We provide first evidence that Matrix Models describe the low lying complex
Dirac eigenvalues in a theory with dynamical fermions at non-zero density.
Lattice data for gauge group SU(2) with staggered fermions are compared to
detailed analytical results from Matrix Models in the corresponding symmetry
class, the complex chiral Symplectic Ensemble. They confirm the predicted
dependence on chemical potential, quark mass and volume.Comment: 6 pages, 8 fig., talk given at Lattice 2005 (Finite Temperature and
  Density) Dublin and Extreme QCD Swanse

Akemann, Gernot

Bittner, Elmar

English

arXiv

Akemann G, Bittner E. Two-colour Lattice QCD with dynamical fermions at non-zero density versus Matrix Models. In:  PoS. Vol LAT2005. 2005: 197.We provide first evidence that Matrix Models describe the low lying complexDirac eigenvalues in a theory with dynamical fermions at non-zero density.Lattice data for gauge group SU(2) with staggered fermions are compared todetailed analytical results from Matrix Models in the corresponding symmetryclass, the complex chiral Symplectic Ensemble. They confirm the predicteddependence on chemical potential, quark mass and volume

Publications at Bielefeld University

Two-colour Lattice QCD with dynamical fermions at non-zero density versus Matrix Models

We provide first evidence that Matrix Models describe the low lying complex Dirac eigenvalues in a theory with dynamical fermions at non-zero density. Lattice data for gauge group SU(2) with staggered fermions are compared to detailed analytical results from Matrix Models in the corresponding symmetry class, the complex chiral Symplectic Ensemble. They confirm the predicted dependence on chemical potential, quark mass and volume

Akemann, G

Bittner, E

Brunel University Research Archive

PoS(LAT2005)197Two-colour Lattice QCD with dynamical fermions atnon-zero density versus Matrix ModelsGernot Akemann∗Department of Mathematical SciencesBrunel University West LondonUxbridge UB8 3PH, United KingdomE-mail: gernot.akemann@brunel.ac.ukElmar BittnerInstitut für Theoretische PhysikUniversität LeipzigAugustusplatz 10/11D-04109 Leipzig, GermanyE-mail: elmar.bittner@itp.uni-leipzig.deWe provide rst evidence that Matrix Models describe the low lying complex Dirac eigenvaluesin a theory with dynamical fermions at non-zero density. Lattice data for gauge group SU(2) withstaggered fermions are compared to detailed analytical results from Matrix Models in the corre-sponding symmetry class, the complex chiral Symplectic Ensemble. They conrm the predicteddependence on chemical potential, quark mass and volume.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2005)197Nc = 2 Lattice QCD with dynamical fermions at µ 6= 0 vs. Matrix Models Gernot Akemann1. IntroductionIn the past few years many new analytical results have emerged from Matrix Models (MM)for the Dirac operator spectrum with non-vanishing chemical potential µ 6= 0. There are 3 differentpossible chiral symmetry breaking patterns [1] and in the MM picture they differ crucially in theway the eigenvalues are depleted from the imaginary axis through µ 6= 0 [2]. Today we havedetailed predictions for microscopic Dirac spectra in 2 of these symmetry classes, both quenchedand unquenched: QCD and the adjoined representation, where the latter is replaced by SU(2)colour in the fundamental when using staggered fermions as we will do. After first approximateresults for quenched QCD [3] the exact quenched density was derived in [4] and related to chiralPerturbation Theory in the epsilon regime (εχPT). Unquenched partition functions were computedin [5] and related to εχPT, and finally fully unquenched Dirac spectra for QCD became available[6, 7]. Results for the adjoint (or SU(2) staggered) class followed very recently [8] and we confrontits unquenched predictions including dependence on µ and quark mass m with Lattice data here.Not all the above MM results have been compared to the Lattice so far, precisely due to thesign problem in unquenched QCD. Up to now only quenched simulations at µ 6= 0 using staggeredfermions have been successfully described: for QCD [9] (see also [10] in these proceedings) andfor SU(2) [11], which we will extend here. Previous comparisons [12] were lacking analytic pre-dictions at the time, they could only be done in the bulk of the spectrum [13] for MM withoutchiral symmetry. Very recently it has been shown at µ = 0 that the previous topology-blindness ofstaggered fermions can be cured by improvement, as reviewed in [14].In the last years different ways of attacking the sign problem in QCD were developed: multi-parameter reweighting, Taylor-expansion and imaginary µ (see [15] for a review and references).However, none of these have been applied so far to the region of εχPT where a comparison to MMis expected to hold. We purse a different avenue here, choosing an SU(2) gauge theory without signproblem where dynamical simulations can be performed in a standard way [16]. This permits us toextend previous MM comparisons [17, 18] to µ 6= 0. It is of principle interest to test the validity ofall MM predictions for complex Dirac spectra on the Lattice including dynamical fermions.2. Predictions from Matrix Models with µ 6= 0In this section we briefly introduce the relevant MM used and give its results for the spectraldensity. For more details and references we refer to [8]. The MM partition function is given byZ(2N f )N ({m f }; µ)≡∫dΦdΨexp[−NTr(Φ†Φ + Ψ†Ψ)] N f∏f =1det(m f 1 iΦ + µΨiΦ† + µΨ† m f 1), (2.1)where 1 is the quaternion unity element. The two rectangular (N + ν)×N matrices, Φ and Ψ,contain quaternion real elements and replace the off-diagonal blocks of the Dirac operator, averagedwith a Gaussian matrix weight instead of the gauge action. Here we model the µγ0 part with arandom matrix of the same symmetry as the kinetic part, assuming it is non-diagonal in matrixspace (as in [6] for QCD). If universality holds this choice of basis should not matter, comparedto Ψ replaced by unity as [2]. This assertion is true in the QCD symmetry class, see [4, 5] vs.[6, 7]. The size of the Dirac matrix 2N ∼V relates to the volume, and we have chosen rectangular197 / 2PoS(LAT2005)197Nc = 2 Lattice QCD with dynamical fermions at µ 6= 0 vs. Matrix Models Gernot Akemannmatrices to be in the fixed sector of ν ≥ 0 zero eigenvalues or topological charge. Because of usingunimproved staggered fermions we set ν = 0 in our comparison to the data later.After transforming the linear combinations iΦ(†) + µΨ(†) to be triangular we can change tocomplex eigenvalues zk=1,...,N (rotated to lie on the real axis for µ = 0). All their spectral correlationfunctions can be computed using skew orthogonal Laguerre polynomials in the complex plane [8].In the large-N limit the complex eigenvalues, masses and chemical potential have to be rescaled√2 Nz ≡ ξ ,√2 Nm ≡ η , and limN→∞, µ→02Nµ2 ≡ α2 . (2.2)This limit is called weakly non-hermitian as µ 2 is rescaled with the volume. The same scaling isfound in the static part of the chiral Lagrangian [19] dominating at εχPT. While the latter containstwo parameters, the chiral condensate Σ and the decay constant Fpi , the MM contains only one, thevariance fixed to be Σ =√2 here. Thus the constant multiplying α is not know and we are left with1 free fit parameter, in contrast to the parameter-free MM prediction at µ = 01.The unquenched microscopic spectral density, normalised to ℑm(ξ )δ ′(ℑm(ξ )) for α → 0,ρ (4)weak(ξ ;η) = ∆ρ(4)weak(ξ ;η) +132α4(ξ ∗2−ξ 2) |ξ |2 K2ν( |ξ |22α2)exp[+14α2(ξ 2 + ξ ∗2)](2.3)×∫ 10ds∫ 10dt√te−2s(1+t)α2 (J2ν(2√st ξ )J2ν(2√s ξ ∗) − J2ν(2√s ξ )J2ν(2√st ξ ∗)),splits into the quenched part and a correction term ∆ρ (4)weak(ξ ;η) depending on the mass η :∆ρ (4)weak(ξ ;η) =132α4(ξ ∗2−ξ 2) |ξ |2 K2ν( |ξ |22α2)exp[+14α2(ξ 2 + ξ ∗2)](2.4)×{(∫ 10ds∫ 10dt√ste−2s(1+t)α2(J2ν(2√st ξ )I2ν+1(2√s η)−√tI2ν+1(2√st η)J2ν(2√s ξ )))×(∫ 10ds∫ 10dt√te−2s(1+t)α2(J2ν(2√st ξ ∗)I2ν(2√sη)− I2ν(2√st η)J2ν(2√sξ ∗)))− (ξ ↔ ξ ∗)}×[∫ 10ds∫ 10dt√ste−2s(1+t)α2 (√tI2ν+1(2√st η)I2ν(2√s η)− I2ν(2√st η)I2ν+1(2√s η))]−1.We only give the result for N f = 2 staggered flavours of equal mass here, see [8] for more flavoursand higher correlation functions, as well as [17] for matching MM and staggered flavours. Fig.1 shows this density in the complex plane at weak non-Hermiticity, as we shall compare to databelow. The eigenvalues are repelled from the real axis for µ 6= 0, being a distinct feature of thissymmetry class (compare to [9] for QCD). We have observed this repulsion in the data for µ assmall as 10−6. The individual eigenvalues located at the maxima previously are now split into adouble peak in the complex plane.Increasing the mass η → ∞ moves the density to the left, bringing it back to the quenchedexpression, eq. (2.3) at ∆ρ (4)weak(ξ ;η) = 0. Decreasing η → 0 pushes the eigenvalues further awayfrom the origin, approaching the quenched density at ν = 2 approximately. Increasing α rapidlywashes out the oscillations and leads to the formation of a plateau. The limit α→∞ takes us to theMM at strong non-Hermiticity, with µ unscaled (see [8]). A comparison to quenched Lattice datain this regime was given previously in [11], our unquenched results will be reported elsewhere.1This observation was missed in [9, 11].197 / 3PoS(LAT2005)197Nc = 2 Lattice QCD with dynamical fermions at µ 6= 0 vs. Matrix Models Gernot Akemann510-0.0200.02051015200.00.20.40.60.81.01.21.41.61.8 0  2  4  6  8  10  12  14ρ weak(ζ)Re(ζ)η=14η=33quenchedℜe(ξ )ℑm(ξ )ρ (4)weak(ξ )Figure 1: The N f = 2 avour spectral density at α = 0.012 with masses η = 8.74 and ν = 0 (left) and a cutthrough the density normalised to unity for different mass values compared to quenched (right).3. Lattice data with dynamical fermions at µ 6= 0Our data were generated for gauge group SU(2) with coupling β = 4/g2 = 1.3 and N f = 2staggered flavours, using the code of [16]. In this setup the fermion determinant remains real (as itdoes in the MM, see [8]) and standard Monte Carlo applies. We have studied two different Volumes64 and 84 for various values of µ = 10−6− 0.4 and values of the quark masses ma = 0.025− 20.Because the eigenvalues lie in the complex plane of the order of 5-10k configurations are needed.In order to have a window where a MM description applies for these small lattices we have to goto relatively strong coupling (see [10] for a discussion of this issue in QCD).To compare with the prediction (2.3) we have taken cuts through the density: along the maximaparallel to the real axis, Fig. 1 right, and perpendicular to that over the first maximum pair. Theeffect of dynamical fermions is most clearly seen in the shift in the first cut where we choose thevalues η = 14 (blue) and η = 33 (pink) to be used in our data below, compared to quenched (black).Being very costly we did not to go to smaller η .The parameters η and α were obtained as follows from our data with input ma and µ . First,we determined the rescaling of the masses and eigenvalues by measuring the mean level-spacingd ∼ 1/ρ(0), using the Banks-Casher relation from µ = 0: piρ(0) = ΣV where ρ(0) is the meanspectral density. Due to µ  1 the geometric distance between eigenvalues agrees within errorswith the distance obtained by a projection onto the real axis. This provides us with the rescaling ofthe eigenvalues and masses,zapi/d ≡ ξ and mapi/d ≡ η . (3.1)At the same time the spacing d contains the volume factor for the rescaling of µ 2: α2 = Cµ2pi/d.The constant C∼ F2pi /Σ of order 1 is obtained by fitting the data to the cut parallel to the imaginaryaxis on the first maximum. Since the surface under this curve is finite we can fit to its integralfunction, being independent of the choice of histogram widths in imaginary direction. This alsofixes the normalisation. For illustration purposes we show the cut and not its integral. We can nowtest the scaling hypothesis of µ 2 eq. (2.2). For this purpose we have kept µ 2V fixed for the twovolumes V = 64 and 84: µ = 1 · 10−3 and µ = 5.625 · 10−4 , respectively. Since the level spacingd also depends on the mass we have kept ma fixed, leading automatically to different η-values for197 / 4PoS(LAT2005)197Nc = 2 Lattice QCD with dynamical fermions at µ 6= 0 vs. Matrix Models Gernot Akemann0.00.20.40.60.81.01.21.41.61.8 0  2  4  6  8  10  12  14ρ weak(ζ)Re(ζ)0.00.20.40.60.81.01.21.41.61.82.0-0.04 -0.03 -0.02 -0.01  0  0.01  0.02  0.03  0.04ρ weak(ζ)Im(ζ)0.00.20.40.60.81.01.21.41.61.8 0  2  4  6  8  10  12  14ρ weak(ζ)Re(ζ)0.00.20.40.60.81.01.21.41.61.82.0-0.04 -0.03 -0.02 -0.01  0  0.01  0.02  0.03  0.04ρ weak(ζ)Im(ζ)α=0.01244α=0.01308Figure 2: The scaling µ2V for dynamical fermions: V = 64, ma = 0.07 and µ = 1 · 10−3 ⇒ α = 0.013,η = 33 (upper plots) vs. V = 84, ma = 0.07 and µ = 5.625 ·10−4⇒ α = 0.012, η = 102 (lower plots) .different volumes2 . The data (histograms) in Fig. 2 right confirm this scaling very well: the fittedα = 0.013 from V = 64 (blue curve) describes the V = 84 data as well (green). For V = 84 we alsodisplay the α = 0.012 (blue) obtained from an independent fit, they agree within 5%. In the cutsin Fig. 2 left these small variations in α cannot be seen. If we were to compare to the quencheddensity a different fit value for α ∼ 0.0185 (64) and α ∼ 0.0175 (84) would be obtained instead,describing the right curves equally well. However, in the left plots the quenched MM curve (grey)deviates from the unquenched (blue) one. In the upper V = 64 plot the discrepancy from η = 33 toquenched can be clearly seen in the data as we capture up to the 4th maximum, whereas in V = 84keeping ma fixed implies η = 102, taking us back to almost quenched.In order to see the difference from quenched at smaller masses we compare to data correspond-ing to a rescaled mass η = 14 (blue curve) in Fig. 3 below. Although we can only resolve well thefirst 2-3 maxima in the left picture, the mismatch with the quenched curve (grey) is evident.To summarise we have shown that the MM correctly predicts complex SU(2) Lattice data withN f = 2 dynamical staggered fermions at µ 6= 0, describing the effect of small quark masses. Wehave also confirmed the scaling of µ 2 with the volume at weak non-Hermiticity from unquenchedLattice data.Acknowledgements: We thank Maria-Paola Lombardo, Harald Markum, Rainer Pullirschand Tilo Wettig for helpful discussions and previous collaboration. This work was supported by2For the comparison in [11] the different η  1 were both close enough to quenched.197 / 5PoS(LAT2005)197Nc = 2 Lattice QCD with dynamical fermions at µ 6= 0 vs. Matrix Models Gernot Akemann0.00.20.40.60.81.01.21.41.61.8 0  2  4  6  8  10  12  14ρ weak(ζ)Re(ζ)0.00.20.40.60.81.01.21.41.61.8-0.04 -0.03 -0.02 -0.01  0  0.01  0.02  0.03  0.04ρ weak(ζ)Im(ζ)Figure 3: Data (histograms) vs. MM (blue curve=unquenched, grey=quenched) for V = 64 ma = 0.035 andµ = 1 ·10−3⇒ η = 14 and α = 0.012.a BRIEF Award No. 707 from Brunel University (G.A.) and by the Deutsche Forschungsgemein-schaft under grant No. JA483/17-3 (E.B.).References[1] J.J.M. Verbaarschot, Phys. Rev. Lett. 72 (1994) 2531.[2] M.A. Halasz, J.C. Osborn and J.J.M. Verbaarschot, Phys. Rev. D56 (1997) 7059.[3] G. Akemann, Phys. Rev. Lett. 89 (2002) 072002; J. Phys. A: Math. Gen. 36 (2003) 3363.[4] K. Splittorff and J.J.M. Verbaarschot, Nucl. Phys. B683 (2004) 467.[5] G. Akemann, Y.V. Fyodorov and G. Vernizzi, Nucl. Phys. B694 (2004) 59.[6] J.C. Osborn, Phys. Rev. Lett. 93 (2004) 222001; private communication.[7] G. Akemann, J.C. Osborn, K. Splittorff and J.J.M. Verbaarschot Nucl. Phys. B712 (2005) 287.[8] G. Akemann, Nucl. Phys. B to appear [hep-th/0507156].[9] G. Akemann and T. Wettig, Phys. Rev. Lett. 92 (2004) 102002.[10] J.C. Osborn, Eigenvalue correlations in quenched QCD at nite density, these proceedingsPoS(LAT2005)200.[11] G. Akemann, E. Bittner, M.-P. Lombardo, H. Markum and R. Pullirsch, Nucl. Phys. B140 Proc.Suppl. (2005) 568.[12] E. Bittner, M.-P. Lombardo, H. Markum and R. Pullirsch, Nucl. Phys. B106 Proc. Suppl. (2002) 468.[13] H. Markum, R. Pullirsch and T. Wettig, Phys. Rev. Lett. 83 (1999) 484.[14] E. Follana, Nucl. Phys. B140 Proc. Suppl. (2005) 141.[15] F. Karsch, Prog. Theor. Phys. Suppl. 153 (2004) 106.[16] S. Hands, J.B. Kogut, M.-P. Lombardo, and S.E. Morrison, Nucl. Phys. B 558 (1999) 327.[17] M.E. Berbenni-Bitsch, S. Meyer and T. Wettig, Phys. Rev. D58 (1998) 071502.[18] G. Akemann and E. Kanzieper, Phys. Rev. Lett. 85 (2000) 1174.[19] J. B. Kogut, M. A. Stephanov, D. Toublan, J. J. M. Verbaarschot and A. Zhitnitsky, Nucl. Phys. B582(2000) 477.197 / 6

Two-colour Lattice QCD with dynamical fermions at non-zero density versus Matrix Models

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