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Sign problem in $Z_3$-symmetric effective Polyakov-line model

Takehiro Hirakida, Junpei Sugano, Hiroaki Kouno, Junichi Takahashi, Masanobu Yahiro

Abstract

As an effective model corresponding to $Z_3$-symmetric QCD ($Z_3$-QCD), we construct a $Z_3$-symmetric effective Polyakov-line model ($Z_3$-EPLM) by using the logarithmic fermion effective action. Since $Z_3$-QCD tends to QCD in the zero temperature limit, $Z_3$-EPLM also agrees with the ordinary effective Polyakov-line model (EPLM) there; note that ordinary EPLM does not possess $Z_3$ symmetry. Our main purpose is to discuss a sign problem appearing in $Z_3$-EPLM. The action of $Z_3$-EPLM is real, when the Polyakov line is not only real but also its $Z_3$ images. This suggests that the sign problem becomes milder in $Z_3$-EPLM than in EPLM. In order to confirm this suggestion, we do lattice simulations for both EPLM and $Z_3$-EPLM by using the reweighting method with the phase quenched approximation. In the low-temperature region, the sign problem is milder in $Z_3$-EPLM than in EPLM. We also propose a new reweighting method. This makes the sign problem very weak in $Z_3$-EPLM.

Sign problem in $Z_3$-symmetric effective Polyakov-line model

Abstract

As an effective model corresponding to -symmetric QCD (-QCD), we construct a -symmetric effective Polyakov-line model (-EPLM) by using the logarithmic fermion effective action. Since -QCD tends to QCD in the zero temperature limit, -EPLM also agrees with the ordinary effective Polyakov-line model (EPLM) there; note that ordinary EPLM does not possess symmetry. Our main purpose is to discuss a sign problem appearing in -EPLM. The action of -EPLM is real, when the Polyakov line is not only real but also its images. This suggests that the sign problem becomes milder in -EPLM than in EPLM. In order to confirm this suggestion, we do lattice simulations for both EPLM and -EPLM by using the reweighting method with the phase quenched approximation. In the low-temperature region, the sign problem is milder in -EPLM than in EPLM. We also propose a new reweighting method. This makes the sign problem very weak in -EPLM.

Paper Structure

This paper contains 17 sections, 33 equations, 20 figures.

Table of Contents

  1. Introduction
  2. $Z_3$-QCD
  3. $Z_3$-symmetric effective Polyakov-line model
  4. Effective Polyakov-line model
  5. Particle-hole symmetry
  6. $Z_3$-symmetric effective Polyakov-line model ($Z_3$-EPLM)
  7. Fermion potential in EPLM
  8. Reweighting methods
  9. Phase quenched approximation and reweighting method
  10. Sign problem in EPLM
  11. Improved reweighting method
  12. Observables
  13. Numerical results
  14. Results at $\mu =0$
  15. Phase quenched reweighting
  16. ...and 2 more sections

Figures (20)

  • Figure 1: Allowed regions of (a) $P_{\hbox{\boldmath ${ x}$}}$ and (b) $Q_{\hbox{\boldmath ${ x}$}}$ in the complex plane. In (a), three vertices correspond to the deconfinement points $(1,0)$, $(-1/2,\sqrt{3}/2)$ and $(-1/2,-\sqrt{3}/2)$, while they are degenerate at the point (1,0) in (b). In (b), the confinement point is also degenerate with the three deconfinement points. In $Z_3$-EPLM, the sign problem is severest at the blue circles.
  • Figure 2: ${\rm Re}[{L}_{\rm F}]$ in $\varphi_{r,{\hbox{\boldmath ${ x}$}}}$--$\varphi_{g,{\hbox{\boldmath ${ x}$}}}$ plane for the case of EPLMWO.The fermion potential takes minimum at the origin. In the calculation, we set $M/T=10$, and set (a) $\mu =0.5M$, (b) $\mu =M$, and (c) $\mu =1.5M$, respectively. Note $\varphi_{b,{\hbox{\boldmath ${ x}$}}}=-\varphi_{r_{\hbox{\boldmath ${ x}$}}}-\varphi_{g,{\hbox{\boldmath ${ x}$}}}$. Due to the P-H symmetry, the result in (c) is (almost) the same as that in (a) up to the total scale factor.
  • Figure 3: ${\rm Re}[{L}_{{\rm F},Z_3}]$ in $\varphi_{r,{\hbox{\boldmath ${ x}$}}}$--$\varphi_{g,{\hbox{\boldmath ${ x}$}}}$ plane for the case of $Z_3$-EPLM. The fermion potential Largangian takes minimum at the confinement and deconfinement points. In the calculation, we set $M/T=10$, and set (a) $\mu =0.5M$, (b) $\mu = M$, and (c) $\mu =1.5M$, respectively. Note $\varphi_{b,{\hbox{\boldmath ${ x}$}}}=-\varphi_{r_{\hbox{\boldmath ${ x}$}}}-\varphi_{g,{\hbox{\boldmath ${ x}$}}}$. Due to the P-H symmetry, the result in (c) is (almost) the same as that in (a) up to the total scale factor.
  • Figure 4: The effective potential $L_{\rm H}$ induced from the Haar measure is shown in $\varphi_{r,{\hbox{\boldmath ${ x}$}}}$--$\varphi_{g,{\hbox{\boldmath ${ x}$}}}$ plane. $L_{\rm H}$ takes minimum at the confinement points.
  • Figure 5: The ${\rm Re}[{L}_{\rm F}]$-${\rm Im}[{L}_{\rm F}]$ relation in the 3 flavor EPLMWO. We set $M/T=10$ and $\mu /M=0.95$.
  • ...and 15 more figures