[Paper Review] Probing the Aoki phase with N_f=2 Wilson fermions at finite temperature

E.‐M. Ilgenfritz, Werner Kerler+3 more|ArXiv.org|Nov 28, 2005

Theoretical and Computational Physics3 citations

TL;DR

This study investigates the Aoki phase in QCD with N_f=2 Wilson fermions at finite temperature using lattice simulations at β=4.6 and N_τ=4. By computing the parity-flavor breaking order parameter ⟨ψ̄iγ₅τ³ψ⟩ and extrapolating to zero source mass h→0, the authors confirm the existence of the Aoki phase in a narrow κ interval (0.1968 ≤ κ ≤ 0.19715), shifted to lower κ compared to the zero-temperature case, though it remains indistinguishable from the thermal transition line.

ABSTRACT

In this letter we report on a numerical investigation of the Aoki phase in the case of finite temperature which continues our former study at zero temperature. We have performed simulations with Wilson fermions at $β=4.6$ using lattices with temporal extension $N_τ=4$. In contrast to the zero temperature case, the existence of an Aoki phase can be confirmed for a small range in $κ$ at $β=4.6$, however, shifted slightly to lower $κ$. Despite fine-tuning $κ$ we could not separate the thermal transition line from the Aoki phase.

Motivation & Objective

• To investigate the existence and structure of the Aoki phase in QCD with N_f=2 dynamical Wilson fermions at finite temperature.
• To determine whether the Aoki phase persists at finite temperature and how it is positioned relative to the thermal deconfinement transition.
• To clarify the relationship between the Aoki phase and the finite-temperature phase transition line, especially whether they can be separated.
• To examine the behavior of the parity-flavor breaking order parameter ⟨ψ̄iγ₅τ³ψ⟩ in the thermodynamic limit via extrapolation to h→0.

Proposed method

• Simulations performed on 8³×4 and 10³×4 lattices at β=4.6 and N_τ=4 using Hybrid Monte Carlo (HMC) with a small explicit parity-breaking source h.
• The order parameter ⟨ψ̄iγ₅τ³ψ⟩ was computed at multiple h values (0.001 ≤ h ≤ 0.02) and extrapolated to h→0 using a power-law fit: σ(h) = A + Bh^C + …
• Fisher plots were used to visualize the extrapolation and assess the non-zero intercept A, indicating the presence of the Aoki phase.
• Autocorrelation times of the Polyakov loop were measured to assess thermalization and critical slowing down near the phase boundary.
• The analysis included volume dependence checks using both 8³×4 and 10³×4 lattices to confirm the signal is not finite-size artifact.
• The results were compared to previous studies at zero temperature and finite temperature (Aoki et al., 1996; 1998), particularly regarding the location of the Aoki phase endpoint.

Experimental results

Research questions

• RQ1 Does the Aoki phase persist at finite temperature in the N_f=2 Wilson fermion formulation, and if so, in what κ range?
• RQ2 How does the finite-temperature Aoki phase compare to the zero-temperature phase in terms of κ and β?
• RQ3 Is the Aoki phase's boundary separable from the finite-temperature deconfinement transition line at β=4.6 and N_τ=4?
• RQ4 What is the behavior of the order parameter ⟨ψ̄iγ₅τ³ψ⟩ as h→0, and does it show a non-zero intercept indicating spontaneous parity-flavor symmetry breaking?
• RQ5 Does the Aoki phase exhibit volume dependence, and is it resolved only on larger lattices?

Key findings

• The Aoki phase is confirmed at finite temperature (N_τ=4, β=4.6) in the κ interval 0.1968 ≤ κ ≤ 0.19715, with a non-zero order parameter at h→0.
• The Aoki phase is shifted to lower κ compared to the zero-temperature case, where it was found to end near κ=0.1984 at β=4.6.
• The order parameter ⟨ψ̄iγ₅τ³ψ⟩ at h→0 is non-zero in the interval 0.1968 ≤ κ ≤ 0.19715, with the intercept A from the fit σ(h) = A + Bh^C + … being non-zero and volume-dependent.
• At κ=0.19720, the order parameter vanishes at h→0, indicating the phase boundary is located beyond this value.
• Critical slowing down is observed near κ≈0.19705, with large exponential autocorrelation times (τ_exp > 4000 HMC trajectories), indicating proximity to a phase transition.
• Despite fine-tuning and high statistics, the Aoki phase boundary cannot be separated from the finite-temperature transition line within the current resolution.