Phase transitions on the dark side of the Gross-Neveu model

TL;DR

The paper investigates phase transitions in the 2+1D Gross-Neveu model, focusing on the dark side where an O($M$) symmetry may spontaneously break at repulsive couplings and connect to a quantum anomalous Hall state. It constructs a lattice model with exact O($2N$) symmetry and analyzes it with sign-problem-free determinant Quantum Monte Carlo to explore the repulsive regime, revealing an emergent O($4N$) symmetry structure in the low-energy theory. The results show a symmetry-breaking transition from a Dirac semimetal to an ordered insulator with a transition that is weakly first-order for $N=2$ and becomes more strongly first-order as $N$ increases; finite chemical potential drives superconductivity, breaking O($2N$) to a subgroup. Overall, the findings support the unified field-theory picture that organizes mass terms into O($4N$) irreps and delineate the distinct roles of QAH (attractive side) and the repulsive GN transition, while elucidating lattice versus low-energy symmetry breaking patterns.

Abstract

Gross-Neveu model in 2+1 dimensions exhibits a continuous transition from gapless Dirac semimetal to the gapped quantum anomalous Hall (QAH) insulator at a finite (attractive) coupling, at which the inversion and time-reversal symmetry become spontaneously broken, and the flavor O($M$) symmetry remains preserved. A unification of leading order parameters of 2+1 dimensional $N$ four-component Dirac fermions collects all Lorentz-singlet mass-like fermion bilinears, except the one condensing in the QAH state, into an irreducible representation of the O($M=4N$), and predicts another phase transition in the Gross-Neveu model to occur at a strong (repulsive) coupling. Here, a fermionic auxiliary-field quantum Monte Carlo algorithm is employed in order to study a lattice realization of the Gross-Neveu field theory in the repulsive regime, where the sign problem is absent. We indeed find the O($4N$) symmetry breaking transition out of Dirac semimetal to occur and to be weakly first-order for $N=2$, relevant to graphene. The size of the discontinuity and the magnitude of the critical coupling, however, both grow with $N$. Adding a finite chemical potential is found to break the symmetry and cause superconductivity. These results are in broad agreement with the predictions of the unified field theory. Our lattice model also displays an interesting exact O($2N$) symmetry, a subgroup of the low-energy O($4N$), and has the ordered ground state with the order parameter that belongs to its $N(2N-1)$-dimensional representation. Other order parameters are also examined, and a certain hierarchy among those that belong to different representations of the exact $O(2N)$ is observed.

Figures (4)

• Figure 1: Spin structure factor at $\boldsymbol{q}=0$ (top) and scaled potential energy (bottom) as a function of $\lambda/t$ for different values of $N$. The structure factor indicates a transition from a Dirac SM to an AFM insulator that gets shifted towards larger $\lambda$ with increasing $N$. The potential energy shows a discontinuity, which becomes more pronounced as $N$ increases. In (c), (d), (e) and (f) the scale of the $x$-axis is adjusted in order to better resolve the split of the data with different lattice sizes.
• Figure 2: Spin-structure factor at $\boldsymbol{q}=0$ (a), derivative of free energy with respect to $\lambda$ (b) and correlation value $R$ at $\boldsymbol{q}=0$ (c) as a function of interaction strength $\lambda/t$ for $N=2$ at $\beta t=L$. The inset in (c) shows the crossing point $\lambda_c$ between the correlation ratio for a respective lattice size $L$ and $L+3$ as a function of $1/L$ together with a power law fit. Thereby, in the thermodynamic limit we extract a crossing point $\lambda_c=0.955(3)$. With this and a correlation length exponent $\nu=0.529$ we obtain our best data collapse shown in (d).
• Figure 3: Particle number (a), Spin structure factor (b) and SC structure factor (c) at $\boldsymbol{q}=0$ as a function of chemical potential $\mu/t$ for $N=2$ at $\lambda=0.4t$ and $\beta t=L^2$.
• Figure 4: Normalized structure factor of (a) sSC, (b) QSH, (c) fSC, (d) singlet Kekulé, (e) triplet Kekulé and (f) QAH at various wave vectors along the $k_x$-axis as a function of interaction $\lambda/t$ for $N=2$ and $L=12$ at half-filling.