Microscopic Spectrum of the Wilson Dirac Operator

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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 ... mˆ ) and diag(ˆz ... zˆ ). M≡ 1 k+l Z ≡ 1 k+l This graded partition function differs in a subtle way 1 ρν (ˆx, mˆ ;â) = Im[Gν (ˆx, mˆ ;â))], (9) from the one introduced in [12]. As discussed in [6], the 5 π 3

1.5 divergent. However, the graded partition function a=0.125, m=3, ν=2 a=0.250, m=3, ν=2 a=0.500, m=3, ν=2 ¯ν ν Z1|1( , ;ˆa) = dU Sdet(U) (11) 1 M Z Z 1 −1 1 −1 2 2 −2 e 2 Str(M[U+U ])+ 2 Str(Z[U−U ])−aˆ Str(U +U ) (x)

5 × ρ

0.5 is now convergent. Repeating the steps leading to (7) with this convergent integral, we find a resolvent for an operator that, unlike D5, is not Hermitian. We believe that the absence of solutions in the p-regime [12] for W8 < 0 -16 -12 -8 -4 0 4 8 12 16 x 0 has the same origin. The ensemble with the structure of the matrix D˜ W FIG. 2: The microscopic spectrum of D5 form ˆ = 3, ν = 2 belongs to one of the classes in the non-Hermitian clas- a . a . a . and ˆ = 0 125, ˆ = 0 250 and ˆ = 0 500 respectively. sification of [19] (the γ5-Hermiticity is there referred to as Q-symmetry). In the microscopic scaling limit, the partition function (10) has the determinantal structure is, however, uncontaminated by these ultraviolet pieces. Plots of ρν are shown in Fig. 1 for ν = 0, and in Fig. 2 for Zν (ˆm, zˆ;â) = det[Zν+i−j (ˆm, zˆ;â)] (12) 5 Nf Nf =1 i,j=1...Nf ν = 2. The eigenvalues that converge towards the end- points of the spectrum at sign(ν)m in the a 0 limit are where clearly visible. The sum over ν of the spectral→ density, π 2 for which the continuum limit has been established rig- ν dθ iθν mˆ cos(θ)+izˆ sin(θ)−2â cos(2θ) Z1 = e e . (13) orously [18], can be evaluated in a straightforward way. Z−π 2π Random Matrix Theory. An efficient alternative way to extend the above results to all spectral correlation func- This form suggests that the partition function is a τ- tions and individual eigenvalue distributions is to con- function of an integrable system of Toda type and that struct a chiral Random Matrix Theory that is equiva- an eigenvalue representation can be obtained. lent to the chiral Lagrangian in the same scaling regime. The simplest quantity to compute from (12) is the chi- This the case if the chiral Random Matrix Theory has ral condensate, Σ(ˆm) = ∂mˆ log Z. For ν = 0 there is the same global symmetries and transformation proper- a striking similarity to the chiral condensate for QCD ties as the QCD partition function with the Wilson Dirac at non-zero isospin chemical potential µ. The conden- operator. The chiral Random Matrix Theory is sate is constant for largem ˆ and drops roughly linearly to zero inside a well defined region (the Aoki phase and the pion condensed phase, respectively). This is not ac- ˜ν ˜ ˜ ˜ Nf ˜ ˜ ZN = dAdBdW det (DW +m ˜ +˜zγ˜5) P (DW ),(10) f Z cidental: The microscopic spectrum of DW at a = 0 forms a thin strip along the imaginary axis just as6 the where D˜ is of the same block form as (1) and the inte- continuum Dirac operator does at µ = 0. In both cases, W 6 gration is over the real and imaginary parts of the matrix the chiral condensate can be interpreted as the electric elements of the Hermitian n n matrix, A˜, the Hermitian field at m of point charges at the position of the eigen- (n + ν) (n + ν) matrix B˜ ×and the complex n (n + ν) values. Also the convergence requirements of the graded × × matrix W˜ . This D˜ W has ν real eigenvalues. We have partition function (5) have direct analogues at nonzero added tildes to stress that| this| is a zero-dimensional ma- chemical potential [20]. trix integral with parametersm ˜ andz ˜ instead of m and The Density of Real Modes. The analytical result for z. In the universal scaling limit there is a one-to-one cor- the quenched average spectral density of the real eigen- ν respondence between the two pairs, just as in the a = 0 values, denoted by ρWreal, also follows from the generat- case. The precise form of the distribution of the matrix ing function (5) (a derivation will be given in [21]). A elements P (D˜ ) is not important on account of univer- plot of ρν for ν = 4 versus ζˆ ζΣV is shown in W Wreal ≡ sality. The partition function (3) (with W6 = W7 = 0) is Fig. 3. The real eigenvalues, ζi, of DW repel each other recovered in the microscopic scaling limit. For a Gaus- and the ν = 4 real modes are clearly visible. We have sian distribution, this can be shown by a simple explicit checked| that| the distribution is in agreement with the calculation. It also follows that W8 > 0. This is a con- chiral Random Matrix Theory (10). This illustrates that sequence of the γ5-Hermiticity: Changing the sign of W8 in the label ν introduced in (3) corresponds to the num- ˜ ν is equivalent to a ia, violating γ5-Hermiticity of DW . ber of real eigenvalues. For large ν we find that ρWreal This suggests that→ the Hermiticity properties of the Dirac approaches a semi-circle. We suggest that analyzing just operator can restrict the coefficients of the effective La- these real eigenmodes of DW may provide a new useful grangian. In fact, with W8 < 0 the integrals in (5) are tool in lattice QCD. 4

tial to this study is an analysis of the gap of the Wilson ν=4 , a=0.2 1 Dirac operator at ﬁnite mass. Lattice QCD simulations depend crucially on control of this gap and its variation 0.8 as a function of the lattice spacing. As we have stressed, ;a) ξ

( our results are not only important for understanding the 0.6 ﬁnite lattice spacing eﬀects of Wilson fermions near the Wreal ν 0.4 chiral limit, but may have interesting applications to tail ρ states in condensed matter systems. 0.2

0 -4 -2 0 2 4 Acknowledgments: This work was supported by U.S. ξ DOE Grant No. DE-FG-88ER40388 (JV) and the Dan- ish Natural Science Research Council (KS). We thank B. FIG. 3: The quenched density of the real eigenvalues of DW . Simons, M. L¨uscher and P. de Forcrand for discussions.

Distribution of Tail States. For xˆ mˆ /aˆ2 1 and 8â2 1 the tail of the spectral density| − | inside≫ the gap ≪ follows from a saddle point analysis. Forx> ˆ 0 we find [1] I.M. Lifshitz, Sov. Phys. Usp. 7, 549 (1965). [2] A. Lamacraft and B. D. Simons, Phys. Rev. Lett. 85, 2 2 ρ5(ˆx) exp[ (ˆx mˆ ) /16â ], (14) 4783 (2000); Phys. Rev. B 64, 014514 (2001). ∼ − − [3] C.W.J. Beenakker, Lect. Notes in Phys. 667 131 (2005). and a similar result forx< ˆ 0. This result applies to the [4] E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. tail of the blue and (marginally) of the red curve in Fig. 1. A560, 306 (1993); J.J.M. Verbaarschot, Phys. Rev. Lett. 72 Reinstating physical parameters we find that the width , 2531 (1994). [5] G. Akemann, P. H. Damgaard, U. Magnea and S. Nishi- parameter, σ, in (14) is given by σ2 = 8a2W /(V Σ2) so 8 gaki, Nucl. Phys. B 487, 721 (1997). that, in the microscopic domain and sufficiently smalla ˆ, [6] P. H. Damgaard, J. C. Osborn, D. Toublan and σ scales with 1/√V . Such a scaling has been observed J. J. M. Verbaarschot, Nucl. Phys. B 547, 305 (1999). for Nf = 2 in [13, 22]. [7] J. Gasser and H. Leutwyler, Phys. Lett. B 188, 477 Tail states may also be studied in their own right and (1987); H. Leutwyler and A. V. Smilga, Phys. Rev. D for applications in condensed matter physics. In particu- 46, 5607 (1992). 30 lar, in the thermodynamic limit,m, ˆ x,ˆ aˆ2 1, the aver- [8] S. Aoki, Phys. Rev. D , 2653 (1984). ≫ [9] S. R. Sharpe and R. L. Singleton, Phys. Rev. D 58, age level density of D5 can be obtained by a saddle point 2 074501 (1998); Rupak and N. Shoresh, Phys. Rev. D analysis. For 8â /m<ˆ 1 it vanishes inside [ xˆc, xˆc] with 66 − , 054503 (2002); O. Bar, G. Rupak and N. Shoresh, 2 2 2/3 3/2 70 xˆc given byx ˆc = 8â (ˆm/8â 1] . It has exactly Phys. Rev. D , 034508 (2004); S. R. Sharpe and − 70 the same form as for superconductors with magnetic im- J. M. S. Wu, Phys. Rev. D , 094029 (2004); M. Golter- 2/3 man, S. R. Sharpe and R. L. Singleton, Phys. Rev. D 71, purities [2]. In the scaling limit where V (xc x) is kept fixed, the spectral density can be computed− inside 094503 (2005). [10] F. Farchioni et al., Eur. Phys. J. C 39, 421 (2005); the gap by a saddle point approximation of (5), and it C. Michael and C. Urbach, PoS LAT2007, 122 (2007); agrees with universal Random Matrix Theory results for P. Boucaud et al., Comput. Phys. Com. 179, 695 (2008). the so-called soft edge. Such universal behaviour has also [11] K. M. Bitar, U. M. Heller and R. Narayanan, Phys. Lett. been found in condensed matter systems [2, 3]. B 418, 167 (1998). Conclusions. Using a graded chiral Lagrangian for Wil- [12] S. R. Sharpe, Phys. Rev. D 74, 014512 (2006). son fermions at finite lattice spacings, we have obtained [13] L. Del Debbio, L. Giusti, M. Luscher, R. Petronzio and N. Tantalo, JHEP 0602, 011 (2006). an analytical form for the Wilson Dirac spectrum at fixed [14] A. Shindler, Phys. Lett. B 672, 82 (2009) O. Bar, number, ν, of real eigenvalues. These results, and their S. Necco and S. Schaefer, JHEP 0903, 006 (2009). extensions to dynamical fermions, should be useful for [15] R. Kaiser, H. Leutwyler, Eur. Phys. J. C 17, 623 (2000). lattice simulations at finite volume. We have shown how [16] S. R. Sharpe, Phys. Rev. D 79, 054503 (2009). the leading low-energy constant for Wilson fermions, W8, [17] K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. can be extracted from lattice spectra of the Wilson Dirac 90, 041601 (2003). 0903 operator in the ǫ-regime. [18] L. Giusti and M. Luscher, JHEP , 013 (2009). [19] U. Magnea, arXiv:0707.0418v2 [math-ph]. The problem can also be reformulated in terms of a [20] K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B new chiral Random Matrix Theory that describes spec- 683, 467 (2004); Nucl. Phys. B 757, 259 (2006). tral correlation functions of the Wilson Dirac operator [21] G. Akemann, P.H. Damgaard, K. Splittorff and J.J.M. in the appropriate scaling regime. These results open Verbaarschot, to appear. a new domain of Random Matrix Theory where chiral [22] L. Del Debbio, L. Giusti, M. Luscher, R. Petronzio and ensembles merge with Wigner-Dyson ensembles. Essen- N. Tantalo, JHEP 0702, 082 (2007).