We study the effects of a topological Theta term on 2+1-dimensional principal chiral models, which are nonlinear sigma models defined on Lie group manifolds. We find that when Θ=π, the nature of the disordered phase of the principal chiral model is strongly affected by the topological term: it is either a gapless conformal field theory, or it is gapped and twofold degenerate. The result of our Letter can be used to analyze the boundary states of three-dimensional symmetry protected topological phases.
We canonically quantize
O(D+2) nonlinear sigma models (NLSMs) with a theta term on arbitrary smooth, closed, connected, oriented
D-dimensional spatial manifolds
M, with the goal of proving the suitability of these models for describing symmetry-protected topological (SPT) phases of bosons in
D spatial dimensions. We show that in the disordered phase of the NLSM, and when the coefficient $\ensuremath{\theta}$ of the theta term is an integer multiple of $2\ensuremath{\pi}$, the theory on
M has a unique ground state and a finite energy gap to all excitations. We also construct the ground state wave functional of the NLSM in this parameter regime, and we show that it is independent of the metric on
M and given by the exponential of a Wess-Zumino term for the NLSM field, in agreement with previous results on flat space. Our results show that the NLSM in the disordered phase and at $\ensuremath{\theta}=2\ensuremath{\pi}k$, $k\ensuremath{\in}\mathbb{Z}$, has a symmetry-preserving ground state but no topological order (i.e., no topology-dependent ground state degeneracy), making it an ideal model for describing SPT phases of bosons. Thus, our work places previous results on SPT phases derived using NLSMs on solid theoretical ground. To canonically quantize the NLSM on
M, we use Dirac's method for the quantization of systems with second class constraints, suitably modified to account for the curvature of space. In a series of four Appendixes, we provide the technical background needed to follow the discussion in the main sections of the paper.
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
Sp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ 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 $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. 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)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 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 $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable
(2+1)D conformal field theory with gapless bosonic modes.
Symmetry-protected topological phase is one type of nontrivial quantum disordered many-body state of matter. In this paper we study one class of symmetry-protected topological phases in two-dimensional space with both
PSU(N) and time-reversal symmetry. These states can be described by a principal chiral model with a topological $\ensuremath{\Theta}$ term. As long as the time-reversal symmetry and
PSU(N) symmetry are both preserved, the (1+1)-dimensional boundary of this system must be either gapless or degenerate. We will also construct a wave function of a spin-1 system on the honeycomb lattice, which is a candidate for the symmetry-protected topological phase with both SO(3) and time-reversal symmetry.
Bosonic symmetry-protected topological (SPT) states are gapped disordered phases of matter possessing symmetry-preserving boundary excitations. It has been proposed that, at long wavelengths, the universal properties of an SPT system are captured by an effective nonlinear sigma model field theory in the presence of a quantized topological $\ensuremath{\theta}$ term. By studying lattice models of bosonic SPT states, we are able to identify, in their Euclidean path integral formulation, (discrete) Berry phases that hold relevant physical information on the nature of the SPT ground states. These discrete Berry phases are given intuitive physical interpretation in terms of instanton effects that capture the presence of a $\ensuremath{\theta}$ term on the microscopic scale.
In this paper we systematically classify and describe bosonic symmetry protected topological (SPT) phases in all physical spatial dimensions using semiclassical nonlinear Sigma model (NLSM) field theories. All the SPT phases on a
d−dimensional lattice discussed in this paper can be described by the same NLSM, which is an O(d+2) NLSM in
(d+1)−dimensional space-time, with a topological
Θ−term. The field in the NLSM is a semiclassical Landau order parameter with a unit length constraint. The classification of SPT phases discussed in this paper based on their NLSMs is consistent with the more mathematical classification based on group cohomology. Besides the classification, the formalism used in this paper also allows us to explicitly discuss the physics at the boundary of the SPT phases, and it reveals the relation between SPT phases with different symmetries. For example, it gives many of these SPT states a natural "decorated defect" construction.