Symmetric Mass Generation in the $1+1$ Dimensional Chiral Fermion 3-4-5-0 Model

Lattice regularization of chiral fermions has been a long-standing problem in physics. In this Letter, we present the density matrix renormalization group simulation of the 3-4-5-0 model of $(1+1)D$ chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of $(2+1)D$ lattice model of multilayer Chern insulator, whose finite width implies the quantum system is effectively $(1+1)D$. By introducing two sets of carefully designed six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions.

Figure 1

(a) The fermion hopping pattern on the two-leg ladder lattice for the first layer. Arrow link: $t_1{e}^{i\pi /4}$ (along the arrow direction); solid link: $t_2$; dashed link: $-t_2$. This $(2+1)D$ thin strip is effectively the same as $(1+1)D$ by regarding the finite-width dimension as internal degrees of freedom of the $(1+1)D$ system. (b) Energy dispersion for $t_1=1$, $t_2=0.5$. Gapless edge modes are strictly localized on the two boundaries of the ladder. (c) Schematic diagram showing the configuration of the four flavors of chiral fermions on the edges.

Figure 2

Ground state (GS) energy per unit cell as a function of interaction strength $g$. The inset shows the first-order derivative of the GS energy with respect to $g$. The features around $g\approx 5.7$ (indicated by the gray dashed line) signal a quantum phase transition.

Figure 3

Correlations on both edges before and after transition. Linear fit (red line) is performed for intermediate distances from $r=2$ to $r=6$ in each case, in order to faithfully extract the low energy physics while avoiding the artifacts due to the gap caused by finite bond dimension in the matrix product state representation. (a) $g=5.0<g_c$ for edge $A$. The log-log plot shows a power-law decay for intermediate distances. (b) $g=7.0>g_c$ for edge $A$. The log-log plot again shows a power-law decay. (c) $g=5.0<g_c$ for edge $B$. The log-log plot shows a power-law decay. (d) $g=7.0>g_c$ for edge $B$. The semilog plot indicates an exponential decay, i.e., edge $B$ becomes gapped.

Figure 4

(a) The evolution of fermion scaling dimension ${\mathrm{\Delta }}_{\psi }$ on both edges as the interaction strength $g$ approaches the critical point. The scaling dimension is obtained from the power-law fitting as in Fig. 3. The horizontal dashed line indicates the free fermion limit. The gray stripe shows the estimated critical interaction strength $g_c$ with some uncertainty. (b) The solved scaling dimension for the interaction terms on edge $B$ based on the scaling dimensions of multiple operators (refer to SM, Sec. III [66] for details). The horizontal dashed line indicates the marginal value 2 of ${\mathrm{\Delta }}_{\mathrm{int}}$, across which the phase transition is expected to happen.