Phase diagram of the single-flavor Gross--Neveu--Wilson model from the Grassmann corner transfer matrix renormalization group

Abstract

We investigate the phase structure of the single-flavor Gross--Neveu model with Wilson fermions using the Grassmann corner transfer matrix renormalization group (CTMRG). The path integral is formulated as a two-dimensional Grassmann tensor network and approximately contracted by the Grassmann CTMRG algorithm. We investigate the phase diagram by varying the fermion mass and the four-fermion coupling, using the pseudoscalar condensate as an order parameter for the $\mathbb{Z}_{2}$ parity symmetry breaking phase. The universality classes of the phase boundaries are identified through the central charge $c$ obtained via scaling analysis of the entanglement entropy. Furthermore, we extract the quantity related to the entanglement spectrum from the converged CTMRG environments, allowing us to distinguish the topological insulator phase and the trivial phase. The resulting phase structure suggests that the Aoki phase is separated from the other phases by critical lines characterized by $c=1/2$, while the critical lines with $c=1$ separate the topological insulating and trivial phases. Our numerical results also indicate that the Aoki phase does not persist in the strong-coupling regime for the single-flavor theory.

Figures (20)

• Figure 1: Phase diagram of the GNW model based on the large-$N_{f}$ method. We set $N_{1}=N_{2}=512$, which is sufficiently large to suppress finite-size effects, when solving Eqs. \ref{['eq:gap_pi_tobesolved']} and \ref{['eq:gap_sigma_tobesolved']}.
• Figure 2: Original infinite Grassmann tensor network (left) and its effective representation by using the environment tensors (right). Corner matrices and three-leg edge tensors are introduced to approximate an infinite environment surrounding a local Grassmann tensor $\mathcal{T}$. The local Grassmann tensor has bond dimension $d$, while the bond dimensions connecting the environment tensors are denoted by $D$.
• Figure 3: The left move in the Grassmann CTMRG algorithm consists of two steps: (a) inserting a column of bulk tensors and absorbing it into the left environment; and (b) constructing the Grassmann projectors $\mathcal{P}$ and $\mathcal{Q}$ to truncate the virtual bond dimension and update the left environment tensors $\mathcal{C}^{lu}$, $\mathcal{E}^{l}$, and $\mathcal{C}^{ld}$.
• Figure 4: The relative error $\delta f$ against the bond dimension $D$ for different Grassmann tensor network algorithms. When the system is away from criticality, the CTMRG outperforms other algorithms (a). Although the finite-$D$ effect is enhanced in all the algorithms near criticality, the CTMRG still shows the highest accuracy among these algorithms (b).
• Figure 5: Phase diagram of the $N_{f}=1$ GNW model on the $(M, g^{2})$ plane. The model exhibits three distinct phases: the Aoki phase, the topological insulator phase, and the trivial phase. The heat map represents the magnitude of the pseudoscalar condensate, which serves as the order parameter of the Aoki phase. White crosses indicate the parameter points at which Grassmann CTMRG detects critical behavior via peaks in the correlation length and entanglement entropy. Yellow circles denote the points where the extracted central charge is consistent with $c=1/2$, while red squares indicate the points consistent with $c=1$. The Aoki phase is separated from the other phases by critical lines characterized by $c=1/2$. The topological insulator and trivial phases are separated by critical lines with $c=1$.