Quantum phase transitions between bosonic symmetry-protected topological states without sign problem: Nonlinear sigma model with a topological term

We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a $\mathrm{Sp}(N$) principal chiral model with a topological $\mathrm{\Theta }$ term, whose boundary is described by a $\mathrm{Sp}(N$) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\Theta }$ in the field theory, and the phase transition occurs at $\mathrm{\Theta }=\pi$. The simplest version of these models with $N=1$ is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a $(1+1)\text{D}$ conformal field theory with central charge $c=1$. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or $Z_2$ symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\Theta }=\pi$ is a stable $(2+1)\text{D}$ conformal field theory with gapless bosonic modes.

Figure 1

Honeycomb lattice with the nearest-neighboring hopping $t_{ij}$ and the second-nearest-neighboring hopping ${\lambda }_{ij}$. ${\lambda }_{ij}=-{\lambda }_{ji}=\lambda$ if $i$ follows the bound orientation to $j$. The lattice can be divided into $A$ and $B$ sublattices.

Figure 2

A schematic phase diagram of the interacting bilayer QSH model.

Figure 3

RG flow near the KT transition point. The free-fermion fixed point $({\mathrm{\Delta }}_0,{\lambda }_{\pm })=(2,0)$ is marked by a red point. The red arrows illustrate the RG flow after small ${\lambda }_{\pm }$ perturbation.