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Symmetry Protected Topological order (SPT order)〔 Zheng-Cheng Gu, Xiao-Gang Wen, ( Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order ), Phys. Rev. B80, 155131 (2009); Frank Pollmann, Erez Berg, Ari M. Turner, Masaki Oshikawa, (Symmetry protection of topological order in one-dimensional quantum spin systems ), Phys. Rev. B85, 075125 (2012). 〕 is a new kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap. To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points).〔 The SPT order has the following defining properties: (a) ''distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry''. (b) ''however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation''. Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states ''with a symmetry'' (by contrast: for long-range entanglement see topological order, which is not related to the famous EPR paradox). Since short-range entangled states have only trivial topological orders we may also refer the SPT order as Symmetry Protected "Trivial" order. ==Characteristic properties of SPT order== # The boundary effective theory of a non-trivial SPT state always has pure gauge anomaly or mixed gauge-gravity anomaly for the symmetry group.〔 Xiao-Gang Wen, ''Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders'' Phys. Rev. D 88, 045013 (2013); arXiv:1303.1803. 〕 As a result, the boundary of a SPT state is either gapless or degenerate, regardless how we cut the sample to form the boundary. A gapped non-degenerate boundary is impossible for a non-trivial SPT state. If the boundary is a gapped degenerate state, the degeneracy may be caused by spontaneous symmetry breaking and/or (intrinsic) topological order. # Monodromy defects in non-trivial 2+1D SPT states carry non-trival statistics〔 Michael Levin, Zheng-Cheng Gu, ''Braiding statistics approach to symmetry-protected topological phases'', Phys. Rev. B 86, 115109 (2012), arXiv:1202.3120. 〕 and fractional quantum numbers〔 Xiao-Gang Wen, ''Topological invariants of symmetry-protected and symmetry-enriched topological phases of interacting bosons or fermions'', arXiv:1301.7675. 〕 of the symmetry group. Monodromy defects are created by twisting the boundary condition along a cut by a symmetry transformation. The ends of such cut are the monodromy defects. For example, 2+1D bosonic Zn SPT states are classified by a Zn integer ''m''. One can show that ''n'' identical elementary monodromy defects in a Zn SPT state labeled by ''m'' will carry a total Zn quantum number ''2m'' which is not a multiple of ''n''. # 2+1D bosonic U(1) SPT states have a Hall conductance that is quantized as an even integer.〔 Yuan-Ming Lu, Ashvin Vishwanath, ''Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach'', Phys. Rev. B 86, 125119 (2012), arXiv:1205.3156.〕〔Peng Ye and Xiao-Gang Wen, "Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries", Phys. Rev. B 87, 195128 (2013). arXiv:1212.2121.〕〔 Zheng-Xin Liu, Jia-Wei Mei, Peng Ye, and Xiao-Gang Wen, "U(1)×U(1) symmetry protected topological order in Gutzwiller wave functions", Phys. Rev. B 90, 235146 (2014), arXiv:1408.1676.〕 2+1D bosonic SO(3) SPT states have a quantized spin Hall conductance.〔 Zheng-Xin Liu, Xiao-Gang Wen, ''Symmetry protected Spin Quantum Hall phases in 2-Dimensions'', Phys. Rev. Lett. 110, 067205 (2013), arXiv:1205.7024. 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Symmetry protected topological order」の詳細全文を読む スポンサード リンク
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