Phase diagram of a lattice fermion model with symmetric mass generation

Abstract

We study the phase structure of a model containing two flavors of massless staggered fermions interacting through two independent four-fermion couplings, UI and UB, formulated on a three-dimensional Euclidean space-time lattice. At UB = 0, this model is known to exhibit a direct second-order quantum phase transition between a massless fermion (MF) phase and a phase in which fermions acquire masses through the mechanism commonly referred to as symmetric mass generation (SMG). We demonstrate that introducing a small nonzero value of UB qualitatively alters this structure: the single exotic transition at UB = 0 splits into two distinct, conventional transitions, separated by an intermediate phase in which fermion masses arise through the standard mechanism of spontaneous symmetry breaking (SSB). The first of these is a Gross-Neveu transition separating the MF phase from the SSB-induced massive phase, while the second is a three-dimensional XY transition between the SSB phase and the SMG phase. Using the fermion-bag Monte Carlo method, we verify that the critical exponents associated with both transitions are consistent with the literature, thereby yielding a quantitative characterization of the resulting phase structure of the model.

Figures (12)

• Figure 1: An illustration of an instanton--dimer configuration $[n_i,n_b^u,n_b^d]$ on an $L=2$ lattice. Instantons are shown as solid circles, and dimers are shown as thick bonds. The configuration on the left depicts the instantons and dimers on the $u$ layer, while the configuration on the right shows the corresponding structure on the $d$ layer. Instantons occupy the same lattice sites on both layers, whereas the dimer configurations may differ. Sites containing neither instantons nor dimers are also present.
• Figure 2: Begin--End step of the worm algorithm restricted to a single layer. During the Begin step, the initially chosen site may be either a free site (configuration $A$) or a site occupied by a dimer (configuration $B$). The solid arrow indicates the site selected at the start of the Begin step. Configurations $A$ and $B$ belong to the partition-function sector. From either configuration, a proposal is made to transition to configuration $C$ in the worm sector, with the Tail located at the initially chosen site and the Head placed on a neighboring site. For configuration $A$, the Head is chosen at random from the ${\mathcal{D}_s}$ nearest-neighbor sites on the same layer, while for configuration $B$ it is placed on the fixed neighboring site shown in the figure. The worm update may also terminate immediately, transitioning directly from configuration $A$ to $B$ or from $B$ to $A$. If the Begin step produces configuration $C$, the End step removes the worm by transforming configuration $C$ back into either configuration $A$ or $B$. The same configurations appear in the End step as in the Begin step, but with the arrow indicating the reverse direction of the update. In this figure, both the Tail and the Head are created on the same fermion layer, either the $u$ layer or the $d$ layer.
• Figure 3: Begin--End step of the worm algorithm involving opposite layers. During this Begin step, the initially selected site contains an instanton (configuration B). The solid arrow indicates the site chosen at the start of the Begin step. Configuration B belongs to the partition-function sector. A proposal is then made to transition to configuration C in the worm sector, with the Tail located at the initially selected site and the Head placed on the site with the same space--time coordinates on the opposite layer. If the proposal is accepted, the instanton is broken into two monomers. The End step is the reverse process, in which the two monomers on the two layers are fused to form an instanton.
• Figure 4: First class of configurations among which the Move step transitions. In this class, the chosen neighboring site $k$ of the Head lies on the same layer and is either a free site or a site occupied by a dimer. If $k$ is free, the site $j$ is chosen at random from the remaining nearest-neighbor sites on the same layer, excluding the Head site.
• Figure 5: Second class of configurations among which the Move step transitions. In this class, the chosen neighboring site $k$ of the Head lies on the opposite layer and is either a free site or a site occupied by a dimer. The site $k$ may also lie on the same layer as the Head if it is occupied by an instanton. If $k$ is free, the site $j$ is chosen at random from the nearest-neighbor sites on the same layer.