[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/symmetry-protected-topological-order-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki40\/symmetry-protected-topological-order-wikipedia\/","headline":"Symmetry-protected topological order – Wikipedia","name":"Symmetry-protected topological order – Wikipedia","description":"before-content-x4 Type of topological order in condensed matter physics after-content-x4 Symmetry-protected topological (SPT) order[1][2] is a kind of order in","datePublished":"2022-07-01","dateModified":"2022-07-01","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki40\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f8f5bbb16baf17b66a8c7dc486cdc6e97403ccc7","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f8f5bbb16baf17b66a8c7dc486cdc6e97403ccc7","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki40\/symmetry-protected-topological-order-wikipedia\/","wordCount":10472,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Type of topological order in condensed matter physics       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Symmetry-protected topological (SPT) order[1][2] is a 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).[1] 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.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order.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.Table of Contents       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Characteristic properties[edit]Relation between SPT order and (intrinsic) topological order[edit]Examples[edit]Group cohomology theory for SPT phases[edit]A complete classification of 1D gapped quantum phases (with interactions)[edit]See also[edit]References[edit]Characteristic properties[edit]The boundary effective theory of a non-trivial SPT state always has pure gauge anomaly or mixed gauge-gravity anomaly for the symmetry group.[3] 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[4] and fractional quantum numbers[5] 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.[6][7] 2+1D bosonic SO(3) SPT states have a quantized spin Hall conductance.[8]Relation between SPT order and (intrinsic) topological order[edit]SPT states are short-range entangled while topologically ordered states are long-range entangled.Both intrinsic topological order, and also SPT order, can sometimes have protected gapless boundary excitations. The difference is subtle: the gapless boundary excitations in intrinsic topological order can be robust against any local perturbations, while the gapless boundary excitations in SPT order are robust only against local perturbations that do not break the symmetry. So the gapless boundary excitations in intrinsic topological order are topologically protected, while the gapless boundary excitations in SPT order are symmetry protected.[9]We also know that an intrinsic topological order has emergent fractional charge, emergent fractional statistics, and emergent gauge theory. In contrast, a SPT order has no emergent fractional charge\/fractional statistics for finite-energy excitations, nor emergent gauge theory (due to its short-range entanglement). Note that the monodromy defects discussed above are not finite-energy excitations in the spectrum of the Hamiltonian, but defects created by modifying the Hamiltonian.Examples[edit]The first example of SPT order is the Haldane phase of odd-integer spin chain.[10][11][12][13][14] It is a SPT phase protected by SO(3) spin rotation symmetry.[1]  Note that Haldane phases of even-integer-spin chain do not have SPT order.A more well known example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry.On the other hand, fractional quantum Hall states are not SPT states. They are states with (intrinsic) topological order and long-range entanglements.Group cohomology theory for SPT phases[edit]Using the notion of quantum entanglement, one obtains the following general picture of gappedphases at zero temperature. All gapped zero-temperature phases can be divided into two classes: long-range entangled phases (ie phases with intrinsic topological order) and short-range entangled phases (ie phases with no intrinsic topological order). All short-range entangled phases can be further divided into three classes:  symmetry-breaking phases, SPT phases, and their mix (symmetry breaking order and SPT order can appear together).It is well known that symmetry-breaking orders are described by group theory. For bosonic SPT phases with pure gauge anomalous boundary, it was shown that they are classified by group cohomology theory:[15][16] those (d+1)D SPT states with symmetry G are labeled by the elements in group cohomology classHd+1[G,U(1)]{displaystyle H^{d+1}[G,U(1)]}.For other (d+1)D SPT states[17][18][19][20] with mixed gauge-gravity anomalous boundary, they can be described by \u2295k=1dHk[G,iTOd+1\u2212k]{displaystyle oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]},[21] where iTOd+1{displaystyle iTO^{d+1}} is the Abelian group formed by (d+1)D topologically ordered phases that have no non-trivial topological excitations (referred as iTO phases).From the above results, many new quantum states of matter are predicted, including bosonic topological insulators (the SPT states protected by U(1) and time-reversal symmetry) and bosonic topological superconductors (the SPT states protected by time-reversal symmetry), as well as many other new SPT states protected by other symmetries.A list of bosonic SPT states from group cohomology Hd+1[G,U(1)]\u2295k=1dHk[G,iTOd+1\u2212k]{displaystyle H^{d+1}[G,U(1)]oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]} (Z2T{displaystyle Z_{2}^{T}} = time-reversal-symmetry group)The phases before “+” come from Hd+1[G,U(1)]{displaystyle H^{d+1}[G,U(1)]}. The phases after “+” come from \u2295k=1dHk[G,iTOd+1\u2212k]{displaystyle oplus _{k=1}^{d}H^{k}[G,iTO^{d+1-k}]}.Just like group theory can give us 230 crystal structures in 3+1D, group cohomology theory can give us various SPT phases in any dimensions with any on-site symmetry groups.On the other hand, the fermionic SPT orders are described by group super-cohomology theory.[22] So the group (super-)cohomology theory allows us to construct manySPT orders even for interacting systems, which include  interacting topological insulator\/superconductor.A complete classification of 1D gapped quantum phases (with interactions)[edit]Using the notions of quantum entanglement and SPT order, one can obtaina complete classification of all 1D gapped quantum phases.First, it is shown that there is no (intrinsic) topological order in 1D (ie all 1D gapped statesare short-range entangled).[23]Thus, if the Hamiltonians have no symmetry, all their 1D gapped quantum states belong to one phase\u2014the phase of trivial product states.On the other hand, if the Hamiltonians do have a symmetry,  their 1D gapped quantum statesare either symmetry-breaking phases, SPT phases, and their mix.Such an understanding allows one to classify all 1D gapped quantum phases:[15][24][25][26][27] All 1D gapped phases are classified bythe following three mathematical objects: (GH,G\u03a8,H2[G\u03a8,U(1)]){displaystyle (G_{H},G_{Psi },H^{2}[G_{Psi },U(1)])} , where GH{displaystyle G_{H}} is the symmetry group of the Hamiltonian, G\u03a8{displaystyle G_{Psi }} the symmetry group of the ground states, and H2[G\u03a8,U(1)]{displaystyle H^{2}[G_{Psi },U(1)]} the second group cohomology class of G\u03a8{displaystyle G_{Psi }}. (Note that H2[G,U(1)]{displaystyle H^{2}[G,U(1)]} classifies the projective representations of G{displaystyle G}.) If there is no symmetry breaking (ie G\u03a8=GH{displaystyle G_{Psi }=G_{H}}), the 1D gapped phases are classified by the projective representations of symmetry group GH{displaystyle G_{H}}.See also[edit]References[edit]^ a b c Gu, Zheng-Cheng; Wen, Xiao-Gang (26 October 2009). “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order”. Physical Review B. 80 (15): 155131. arXiv:0903.1069. Bibcode:2009PhRvB..80o5131G. doi:10.1103\/physrevb.80.155131. ISSN\u00a01098-0121. S2CID\u00a015114579.^ Pollmann, Frank; Berg, Erez; Turner, Ari M.; Oshikawa, Masaki (22 February 2012). “Symmetry protection of topological phases in one-dimensional quantum spin systems”. Physical Review B. 85 (7): 075125. arXiv:0909.4059. Bibcode:2012PhRvB..85g5125P. doi:10.1103\/physrevb.85.075125. ISSN\u00a01098-0121. S2CID\u00a053135907.^ Wen, Xiao-Gang (9 August 2013). “Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders”. Physical Review D. 88 (4): 045013. arXiv:1303.1803. Bibcode:2013PhRvD..88d5013W. doi:10.1103\/physrevd.88.045013. ISSN\u00a01550-7998. S2CID\u00a018250786.^ Levin, Michael; Gu, Zheng-Cheng (10 September 2012). “Braiding statistics approach to symmetry-protected topological phases”. Physical Review B. 86 (11): 114109. arXiv:1202.3120. Bibcode:2012PhRvB..86k5109L. doi:10.1103\/physrevb.86.115109. ISSN\u00a01098-0121. S2CID\u00a0118688476.^ Wen, Xiao-Gang (31 January 2014). “Symmetry-protected topological invariants of symmetry-protected topological phases of interacting bosons and fermions”. Physical Review B. 89 (3): 035147. arXiv:1301.7675. Bibcode:2014PhRvB..89c5147W. doi:10.1103\/physrevb.89.035147. ISSN\u00a01098-0121. S2CID\u00a055842699.^ Lu, Yuan-Ming; Vishwanath, Ashvin (14 September 2012). “Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach”. Physical Review B. 86 (12): 125119. arXiv:1205.3156. Bibcode:2012PhRvB..86l5119L. doi:10.1103\/physrevb.86.125119. ISSN\u00a01098-0121. S2CID\u00a086856666.^ Liu, Zheng-Xin; Mei, Jia-Wei; Ye, Peng; Wen, Xiao-Gang (24 December 2014). “U(1)\u00d7U(1)symmetry-protected topological order in Gutzwiller wave functions”. Physical Review B. 90 (23): 235146. arXiv:1408.1676. Bibcode:2014PhRvB..90w5146L. doi:10.1103\/physrevb.90.235146. ISSN\u00a01098-0121. S2CID\u00a014800302.^ Liu, Zheng-Xin; Wen, Xiao-Gang (7 February 2013). “Symmetry-Protected Quantum Spin Hall Phases in Two Dimensions”. Physical Review Letters. 110 (6): 067205. arXiv:1205.7024. Bibcode:2013PhRvL.110f7205L. doi:10.1103\/physrevlett.110.067205. ISSN\u00a00031-9007. PMID\u00a023432300. S2CID\u00a012995741.^ One should also note the semantical subtleness of the name SPT: “symmetry protected”  does not mean that the stability of the state is  conserved “because of the symmetry”, but it is  just meant that the symmetry is kept by the interactions corresponding to the process.^ Haldane, F. D. M. (11 April 1983). “Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis N\u00e9el State”. Physical Review Letters. American Physical Society (APS). 50 (15): 1153\u20131156. Bibcode:1983PhRvL..50.1153H. doi:10.1103\/physrevlett.50.1153. ISSN\u00a00031-9007.^ Haldane, F.D.M. (1983). “Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model”. Physics Letters A. Elsevier BV. 93 (9): 464\u2013468. Bibcode:1983PhLA…93..464H. doi:10.1016\/0375-9601(83)90631-x. ISSN\u00a00375-9601.^ Affleck, Ian; Haldane, F. D. M. (1 September 1987). “Critical theory of quantum spin chains”. Physical Review B. American Physical Society (APS). 36 (10): 5291\u20135300. Bibcode:1987PhRvB..36.5291A. doi:10.1103\/physrevb.36.5291. ISSN\u00a00163-1829. PMID\u00a09942166.^ Affleck, I (15 May 1989). “Quantum spin chains and the Haldane gap”. Journal of Physics: Condensed Matter. IOP Publishing. 1 (19): 3047\u20133072. Bibcode:1989JPCM….1.3047A. doi:10.1088\/0953-8984\/1\/19\/001. ISSN\u00a00953-8984. S2CID\u00a0250850599.^ Mishra, Shantanu; Catarina, Gon\u00e7alo; Wu, Fupeng; Ortiz, Ricardo; Jacob, David; Eimre, Kristjan; Ma, Ji; Pignedoli, Carlo A.; Feng, Xinliang; Ruffieux, Pascal; Fern\u00e1ndez-Rossier, Joaqu\u00edn; Fasel, Roman (13 October 2021). “Observation of fractional edge excitations in nanographene spin chains”. Nature. 598 (7880): 287\u2013292. arXiv:2105.09102. Bibcode:2021Natur.598..287M. doi:10.1038\/s41586-021-03842-3. PMID\u00a034645998. S2CID\u00a0234777902.^ a b Chen, Xie; Liu, Zheng-Xin; Wen, Xiao-Gang (22 December 2011). “Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations”. Physical Review B. 84 (23): 235141. arXiv:1106.4752. Bibcode:2011PhRvB..84w5141C. doi:10.1103\/physrevb.84.235141. ISSN\u00a01098-0121. S2CID\u00a055330505.^ Chen, Xie; Gu, Zheng-Cheng; Liu, Zheng-Xin; Wen, Xiao-Gang (4 April 2013). “Symmetry protected topological orders and the group cohomology of their symmetry group”. Physical Review B. 87 (15): 155114. arXiv:1106.4772. Bibcode:2013PhRvB..87o5114C. doi:10.1103\/physrevb.87.155114. ISSN\u00a01098-0121. S2CID\u00a0118546600.^ Vishwanath, Ashvin; Senthil, T. (28 February 2013). “Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect”. Physical Review X. 3 (1): 011016. arXiv:1209.3058. Bibcode:2013PhRvX…3a1016V. doi:10.1103\/physrevx.3.011016. ISSN\u00a02160-3308.^  Anton Kapustin, “Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology” arXiv:1403.1467^ Wang, Juven C.; Gu, Zheng-Cheng; Wen, Xiao-Gang (22 January 2015). “Field-Theory Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond”. Physical Review Letters. 114 (3): 031601. arXiv:1405.7689. Bibcode:2015PhRvL.114c1601W. doi:10.1103\/physrevlett.114.031601. ISSN\u00a00031-9007. PMID\u00a025658993. S2CID\u00a02370407.^ Kapustin, Anton; Thorngren, Ryan; Turzillo, Alex; Wang, Zitao (2015). “Fermionic symmetry protected topological phases and cobordisms”. Journal of High Energy Physics. 2015 (12): 1\u201321. arXiv:1406.7329. Bibcode:2015JHEP…12..052K. doi:10.1007\/jhep12(2015)052. ISSN\u00a01029-8479. S2CID\u00a042613274.^ Wen, Xiao-Gang (4 May 2015). “Construction of bosonic symmetry-protected-trivial states and their topological invariants via G\u00d7SO(\u221e) nonlinear \u03c3 models”. Physical Review B. 91 (20): 205101. arXiv:1410.8477. Bibcode:2015PhRvB..91t5101W. doi:10.1103\/physrevb.91.205101. ISSN\u00a01098-0121. S2CID\u00a013950401.^ Gu, Zheng-Cheng; Wen, Xiao-Gang (23 September 2014). “Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear \u03c3 models and a special group supercohomology theory”. Physical Review B. 90 (11): 115141. arXiv:1201.2648. Bibcode:2014PhRvB..90k5141G. doi:10.1103\/physrevb.90.115141. ISSN\u00a01098-0121. S2CID\u00a0119307777.^ Verstraete, F.; Cirac, J. I.; Latorre, J. I.; Rico, E.; Wolf, M. M. (14 April 2005). “Renormalization-Group Transformations on Quantum States”. Physical Review Letters. 94 (14): 140601. arXiv:quant-ph\/0410227. Bibcode:2005PhRvL..94n0601V. doi:10.1103\/physrevlett.94.140601. ISSN\u00a00031-9007. PMID\u00a015904055. S2CID\u00a021362387.^ Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (13 January 2011). “Classification of gapped symmetric phases in one-dimensional spin systems”. Physical Review B. 83 (3): 035107. arXiv:1008.3745. Bibcode:2011PhRvB..83c5107C. doi:10.1103\/physrevb.83.035107. ISSN\u00a01098-0121. S2CID\u00a09139955.^ Turner, Ari M.; Pollmann, Frank; Berg, Erez (8 February 2011). “Topological phases of one-dimensional fermions: An entanglement point of view”. Physical Review B. 83 (7): 075102. arXiv:1008.4346. Bibcode:2011PhRvB..83g5102T. doi:10.1103\/physrevb.83.075102. ISSN\u00a01098-0121. S2CID\u00a0118491997.^ Fidkowski, Lukasz; Kitaev, Alexei (8 February 2011). “Topological phases of fermions in one dimension”. Physical Review B. American Physical Society (APS). 83 (7): 075103. arXiv:1008.4138. Bibcode:2011PhRvB..83g5103F. doi:10.1103\/physrevb.83.075103. ISSN\u00a01098-0121. S2CID\u00a01201670.^ Schuch, Norbert; P\u00e9rez-Garc\u00eda, David; Cirac, Ignacio (31 October 2011). “Classifying quantum phases using matrix product states and projected entangled pair states”. Physical Review B. 84 (16): 165139. arXiv:1010.3732. Bibcode:2011PhRvB..84p5139S. doi:10.1103\/physrevb.84.165139. ISSN\u00a01098-0121. S2CID\u00a074872240.     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki40\/symmetry-protected-topological-order-wikipedia\/#breadcrumbitem","name":"Symmetry-protected topological order – Wikipedia"}}]}]