We study the density of states method to explore the phase diagram of the
chiral transition on the tempeature and quark chemical potential plane. Four
quark flavours 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: Talk presented on Lattice 2005 (non-zero temperature and density), 6
  pages, 6 figure

Fodor, Zoltan

Katz, Sandor

Schmidt, Christian

English

arXiv

We study the density of states method to explore the phase diagram of the chiral transition on the tempeature and quark chemical potential plane. Four quark flavours 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

Schmidt, C

Fodor, Z

Katz, SD

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-lat/0510087v1  18 Oct 2005The QCD phase diagram at finite densityBNL-NT-05/38WUB-05-12ITP-Budapest 623Christian Schmidt ∗Physics Department, Brookhaven National Laboratory, Upton, NY, 11973, USAE-mail: cschmidt@bnl.govZoltan FodorDepartment of Physics, University of Wuppertal, Wuppertal, GermanyInstitute for Theoretical Physics, Eötvös University, Budapest, HungaryE-mail: fodor@bodri.elte.huSandor KatzInstitute for Theoretical Physics, Eötvös University, Budapest, HungaryE-mail: katz@bodri.elte.huWe study the density of states method to explore the phase diagram of the chiral transition onthe tempeature and quark chemical potential plane. Four quark flavours are used in the analy-sis. Though the method is quite expensive small lattices show an indication for a triple-pointconnecting three different phases on the phase diagram.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.PoS(LAT2005)163c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licen. http://pos.sissa.it/The QCD phase diagram at finite density Christian Schmidt1. IntroductionTo clarify the phase diagram of QCD and thus the nature of matter under extreme conditionsis one of the most interesting and fundamental tasks of high energy physics. Lattice QCD has beenshown 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 eversince its inception. Forµ > 0 the determinant of the fermion matrix (detM) becomes complex.Standard Monte Carlo techniques using importance samplingare thus no longer applicable whencalculating observables in the grand canonical ensemble according to the partition functionZGC(µ) =∫DU detM[U](µ)exp{−SG[U]}. (1.1)Recently many different methods have been developed to cirumvent the complex action problemfor smallµ/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. The expectation value of a thermodynamic observable (O), according to the usual grandcanonical partition function (1.1), can be recovered by theint gral< O>=∫dφ 〈O f(U)〉φ ρ(φ)/∫dφ 〈 f (U)〉φ ρ(φ) (2.1)where the density of states (ρ) is given by the constrained partition function:ρ(x)≡ Zφ (x) =∫DU g(U)δ (φ −x). (2.2)With 〈 〉φ we denote the expectation value with respect to the constraied partition function. Inaddition, the product of the weight functionsf ,g has to give the correct measure ofZGC: f g =detMexp{−SG}. This idea of reordering the partition functions is rather old and was used inmany different cases [4, 5, 6] The advantages of this additional integration becomes clear, whenchoosingφ = P andg(U) = 1. In this caseρ(φ) is independent of all simulation parameters. Theobservable can be calculated as a function of all values of the lattice couplingβ . If one has storedall eigenvalues of the fermion matrix for all configurations, the observable can also be calculatedas a function of quark mass (m) and number of flavors[5] (Nf ). In this work we choseφ = P and g= |detM|exp{−SG}, f = exp{iθ}. (2.3)In other words we constrain the plaquette and perform simulations with measureg. In practice,we replace the delta function in Equation (2.2) by a sharply peaked potential [6]. The constrainedpartition function for fixed values of the plaquette expectation value can then be written asρ(x)≈∫DU g(U)exp{−V(x)} , (2.4)PoS(LAT2005)163163 / 2The QCD phase diagram at finite density Christian Schmidtwhere exp{−V(x)} is a Gaussian potential withV(x) =12γ (x−P)2 . (2.5)We obtain the density of states (ρ(x)) by the fluctuations of the actual plaquetteP around theconstraint valuex. The fluctuation dissipation theorem givesddxlnρ(x) =< x−P>x . (2.6)Before performing the integrals in Equation (2.1) we compute from an ensemble generated at(µ0,β0):〈O f(U)〉x (µ ,β ) = 〈O f(U)R(µ ,µ0,β ,β0)〉x/〈R(µ ,µ0,β ,β0)〉x , (2.7)〈 f (U)〉x (µ ,β ) = 〈 f (U)R(µ ,µ0,β ,β0)〉x/〈R(µ ,µ0,β ,β0)〉x , (2.8)ddxlnρ(x,µ ,β ) = 〈(x−P)R(µ ,µ0,β ,β0)〉x . (2.9)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)}. (2.10)Heaving calculated the expressions (2.7)-(2.9), we are ablto extrapolate the expectation valueof the observable (2.1) 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 (2.1). Wealsocombine the data from several simulation points to interpolate between them.3. Simulations with constrained plaquetteThe value we want to constrain is the expectation value of theglobal plaquette, which is givenon every gauge configuration by the sum over all lattice points (y) and directions (µν) of the localplaquettePµν(y) and its adjointP†µν(y),P= ∑y∑1≤µ<ν≤416[TrPµν(y)+TrP†µν(y)]. (3.1)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}, (3.2)the additional potentialV can be easily introduced in the hybrid Monte Carlo update procedure ofthe hybrid-R algorithm [7]. After calculating the equationf motion for the link variablesUµ(y),we find for the gauge part of the forceiḢµ(y) =[β3Uµ(y)Tµ (y)(1+γ(x−P)β)]TA. (3.3)Here the subscript TA indicates the traceless anti-Hermitian part of the matrix. We see that ineach molecular dynamical step the measurement of the plaquette is required. However, the onlymodification in the gauge force is the factor in round brackets.PoS(LAT2005)163163 / 3The QCD phase diagram at finite density Christian Schmidt(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)2.72.752.82.852.92.9534.96 4.965 4.97 4.975 4.98 4.985 4.99 4.995 5βP with phasewithout phase(c)1.21.31.41.51.61.71.81.924.96 4.965 4.97 4.975 4.98 4.985 4.99 4.995 5βχP with phasewithout phase(d)4.934.944.954.964.974.984.9950 0.1 0.2 0.3 0.4 0.5 0.6λ/mβ(2002)without phasewith phaseextrapolatedFodor und KatzFigure 1: Results for Simulations atβ = 4.98,µ = 0.3, λ = 0.02,nf = 4, am= 0.05, and number of latticepoints: 44. Shown are: (a) the density of statesρ(x), the phase factor〈cos(θ )〉, and their product, (b) thePlaquette as a function of the couplingβ , with and without the phase factor, (c) the Susceptibility of thePlaquette as a function of the couplingβ , with and without the phase factor, and (d) the extrapolation βc(λ )to λ = 0, with and without the phase factor.4. The critical line and the determination of a triple-pointSimulations have been performed with staggered fermions and Nf = 4. We chose 9 differedpoints in the(β ,µ)-plane for the 44 lattice and 8 points for the 64 lattice. On each of these points wedid simulations with 20-40 constrained plaquette values, all with quark massam= 0.05. Furthersimulations has been done with(β ,µ) = (5.1,0.3) on the 63×8 lattice foram= 0.05 andam=0.03.Fist of all we check, whether we can reproduce old results with our new method. We showin Figure 1(a) results from a Simulation atµ = 0.3, β = 4.98 andλ = 0.02. Here we plot thedensity of states (ρ) and the real part of the phase factor〈cos(θ)〉 as a function of the constrainedPoS(LAT2005)163163 / 4The QCD phase diagram at finite density Christian Schmidt(a)901001101201301401501601700 50 100 150 200 250 300 350 400µq [MeV]Τ [MeV] multiparameter reweightingDOS method, am=0.05DOS method, am=0.03(b)-0.500.511.522.533.50 50 100 150 200 250 300 350nB [fm-3]µq [MeV]T=143 MeVT=124 MeVT=93 MeVFigure 2: 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).plaquette value. The results have been interpolated inP, order to obtain a better result for thenecessary integration overP. The distributionρ shows a clear double peak structure, which signalsthe transition. The phase factor is smaller in the low temperature phase (P<∼2.8). Hence in theproductρ 〈cos(θ)〉 the low temperature peak is suppressed. Now we perform the inegrals〈P〉=∫dx xρ(x)〈cos(θ)〉x ,〈P2〉=∫dx x2ρ(x)〈cos(θ)〉x . (4.1)In Figure 1(b) we plot the plaquette expectation value〈P〉 as a function of the couplingβ . Theβ -dependence is given by Equations (2.7)-(2.9). We indeed find that including the phase factordoes shift the transition to lower values of the coupling, which also means to lower temperatures.This can also be seen in a shift of the peak of the susceptibility of the plaquetteχP ≡〈P2〉−〈P〉2,which we plot in Figure 1(c). Since theλ parameter introduces a systematic error, which can beseen by the relative large critical coupling ofβc = 4.976(4) in comparison to the result form multi-parameter reweightingβc = 4.938(2) [1], we perform the a linear extrapolation ofλ → 0, fromλ = 0.02, λ = 0.015 andλ = 0.01. We show the extrapolation in Figure 1(d). The extrapolatedresultβ = 4.938(4) (including the phase factor) and the result from multi-parameter reweightingare in very good agreement. From now on we only give results for λ/m= 0.2, theλ dependenceis however expected to be smaller for largerµ .In the range of 0.4<∼aµ<∼0.5 for the 44 lattice, as well as 0.3<∼aµ<∼0.4 for the 64 lattice wefound two transitions in the plaquette expectation value〈P〉(β ). The two critical couplings resultin two transition lines in the phase diagram. The two transition lines are almost perpendicular in the(β ,µ)-diagram, and join in a triple-point of the phase diagram. in Figure 2(a) we show the phasediagram in physical units. The scale was set by the Sommer radius r0, measured on a 103 × 20lattice. In both cases, the triple point is located aroundµ triq ≈ 300 MeV, however its temperature(T tri) decreases fromT tri ≈ 148MeV on the 44 lattice toT tri ≈ 137MeV on the 64 lattice.Also shown in Figure 2(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.PoS(LAT2005)163163 / 5The QCD phase diagram at finite density Christian Schmidt5. The quark number densityTo reveal the properties of the new phase located in the lowerright corner of the phase dia-gram, we calculated the quark number density, at constant coupling β and at constant temperaturerespectively. To obtain the densitynq we perform the following integration〈dlndetMd(aµ)〉=∫dx〈dlndetMd(aµ)cos(θ)〉xρ(x) (5.1)The thermodynamic quantitynq are given as usual bynq =1a3N3s Nt〈dlndetMd(aµ)〉(5.2)In Figure 2(b) we show the baryon number density, which is related to the quark number densityby nB = nq/3. The results are plotted in physical units and correspond ta constant temperature ofT ≈ 143 MeV (44 lattice),T ≈ 124 MeV (64 lattice) andT ≈ 93 MeV (64×8 lattice). In order todivide 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 givenvalue ofNt , divided by its continuum Stefan-Boltzmann value. At the same value of the chemicalpotential where we find also a peak in the susceptibility of the plaquette (µc), we see a sudden risein the baryon number density. Thus forµ > µc we enter a phase of dense matter. The transitionoccurs at a density of(2−3)×nN, wherenN denotes nuclear matter density. Above the transition,the density reaches values of(10−20)×nN. Quite similar results have been obtained recently bysimulations in the canonical ensemble [8].References[1] Z. Fodor and S. D. Katz,Phys. Lett.B534 (2002) 87 [hep-lat/0104001].[2] Z. Fodor and S. D. Katz,JHEP0203 (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. Crompton, [hep-lat/0301001];M. D’Elia and M. P. Lombardo,Phys. Rev.D67 (2003) 014505; P. de Forcrand and O. Philipsen,Nucl.Phys.B642 (2002) 290;Nucl. Phys.B673 (2003) 170; V. Azcotiti,et al., [hep-lat/0503010].[3] O. Philipsen, these proceedings.[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 andA. F. Grillo, Phys. Rev. Lett.65 (1990) 2239; A. Gocksch,Phys. Rev. Lett.61 (1988) 2054.[5] X. Q. Luo, Mod. Phys. Lett.A16 (2001) 1615.[6] J. Ambjorn, K. N. Anagnostopoulos, J. Nishimura and J. J.M. Verbaarschot,JHEP0210 (2002) 062.[7] S. Gottlieb, W. Liu, D. Toussaint, R. L. Renken and R. L. Sugar,Phys. Rev.B35 (1987) 2531.[8] A. Alexandru, M. Faber, I. Horvath and K. F. Liu, [hep-lat/0410002].PoS(LAT2005)163163 / 6

The QCD phase diagram at finite density

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 flavours 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

Schmidt, C.

Fodor, Z.

Katz, S.

UNT Digital Library

BROOKHAWEN NATI O N  A L LAB URATO RY BNL-75124-2005-CP The QCD Phase Diagram at Finite Density Christian Schmidt', Zoltan FodorZ9 Sandor Katz3 'Brookhaven National Laboratory, Upton, NY 11973 USA 'Department of Physics, University of Wuppertal, Wuppertal, Germany Institute for Theoretical Physics, Eotvos University, Budapest, Hungary 31nstitute for Theoretical Physics, Eotvos University, Budapest, Hungary Presented at the 23rd International Syrnposiiiin on Lattice Field Theory Trinity College, Dublin, Ireland July 25-30,2005 October 2005 Physics Department Brookhaven National Laboratory P.O. Box 5000 Upton, NY 1 1973-5000 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. C 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. 0 0 BNL-NT-05/38 WUB- 0 5 - 12 ITP-Budapest 6 2 3  The QCD phase diagram at finite density Christian Schmidt* Physics Department, Brookhaven National L.uboratory, Upton, NI: 11973, USA E-mail: cschmidt@bnl . gov Zoltan Fodor Department of Physics, University of Wuppertal, Wuppertal, Germany Institute for Theoretical Physics, Eotvos University, Budapest, Hungary E-mail: fodor@bodri. elte. hu Sandor Katz Institute for Theoretical Physics, Eotvos University, Budapest, Hungary E-nlail: katz@bodri . elte. hu We study the density of states method to explore the phase diagram of the chiral transition on the tempeature and quark chemical potential plane. Four quark flavours are used in the analy- sis. Though the method is quite expensive small lattices show an indication for a triple-point connecting three different phases on the phase diagram. XXIIIrd International Syrnposiurn on Lattice Field Theory Trinity College, Dublin, Ireland 25-30 July 2005 *Speaker. 0 Copyright owned by die author(s1 under the terms nf the Creative Commons Attribution-NonCommercial-Sh~eAlike~icencc. http://pos.sissa.it/ The QCD phase diagram at finite density Christian Schmidt 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 zCC(P)  = 1.U detM[U](p)exp{-SG[U]}. ( 1 . 1 2  Recently many different methods have been developed to cirumvent the complex action problem for small p / T  [ 1,2]. For a recent overview see also [3]. 2. Formulation of the method A very general formulation of the DOS method is the following: One exposed parameter ($) is fixed. The expectation value of a thermodynamic observable (O), according to the usual grand canonical partition function (1. l), can be recovered by the integral (2.1) where the density of states ( p )  is given by the constrained partition function: p(x) = Z&) = .Ug(V) S ( $  -x). (2.2) s 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: f g  = detMexp{-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 $ = P and g ( U )  = 1. In this case p ( $ )  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 configurations, the observable can also be calculated as a function of quark mass (m) and number of flavors[5] ( N f ) .  In this work we chose $ = P and g = ldetMlexp{-Sc}, f = exp{i@}. (2.3) In other words we constrain the plaquette and perform simulations with measure g. In practice, we replace the delta function in Equation (2.2) by a sharply peaked potential [6] .  The constrained partition function for fixed values of the plaquette expectation value can then be written as 163 I 2  The QCD phase diagram atfinite density Christian Schmidt where exp{ -V(x)}  is a Gaussian potential with 2 (2.5) V ( x )  = - y (x -P)  1 2 . We obtain the density of states (p(x)) by the fluctuations of the actual plaquette P around the constraint value x. The fluctuation dissipation theorem gives d -lnp(x) dx =<x-P>,  . (2.6) Before performing the integrals in Equation (2.1) we compute from an ensemble generated at ( P o I P O ) :  ( O f ( W ,  (k P )  = (Of(U)R(PI h 7 P ,  Po>>,/ (R(PIlso7P I Po)) ,  I (f(W,(PIP) = (f(U)R(pu,Pouo,P7Po)),/(R(C17Po7P7Po)),I (2.8) - dx lnP(x7P7P) = ((.-P)R(ruIPoIP7Po)),* (2.7) d (2.9) Here R is given by the quotient of the measure g at the point ( p , p )  and at the simulation point (Po 7 P O  1, ~(PIPoO,P>PO) =s(PU,P)/g(hFLo,Po) = I d e t ( h ) I  ' et(p)' exp { SG (p ) - SG ( P o ) } .  (2.10) Heaving calculated the expressions (2.7)-(2.9), we are able to extrapolate the expectation value of the observable (2.1) to any point (p,P)  in a small region around the simulation point  PO). For any evaluation of (0) ( p , P ) ,  we numerically perform the integrals in Equation (2.1). 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 (r) and directions (pv) of the local plaquette Ppv(y) and its adjoint Ppv(y), -I 1 P = c c 6 [TrPpdY) +Tr$,(y)] y l<p<v14 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 Vp(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. 163 1 3  The QCD phase diagram atJinite density Christian Schmidt 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 (c> 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 P 4.96 4.965 4.97 4.975 4.98 4.985 4.99 4.995 5 (b) 3 2.95 2.9 2.85 2.8 2.75 2.7 P 4.96 4.965 4.97 4.975 4.98 4.985 4.99 4.995 5 (d) 4.99 4.98 4.97 4.96 4.95 4.94 4.93 + 0 0.1 0.2 0.3 0.4 0.5 0.6 Urn Figure 1: Results for Simulations at p = 4.98, p = 0.3, A = 0.02, nf = 4, am = 0.05, and number of lattice points: 44. Shown are: (a) the density of states p ( x ) ,  the phase factor (cos(O)), and their product, (b) the Plaquette as a function of the coupling p, with and without the phase factor, (c) the Susceptibility of the Plaquette as.a function of the coupling p, with and without the phase factor, and (d) the extrapolation & ( A )  to A = 0, with and without the phase factor. 4. The critical line and the determination of a triple-point 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. Fist of all we check, whether we can reproduce old results with our new method. We show in Figure l(a) results from a Simulation at p = 0.3, p = 4.98 and A = 0.02. Here we plot the density of states ( p )  and the real part of the phase factor (cos(€))) as a function of the constrained 163 14 The QCD phase diagram atjnite density Christian Schmidt (a) 170 160 150 140 130 120 110 100 90 (b) 3.5 3 2.5 2 1.5 1 0.5 0 .n s 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 Figure 2: The phase diagram in physical (a), and the quark number density at constant temperature T = 143 MeV (44 lattice), T = 124MeV (64 lattice) and T = 93 MeV (63 x 8 lattice). plaquette value. The results have been interpolated in P,  in order to obtain a better result for the necessary integration over P. The distribution p shows a clear double peak structure, which signals the transition. The phase factor is smaller in the low temperature phase (P52.8). Hence in the product p (cos(6)) the low temperature peak is suppressed. Now we perform the integrals In Figure l(b) we plot the plaquette expectation value ( P )  as a function of the coupling p. The p-dependence is given by Equations (2.7)-(2.9). We indeed find that including the phase factor does shift the transition to lower values of the coupling, which also means to lower temperatures. This can also be seen in a shift of the peak of the susceptibility of the plaquette x p  (P') - (P)2 ,  which we plot in Figure l(c). Since the A parameter introduces a systematic error, which can be seen by the relative large critical coupling of pc = 4.976(4) in comparison to the result form multi- parameter reweighting pc = 4.938(2) [l], we perform the a linear extrapolation of A --+ 0, from A = 0.02, A = 0.015 and A = 0.01. We show the extrapolation in Figure l(d). The extrapolated result p = 4.938(4) (including the phase factor) and the result from multi-parameter reweighting are in very good agreement. From now on we only give results for A/m = 0.2, the A dependence is however expected to be smaller for larger p. In the range of 0 .42ap50.5  for the 44 lattice, as well as 0 .35ap50.4 for the 64 lattice we found two transitions in the plaquette expectation value ( P )  (p) .  The two critical couplings result in two transition lines in the phase diagram. The two transition lines are almost perpendicular in the (p  , p)-diagram, and join in a triple-point of the phase diagram. in Figure 2(a) we show the phase diagram in physical units. The scale was set by the Sommer radius ro, measured on a lo3 x 20 lattice. In both cases, the triple point is located around pp M 300 MeV, however its temperature ( T t ~ )  decreases from T t ~  M 148MeV on the 44 lattice to Ttr' M 137MeV on the 64 lattice. Also shown in Figure 2(a) are points from simulations with quark mass nnz = 0.03. The phase boundary turned out to be - within our statistical uncertainties - independent of the the mass. 163 1 5  The QCD phase diagram atjnite density Christian Schmidt 5. The quark number density To reveal the properties of the new phase located in the lower right corner of the phase dia- gram, we calculated the quark number density, at constant coupling p and at constant temperature respectively. To obtain the density ny we perform the following integration dln detM dln detM ( d(a,u) ) = Sdx ( d(ap)  cor(8))xp(x) The thermodynamic quantity n4 are given as usual by (5.1) 1 dlndetM nq= =( d(ap)  ) In Figure 2(b) we show the baryon number density, which is related to the quark number density by n~ = n,/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 (64 lattice) and T M 93 MeV (64 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(N,)/SB, which is the Stefan-Boltzmann value of a free lattice gas of quarks at a given value of N,,  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 (&), 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 nN, where nN denotes nuclear matter density. Above the transition, the density reaches values of (10 - 20) x nN. Quite similar results have been obtained recently by simulations in the canonical ensemble [8]. 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, Chep-latlO3 OlOOl]; M. D’Elia and M. P. Lombardo, Phys. Rev. D67 (2003) 014505; P. de Forcrand and 0. Philipsen, N u l .  Phys. B642 (2002) 290; Nucl. Phys. B673 (2003) 170; V. Azcotiti, et al., [hep-lat/0503010]. [3] 0. Philipsen, these proceedings. [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, N L ~ .  Phys. B302 (1988) 204; V. Azcoiti, G. di Carlo and A. E Grillo, Phys. Rev. Lett. 65 (1990) 2239; A. Gocksch, Phys. Rev. Lett. 61 (1988) 2054. [5] X. Q. Luo, Mod. Phys. Lett. A16 (2001) 1615. [6] J. Arnbjorn, K. N. Anagnostopoulos, J. 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. [8] A. Alexandru, M. Faber, I. Horvath and K. F. Liu, [hep- lat / 0 4 10 0 0 2 ] . 163 I 6  

The Qcd Phase Diagram at Finite Density.

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The QCD phase diagram at finite density

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