Symmetry-Protected Topological Orders in Interacting Bosonic Systems

Symmetry Semantics Topological insulators (TIs) are characterized by boundary states that are protected by time-reversal symmetry. A systematic study of this, and other symmetry-protected states, is possible in noninteracting systems, but complications arise when interactions are present. Chen et al. (p. 1604; see the Perspective by Qi) used group cohomology theory to predict symmetry-protected phases of interacting bosons. The analysis enabled the generalization of a known result in one dimension by using a path-integral formulation and suggests the existence of three counterparts of TIs in three dimensions, and one in two dimensions, as well as phases protected by other symmetries. The formalism is applicable to any symmetry group and dimension and is valid for interactions of arbitrary strength. Counterparts of topological insulators are predicted to exist in interacting bosonic systems. Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.

[1]  Michael Levin,et al.  Braiding statistics approach to symmetry-protected topological phases , 2012, 1202.3120.

[2]  David Pérez-García,et al.  Classifying quantum phases using matrix product states and projected entangled pair states , 2011 .

[3]  Xiao-Gang Wen,et al.  Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations , 2011, 1106.4752.

[4]  Frank Pollmann,et al.  Topological Phases of One-Dimensional Fermions: An Entanglement Point of View , 2010, 1008.4346.

[5]  Alexei Kitaev,et al.  Topological phases of fermions in one dimension , 2010, 1008.4138.

[6]  Xiao-Gang Wen,et al.  Classification of gapped symmetric phases in one-dimensional spin systems , 2010, 1008.3745.

[7]  Xiao-Gang Wen,et al.  Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order , 2010, 1004.3835.

[8]  Shinsei Ryu,et al.  Topological insulators and superconductors: tenfold way and dimensional hierarchy , 2009, 0912.2157.

[9]  Xiao-Gang Wen,et al.  Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order , 2009, 0903.1069.

[10]  Alexei Kitaev,et al.  Periodic table for topological insulators and superconductors , 2009, 0901.2686.

[11]  Shinsei Ryu,et al.  Classification of topological insulators and superconductors in three spatial dimensions , 2008, 0803.2786.

[12]  Liang Fu,et al.  Topological insulators in three dimensions. , 2006, Physical review letters.

[13]  J. E. Moore,et al.  Topological invariants of time-reversal-invariant band structures , 2006, cond-mat/0607314.

[14]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[15]  J. Preskill,et al.  Topological entanglement entropy. , 2005, Physical review letters.

[16]  Shou-Cheng Zhang,et al.  Quantum spin Hall effect. , 2005, Physical review letters.

[17]  C. Kane,et al.  Z2 topological order and the quantum spin Hall effect. , 2005, Physical review letters.

[18]  C. Kane,et al.  Quantum spin Hall effect in graphene. , 2004, Physical review letters.

[19]  Ng.  Edge states in antiferromagnetic quantum spin chains. , 1994, Physical review. B, Condensed matter.

[20]  Lee,et al.  Observation of fractional spin S=1/2 on open ends of S=1 linear antiferromagnetic chains: Nonmagnetic doping. , 1991, Physical review letters.

[21]  Hagiwara,et al.  Observation of S=1/2 degrees of freedom in an S=1 linear-chain Heisenberg antiferromagnet. , 1990, Physical review letters.

[22]  Xiao-Gang Wen,et al.  Topological Orders in Rigid States , 1990 .

[23]  E. Lieb,et al.  Valence bond ground states in isotropic quantum antiferromagnets , 1988 .

[24]  N. Seiberg.  Topology in strong coupling , 1984 .

[25]  E. Witten.  Global Aspects of Current Algebra , 1983 .

[26]  F. Haldane.  Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State , 1983 .

[27]  F. Haldane.  Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model , 1983 .

[28]  B. Zumino,et al.  Consequences of anomalous ward identities , 1971 .