We present a quantitative analysis of the microscopic Dirac spectrum which is complex in the presence of a non-vanishing quark chemical potential. Data from quenched SU(3) lattice simulations for different volumes V and small values of the chemical potential are compared to analytical predictions from matrix models. We confirm the existence of two distinct limits for weakly and strongly nonhermitian Dirac operators. Good agreement is found in both limits, confirming the different scaling of chemical potential and eigenvalues with the volume

Akemann

Allton

Ambjørn

Berbenni-Bitsch

D'Elia

de Forcrand

Fodor

Fyodorov

Gernot Akemann

Shuryak

Stephanov

Tile Wettig

Verbaarschot

English

arXiv

Akemann, G

Wettig, T

Brunel University Research Archive

arXiv:hep-lat/0309037 v1   10 Sep 20031Comparing matrix models and QCD lattice data with chemical potentialGernot Akemanna∗ and Tilo Wettigb†aService de Physique The´orique, CEA/DSM/SPhT Saclay,Unite´ associe´e CNRS/SPM/URA 2306, F-91191 Gif-sur-Yvette Cedex, FrancebDepartment of Physics, Yale University, New Haven, CT 06520-8120, USA andRIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USAWe present a quantitative analysis of the microscopic Dirac spectrum which is complex in the presence of a non-vanishing quark chemical potential. Data from quenched SU(3) lattice simulations for different volumes V andsmall values of the chemical potential µ are compared to analytical predictions from matrix models. We confirmthe existence of two distinct limits for weakly and strongly nonhermitian Dirac operators. Good agreement isfound in both limits, confirming the different scaling of chemical potential and eigenvalues with the volume.1. IntroductionLattice QCD at nonzero density remains a diffi-cult topic. Several methods to solve the problemsrelated to the complex phase of the Dirac oper-ator determinant have recently been suggested:multiparameter reweighting, Taylor expansion,and analytic continuation from imaginary µ [1].So far, all three methods work only in the vicinityof the phase transition close to the temperature-axis. Here, we instead explore the µ-axis at zerotemperature and small µ. In this region matrixmodels analytically predict the complex micro-scopic Dirac spectrum [2]. We test these predic-tions for pure SU(3) gauge theory [3], where µ 6= 0is easily implemented. Although the quenchedapproximation is problematic [4], our test is animportant check, as the matrix model remainsvalid and predictive forNf > 0 flavors. For exam-ple, the 1-flavor matrix model has been used totest a new factorization algorithm for the com-plex action problem [5]. In fact, at µ = 0 thesituation is analogous: while most tests of matrixmodel predictions have been performed only forpure gauge theories, dynamical fermions are alsowell described [6]. For a review of matrix modelsin QCD we refer to Ref. [7].∗Supported by a Heisenberg fellowship of the DFG.†Supported in part by DOE grant DE-FG02-91ER40608.2. Matrix model predictionsIn Ref. [2] a matrix model has been formulatedin terms of N complex eigenvalues and solved inthe limit N ∝ V → ∞. It has the same globalsymmetries as QCD at µ 6= 0 due to an equiva-lence [8] to the matrix model of Ref. [4] at smallµ in the phase with broken chiral symmetry. Wegive here only the quenched results for the mi-croscopic density and refer to Ref. [2] for moredetails and higher-order correlation functions.In Ref. [2] two different large-N limits werefound, those of weak and of strong nonhermiticity.In QCD at µ = 0, the lowest Dirac eigenvaluesscale with 1/V to build up a finite condensateaccording to the Banks-Casher relation. In theweak-nonhermiticity limit, first introduced in [9],this scaling remains unchanged. The support ofthe density remains quasi one-dimensional as wesend µ → 0, keeping limN→∞ 2Nµ2 ≡ α2 fixed.In the strong-nonhermiticity limit, the eigenval-ues fill a two-dimensional surface, and thus thescaling is modified to 1/√V . We find for thequenched microscopic density at weak nonher-miticity in the sector of topological charge ν [2]ρweak(ξ) =√piα2erf(α)|ξ| exp[− (ℑmξ)2α2](1)×∫ 10dt e−α2tJ|ν|(√tξ)J|ν|(√tξ∗) .2Here, we have rescaled the complex eigenvaluesaccording to ξ ∼ V z. At strong nonhermiticity,µ is kept fixed as V →∞, and one obtains [2]ρstrong(ξ) =√piµ2|ξ| exp[−|ξ|22µ2]I|ν|( |ξ|22µ2), (2)rescaling the eigenvalues according to ξ ∼ √V z.The microscopic density Eq. (2) is rotationallyinvariant in contrast to the macroscopic support.3. Lattice Data at µ 6= 0Our lattice calculations were done using thestaggered Dirac operator with chemical potentialµ 6= 0 for a pure SU(3) gauge theory, correspond-ing to Nf = 0. There are two reasons for thischoice. First, we need high statistics (≈ 20, 000configurations1) to test the matrix model predic-tions of Eqs. (1,2). Ginsparg-Wilson type opera-tors would be too expensive here. Second, Wil-son fermions break chiral symmetry explicitly andhave complex eigenvalues already at µ = 0.In the simulations, we have chosen β = 6/g2 =5.0 in the strong-coupling regime for the follow-ing reason. At zero temperature and µ = 0 thematrix model [10] is equivalent to a low-energyeffective theory of QCD (the effective chiral La-grangian) in the so-called ε-regime, where thezero-momentum modes of the pseudo-Goldstonebosons dominate (see e.g. [7]). When the nonzero-momentum modes start contributing, this equiv-alence breaks down. The corresponding scale, theso-called Thouless energy, is a function of both βand V . As β is increased, fewer eigenvalues aredescribed by the matrix model, an effect that canbe compensated by increasing V . For our smallvolumes V = 64, 84, 104, however, we are limitedto relatively small values of β (assuming a similarbehavior of the Thouless energy for small µ 6= 0).At strong coupling and thus away from the con-tinuum limit, staggered fermions have the disad-vantage of shifting the (topological) Dirac zeromodes and mixing them with the nonzero modes.We have accounted for this by setting ν = 0 inEqs. (1) and (2) above.1The V =104 data in Figs. 1,2 contain only 4,000 config-urations, therefore the statistical errors are larger.Figure 1. Densities of small Dirac eigenvalues cutalong the real axis (left) and parallel to the imag-inary axis at the first maximum (right). Data(histograms) for V = 64 at µ = 0.006 (top),V = 84 at µ = 0.003375 (middle), and V = 104at µ = 0.00216 (bottom) vs. Eq. (1) for weaknonhermiticity.The three different lattice volumes are cho-sen in order to test the different scaling behaviorof eigenvalues and chemical potential for weaklyand strongly nonhermitian lattice Dirac opera-tors. For more simulation details we refer to [3].Let us first discuss the data at weak nonher-miticity. In Eq. (1) the average level spacing be-tween consecutive eigenvalues is pi. In order tocompare with lattice data, we first determine themean level spacing d from the data averaged overmany configurations and then rescale the latticeeigenvalues z by ξ = piz/d. At the same time thespacing d ∝ 1/V provides us with the weak non-hermiticity parameter α = µ√2/d ≈ 0.19 to be3Figure 2. Densities of small Dirac eigenvalues cutalong the real (left) and imaginary (right) axes.Data (histograms) for V = 64 (top), V = 84 (mid-dle), and V = 104 (bottom), all at µ = 0.2, vs.Eq. (2) for strong nonhermiticity.used in Eq. (1). After normalizing the histogramsto unity we thus obtain the parameter-free com-parison shown in Fig. 1. The three different val-ues of µ are such that the product V µ2 is con-stant. All data are well described by the samevalue2 of α determined from the volume V = 64.Apart from the scaling V µ2 ∝ α2 we have thusalso confirmed the scaling of the complex Diraceigenvalues with the volume.At strong nonhermiticity we have chosen µ =0.2 for all lattice volumes. In this case, the eigen-values are rescaled with the square root of the vol-ume, ξ = piz/√d, where the level spacing is now2The values of α obtained from the V = 84 and 104 dataagree within errors.obtained by averaging over the smallest geomet-ric distance between complex eigenvalues. Thenormalized data confirm the prediction of Eq. (2)together with the different scaling.In conclusion, we have confirmed the analyticalmatrix model predictions in two different scalingregimes from quenched QCD lattice data. Forthe small values of µ at weak nonhermiticity, un-quenched simulations should be feasible as well,keeping V µ2 fixed. On the other hand, the lat-tice simulations [1] close to the phase transitionare at strong nonhermiticity, since µ is kept con-stant. So far, our analytical predictions are re-stricted to the broken phase and, unfortunately,are not suitable to describe the transition region.It would be very interesting to compute predic-tions for the unbroken phase as well.REFERENCES1. Z. Fodor and S.D. Katz, Phys. Lett. B 534(2002) 87; C.R. Allton et al., Phys. Rev. D66 (2002) 074507; P. de Forcrand and O.Philipsen, Nucl. Phys. B 642 (2002) 290; M.D’Elia and M.-P. Lombardo, Phys. Rev. D67(2003) 0145052. G. Akemann, Phys. Rev. Lett. 89 (2002)072002, J. Phys. A 36 (2003) 33633. G. Akemann and T. Wettig, hep-th/03080034. M.A. Stephanov, Phys. Rev. Lett. 76 (1996)44725. J. Ambjørn, K.N. Anagnostopoulos, J.Nishimura, and J.J.M. Verbaarschot, JHEP0210 (2002) 0626. M.E. Berbenni-Bitsch, S. Meyer, and T. Wet-tig, Phys. Rev. D 58 (1998) 071502; G. Ake-mann and E. Kanzieper, Phys. Rev. Lett. 85(2000) 11747. J.J.M. Verbaarschot and T. Wettig, Ann.Rev. Nucl. Part. Sci. 50 (2000) 3438. G. Akemann, hep-th/03071169. Y.V. Fyodorov, B.A. Khoruzhenko, and H.-J.Sommers, Phys. Lett. A 226 (1997) 46, Phys.Rev. Lett. 79 (1997) 55710. E.V. Shuryak and J.J.M. Verbaarschot, Nucl.Phys. A 560 (1993) 306; J.J.M. Verbaarschot,Phys. Rev. Lett. 72 (1994) 2531

87; C.R. Allton et al.,

Comparing matrix models and QCD lattice data with chemical potential

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Akemann G, Wettig T. Comparing matrix models and QCD lattice data with chemical potential. In:  Nuclear Physics B - Proceedings Supplements. Nuclear Physics B, Proceedings Supplements. Vol 129-130. Elsevier BV;  2004: 527-529.We present a quantitative analysis of the microscopic Dirac spectrum which iscomplex in the presence of a non-vanishing quark chemical potential. Data fromquenched SU(3) lattice simulations for different volumes V and small values ofthe chemical potential are compared to analytical predictions from matrixmodels. We confirm the existence of two distinct limits for weakly and stronglynonhermitian Dirac operators. Good agreement is found in both limits,confirming the different scaling of chemical potential and eigenvalues with thevolume

Akemann, Gernot

Wettig, T.

Publications at Bielefeld University

3 pages, 2 figures, presented at Lattice2003(non-zero temperature and density)We present a quantitative analysis of the microscopic Dirac spectrum which is complex in the presence of a non-vanishing quark chemical potential. Data from quenched SU(3) lattice simulations for different volumes V and small values of the chemical potential are compared to analytical predictions from matrix models. We confirm the existence of two distinct limits for weakly and strongly nonhermitian Dirac operators. Good agreement is found in both limits, confirming the different scaling of chemical potential and eigenvalues with the volume

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Comparing matrix models and QCD lattice data with chemical potential

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