A numerical reinvestigation of the Aoki phase with N_f=2 Wilson fermions at zero temperature

TL;DR

The paper numerically reinvestigates the Aoki phase in lattice QCD with $N_f=2$ Wilson fermions at zero temperature by introducing an explicit symmetry-breaking source term $h\bar{\psi} i \gamma_{5} \tau^{3} \psi$ and measuring the order parameter $<\bar{\psi} i \gamma_{5} \tau^{3} \psi>_h$ across a range of $\beta$ and $\kappa$, extrapolating to $h\to 0$. Using Hybrid Monte Carlo on lattices from $4^{4}$ to $12^{4}$ and $eta=4.0,4.3,4.6,5.0$, they find evidence for a parity-flavor breaking phase at $(\beta,\kappa)=(4.0,0.22)$ and $(4.3,0.21)$, but no such phase at higher $\beta$ values, with $\langle\bar{\psi} i \gamma_{5} \tau^{3} \psi\rangle_{h=0}$ vanishing as $\beta$ increases. The results indicate the Aoki phase is confined to $\beta\lesssim 4.6$, providing bounds on the phase boundaries and highlighting finite-volume and scaling limitations that motivate further, larger-scale studies. This has implications for simulations using Wilson-type fermions and for the interface with overlap/domain-wall formulations in the continuum limit.

Abstract

We report on a numerical reinvestigation of the Aoki phase in lattice QCD with two flavors of Wilson fermions where the parity-flavor symmetry is spontaneously broken. For this purpose an explicitly symmetry-breaking source term $h\barψ i γ_{5} τ^{3}ψ$ was added to the fermion action. The order parameter $<\barψ i γ_{5}τ^{3}ψ>$ was computed with the Hybrid Monte Carlo algorithm at several values of $(β,κ,h)$ on lattices of sizes $4^4$ to $12^4$ and extrapolated to $h=0$. The existence of a parity-flavor breaking phase can be confirmed at $β=4.0$ and 4.3, while we do not find parity-flavor breaking at $β=4.6$ and 5.0.

Figures (5)

• Figure 1: The phase diagram proposed by Aoki et al. in the $(g^2, m_0)$ plane (left-hand side) and in the $(\beta, \kappa)$ plane (right-hand side). The shaded region labeled $B$ denotes the phase where flavor and parity are spontaneously broken. Both symmetries are conserved in regions labeled $A$.
• Figure 2: Results for $\langle\bar{\psi}i\gamma_{5}\tau^{3}\psi\rangle$ as a function of $\kappa$ and $h$ at $\beta=4.0$ on a $6^4$ lattice.
• Figure 3: In the left column data for $\langle\bar{\psi}i\gamma_{5}\tau^{3}\psi\rangle$ from a $6^4$ lattice are shown as a function of $\kappa$ at several values of $h$ (the lines are spline interpolations to guide the eye). The extrapolation to $h=0$ in the infinite volume limit is shown in the center column of this figure. The right column shows the Fisher plots with the corresponding fitting function. The upper row shows results for $\beta=4.0$, the lower one for $\beta=4.3$.
• Figure 4: In the left column data for $\langle\bar{\psi}i\gamma_{5}\tau^{3}\psi\rangle$ from lattices of various sizes are shown as a function of $\kappa$ at $h=0.005$ fixed (the lines are drawn to guide the eye). The extrapolation to $h=0$ in the infinite volume limit is shown in the center column. In the right column the corresponding Fisher plots are shown. The upper row contains data at $\beta=4.6$ from lattices of sizes $6^4$, $8^4$ and $10^4$. The lower row shows measurements for $\beta=5.0$ from lattices of size from $6^4$ to $12^4$.
• Figure 5: The part of the phase diagram studied in this work. Squares denote ($\beta,\kappa$) pairs where a finite value of $\langle\bar{\psi}i\gamma_{5}\tau^{3}\psi\rangle_{h=0}$ is found. Diamonds refer to points where $\langle\bar{\psi}i\gamma_{5}\tau^{3}\psi\rangle_{h=0}=0$. Stars denote points where a finite value of $\langle\bar{\psi}i\gamma_{5}\tau^{3}\psi\rangle_{h=0}$ is uncertain. The lines indicate the position of the critical lines $\kappa^{(l)}_c(\beta)$ and $\kappa^{(u)}_c(\beta)$. The shaded region labeled $B$ refers to the parity-flavor breaking phase. The point ($\beta,\kappa$)=($4.0,0.215$) marked by a circle seems to lie very close to the border of the broken phase.