potential. However, for QCD-like theories without a sign problem the Banks-Casher relation can be extended to the case of complex Dirac eigenvalues. We derive such extensions for the zero-temperature, high-density limits of two-color QCD, QCD at nonzero isospin chemical potential, and adjoint QCD. In all three cases the density of the complex Dirac eigenvalues at the origin is proportional to the BCS gap squared.
We show how all the contributions to the determinant of the Dirac-Kogut-Susskind operator of QCD at finite temperature containing a net number of Polyakov loops become irrelevant in the infinite volume limit. We discuss also on two of the most interesting physical implications of this result: i) the restoration of the Polyakov symmetry in the full theory with dynamical fermions and ii) the total suppression of baryonic thermal fluctuations in QCD at finite temperature.
We analyze the eigenvalues and eigenvectors of the staggered Dirac operator in quenched lattice QCD in the vicinity of the deconfinement phase transition using the Lüscher-Weisz gauge action. The spectral and localization properties of the low-lying eigenmodes show characteristic differences between the Z_3 sectors above the critical temperature T_c. These findings can be interpreted in terms of calorons.
We compute by Monte Carlo methods the individual distributions of the $k$th smallest Dirac operator eigenvalues in QCD, and compare them with recent analytical predictions. We do this for both massless and massive quarks in an SU(3) gauge theory with staggered fermions. Very precise agreement is found in all cases. As a simple by-product we also extract the microscopic spectral density of the Dirac operator in SU(3) gauge theory with dynamical massive fermions for $N_f=1$ and 2, and obtain high-accuracy agreement with analytical expressions.
The spectral flow of the low-lying eigenvalues of the improved and unimproved Wilson-Dirac operator is studied on instanton-like configurations and on thermalized quenched configurations at various $β$-values and lattice sizes. We also investigate the space-time localisation and chirality of the corresponding eigenvectors.
We present a new multigrid solver that is suitable for the Dirac operator in the presence of disordered gauge fields. The key behind the success of the algorithm is an adaptive projection onto the coarse grids that preserves the near null space. The resulting algorithm has weak dependence on the gauge coupling and exhibits mild critical slowing down in the chiral limit. Results are presented for the Wilson Dirac operator of the 2d U(1) Schwinger model.
We investigate the eigenvalue spectrum of the staggered Dirac matrix in SU(3) and U(1) gauge theory as well as in full QCD with two colors and finite chemical potential. Along the strong-coupling axis up to the phase transition, the low-lying Dirac spectrum of these quantum field theories is well described by random matrix theory and exhibits universal behavior. Related results for gauge theories with minimal coupling are discussed in the chirally symmetric phase and no universality is seen for the microscopic spectral densities.
We outline the essential features of a Linux PC cluster which is now being developed at National Taiwan University, and discuss how to optimize its hardware and software for lattice QCD with overlap Dirac quarks. At present, the cluster constitutes of 30 nodes, with each node consisting of one Pentium 4 processor (1.6/2.0 GHz), one Gbyte of PC800 RDRAM, one 40/80 Gbyte hard disk, and a network card. The speed of this system is estimated to be 30 Gflops, and its price/performance ratio is better than $1.0/Mflops for 64-bit (double precision) computations in quenched lattice QCD with overlap Dir…
We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.
Random matrix theory (RMT) is a powerful statistical tool to model spectral fluctuations. In addition, RMT provides efficient means to separate different scales in spectra. Recently RMT has found application in quantum chromodynamics (QCD). In mesoscopic physics, the Thouless energy sets the universal scale for which RMT applies. We try to identify the equivalent of a Thouless energy in complete spectra of the QCD Dirac operator with staggered fermions and $SU_c(2)$ lattice gauge fields. Comparing lattice data with RMT predictions we find deviations which allow us to give an estimate for this …
We calculate the spectral function of the QCD Dirac operator using the four-dimensional effective operator constructed from the Mobius domain-wall implementation. We utilize the eigenvalue filtering technique combined with the stochastic estimate of the mode number. The spectrum in the entire eigenvalue range is obtained with a single set of measurements. Results on 2+1-flavor ensembles with Mobius domain-wall sea quarks at lattice spacing ~ 0.08 fm are shown.
We analyze the eigenvalue spectrum of the staggered Dirac matrix in two-color QCD at nonzero baryon density when the eigenvalues become complex. The quasi-zero modes and their role for chiral symmetry breaking and the deconfinement transition are examined. The bulk of the spectrum and its relation to quantum chaos is considered. A comparison with predictions from random matrix theory is presented. An analogous analysis is performed for the spectrum of the Gor'kov representation of the fermionic action.
The axial $U(1)$ symmetry in the high-temperature phase is investigated with $N_f = 2$ lattice QCD simulations. The gauge ensembles are generated with Möbius domain-wall fermions, and the overlap/domain-wall reweighting is applied. We find that the $U(1)_A$ susceptibility evaluated from the spectrum of overlap-Dirac eigenvalues is strongly suppressed in the chiral limit. We also study its volume dependence.
The eigenvalue spectrum $ρ(λ)$ of the Dirac operator is numerically calculated in lattice QCD with 2+1 flavors of dynamical domain-wall fermions. In the high-energy regime, the discretization effects become significant. We subtract them at the leading order and then take the continuum limit with lattice data at three lattice spacings. Lattice results for the exponent $\partial\lnρ/\partial\lnλ$ are matched to continuum perturbation theory, which is known up to $O(α_s^4)$, to extract the strong coupling constant $α_s$.
Efficient algorithms for the solution of partial differential equations on parallel computers are often based on domain decomposition methods. Schwarz preconditioners combined with standard Krylov space solvers are widely used in this context, and such a combination is shown here to perform very well in the case of the Wilson--Dirac equation in lattice QCD. In particular, with respect to even-odd preconditioned solvers, the communication overhead is significantly reduced, which allows the computational work to be distributed over a large number of processors with only small parallelization los…
I describe a mechanism to understand the relation between chiral-symmetry breaking and eigenmodes of the Dirac operator in lattice QCD with Kogut-Susskind sea quarks. It can be shown that if chiral symmetry is spontaneously broken, the eigenvalues $λ_i$ should behave as $z(i)/V$ for large volume $V$, where $z(i)$ is the i-th zero of the Bessel function. With neither chiral nor $λ$ extrapolation, one can precisely calculate the chiral condensate using only a small set of eigenvalues. Therefore, it is economical and free of systematic uncertainties. I present the first QCD data to support this m…
Using the eigenmodes of the overlap-Dirac operator, we study the topological structure of the QCD vacuum. We investigate the space-time profile of the low-lying eigenmodes and their contribution to the vacuum action density and chiral condensate under the existence of static color sources. We demonstrate that the low-lying Dirac eigenmode shows the flux-tube structure, which suggests the relevance to confinement. We also analyze the chiral condensate in the flux-tube. Chiral symmetry is partially restored inside the flux, and the reduction of the condensate is about 20% at the center of the tu…
We show how the integrators used for the molecular dynamics step of the Hybrid Monte Carlo algorithm can be further improved. These integrators not only approximately conserve some Hamiltonian $H$ but conserve exactly a nearby shadow Hamiltonian $\tilde{H}$. This property allows for a new tuning method of the molecular dynamics integrator and also allows for a new class of integrators (force-gradient integrators) which is expected to reduce significantly the computational cost of future large-scale gauge field ensemble generation.
Using the Dirac-mode expansion method, which keeps the gauge invariance, we analyze the Polyakov loop in terms of the Dirac modes in SU(3) quenched lattice QCD in both confined and deconfined phases. First, to investigate the direct correspondence between confinement and chiral symmetry breaking, we remove low-lying Dirac-modes from the confined vacuum generated by lattice QCD. In this system without low-lying Dirac modes, while the chiral condensate $\langle \bar{q} q\rangle$ is extremely reduced, we find that the Polyakov loop is almost zero and $Z_3$-center symmetry is unbroken, which indic…
Approximating the sum over all gauge field configurations in the QCD partition function by a liquid of instantons, we calculate the spectrum of the Dirac operator for two and three colors and for 0, 1 and 2 flavors. We find a remarkable difference in the spectrum near zero virtuality between 2 and 3 colors, which can be explained in terms of chiral random matrix theory. For two colors the Dirac operator is real, and the appropriate random matrix ensemble has real matrix elements. For three colors the Dirac operator is complex, and the spectrum can be described by a random matrix ensemble with …
We report on an ongoing project to parametrize the Fixed-Point Dirac operator for massless quarks, using a very general construction which has arbitrarily many fermion offsets and gauge paths, the complete Clifford algebra and satisfies all required symmetries. Optimizing a specific construction with hypercubic fermion offsets, we present some preliminary results.
We compute the Dirac spectral density of QCD in a wide range of eigenvalues by using a stochastic method. We use 2+1 flavor lattice ensembles generated with Mobius domain-wall fermion at three lattice spacings ($a=0.083, 0.055, 0.044$ fm) to estimate the continuum limit. The discretization effect can be minimized by a generalization of the valence domain-wall fermion. The spectral density at relatively high eigenvalues can be matched with perturbation theory. We compare the lattice results with the perturbative expansion available to $O(α_s^4)$.
Recently various new concepts for the construction of Dirac operators in lattice Quantum Chromodynamics (QCD) have been introduced. These operators satisfy the so-called Ginsparg-Wilson condition (GWC), thus obeying the Atiyah-Singer index theorem and violating chiral symmetry only in a modest and local form. Here we present studies in 4-d for SU(3) gauge configurations with non-trivial topological content. We study the flow of eigenvalues and we compare the numerical stability and efficiency of a recently suggested chirally improved operator with that of others in this respect.
Recent studies on lattice QCD have shown the emergence of large symmetries at high temperature. This includes not only the restoration $SU(n_F)_L \times SU(n_F)_R$, but also the effective emergence of an unexpected symmetry group, namely $SU(2)_{CS}$, which contains $U(1)_A$ as subgroup. At the same time, at high $T$, a gap in Dirac spectrum appears. As it is argued in several works of \textit{L. Glozman et al.}, there should be a connection between a gap in the Dirac spectrum and the presence of $SU(2)_{CS}$.In this paper, we analyze whether the quark propagator can be invariant under $SU(n_F…
We construct a parametrization of a Fixed Point (FP) Dirac operator and apply it in quenched lattice QCD. The symmetry requirements for a general lattice Dirac operator are discussed and an efficient way to make a practical construction of general lattice Dirac operators is provided. We use such a lattice Dirac operator to approximately solve the Renormalization Group equation that defines the FP Dirac operator in an iterative procedure. We discuss the properties of this parametrization and show that its breaking of chiral symmetry and its scaling violations in hadron specroscopy are much redu…
We discuss the eigenvalue distribution of the overlap Dirac operator in quenched QCD on lattices of size 8^{4}, 10^{4} and 12^{4} at β= 5.85 and β= 6. We distinguish the topological sectors and study the distributions of the leading non-zero eigenvalues, which are stereographically mapped onto the imaginary axis. Thus they can be compared to the predictions of random matrix theory applied to the ε-expansion of chiral perturbation theory. We find a satisfactory agreement, if the physical volume exceeds about (1.2 fm)^{4}. For the unfolded level spacing distribution we find an accurate agreement…
Dirac spectrum representations of the Polyakov loop fluctuations are derived on the temporally odd-number lattice, where the temporal length is odd with the periodic boundary condition. We investigate the Polyakov loop fluctuations based on these analytical relations. It is semianalytically and numerically found that the low-lying Dirac eigenmodes have little contribution to the Polyakov loop fluctuations, which are sensitive probe for the quark deconfinement. Our results suggest no direct one-to-one corresponding between quark confinement and chiral symmetry breaking in QCD.
We compute the low-lying spectrum of the staggered Dirac operator above and below the finite temperature phase transition in both quenched QCD and in dynamical four flavor QCD. In both cases we find, in the high temperature phase, a density with close to square root behavior, $ρ(λ) \sim (λ-λ_0)^{1/2}$. In the quenched simulations we find, in addition, a volume independent tail of small eigenvalues extending down to zero. In the dynamical simulations we also find a tail, decreasing with decreasing mass, at the small end of the spectrum. However, the tail falls off quite quickly and does not see…
There exist two major problems in application of the conventional block BiCGSTAB method to the O(a)-improved Wilson-Dirac equation with multiple right-hand-sides: One is the deviation between the true and the recursive residuals. The other is the convergence failure observed at smaller quark masses for enlarged number of the right-hand-sides. The block BiCGGR algorithm which was recently proposed by the authors succeeds in solving the former problem. In this article we show that a preconditioning technique allows us to improve the convergence behavior for increasing number of the right-hand-si…
Using lattice QCD simulations with $N_f = 2$ dynamical fermions, we study the axial $U(1)$ symmetry, topological charge, and Dirac eigenvalue spectra in the high-temperature phase in which the chiral symmetry is restored. Our gauge ensembles are generated with Möbius domain-wall fermions, but the measurements such as susceptibilities are reweighted to those for the overlap fermions by using overlap/domain-wall reweighting technique. We find that the $U(1)_A$ and topological susceptibilities are strongly suppressed in the small quark mass region, which is related to the reduction of chiral-zero…
The structure of QCD vacuum can be studied from first principles using lattice-regularized theory. This line of research entered a qualitatively new phase recently, wherein the space-time structure (at least for some quantities) can be directly observed in configurations dominating the QCD path integral, i.e. without any subjective processing of typical configurations. This approach to QCD vacuum structure does not rely on any proposed picture of QCD vacuum but rather attempts to characterize this structure in a model-independent manner, so that a coherent physical picture of the vacuum can em…
We compute the ratio between the scale $Λ_L$ associated with a lattice formulation of QCD using the overlap-Dirac operator, and $Λ_{MS-bar}$. To this end, the one-loop relation between the lattice coupling $g_0$ and the coupling renormalized in the MS-bar scheme is calculated, using the lattice background field technique. We also compute the one-loop renormalization $Z_Γ$ of the two-quark operators $\barψ Γψ$, where $Γ$ denotes a generic Dirac matrix. Furthermore, we study the renormalization of quark bilinears which are more extended and have better chiral properties. Finally, we present impr…
The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with controlled numerical errors by a conjugate gradient (CG) method. This CG algorithm is accelerated by alternating it with exact diagonalisations in the subspace spanned by the numerically computed eigenvectors. We study this combined algorithm in case of the Dirac operator with (dynamical) Wilson fermions in four-dimensional $\SUtwo$ gauge fields. The algorithm is numerically very stable and can be parallelized in an efficient way. On lattices of sizes $4^4-16^4$ an acceleration of the pure CG method by a factor of~$4…
At low temperature in the epsilon regime of QCD the low-end of the Dirac spectrum is described by random matrix theory. In contrast, there has been no similarly well established staistical description in the high temperature, chirally symmetric phase. Using lattice simulations we show that at high temperature a band of extremely localized eigenmodes appear at the low-end of the Dirac spectrum. The corresponding eigenvalues are statistically independent and obey a generalized Poisson distribution. Higher up in the spectrum the Poisson distribution rapidly crosses over into the bulk distribution…
We show that the spectrum of the Dirac operator in complex Langevin simulations of QCD at non-zero chemical potential must behave in a way which is radically different from the one in simulations with ordinary non-complexified gauge fields: At low temperatures the small eigenvalues of the Dirac operator must be inside the quark mass for chemical potentials as large as a third of the nucleon mass. In particular, in the chiral limit the Dirac eigenvalues of complex Langevin simulations must accumulate at the origin.
The relation between the baryon number in QCD at nonzero chemical potential and the spectral density of the baryon number Dirac operator, $γ_0(D+m)$, is examined. We show that extreme oscillations of the spectral density, caused by the QCD sign problem, are essential for the formation of the average baryon number when $μ>m_π/2$. We compute the oscillating region of the spectral density using chiral perturbation theory. The extreme oscillations have a microscopic period and are resolved using random matrix theory.
We analyze the relation between the Dirac spectrum and the gauge field in SU(3) lattice QCD. We focus on how components of the gauge field are related to the Dirac spectrum. First, we consider momentum components of the gauge field. It turns out that the broad region of momentum components is relevant for the low-lying Dirac spectrum and zero modes, i.e., topological charges. The connection with chiral random matrix theory is also discussed. Second, we consider an SU(2) subgroup component of the SU(3) gauge field. The SU(2) subgroup component behaves like the SU(2) gauge field in the low-lying…
Recently, a non-Hermitian chiral random matrix model was proposed to describe the eigenvalues of the QCD Dirac operator at nonzero chemical potential. This matrix model can be constructed from QCD by mapping it to an equivalent matrix model which has the same symmetries as QCD with chemical potential. Its microscopic spectral correlations are conjectured to be identical to those of the QCD Dirac operator. We investigate this conjecture by comparing large ensembles of Dirac eigenvalues in quenched SU(3) lattice QCD at nonzero chemical potential to the analytical predictions of the matrix model.…
We analyze the smallest Dirac eigenvalues by formulating an effective theory for the QCD Dirac spectrum. We find that in a domain where the kinetic term of the effective theory can be ignored, the Dirac eigenvalues are distributed according to a Random Matrix Theory with the global symmetries of the QCD partition function. The kinetic term provides information on the slope of the average spectral density of the Dirac operator. In the second half of this lecture we interpret quenched QCD Dirac spectra at nonzero chemical potential (with eigenvalues scattered in the complex plane) in terms of an…
Chiral symmetry breaking in Quantum Chromodynamics is associated with the low lying spectral modes of the Dirac operator according to the Banks-Casher relation. Here we study how removal of a variable number of low lying modes from the valence quark sector affects the masses of the ground states and first excited states of baryons and mesons in two flavor lattice QCD.
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 …
The zero momentum sectors in effective theories of QCD coupled to pseudoreal (two colors) and real (adjoint) quarks have alternative descriptions in terms of chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute correlation functions of Dirac operator eigenvalues within a sector with an arbitrary topological charge in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of Pfaffians, interpolate between known results for the chiral and quenched limits as quark masses…
I propose a simple model of the distribution of the small eigenvalues of the QCD Dirac operator well above the finite temperature phase transition where chiral symmetry is restored and the spectral density at zero vanishes. Assuming the absence of correlations between different regions of the low lying spectrum I derive analytic formulas for the distribution of the first two eigenvalues. I find good agreement with data obtained using the overlap Dirac operator in quenched SU(2) lattice simulations. This suggests that if chiral symmetry is restored spectral correlations are not important and al…
