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

Abstract

We propose a series of simple 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 Sp(N ) principal chiral model with a topological Θ-term, whose boundary is described by a 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 Θ in the field theory and the phase transition occurs at Θ = π. 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)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 Z2 symmetries, etc. All these fermion models including the bulk quantum phase transitions can be simulated with 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 Θ = π is a stable (2 + 1)d conformal field theory (CFT) with gapless bosonic modes.

Topological phase transitions with SO(4) symmetry in (2+1)D interacting Dirac fermions

“…Dirac fermions in (2+1)d emerge in a number of solid state systems such as graphene [1], surface states of 3D topological insulators [2] and d-wave superconductors [3][4][5]. A parallel research track involves interaction-driven topological phase transitions in (2+1)d systems [6][7][8][9][10][11][12][13][14][15][16][17], which exhibit exotic quantum critical points. In this context, an important question is what new physics would emerge if Dirac fermions and topological phase transitions were brought together via electronic interactions?…”

Visualizing a bosonic symmetry protected topological phase in an interacting fermion model

“…1 (a)). The third term J is the interlayer antiferromagnetic Heisenberg (approximated) interaction [24], and the last term J z denotes the interlayer antiferromagnetic Ising (approximated) interaction [27]. When J/t > 0 and J z /t > 0, we can prove that there is no fermion sign problem in the QMC calculations [27].…”

“…Model and Method. The Hamiltonian [24,27] of the AA-stacked bilayer Kane-Mele-Hubbard model is given byĤ…”

UV perspective on mixed anomalies at critical points between bosonic symmetry-protected phases

“…(16) holds, with ν 1 defined as in Eq. (14). This immediately implies that β can only be a function of the group elements of G. By taking h 1 to be the trivial group element such that γ(h 1 ) = 1, one finds that β is a (d + 1)-cocycle of G. By taking h 1 to be the non-trivial group element of Z T 2 it then follows that ω = β −2 .…”