Construction of bosonic symmetry-protected-trivial states and their topological invariants viaG×SO(∞)nonlinearσmodels

2014/10/31 by Xiao-Gang Wen · 5 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Boundary (topology) #Cohomology #Combinatorics #Computer science #Group (periodic table) #Group cohomology #Mathematical analysis #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Quantum mechanics #Space (punctuation) #Topological Materials and Phenomena #Topology (electrical circuits) #cond-mat.str-el #hep-th

paper · pdf · doi:10.1103/physrevb.91.205101

published as Phys. Rev. B 91, 205101 (2015) · 39 pages, 10 figure. PRB version

arxiv created 2015/03/23 · openalex publication_date 2015/05/04 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear \ensuremathσ models (NL\ensuremathσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H^d(G,ℝ/ℤ). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G\ifmmode×\else\texttimes\fiSO(\ensuremath∞) NL\ensuremathσMs with various topological terms. Through the group cohomology H^d[G\ifmmode×\else\texttimes\fiSO(\ensuremath∞),ℝ/ℤ], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E^d(G)\ensuremath\rtimes\ensuremath\bigoplus_k=1^d\ensuremath-1H^k(G,iTO_L^d\ensuremath-k)\ensuremath\bigoplusH^d(G,ℝ/ℤ) where G may contain time reversal. Here iTO_L^d is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO_L¹=iTO_L²=iTO_L⁴=iTO_L⁶=0,\phantom\rule0.16em0exiTO_L³=ℤ,\phantom\rule0.16em0exiTO_L⁵=ℤ₂,\phantom\rule0.16em0exiTO_L⁷=2ℤ. For G=U(1)\ensuremath\rtimesZ₂^T (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H^d[U(1)\ensuremath\rtimesZ₂^T,ℝ/ℤ] are described by ℤ₂ in 3 + 1D, ℤ in 4 + 1D, 3ℤ₂ in 5 + 1D, and 4ℤ₂ in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.

Citations

• Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
• Topological properties of Abelian and non-Abelian quantum Hall states classified using patterns of zeros
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Anomalies of discrete symmetries in three dimensions and group cohomology
• A calculus for branched spines of 3-manifolds
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Entanglement spectrum of a topological phase in one dimension
• String-net condensation: A physical mechanism for topological phases
• Models for gapped boundaries and domain walls
• Nonabelions in the fractional quantum hall effect
• Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model

Cited by

• Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries
• Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface
• Choreographed entanglement dances: Topological states of quantum matter
• The emergence of 3+1D Einstein gravity from topological gravity
• Group cohomology [wikipedia]

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