Spectral Properties of the Wilson Dirac Operator and random matrix theory Mario Kieburg1,2 ,∗ Jacobus J. M. Verbaarschot1 ,† and Savvas Zafeiropoulos1,3‡ 1
3
Department of Physics and Astronomy, State University of New York at Stony Brook, NY 11794-3800, USA 2 Fakult¨ at f¨ ur Physik, Postfach 100131, 33501 Bielefeld, Germany and Laboratoire de Physique Corpusculaire, Universit´e Blaise Pascal, CNRS/IN2P3, 63177 Aubi`ere Cedex, France (Dated: November 13, 2013)
arXiv:1307.7251v2 [hep-lat] 12 Nov 2013
Random Matrix Theory has been successfully applied to lattice Quantum Chromodynamics. In particular, a great deal of progress has been made on the understanding, numerically as well as analytically, of the spectral properties of the Wilson Dirac operator. In this paper, we study the infra-red spectrum of the Wilson Dirac operator via Random Matrix Theory including the three leading order a2 correction terms that appear in the corresponding chiral Lagrangian. A derivation of the joint probability density of the eigenvalues is presented. This result is used to calculate the density of the complex eigenvalues, the density of the real eigenvalues and the distribution of the chiralities over the real eigenvalues. A detailed discussion of these quantities shows how each low energy constant affects the spectrum. Especially we consider the limit of small and large (which is almost the mean field limit) lattice spacing. Comparisons with Monte Carlo simulations of the Random Matrix Theory show a perfect agreement with the analytical predictions. Furthermore we present some quantities which can be easily used for comparison of lattice data and the analytical results.
∗ † ‡
[email protected] [email protected] [email protected]
2 Keywords: Wilson Dirac operator, epsilon regime, low energy constants, chiral symmetry, random matrix theory PACS: 12.38.Gc, 05.50.+q, 02.10.Yn, 11.15.Ha
I.
INTRODUCTION
The drastically increasing computational power as well as algorithmic improvements over the last decades provide us with deep insights in non-perturbative effects of Quantum Chromodynamics (QCD). However, the artefacts of the discretization, i.e. a finite lattice spacing, are not yet completely under control. In particular, in the past few years a large numerical [1–7] and analytical [8–14] effort was undertaken to determine the low energy constants of the terms in the chiral Lagrangian that describe the discretization errors. It is well known that new phase structures arise such as the Aoki phase [15] and the Sharpe-Singleton scenario [16]. A direct analytical understanding of lattice QCD seems to be out of reach. Fortunately, as was already realized two decades ago, the low lying spectrum of the continuum QCD Dirac operator can be described in terms of Random Matrix Theories (RMTs) [17, 18]. Recently, RMTs were formulated to describe discretization effects for staggered [19] as well as Wilson [9, 10] fermions. Although these RMTs are more complicated than the chiral Random Matrix Theory formulated in [17, 18], in the case of Wilson fermions a complete analytical solution of the RMT has been achieved [9–11, 13, 20–23]. Since the Wilson RMT shares the global symmetries of the Wilson Dirac operator it will be equivalent to the corresponding (partially quenched) chiral Lagrangian in the microscopic domain (also known as the -domain) [24–29]. Quite recently, there has been a breakthrough in deriving eigenvalue statistics of the infra-red spectrum of the Hermitian [20] as well as the non-Hermitian [21–23] Wilson Dirac operator. These results explain [13] why the Sharpe-Singleton scenario is only observed for the case of dynamical fermions [1, 5, 30–36] and not in the quenched theory [37, 38] while the Aoki phase has been seen in both cases. First comparisons of the analytical predictions with lattice data show a promising agreement [4, 6, 7]. Good fits of the low energy constants are expected for the distributions of individual eigenvalues [9, 10, 39]. Up to now, mostly the effects of W8 [10, 20, 23], and quite recently also of W6 [12, 13, 40], on the Dirac spectrum were studied in detail. In this article, we will discuss the effect of all three low energy constants. Thereby we start from the Wilson RMT for the non-Hermitian Wilson Dirac operator proposed in Refs. [9]. In Sec. II we recall this Random Matrix Theory and its properties. Furthermore we derive the joint probability density of the eigenvalues which so far was only stated without proof in Refs. [13, 22]. We also discuss the approach to the continuum limit in terms of the Dirac spectrum. In Sec. III, we derive the level densities of DW starting from the joint probability density. Note that due to its γ5 -Hermiticity DW has complex eigenvalues as well as exactly real eigenvalues. Moreover, the real modes split into those corresponding to eigenvectors with positive and negative chirality. In Sec. IV, we discuss the spectrum of the quenched non-Hermitian Wilson Dirac operator in the microscopic limit in detail. In particular the asymptotics at small and large lattice spacing is studied. The latter limit is equal to a mean field limit for some quantities which can be trivially read off. In Sec. V we summarize our results. In particular we present easily measurable quantities which can be used for fitting the three low energy constants W6/7/8 and the chiral condensate Σ. Detailed derivations are given in several appendices. The joint probability density is derived in Appendix A. Some useful integral identities are given in Appendix B and in Appendix C we perform the microscopic limit of the graded partition function that enters in the distribution of the chiralities over the real eigenvalues of DW . Finally, some asymptotic results are derived in Appendix D.
II.
WILSON RANDOM MATRIX THEORY AND ITS JOINT PROBABILITY DENSITY
In Sec. II A we introduce the Random Matrix Theory for the infra-red spectrum of the Wilson Dirac operator and recall its most important properties. Its joint probability density is given in Sec. II B, and the continuum limit is derived in Sec. II C.
A.
The random matrix ensemble
We consider the random matrix ensemble [9, 10] DW =
A W −W † B
(1)
3 distributed by the probability density 2 n [n2 +(n+ν)2 ]/2 n n(n+ν) a n+ν 2 2 − P (DW ) = exp − µr + µl 2πa2 2π 2 n i h n (2) × exp − 2 (tr A2 + tr B 2 ) − ntr W W † + µr tr A + µl tr B . 2a The Hermitian matrices A and B break chiral symmetry and their dimensions are n × n and (n + ν) × (n + ν), respectively, where ν is the index of the Dirac operator. Both µr and µl are one dimensional real variables. The chiral RMT describing continuum QCD [17] is given by the ensemble (1) with A and B replaced by zero. The Nf flavor RMT partition function is defined by Z ν ZN (m) = D[DW ]P (DW )detNf (DW + m). (3) f Without loss of generality we can assume ν ≥ 0 since the results are symmetric under ν → −ν together with µr ↔ µl . The Gaussian integrals over the two variables µr and µl yield the two low energy constants W6 and W7 [9, 10]. The reason is that the integrated probability density 2 Z ∞ a (µr + µl )2 a2 (µr − µl )2 a2 dµr dµl √ P (DW , W6/7 6= 0) = P (DW ) exp − − 16V |W6 | 16V |W7 | 8πV W6 W7 −∞ generates the terms (tr A + tr B)2 and (tr A − tr B)2 which correspond to the squares of traces in the chiral Lagrangian [24–27]. In the microscopic domain the corresponding partition function for Nf fermionic flavors is then given by Z ΣV ν −1 2 2 −1 ZNf (m) dµ(U ) exp e = tr m(U e +U )−e a V W6 tr (U + U ) 2 U (Nf )
2 × exp −e a V W7 tr 2 (U − U −1 ) − e a2 V W8 tr (U 2 + U −2 ) detν U
(4)
with the physical quark masses m e = diag (m e 1, . . . , m e Nf ), the space-time volume V , the physical lattice spacing e a and the chiral condensate Σ. The low energy constant W8 is generated by the term tr A2 + tr B 2 in Eq. (2) and is a priori positive. We include the lattice spacing a in the standard deviation of A and B, cf. Eq. (2), out of convenience for deriving the joint probability density. We employ the sign convention of Refs. [9, 10] for the low energy constants. The microscopic limit (n → ∞) is performed in Sec. III. In this limit the rescaled lattice spacing b a28 = na2 /2 = b7 = a2 (µr − µl ), and the rescaled eigenvalues Zb = 2nZ = e a2 V W8 , the rescaled parameters m b 6 = a2 (µr + µl ) and λ b7 are distributed with diag (2nz1 , . . . , 2nz2n+ν ) of DW are kept fixed for n → ∞. The mass m b 6 and axial mass λ 2 2 2 2 respect to Gaussians with variance 8b a6 = −8e a V W6 and 8b a7 = −8e a V W7 , respectively. Note the minus sign in front of W6/7 . As was shown in Ref. [13] the opposite sign is inconsistent with the symmetries of the Wilson Dirac operator. The notation is slightly different from what is used in the literature to get rid of the imaginary unit in b a6 and b a7 . The joint probability density p(Z) of the eigenvalues Z = diag (z1 , . . . , z2n+ν ) of DW can be defined by Z Z I[f ] = f (DW )P (DW )d[DW ] = f (Z)p(Z)d[Z], (5) C(2n+ν)×(2n+ν)
C(2n+ν)
where f is an arbitrary U (n, n + ν) invariant function. The random matrix DW is γ5 = diag (11n , −11n+ν ) Hermitian, i.e. † DW = γ5 D W γ5 .
(6)
Hence, the eigenvalues z come in complex conjugate pairs or are exactly real. The matrix DW has ν generic real modes and 2(n − l) additional real eigenvalues (0 ≤ l ≤ n). The index l decreases by one when a complex conjugate pair enters the real axis. B.
The joint probability density of DW
Let Dl be DW if it can be quasi-diagonalized by a non-compact unitary rotation U ∈ U (n, n + ν), i.e. U γ5 U † = γ5 , to x1 0 0 x2 =U 0 −y2 0 0
Dl = U Zl U −1
0 y2 x2 0
0 0 −1 U , 0 x3
(7)
4 (1)
(1)
(2)
(2)
(2)
(2)
where the real diagonal matrices x1 = diag (x1 , . . . , xn−l ), x2 = diag (x1 , . . . , xl ), y2 = diag (y1 , . . . , yl ) and (3)
(3)
x3 = diag (x1 , . . . , xn+ν−l ) have the dimension n − l, l, l and n + ν − l, respectively. The matrices x1 and x3 comprise all real eigenvalues of Dl corresponding to the right-handed and left-handed modes, respectively. We refer to an eigenvector ψ of DW as right-handed if the chirality is positive definite, i.e. hψ|γ5 |ψi > 0,
(8)
and as left-handed if the chirality is negative definite. The eigenvectors corresponding to complex eigenvalues have vanishing chirality. The complex conjugate pairs are (z2 = x2 + ıy2 , z2∗ = x2 − ıy2 ). Note that it is not possible to diagonalize DW with a U(n, n + ν) transformation with complex conjugate eigenvalues. Moreover we emphasize that almost all γ5 -Hermitian matrices can be brought to the form (7) excluding a set of measure zero. The quasi-diagonalization Dl = U Zl U −1 determines U up to a U 2n+ν−l (1) × O l (1, 1) transformation while the set of eigenvalues Zl can be permuted in l!(n − l)!(n + ν − l)!2l different ways. The factor 2l is due to the complex conjugation of each single complex pair. The Jacobian of the transformation to eigenvalues and the coset Gl = U (n, n + ν)/[U 2n+ν−l (1) × O l (1, 1)] is given by |∆2n+ν (Zl )|2 ,
(9)
where the Vandermonde determinant is defined as h i Y ∆2n+ν (Z) = (zi − zj ) = (−1)n+ν(ν−1)/2 det zij−1 1≤i 0 given in Eq. (93). The case b a8 = 0 reduces ρχ (b x) to the result (59) and will not be discussed in this section. We set b a6/7 = 0 to begin with and introduce them later on. b7 = ε = 0. The best way to obtain the asymptotics for large lattice spacing is to start with Eq. (C1) with m b6 = λ The integral does not need a regularization since the b a8 -term guarantees the convergence. We also omit the sign in front of the linear trace terms in the Lagrangian because we can change U → −U . In the first step we substitute η → eıϕ η and η ∗ → eϑ η ∗ . Then the measure is dϕdϑdηdη ∗ and the parametrization of U is given by " U=
eϑ 0 0 eıϕ
#"
1 η∗ η 1
#
" , U −1 =
1 + η ∗ η −η ∗ −η 1 − η∗ η
#"
e−ϑ 0 0 e−ıϕ
# .
(D21)
There are two saddlepoints in the variables ϑ and ϕ, i.e.
e
ϑ0
ıb x2 =− 2 + 8b a8
s
1−
x b2 8b a28
2 ,
ıϕ0
e
s 2 ıb z1 zb1 =− 2 +L 1− 8b a8 8b a28
(D22)
with L = ±1. Moreover, the variables zb1 , x b2 have to be in the interval [−8b a28 , 8b a28 ] else the contributions will be exponentially suppressed. We have no second saddlepoint for the variable ϑ since the real part of the exponential has to be positive definite. Other saddlepoints which can be reached by shifting ϕ and ϑ by 2πı independently are forbidden since they are not accessible in the limit b a8 → ∞ . Notice that the saddlepoint solutions (D22) are phases, i.e. |eϑ0 | = |eıϕ0 | = 1. In the second step we expand the integration variables ϑ
e =e
ϑ0
1+ p
δϑ (8b a28 )2 − x b22
! ,
ıϕ
e
ıϕ0
=e
ıδϕ
1+ p 2 (8b a8 )2 − zb12
! .
(D23)
37 All terms in front of the exponential as well as of the Grassmann variables are replaced by the saddlepoint solutions ϑ0 and ϕ0 . The resulting Gaussian integrals over the variables δϑ and δϕ yield Z h i ı Im dµ(U )Sdet ν U exp −b a28 Str (U − U −1 )2 + Str diag (b x2 , zb1 )(U − U −1 ) 2 2 X x b2 − zb12 exp[ν(ϑ0 − ıϕ0 )] p exp − ∝ Im p 2 16b a28 (8b a8 )2 − x b22 (8b a28 )2 − zb12 L∈{±1} Z × dηdη ∗ (1 − η ∗ η)ν exp[−2b a28 (eϑ0 + e−ıϕ0 )(e−ϑ0 + eıϕ0 )η ∗ η]. (D24) After the integration over the Grassmann variables we have two terms, one is of order one, and the other one of order b a28 which exceeds the first term for b a8 1. Hence we end up with Z h i ı Im dµ(U )Sdet ν U exp −b a28 Str (U − U −1 )2 + Str diag (b x2 , zb1 )(U − U −1 ) 2 !ν p X −ıb x2 + (8b b22 a28 )2 − x 1 p p (D25) ∝ Im p 2 2 (8b 2 2 )2 − z 2−x ) b b (8b a a −ıb z + L (8b a28 )2 − zb12 1 2 1 8 8 L∈{±1} s s 2 2 2 2 2 b2 b2 zb1 x b2 − zb12 zb1 − x 1− x × exp 1 − − . + + L 8b a28 8b a28 8b a28 16b a28 Notice that both saddlepoints, L ∈ {±1}, give a contribution for independent variables zb1 and x b2 . To obtain the resolvent we differentiate this expression with respect to zb1 and put zb1 = x b1 afterwards. The first term between the large brackets and the second term for L = −1 are quadratic in zb1 − x b2 and do not contribute to the resolvent. For L = +1 we obtain Z i h Θ(8b a2 − |b x2 |) ı Im ∂zb1 |zb1 =bx2 dµ(U )Sdet ν U exp −b x2 , zb1 )(U − U −1 ) ∝ ν 2 p 8 4 a28 Str (U − U −1 )2 + Str diag (b (D26) 2 2b a8 64b a8 − x b22 This limit yields the square root singularity. The normalization of ρχ to ν yields an overall normalization constant of 2b a28 /π. The effect of b a6 is introduced by the integral Z Z m b 26 1 m b 26 Θ(8b a28 − |b x−m b 6 |) 1 √ √ p exp − ρ (b x − m b )| d m b = exp − dm b 6 (D27) χ 6 6 b a =0 6 2 2 2 2 16b a6 16b a6 π (8b 4b a6 π 4b a6 π a8 ) − (b x−m b 6 )2 R
R
1 = 4b a6 π 3/2
Zπ 0
"
4b a4 exp − 28 b a6
x b cos ϕ + 2 8b a8
2 #
In the large b a limit this evaluates to Z m b2 Θ(8b a28 − |b x−m b 6 |) Θ(8b a2 − |b x|) 1 b a 1 √ p exp − 62 dm b6 = ν p 8 4 , 2 16b a6 π (8b 4b a6 π a8 )2 − (b x−m b 6 )2 π 64b a8 − x b2
dϕ.
(D28)
R
which is exactly the same Heaviside distribution with the square root singularities in the interval [−8b a28 , 8b a28 ] of 2 2 2 a7 and sum the result over Eq. (D26). The introduction of b a7 follows from Eq. (87). We have to replace b a6 → b a6 + b a7 the index j with the prefactor exp(−8b a27 )[Ij−ν (8b a27 ) − Ij−ν (8b a27 )]. The intermediate result (D28) is independent of b and linear in the index, in the sum this index is j. The sum over j can be performed according to ∞ X
j Ij−ν (8b a27 ) − Ij+ν (8b a27 ) = ν exp(8b a27 )
j=1
resulting in the asymptotic result (93).
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