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Phase diagram of 4D SU(3) Yang-Mills theory at $θ=π$ via imaginary theta simulations

Akira Matsumoto, Mitsuaki Hirasawa, Jun Nishimura, Atis Yosprakob

Abstract

It has been speculated that the CP symmetry of 4D SU(3) Yang-Mills theory at $θ=π$ is spontaneously broken in the confined phase, and it is recovered precisely at the deconfining temperature. The direct simulation of the theory at $θ=π$ is, however, difficult due to the sign problem. We therefore simulate the theory with an imaginary theta parameter and perform analytic continuation to the real theta to explore the phase diagram. We implement the stout smearing technique in the hybrid Monte Carlo simulation to recover the topological property of the gauge field. The smearing-time dependence of the observable is investigated using the reweighting method with respect to the smearing step parameters, and a clear scaling behavior is observed. The order parameter of the CP symmetry is then computed in the scaling region to detect symmetry breaking. We report preliminary results on the expected CP breaking and restoration temperature.

Phase diagram of 4D SU(3) Yang-Mills theory at $θ=π$ via imaginary theta simulations

Abstract

It has been speculated that the CP symmetry of 4D SU(3) Yang-Mills theory at is spontaneously broken in the confined phase, and it is recovered precisely at the deconfining temperature. The direct simulation of the theory at is, however, difficult due to the sign problem. We therefore simulate the theory with an imaginary theta parameter and perform analytic continuation to the real theta to explore the phase diagram. We implement the stout smearing technique in the hybrid Monte Carlo simulation to recover the topological property of the gauge field. The smearing-time dependence of the observable is investigated using the reweighting method with respect to the smearing step parameters, and a clear scaling behavior is observed. The order parameter of the CP symmetry is then computed in the scaling region to detect symmetry breaking. We report preliminary results on the expected CP breaking and restoration temperature.
Paper Structure (8 sections, 16 equations, 6 figures)

This paper contains 8 sections, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. 4D SU(N) Yang-Mills theories with the theta term
  3. Lattice regularization and simulation method
  4. Two-dimensional version of the parallel tempering
  5. Numerical results
  6. CP symmetry at theta = pi
  7. Deconfining temperature for nonzero theta
  8. Summary

Figures (6)

  • Figure 1: The topological charge expectation value is plotted against the flow time $\rho N_\rho$ for several values of $\rho$. The simulation was performed on the $16^3 \times 5$ lattice at $T/T_c = 0.9$ and $\tilde{\theta}_\mathrm{L}/\pi = 0.6$.
  • Figure 2: The histograms of the rescaled topological charge $Q=wQ_\mathrm{L}[\mathcal{U}]$ at $\theta=0$ are shown for several values of $N_\rho$. The simulation was performed on the $16^3 \times 5$ lattice at $T/T_c = 0.90$ and $\rho = 0.04$.
  • Figure 3: The integrated autocorrelation time at $(T/T_c,~ \tilde{\theta}_\mathrm{L}) = (1.0,~ 0.5)$ obtained by the 1d and 2d versions of parallel tempering.
  • Figure 4: The normalized topological charge expectation value $-\braket{Q}_{\tilde{\theta}}/\braket{Q^2}_0$ is plotted against $\tilde{\theta}/\pi$ for various temperatures $0.75 \le T/T_c \le 1.02$. The simulation was performed on the $16^3 \times 5$ lattice with the smearing parameters $\rho = 0.04$ and $N_\rho = 80$.
  • Figure 5: The fitting curves of the data points in Figure \ref{['fig_Q_itheta']} are analytically continued to real $\theta$, and plotted against $\theta/\pi$ with the error bands.
  • ...and 1 more figures