We study the density of states method to explore the phase diagram of the
chiral transition on the temperature and quark chemical potential plane. Four
quark flavors are used in the analysis. Though the method is quite expensive
small lattices show an indication for a triple-point connecting three different
phases on the phase diagram.Comment: 5 pages, 6 figures, talk given at 'Extreme QCD', August 2-5 2005,
  Swansea, Wales, U

Fodor, Z.

Katz, S.

Schmidt, C.

English

arXiv

We study the density of states method to explore the phase diagram of the chiral transition on the temperature and quark chemical potential plane. Four quark flavors are used in the analysis. Though the method is quite expensive small lattices show an indication for a triple-point connecting three different phases on the phase diagram

UNT Digital Library

BNL-75354-2006-CP THE DENSITY OF STATES METHOD AT FINITE CHEMICAL POTENTIAL Christian Schmidt1, Zoltan Fodor2, Sandor Katz3 1Brookhaven National Laboratory, Upton, NY 11973 USA 2Universitat Wuppertal, Wuppertal, Germany 3Eotvos University, Budapest, Hungary Presented at the Workshop on Extreme QCD Swansea, Wales 2-5 August 2005 February 2006 Physics Department Brookhaven National Laboratory P.O. Box 5000 Upton, NY 11973-5000 www.bnl.gov Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This preprint is intended for publication in a journal or proceedings. Since changes may be made before publication, it may not be cited or reproduced without the author’s permission. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. BNL-NT-05/51 THE DENSITY OF STATES METHOD AT FINITE CHEMICAL POTENTIAL Christian Schmidt", Zoltan F0d0?t3 and Sandor Katz3 Physics Department, Brookhaven National Laboratory, Bldg. 510A, Upton, NY, 11973, USA Department of Physics, University of Wuppertal, Gaussstrasse 20, D-42119, Germany Institute for Theoretical Physics, Eotvtjs University, Pazmany 1, H-I117 Budapest, Hungary Abstract We study the density of states method to explore the phase diagram of the chiral transition on the temperature and quark chemical potential plane. Four quark flavors are used in the analysis. Though the method is quite expensive small lattices show an indication for a triple-point connecting three different phases on the phase diagram. 1 Introduction To clarify the phase diagram of QCD and thus the nature of matter under extreme conditions is one of the most interesting and fundamental tasks of high energy physics. Lattice QCD has been shown to provide important and reliable information from first principals on QCD at zero density. However, Lattice QCD at finite densities has been harmed by the complex action problem ever since its inception. For p > 0 the determinant of the fermion matrix (detM) becomes complex. Standard Monte Carlo techniques using importance sampling are thus no longer applicable when calculating observables in the grand canonical ensemble according to the partition function zGC(p) = J' DU detM[U](p) exp(-SG[U]}. (1) Recently many different methods have been developed to circumvent the complex action problem for small p / T [I, 21. For a recent overview see also 131. 2 Formulation of the method A very general formulation of the DOS method is the following: One exposed parameter (4) is fixed. The expectation value of a thermodynamic observable (0), according to the usual grand canonical partition function (l), can be recovered by the integral < 0 >= 1 (@ (Of(U)), P ( 4 ) / I d 4 (f(U)), P(+> s (2) where the density of states (p) is given by the constrained partition function: p(x) = Z&) = D U g ( U ) 6($ - x). (3) With ( ), we denote the expectation value with respect to the constrained partition function. In addition, the product of the weight functions f, g has to give the correct measure of ZGC: fg = detM exp{-Sc}. This idea of reordering the partition functions is rather old and was used in many different cases [4,5. 61 The advantages of this additional integration becomes clear, when choosing 4 = P and g(U) = 1. In this case p(4) is independent of all simulation parameters. The observable can be calculated as a function of all values of the lattice coupling p. If one has stored all eigenvalues of the fermion matrix for all * Speaker 1 2 configurations, the observable can also be calculated as a function of quark mass (m) and number of flavorsl51 ( N f ) . In this work we chose q5 = P and g = IdetMl exp{-SG}, f = exp(i6). (4) In other words we constrain the plaquette and perform simulations with measure 9. In practice, we replace the delta function in Equation (3) by a sharply peaked potential [6]. The constrained partition function for fixed values of the plaquette expectation value can then be written as ( 5 ) where exp{-V(x)} is a Gaussian potential with (6) 2 We obtain the density of states (p(x)) by the fluctuations of the actual plaquette P around the constraint value z. The fluctuation dissipation theorem gives 1 2 V(X) = -T(Z - P ) ~ d dx - Inp(x) =, . (7) (om)), (P, P> = ( O f ( U ) R ( P , Po, P , Po)) , / (R(P, Po, P , D O ) ) , > (8) (f(W), (P, PI = (f(U)R(CL, Po, P , Po)), / (R(CL, Po, PI Po)), 1 (9) (10) - b ( X , P, PI Here R is given by the quotient of the measure g at the point (p, P ) and at the simulation point (po, Po) , Before performing the integrals in Equation (2) we compute from an ensemble generated at (PO, PO): d dX = ((x - P ) W , Po, PI Po)) , * Having calculated the expressions (8)-(lo), we are able to extrapolate the expectation value of the observ- able (2) to any point (p, P ) in a small region around the simulation point (pug, 00). For any evaluation of (0) (p , PI , we numerically perform the integrals in Equation (2). We also combine the data from several simulation points to interpolate between them. 3 Simulations with constrained Plaquette The value we want to constrain is the expectation value of the global plaquette, which is given on every gauge configuration by the sum over all lattice points (y) and directions (pv) of the local plaquette PFv (y) and its adjoint Piu (y), (12) Since the plaquette is also the main part of the gauge action, the additional potential V can be easily introduced in the hybrid Monte Carlo update procedure of the hybrid-R algorithm [7]. After calculating the equation of motion for the link variables U,(y), we find for the gauge part of the force Here the subscript TA indicates the traceless anti-Hermitian part of the matrix. We see that in each molecular dynamical step the measurement of the plaquette is required. However, the only modification in the gauge force is the factor in round brackets. 3 (a> 'lo 1 0.1 0.01 1 10 (b) 1 0.1 0.01 0.001 n nnni -."~". 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 Fig. 1: (a) Results for Simulations at p = 4.98, p = 0.3, X = 0.02, nf = 4, am = 0.05, and number of lattice points: 44. Shown are the density of states p(x), the phase factor (cos(@)), and their product. (b) Results for Simulations .at ,O = 5.1, X = 0.01, nf = 4, am = 0.05, and number of lattice points: 64. Shown is the suppression from the complex phase of the fermion determinant (cos(@)) for different chemical potentials. 4 Simulation details and the strength of the sign problem Simulations have been performed with staggered fermions and N f = 4. We chose 9 differed points in the (p, p)-plane for the 44 lattice and 8 points for the 64 lattice. On each of these points we did simulations with 20-40 constrained plaquette values, all with quark mass am = 0.05. Further simulations has been done with (p, p) = (5.1,0.3) on the 63 x 8 lattice for am = 0.05 and am = 0.03. In order to calculate the plaquette expectation value, or its susceptibility, one has to perform the following integrals: Thus the functions p(z) and (cos(@)), have to be known quite precisely. We plot both functions in Figure l(a>. The transition is signaled in the double peak structure of p(z). The phase factor (cos(O)), suppresses the peak of p(z) at smaller plaquette values, which results in a shift of the critical temperature to smaller values, in comparison with the phase quenched theory. In Figure l(b) we show the phase factor for different chemical potentials. With increasing chemical potential the phase factor becomes compatible with zero within errors. In fact, its average value becomes as low as cos(@) N 0.005. There exist however a small interval around P N 2.85, where the phase factor stays finite. In this way, the Plaquette expectation values is strongly altered by the phase factor. Figure l(b) demonstrates also the advantage of he DOS method over the other approaches of lattice QCD to finite density. Using the DOS method one is able to do simulations at directly those Plaquette values which are relevant at finite density. This solves the so called overlap problem of the reweighting approach. Furthermore we have checked in [SI, that results with have been obtained within the framework of the DOS method agree very well with earlier results from the multi-reweighting approach. 5 The Plaquette expectation value and the phases diagram Performing the integration in Eq. (15) numerically, we calculate the plaquette expectation values as shown in Fig. 2. At chemical potentials p20.36, the plaquette signals the QCD transition through a rapid crossover from a low temperature phase of N 2.9 to a high temperature phase of N 3.1. For p20.36 the plaquette expectation value at small temperatures drops to - 2.85. This new low temperature phase of the plaquette at high chemical potentials is caused by the fermion determinant. 4 (a) 3.2 3.1 3 2.9 2.8 2.7 2.6 I (b) 2.89 2.88 2.87 2.86 2.85 2.84 2.5 5.11 5.115 5.12 5.125 5.13 5.135 5.14 5.145 5.15 P P=5.1 - 0.24 0.26 0.28 0.3 0.32 0.34 0.36 Fig. 2: Results for Simulations at p = 5.1, X = 0.01, nf = 4, am = 0.05, and number of lattice points: 64. Shown is: (a) the Plaquette expectation value as a function of the coupling /3 for different chemical potentials and (b) the plaquette expectation value at fixed coupling, as a function of the chemical potential. (a) j70 160 150 140 130 120 110 100 nn T [MeV] multiparameter reweighting - DOS method, am=0.05 - DOS method, am=0.03 +-e+ (b) 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 "B [fm"l T=143MeV - 1 1 1 T=124 MeV - -10 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 Fig. 3: The phase diagram in physical (a), and the quark number density at constant temperature T = 143 MeV (4* lattice), T = 124 MeV (64 lattice) and T = 93 MeV (63 x 8 lattice). As on can see in Figure l(b) the region around P N 2.85 is the region which is less suppressed by the phase factor. Another interesting observation is that the critical coupling, which is decreasing in p for p < 0.36 starts to increase for p > 0.36. The plaquette expectation value thus suggests the existence of three different phases in the (T,p)-diagram with a triple point, where all those phases coincide. In Figure 3(a) we show the phase diagram in physical units. The phase boundaries were determined by calculating the peaks in the plaquette susceptibility. Note, that we make no statement about the order of the transition lines. To determine the order of the transition one has to perform finite a finite-size-scaling analysis. The scale was set by the Sommer radius TO, measured on a lo3 x 20 lattice. The triple point is located around p p M 300 MeV, however its temperature (Ttri) decreases from Ttri M 148MeV on the 44 lattice to Ttri M 137MeV on the 6" lattice. This shift reflects the relatively large cut-off effects one faces, with standard staggered fermions and temporal extents of 4 and 6. Also shown in Figure 3(a) are points from simulations with quark mass am = 0.03. The phase boundary turned out to be - within our statistical uncertainties - independent of the the mass. 6 The quark number density To reveal the properties of the new phase located in the lower right comer of the phase diagram, calculated the quark number density, at constant coupling ,i3 and at constant temperature respectively. obtain the density nq we perfoim the following integration 5 we To The thermodynamic quantity nq are given as usual by In Figure 3(b) we show the baiyon number density, which is related to the quark number density by ng = n q / 3 . The results are plotted in physical units and correspond to a constant temperature of T M 143 MeV (44 lattice), T M 124 MeV (S4 lattice) and T M 93 MeV (S4 x 8 lattice). In order to divide out the leading order cut-off effect, we multiply we have multiplied the data with the factor c = SB(Nt ) /SB , which is the Stefan-Boltzmann value of a free lattice gas of quarks at a given value of Nt, divided by its continuum Stefan-Boltzmann value. At the same value of the chemical potential where we find also a peak in the susceptibility of the plaquette (pc), we see a sudden rise in the baryon number density. Thus for p > pc we enter a phase of dense matter. The transition occurs at a density of (2 - 3) x n N , where n N denotes nuclear matter density. Above the transition, the density reaches values of (10 - 20) x n N . Quite similar results have been obtained recently by simulations in the canonical ensemble [9]. References [l] Z. Fodor and S. D. Katz, Phys. Lett. B534 (2002) 87 [hep-lat/0104001]. [2] Z. Fodor and S. D. Katz, JHEP 0203 (2002) 014; C. R. Allton et al., Phys. Rev. D66 (2002) 074507; R. V. Gavai and S. Gupta, Phys. Rev. D 68 (2003) 034506; P. R. Crompton, [hep-lat / 03 OlOOl]; M. D’Elia and M. P. Lombardo, Phys. Rev. D67 (2003) 014505; P. de For- crand and 0. Philipsen, Nucl. Phys. B642 (2002) 290; NucL Phys. B673 (2003) 170; V. Azcotiti, et al., [hep- Pat / 0 5 0 3 0 lo]. [3] 0. Philipsen, PoSLAT2005 (2005) 016 [hep-lat/0510077]. [4] G. Bhanot, K. Bitar and R. Salvador, Phys. Lett. B187 (1987) 381; Phys. Lett. B188 (1987) 246; M. Karliner, S.R. Sharpe and Y.F. Chang, Nucl. Phys. B302 (1988) 204; V. Azcoiti, G. di Carlo and A. F. Grillo, Phys. Rev. Lett. 65 (1990) 2239; A. Gocksch, Phys. Rev. Lett. 61 (1988) 2054. [5] X. Q. Euo, Mod. Phys. Lett. A16 (2001) 1615. [6] J. Ambjom, K. N. Anagnostopoulos, 9. Nishimura and J. J. M. Verbaarschot, JHEP 0210 (2002) 062. [7] S. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar, Phys. Rev. B35 (1987) 2531. [SI C. Schmidt, Z. Fodor and S. Katz, PoSLAT2005 (2005) 163 [hep-lat/0510087]. [9] A. Alexandru, M. Faber, I. Horvath and K. F. Liu, [hep-lat/0410002]; S. Kratochvila and P. deForcrand, PoSLAT2005 (2005) 167 [hep-lat/0509143].

THE DENSITY OF STATES METHOD AT FINITE CHEMICAL POTENTAL

Schmidt, C

Fodor, Z

Katz, SD

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-lat/0512032v1 22 Dec 2005THE DENSITY OF STATES METHOD AT FINITE CHEMICALPOTENTIALBNL-NT-05/51Christian Schmidt∗1, Zoltan Fodor2,3 and Sandor Katz31 Physics Department, Brookhaven National Laboratory, Bldg. 510A, Upton, NY, 11973, USA2 Department of Physics, University of Wuppertal, Gaussstrae 20, D-42119, Germany3 Institute for Theoretical Physics, Eötvös University, Pazmany 1, H-1117 Budapest, HungaryAbstractWe study the density of states method to explore the phase diagram of thechiral transition on the temperature and quark chemical potential plane. Fourquark flavors are used in the analysis. Though the method is quite expensivesmall lattices show an indication for a triple-point connecting three differentphases on the phase diagram.1 IntroductionTo clarify the phase diagram of QCD and thus the nature of matter under extreme conditions is one of themost interesting and fundamental tasks of high energy physics. Lattice QCD has been shown to provideimportant and reliable information from first principals onQCD at zero density. However, Lattice QCDat finite densities has been harmed by the complex action problem ever since its inception. Forµ > 0 thedeterminant of the fermion matrix (detM) becomes complex. Standard Monte Carlo techniques usingimportance sampling are thus no longer applicable when calculating observables in the grand canonicalensemble according to the partition functionZGC(µ) =∫DU detM[U](µ) exp{−SG[U]}. (1)Recently many different methods have been developed to circumvent the complex action problem forsmallµ/T [1, 2]. For a recent overview see also [3].2 Formulation of the methodA very general formulation of the DOS method is the following: One exposed parameter (φ) is fixed. Theexpectation value of a thermodynamic observable (O), according to the usual grand canonical partitionfunction (1), can be recovered by the integral< O >=∫dφ 〈Of(U)〉φ ρ(φ)/∫dφ 〈f(U)〉φ ρ(φ) (2)where the density of states (ρ) is given by the constrained partition function:ρ(x) ≡ Zφ(x) =∫DU g(U) δ(φ − x). (3)With 〈 〉φ we denote the expectation value with respect to the constraied partition function. In addition,the product of the weight functionsf, g has to give the correct measure ofZGC : fg = detMexp{−SG}.This idea of reordering the partition functions is rather old and was used in many different cases [4, 5, 6]The advantages of this additional integration becomes clear, when choosingφ = P andg(U) = 1. In thiscaseρ(φ) is independent of all simulation parameters. The observable can be calculated as a functionof all values of the lattice couplingβ. If one has stored all eigenvalues of the fermion matrix for all∗Speaker12configurations, the observable can also be calculated as a function of quark mass (m) and number offlavors[5] (Nf ). In this work we choseφ = P and g = |detM | exp{−SG}, f = exp{iθ}. (4)In other words we constrain the plaquette and perform simulations with measureg. In practice, wereplace the delta function in Equation (3) by a sharply peaked potential [6]. The constrained partitionfunction for fixed values of the plaquette expectation valuecan then be written asρ(x) ≈∫DU g(U) exp {−V (x)} , (5)whereexp{−V (x)} is a Gaussian potential withV (x) =12γ (x− P )2 . (6)We obtain the density of states (ρ(x)) by the fluctuations of the actual plaquetteP around the constraintvaluex. The fluctuation dissipation theorem givesddxln ρ(x) =< x− P >x . (7)Before performing the integrals in Equation (2) we compute from an ensemble generated at(µ0, β0):〈Of(U)〉x (µ, β) = 〈Of(U)R(µ, µ0, β, β0)〉x / 〈R(µ, µ0, β, β0)〉x , (8)〈f(U)〉x (µ, β) = 〈f(U)R(µ, µ0, β, β0)〉x / 〈R(µ, µ0, β, β0)〉x , (9)ddxln ρ(x, µ, β) = 〈(x− P )R(µ, µ0, β, β0)〉x . (10)HereR is given by the quotient of the measureg at the point(µ, β) and at the simulation point(µ0, β0),R(µ, µ0, β, β0) = g(µ, β)/g(µ0, β0) =|det(µ)||det(µ0)|exp{SG(β)− SG(β0)}. (11)Having calculated the expressions (8)-(10), we are able to extrapolate the expectation value of the observ-able (2) to any point(µ, β) in a small region around the simulation point(µ0, β0). For any evaluation of〈O〉 (µ, β), we numerically perform the integrals in Equation (2). We also combine the data from severalsimulation points to interpolate between them.3 Simulations with constrained PlaquetteThe value we want to constrain is the expectation value of theglobal plaquette, which is given on everygauge configuration by the sum over all lattice points (y) and directions (µν) of the local plaquettePµν(y)and its adjointP †µν(y),P =∑y∑1≤µ<ν≤416[TrPµν(y) + TrP†µν(y)]. (12)Since the plaquette is also the main part of the gauge action,SG = −β∑x∑1≤µ<ν≤4{16[TrPµν(x) + TrP†µν(x)]− 1}, (13)the additional potentialV can be easily introduced in the hybrid Monte Carlo update procedure of thehybrid-R algorithm [7]. After calculating the equation of motion for the link variablesUµ(y), we findfor the gauge part of the forceiḢµ(y) =[β3Uµ(y)Tµ(y)(1 +γ(x− P )β)]TA. (14)Here the subscriptTA indicates the traceless anti-Hermitian part of the matrix.We see that in eachmolecular dynamical step the measurement of the plaquette is required. However, the only modificationin the gauge force is the factor in round brackets.3(a)0.010.11101.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6ρ(x)cos(θ)cos(θ)ρ(x)(b)0.00010.0010.010.11102.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15<cos(θ)>Pµ=0.30µ=0.32µ=0.34µ=0.36µ=0.38µ=0.40Fig. 1: (a) Results for Simulations atβ = 4.98, µ = 0.3, λ = 0.02, nf = 4, am = 0.05, and number of lattice points:44.Shown are the density of statesρ(x), the phase factor〈cos(θ)〉, and their product. (b) Results for Simulations atβ = 5.1,λ = 0.01, nf = 4, am = 0.05, and number of lattice points:64. Shown is the suppression from the complex phase of thefermion determinant〈cos(θ)〉 for different chemical potentials.4 Simulation details and the strength of the sign problemSimulations have been performed with staggered fermions andNf = 4. We chose 9 differed points in the(β, µ)-plane for the44 lattice and 8 points for the64 lattice. On each of these points we did simulationswith 20-40 constrained plaquette values, all with quark mass am = 0.05. Further simulations has beendone with(β, µ) = (5.1, 0.3) on the63 × 8 lattice foram = 0.05 andam = 0.03. In order to calculatethe plaquette expectation value, or its susceptibility, one has to perform the following integrals:〈P 〉 =∫dx xρ(x) 〈cos(θ)〉x ,〈P 2〉=∫dx x2ρ(x) 〈cos(θ)〉x . (15)Thus the functionsρ(x) and 〈cos(θ)〉x have to be known quite precisely. We plot both functions inFigure 1(a). The transition is signaled in the double peak structure ofρ(x). The phase factor〈cos(θ)〉xsuppresses the peak ofρ(x) at smaller plaquette values, which results in a shift of the critical temperatureto smaller values, in comparison with the phase quenched theory. In Figure 1(b) we show the phasefactor for different chemical potentials. With increasingchemical potential the phase factor becomescompatible with zero within errors. In fact, its average value becomes as low ascos(θ) ∼ 0.005. Thereexist however a small interval aroundP ∼ 2.85, where the phase factor stays finite. In this way, thePlaquette expectation values is strongly altered by the phase factor. Figure 1(b) demonstrates also theadvantage of he DOS method over the other approaches of lattice QCD to finite density. Using the DOSmethod one is able to do simulations at directly those Plaquette values which are relevant at finite density.This solves the so called overlap problem of the reweightingapproach. Furthermore we have checked in[8], that results with have been obtained within the framework of the DOS method agree very well withearlier results from the multi-reweighting approach.5 The Plaquette expectation value and the phases diagramPerforming the integration in Eq. (15) numerically, we calculate the plaquette expectation values asshown in Fig. 2. At chemical potentialsµ<∼0.36, the plaquette signals the QCD transition through a rapidcrossover from a low temperature phase of∼ 2.9 to a high temperature phase of∼ 3.1.For µ>∼0.36 the plaquette expectation value at small temperatures drops to∼ 2.85. This newlow temperature phase of the plaquette at high chemical potentials is caused by the fermion determinant.4(a)2.52.62.72.82.933.13.25.11 5.115 5.12 5.125 5.13 5.135 5.14 5.145 5.15Pβaµ=0.32aµ=0.34aµ=0.36aµ=0.38(b)2.832.842.852.862.872.882.890.24 0.26 0.28 0.3 0.32 0.34 0.36µP β=5.1Fig. 2: Results for Simulations atβ = 5.1, λ = 0.01, nf = 4, am = 0.05, and number of lattice points:64. Shown is: (a) thePlaquette expectation value as a function of the couplingβ for different chemical potentials and (b) the plaquette expectationvalue at fixed coupling, as a function of the chemical potential.(a)901001101201301401501601700 50 100 150 200 250 300 350 400µq [MeV]Τ [MeV] multiparameter reweightingDOS method, am=0.05DOS method, am=0.03(b)-0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 50 100 150 200 250 300 350nB [fm-3]µq [MeV]T=143 MeVT=124 MeVT=93 MeVFig. 3: The phase diagram in physical (a), and the quark number density at constant temperatureT = 143 MeV (44 lattice),T = 124 MeV (64 lattice) andT = 93 MeV (63 × 8 lattice).As on can see in Figure 1(b) the region aroundP ∼ 2.85 is the region which is less suppressed by thephase factor. Another interesting observation is that the critical coupling, which is decreasing inµ forµ < 0.36 starts to increase forµ > 0.36. The plaquette expectation value thus suggests the existenceof three different phases in the (T ,µ)-diagram with a triple point, where all those phases coincide. InFigure 3(a) we show the phase diagram in physical units. The phase boundaries were determined bycalculating the peaks in the plaquette susceptibility. Note, that we make no statement about the order ofthe transition lines. To determine the order of the transition one has to perform finite a finite-size-scalinganalysis.The scale was set by the Sommer radiusr0, measured on a103 × 20 lattice. The triple point islocated aroundµtriq ≈ 300 MeV, however its temperature (Ttri) decreases fromT tri ≈ 148MeV on the44 lattice toT tri ≈ 137MeV on the64 lattice. This shift reflects the relatively large cut-off efects onefaces, with standard staggered fermions and temporal extents of 4 and 6.Also shown in Figure 3(a) are points from simulations with quark massam = 0.03. The phaseboundary turned out to be — within our statistical uncertainties — independent of the the mass.56 The quark number densityTo reveal the properties of the new phase located in the lowerright corner of the phase diagram, wecalculated the quark number density, at constant couplingβ and at constant temperature respectively. Toobtain the densitynq we perform the following integration〈d ln detMd(aµ)〉=∫dx〈d ln detMd(aµ)cos(θ)〉xρ(x) (16)The thermodynamic quantitynq are given as usual bynq =1a3N3sNt〈d ln detMd(aµ)〉(17)In Figure 3(b) we show the baryon number density, which is related to the quark number density bynB = nq/3. The results are plotted in physical units and correspond toa c nstant temperature ofT ≈ 143 MeV (44 lattice), T ≈ 124 MeV (64 lattice) andT ≈ 93 MeV (64 × 8 lattice). In orderto divide out the leading order cut-off effect, we multiply we have multiplied the data with the factorc = SB(Nt)/SB, which is the Stefan-Boltzmann value of a free lattice gas ofquarks at a given valueof Nt, divided by its continuum Stefan-Boltzmann value. At the same value of the chemical potentialwhere we find also a peak in the susceptibility of the plaquette (µc), we see a sudden rise in the baryonnumber density. Thus forµ > µc we enter a phase of dense matter. The transition occurs at a density of(2− 3)×nN , wherenN denotes nuclear matter density. Above the transition, the density reaches valuesof (10 − 20) × nN . Quite similar results have been obtained recently by simulations in the canonicalensemble [9].References[1] Z. Fodor and S. D. Katz,Phys. Lett. B534 (2002) 87 [hep-lat/0104001].[2] Z. Fodor and S. D. Katz,JHEP 0203 (2002) 014; C. R. Alltonet al., Phys. Rev. D66(2002) 074507; R. V. Gavai and S. Gupta,Phys. Rev. D 68 (2003) 034506; P. R. 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The Density of states method at finite chemical potential

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The Density of States Method at Finite Chemical Potential

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