Microscopic Spectrum of the Wilson Dirac Operator

Microscopic Spectrum of the Wilson Dirac Operator P.H. Damgaard,1 K. Splittorff,2 and J.J.M. Verbaarschot3 1Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark 2Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark 3Department of Physics and Astronomy, SUNY, Stony Brook, New York 11794, USA (Dated: October 23, 2018) We calculate the leading contribution to the spectral density of the Wilson Dirac operator using chiral perturbation theory where volume and lattice spacing corrections are given by universal scaling functions. We find analytical expressions for the spectral density on the scale of the average level spacing, and introduce a chiral Random Matrix Theory that reproduces these results. Our work opens up a novel approach to the infinite volume limit of lattice gauge theory at finite lattice spacing and new ways to extract coefficients of Wilson chiral perturbation theory. Introduction. Spectral gaps and their suppression by dis- Below we make use only of its block structure given by order are essential for a variety of physical phenomena. aA W States that intrude into the band gap, so called Lifshitz D = (1) W W † aB tail states, affect the conductivity of semiconductors [1], − they may lead to gapless superconductivity in supercon- with A† = A and B† = B, whereas W does not have ductors with magnetic impurities [2], and they may show additional symmetry properties. The Wilson Dirac oper- universal fluctuations given by Random Matrix Theory ator is anti-Hermitian in the continuum limit a 0 and → (see [3]). Here we analyze the spectrum of the Wil- the corresponding eigenvalues of DW are complex away son Dirac operator of lattice Quantum Chromodynamics from the continuum limit. (QCD). In the continuum limit, the Hermitian Wilson Although non-Hermitian, the Wilson Dirac operator Dirac operator has a gap equal to twice the quark mass. satisfies γ5-Hermiticity At finite lattice spacing, eigenvalues of tail states intrude † into the gap. When eigenvalues approach the center of D = γ5Dγ5 . (2) the gap, it becomes increasingly difficult to invert the Instead of the Wilson Dirac operator itself, it is therefore Wilson Dirac operator. As a consequence, such tail states often more convenient to work with the Hermitian Dirac can potentially obstruct lattice simulations. It is there- operator D5 = γ5D. At zero lattice spacing, the spec- fore of importance to have an analytical understanding trum of D5 has a gap around the origin of width 2m. At of the properties of these states. non-zero lattice spacing a, states intrude inside the gap and for sufficiently large lattice spacing the gap closes. Then one enters what is known as the Aoki phase [8]. Our results rely on two approaches, chiral Random It is reminiscent of the Gorkov Hamiltonian for super- Matrix Theory and chiral Lagrangians for the pseudo– conductors, where magnetic impurities (see [2, 3]) play Nambu-Goldstone sector of QCD. The relation between the role of the diagonal blocks in Eq. (1) and the Aoki chiral Random Matrix Theory and the Dirac operator phase corresponds to gapless superconductivity. A first in theories with spontaneously broken chiral symmetries order scenario where the condensate jumps as a function [4] has led to a new understanding of the chiral limit arXiv:1001.2937v3 [hep-th] 20 Sep 2010 of m has been suggested [9] and support for this has been of strongly coupled gauge theories. The Random Ma- found on the lattice [10]. trix Theory results are universal [5] and are equivalent The spectral density ρ (x) of D , evaluated at x = 0, is [6] to what is obtained from a chiral Lagrangian in the 5 5 an order parameter [11] for the onset of the Aoki phase. microscopic domain or ǫ-regime [7]. This gives a finite- Discretization effects in the spectrum of the Wilson Dirac volume scaling theory for spectral correlation functions operator were analyzed by means of chiral perturbation as well as individual eigenvalue distributions of the con- theory in [12]. What is new here is that we obtain an ex- tinuum Dirac operator at fixed topological charge ν. In act analytical description in the microscopic scaling limit lattice QCD it has become standard to utilize these re- and show in detail the transition to the Aoki phase. This sults to obtain physical observables from simulations at opens a novel analytical approach to the infinite-volume finite four-volume V . There has for long been a desire limit of lattice gauge theory at finite lattice spacing and to obtain analogous results for Wilson fermions at finite offers new ways to measure the leading coefficients of Wil- lattice spacing a. Here we present a solution to this prob- son chiral perturbation theory. Understanding the distri- lem. butions of the low-lying eigenvalues of the Wilson Dirac operator is also crucial for establishing a stable domain We denote the Wilson Dirac operator by D = DW +m. for numerical simulations [13]. 2 Chiral Lagrangian. The leading-order terms of the chiral 1.5 Lagrangian for Wilson fermions have been listed in [9]. It is a double expansion: the continuum ordering for chiral perturbation theory and an expansion in the lattice 1 spacing a. The corresponding chiral Lagrangian coin- (x) ν cides with the continuum Lagrangian with shifted mass 5 ρ 2 plus terms starting at order a . It is convenient to in- 0.5 troduce a source for ψγ¯ 5ψ, which we denote by z. Here, we shall focus on the microscopic domain where mV , zV and a2V are kept fixed in the infinite-volume limit. Dif- 0 0 2 4 6 8 10 ferent counting rules are possible [14], but the present one x is most useful for elucidating the effects of finite lattice spacing on the low-lying Dirac eigenvalues. The leading FIG. 1: The microscopic spectrum of D5 form ˆ = 3, ν = 0 contribution to the finite-volume QCD partition function anda ˆ = 0, 0.03, and 0.250. The ν = 0 spectrum is reflection then reduces to a unitary matrix integral, which, up to symmetric aboutx ˆ = 0. a few constants, is determined by symmetry arguments. ν We decompose this partition function as ZN = Z f ν Nf integration manifold for non-perturbative computations with P is non-compact for the bosonic sector. While the action ν ν S[U] (5) and the action introduced in [12] break the flavor ZNf (m,z; a)= dU det U e (3) ZU(Nf ) symmetries in exactly the same way, only the former one is consistent with the convergence requirements of the where the action S[U] for degenerate quark masses is graded integral for W8 > 0. For perturbative calcula- m z tions the convergence requirements are immaterial. Here S = ΣV Tr(U + U †)+ ΣV Tr(U U †) (4) 2 2 − and below we focus mainly on the quenched case, corre- 2 † 2 2 † 2 sponding to integration manifold Gl(1 1)/U(1). The gen- a V W6[Tr U + U ] a V W7[Tr U U ] − − − eralization to an arbitrary number of| flavors is straight- 2 2 †2 a V W8Tr(U + U ). forward, and we expect that the underlying integrability − structure will lead to a full analytical solution just as in Below we will demonstrate that in the microscopic do- the a = 0 case [17]. main Zν corresponds to ensembles of gauge field config- Nf The Microscopic Spectrum of D5. For a = 0, the micro- 2 urations with ν real modes of DW . The a -terms are de- scopic spectral density of D5 follows from the expression termined by invariance arguments [9], and Σ, W6, W7 and for the microscopic spectral density of D through 2 W8 are the low-energy constants of (a ) Wilson ChPT. The two terms corresponding to W Oand W are expected xˆ 6 7 ρν (ˆx> m,ˆ mˆ ;â =0)= ρν ( xˆ2 mˆ 2) . (6) 5 2 2 to be suppressed in the large-Nc limit [15], and we shall √xˆ mˆ p − for simplicity ignore them here. The potential impact − To obtain the spectral density of D for a = 0, we eval- of these terms [16] can be studied at the expense of a 5 6 slightly more cumbersome analysis. The leading finite- uate the resolvent ν volume partition function ZN then only depends on the f ν d ν ′ microscopic scaling variablesm ˆ = mΣV ,z ˆ = zΣV , and G (ˆz, mˆ ;â) lim Z1|1(ˆm, m,ˆ z,ˆ zˆ ;â) (7) ≡ zˆ′→zˆ dzˆ aˆ = a√W V which will be kept fixed for V . The 8 → ∞ sign of W8 will be discussed below. and find The Generating Function. A generating function for ∞ π dθ i spectral correlation functions is given by an average of Gν (ˆz, mˆ ;â) = ds cos(θ)eSf +Sb e(iθ−s)ν ratios of determinants. Because of the inverse determi- Z−∞ Z−π 2π 2 nants, it has an extended graded flavor symmetry. The ( mˆ sin(θ)+ imˆ sinh(s)+ izˆcos(θ)+ izˆcosh(s) (8) × − graded generating function for spectral correlations of D5 +4â2[cos(2θ) + cosh(2s) + (eiθ+s + e−iθ−s)]+1 . is 2 Here Sf = mˆ sin(θ)+ izˆcos(θ)+2â cos(2θ) and Sb = ν ν − 2 Zk|l( , ;â) = dU Sdet(U) (5) imˆ sinh(s) izˆcosh(s) 2â cosh(2s). As is well known, M Z Z − − − 1 −1 1 −1 2 2 −2 the resolvent is defined only up to ultraviolet subtrac- ei 2 Str(M[U−U ])+i 2 Str(Z[U+U ])+â Str(U +U ) × tions in the underlying theory. The microscopic quenched spectral density, where diag(m ˆ ..

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