TY - JOUR

T1 - Canonical quantization of nonlinear sigma models with a theta term and applications to symmetry-protected topological phases

AU - Lapa, Matthew F.

AU - Hughes, Taylor L.

N1 - Funding Information:
We thank A. Kapustin and M. Stone for helpful discussions on the content of this paper. M. F. L. and T. L. H. acknowledge support from the ONR YIP Grant No. N00014-15-1-2383. We also gratefully acknowledge the support of the Institute for Condensed Matter Theory at the University of Illinois at Urbana-Champaign.
Publisher Copyright:
© 2017 American Physical Society.

PY - 2017/8/15

Y1 - 2017/8/15

N2 - 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 θ of the theta term is an integer multiple of 2π, 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 θ=2πk, kZ, 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.

AB - 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 θ of the theta term is an integer multiple of 2π, 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 θ=2πk, kZ, 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.

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U2 - 10.1103/PhysRevD.96.045010

DO - 10.1103/PhysRevD.96.045010

M3 - Article

AN - SCOPUS:85029109746

SN - 2470-0010

VL - 96

JO - Physical Review D

JF - Physical Review D

IS - 4

M1 - 043016

ER -