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Thermodynamics at fixed lattice spacing

Thermodynamics at fixed lattice spacing. Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD Collaboration. This talk is based on arXiv:0809.2842 [hep-lat]. JPS meeting, Yamagata Univ., Yamagata , 20-23 Sept. 2008. /13. Introduction. Equation of State (EOS)

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Thermodynamics at fixed lattice spacing

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  1. Thermodynamics at fixed lattice spacing Takashi Umeda (Univ. of Tsukuba) for WHOT-QCD Collaboration This talk is based on arXiv:0809.2842 [hep-lat] JPS meeting, Yamagata Univ., Yamagata , 20-23 Sept. 2008 T.Umeda (Tsukuba) /13

  2. Introduction • Equation of State (EOS) • is important for phenomenological study of QGP, etc. • Methods to calculate the EOS have been established, • e.g. Integral method J. Engels et al. (’90). • Temperature T=1/(Nτa) is varied by a(β)at fixed Nτ • The EOS calculation requires huge computational cost, • in which T=0 calculations dominate despite T>0 study. • Search for a Line of Constant Physics (LCP) • beta functions at each temperature • T=0 subtraction at each temperature T.Umeda (Tsukuba) /13

  3. T-integration method to calculate the EOS We propose a new method (“T-integration method”) to calculate the EOS at fixed scales (*) Temperature T=1/(Nτa) is varied by Nτ at fixed a(β) Our method is based on the trace anomaly (interaction measure), and the thermodynamic relation. (*) fixed scale approach has been adopted in L.Levkova et al. (’06) whose method is based on the derivative method. T.Umeda (Tsukuba) /13

  4. Notable points in T-integration method • Our method can reduce computational cost at T=0 drastically. • Zero temperature subtraction is performed • using a common T=0 calculation. • Line of Constant Physics (LCP) is trivially exact (even in full QCD). • Only the beta functions at the simulation point are required. • However ... • Temperatures are restricted by integer Nτ. •  Sufficiently fine lattice is necessary. Example of Temp. resolution (a=0.07fm) Integer Nτ provides - higher resolution at T~Tc - lower resolution at high T T~Tc is important for EOS T.Umeda (Tsukuba) /13

  5. Simulation parameters (isotropic lattices) We present results from SU(3) gauge theory as a test of our method • plaquette gauge action on Nσ3 x Nτ lattices • Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3-type of lattices. (1)β=6.0, V=(16a)3 a=0.094fm (2)β=6.0, V=(24a)3 a=0.094fm (3)β=6.2, V=(22a)3 a=0.078fm T.Umeda (Tsukuba) /13

  6. Simulation parameters (anisotropic lattice) Anisotropic lattice is useful to increase Temp. resolution, we also test our method on an anisotropic lattice aσ≠ aτ • plaquette gauge action on Nσ3 x Nτ lattices • with anisotropy ξ=aσ/aτ=4 V=(20aσ)3 =(1.95fm)3 V=(30aσ)3 =(2.92fm)3 V=(40aσ)3 =(3.89fm)3 - critical temp. β=6.1, ξ=4 V=(20aσ)3 =(1.95fm)3 a=0.097fm - EOS calculation - static quark free energy T.Umeda (Tsukuba) /13

  7. Trace anomaly ( e - 3p )/T4 on isotropic lattices (1) β=6.0, a=0.094fm, V=(1.5fm)3 (2) β=6.0, a=0.094fm, V=(2.2fm)3 (3) β=6.2, a=0.068fm, V=(1.5fm)3 beta function : G.Boyd et al. (’96) lattice scale r0 : R.Edwards et al. (’98) • Excellent agreement • between (1) and (3) •  scale violation is small • a=0.1fm is good • Finite volume effect • appears below & near Tc •  volume size is important • V=(2fm)3 is necessary. dotted lines : cubic spline T.Umeda (Tsukuba) /13

  8. is required in SU(3) gauge theory. T.R.Klassen (’98) Trace anomaly ( e - 3p )/T4 on aniso. lattice (1) ξ=4, a=0.097fm, V=(2.0fm)3 (2) ξ=1, a=0.094fm, V=(2.2fm)3 beta function : obtained by r0/aσ fit r0/aσdata H.Matsufuru et al. (’01) • Anisotropic lattice is useful • to increase Temp. resolution. dotted lines : cubic spline T.Umeda (Tsukuba) /13

  9. Pressure & Energy density • Integration • is performed with the cubic • spline of (e-3p)/T4 • Cubic spline vs trapezoidal inte. • yields small difference ~ 1σ • Our results are roughly • consistent with previous results. • Unlike the fixed Nτ approach, scale/temp. is not constant. •  Lattice artifacts increase • as temperature increases. T.Umeda (Tsukuba) /13

  10. Transition temperature at fixed scale • T-dependence of • the (rotated) Polyakov loop • and its susceptibility • No renormalization is • required upto overall factor • due to the fixed scale. • Rough estimation of • critical temperature • is possible. • Tc = 280~300 MeV • at β=6.1, ξ=4 • (SU(3) gauge theory) T.Umeda (Tsukuba) /13

  11. Static quark free energy at fixed scale Static quark free energies at fixed scale color singlet static quark free energy V(r) • Due to the fixed scale, • no renomalization constant • is required. •  small thermal effects in V(r) • at short distance • (without any matching) • Easy to distinguish • temperature effect of V(r) • from scale & volume effects T.Umeda (Tsukuba) /13

  12. Conclusion • We studied thermodynamics of SU(3) gauge theory at fixed lattice scale • Our method ( T-integration method ) works well to calculate the EOS • Fixed scale approach is also useful for - critical temperature - static quark free energy - etc. • Our method is also available in full QCD !! Therefore ... T.Umeda (Tsukuba) /13

  13. Toward full QCD calculations • Our method is suited for • already performed high statistics full QCD results. • When beta functions are (able to be) known at a simulation point • and T=0 configurations are open to the public, • our method requires no additional T=0 simulation !! • We are pushing forward in this direction • using CP-PACS/JLQCD results in ILDG • (Nf=2+1 Clover+RG, a=0.07fm, mps/mv=0.6) • Our final goal is to study • thermodynamics on the physical point • with 2+1 flavors of Wilson quarks see PACS-CS talks T.Umeda (Tsukuba) /13

  14. Pressure & Energy density T.Umeda (Tsukuba) /13

  15. G.Boyd et al. (’96) Pressure & Energy density T.Umeda (Tsukuba) /13

  16. Simulation parameters (isotropic lattices) We present results from SU(3) gauge theory as a test of our method • plaquette gauge action on Nσ3 x Nτ lattices • Jackknife analysis with appropriate bin-size To study scale- & volume-dependence, we prepare 3-type of lattices. T.Umeda (Tsukuba) /13

  17. Pressure & Energy density • Integration • is performed with the cubic • spline of (e-3p)/T4 • Our results are roughly • consistent with previous results. -- mild scale violation -- Large volume is important • Unlike the fixed Nτ approach, scale/temp. is not constant. •  Lattice artifacts increase • as temperature increases. T.Umeda (Tsukuba) /13

  18. EOS on an anisotropic lattice beta function : obtained by r0/aσ fit r0/aσdata H.Matsufuru et al. (’01) • Anisotropic lattice is useful • to increase Temp. resolution. • Results are roughly consistent • with previous & isotropic results • Additional coefficients are • required to calculate (e-3p)/T4 is required in SU(3) gauge theory. T.R.Klassen (’98) T.Umeda (Tsukuba) /13

  19. EOS on an anisotropic lattice beta function : obtained by r0/aσ fit r0/aσdata H.Matsufuru et al. (’01) G.Boyd et al. (’96) • Anisotropic lattice is useful • to increase Temp. resolution. • Results are roughly consistent • with previous & isotropic results • Additional coefficients are • required to calculate (e-3p)/T4 is required in SU(3) gauge theory. T.R.Klassen (’98) T.Umeda (Tsukuba) /13

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