Quantum critical behavior in three dimensional lattice Gross-Neveu models

We study quantum critical behavior in three dimensional lattice Gross-Neveu models containing two four-component massless Dirac fermions. We focus on two models with $SU(2)$ flavor symmetry and either a $Z_2$ or a $U(1)$ chiral symmetry. Both models could not be studied earlier due to sign problems. We use the fermion bag approach which is free of sign problems and compute critical exponents at the phase transitions. We estimate $\nu =0.83(1)$, $\eta =0.62(1)$, ${\eta }_{\psi }=0.38(1)$ in the $Z_2$ and $\nu =0.849(8)$, $\eta =0.633(8)$, ${\eta }_{\psi }=0.373(3)$ in the $U(1)$ model.

Figure 1

A pictorial representation of the bond couplings $U_{\mathbb{L}}$ (left), $U_{\mathbb{F}}$ (center) and $U_{\mathbb{B}}$ (right) discussed in the text. Each bond refers to the four-fermion interaction term of the form ${\overline{\chi }}_x{\chi }_x{\overline{\chi }}_y{\chi }_y$.

Figure 2

Plot of the chiral susceptibility at $U=\infty$ for the $Z_2$ (top) and $U(1)$ (bottom) models. The solid curve in the top graph is a fit to the constant for $L\ge 16$, while in the bottom graph it is a fit to the finite size scaling form (5) for $L\ge 10$ obtained from chiral perturbation theory.

Figure 3

Plots of $\chi /L^{2-\eta }$ and $R_f{L}^{2+{\eta }_{\psi }}$ as a function of $U$ for $L$ from 12 to 36. The solid lines show the combined fit which gives $U_c=0.0893(1)$, $\nu =0.83(1)$, $\eta =0.62(1)$ and ${\eta }_{\psi }=0.38(1)$ in the $Z_2$ case (top row) and $U_c=0.1558(4)$, $v=0.82(2)$, $\eta =0.63(2)$, ${\eta }_{\psi }=0.37(1)$ in the $U(1)$ case (bottom row).