Symmetry-protected topological phases, generalized Laughlin argument, and orientifolds

Hsieh, Chang-Tse; Sule, Olabode Mayodele; Cho, Gil Young; Ryu, Shinsei; Leigh, Robert G.

doi:10.1103/physrevb.90.165134

Cited by 60 publications

(74 citation statements)

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“…For example, the chiral fermion in 1 + 1 dimensions indeed appears as the edge state of a two-dimensional quantum Hall system. Ryu et al [10,11] generalized this observation to classification of gapped SPT phases: the edge or surface state of an SPT phase exhibits an anomaly with respect to the relevant symmetry, which implies the "ingappability" of the edge state in the presence of the symmetry [12]. Conversely, such an anomaly can be identified with an edge or surface state, and thus with an SPT phase in higher dimensions.…”

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Symmetry Protection of Critical Phases and a Global Anomaly in1+1Dimensions

Furuya

Oshikawa

2017

Phys. Rev. Lett.

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We derive a selection rule among the (1 + 1)-dimensional SU(2) Wess-Zumino-Witten theories, based on the global anomaly of the discrete Z2 symmetry found by Gepner and Witten. In the presence of both the SU(2) and Z2 symmetries, a renormalization-group flow is possible between level-k and level-k ′ Wess-Zumino-Witten theories only if k ≡ k ′ mod 2. This classifies the Lorentzinvariant, SU(2)-symmetric critical behavior into two "symmetry-protected" categories corresponding to even and odd levels, restricting possible gapless critical behavior of translation-invariant quantum spin chains.PACS numbers: 75.10. Jm, 75.10.Pq, 03.65.Vf, 11.25.Hf Introduction.-The classification of quantum phases is a central problem in condensed-matter and statistical physics. They are first classified into gapped and gapless phases. There has been a significant progress in further classification of gapped phases, which are relatively easy to be handled theoretically. In particular, symmetry protected topological (SPT) phases [1,2] have become an important concept in the classification. That is, even when two states have no long-range entanglement and are indistinguishable in terms of any local observables, they could still belong to distinct phases separated by a quantum phase transition, in the presence of a certain symmetry. In contrast, classification of gapless quantum phases remains very much open. Symmetries are naturally expected to play an important role also in the classification of gapless quantum phases.The issue of classification of quantum phases is also deeply related to that of quantum field theories. Since a quantum field theory can be regarded as an effective description of universal low-energy behaviors of quantum many-body systems, we may expect that they are essentially the same problem. An intriguing feature of quantum field theory is the anomaly [3][4][5][6]. A particularly interesting anomaly is the global anomaly, which emerges after promotion of a global symmetry to gauge symmetry [6]. As it is the case with many connections between quantum field theory and condensed-matter physics, investigation of the consequences of the anomaly in quantum field theory in a condensed-matter physics context has been quite fruitful, including the discovery of the "Chern insulator" [7]. Nevertheless, the exact correspondence in many concrete cases is not yet understood.An anomaly in a field theory may imply that the field theory cannot be realized in a condensed-matter system or a lattice model in the same dimensions. A renowned case is the impossibility of realization of a chiral fermion with (noninteracting) fermions on a lattice, known as the Nielsen-Ninomiya theorem [8,9], which is deeply related to the chiral anomaly, a representative example of the quantum anomaly. Nevertheless, such a field the-

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Symmetry Protection of Critical Phases and a Global Anomaly in1+1Dimensions

Furuya

Oshikawa

2017

Phys. Rev. Lett.

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“…[45] Chern-Simons theory [46][47][48][49][50], nonlinear sigma models [51,52], and an orbifolding approach implementing modular invariance on 1D edge modes [25,28]. The above approaches have their own benefits, but they may be either limited to certain dimensions, or be limited to some special cases.…”

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Field-Theory Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond

2015

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The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are universal SPT invariants, fully characterizing SPTs. In this work, we use gauge fields to represent those symmetry twists in closed spacetimes of any dimensionality and arbitrary topology. This allows us to express the SPT invariants in terms of continuum field theory. We show that SPT invariants of pure gauge actions describe the SPTs predicted by group cohomology, while the mixed gaugegravity actions describe the beyond-group-cohomology SPTs. We find new examples of mixed gaugegravity actions for U(1) SPTs in ð4 þ 1ÞD via the gravitational Chern-Simons term. Field theory representations of SPT invariants not only serve as tools for classifying SPTs, but also guide us in designing physical probes for them. In addition, our field theory representations are independently powerful for studying group cohomology within the mathematical context. [3][4][5]. However, even without symmetry breaking and without topological order, gapped systems can still be nontrivial if there is certain global symmetry protection, known as symmetry-protected topological states (SPTs) [6][7][8][9]. Their nontrivialness can be found in the gapless-topological boundary modes protected by a global symmetry, which shows gauge or gravitational anomalies . More precisely, they are short-range entangled states which can be deformed to a trivial product state by local unitary transformation [31][32][33] if the deformation breaks the global symmetry. Examples of SPTs are Haldane spin-1 chains protected by spin rotational symmetry [34,35] and the topological insulators [36][37][38] protected by fermion number conservation and time reversal symmetry.While some classes of topological orders can be described by topological quantum field theories (TQFT) [39][40][41][42], it is less clear how to systematically construct field theory with a global symmetry to classify or characterize SPTs for any dimension. This challenge originates from the fact that SPTs are naturally defined on a discretized spatial lattice or on a discretized spacetime path integral by a group cohomology construction [6,43] instead of continuous fields. Group cohomology construction of SPTs also reveals a duality between some SPTs and the Dijkgraaf-Witten topological gauge theory [43,44].Some important progress has been recently made to tackle the above question. For example, there are the ð2 þ 1ÞD[45] Chern-Simons theory [46][47][48][49][50], nonlinear sigma models [51,52], and an orbifolding approach implementing modular invariance on 1D edge modes [25,28]. The above approaches have their own benefits, but they may be either limited to certain dimensions, ...

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“…Similarly, for phases of matter with more generic symmetry, one can consider twisting the boundary condition using the symmetry of the system. For SPT phases protected by orientation reversing symmetry, the symmetry-twisted boundary conditions naturally give rise to unoriented spacetime manifolds [24,[30][31][32][33][34]. From the topological arXiv:1607.03896v3 [cond-mat.str-el] 29 May 2017 2 quantum field theory description of topological superconductors [30,34], one expects that the complex phase of the partition function, when the system is put on an appropriate unoriented manifold, is quantized and serves as a topological invariant.…”

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Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

2017

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We define and compute many-body topological invariants of interacting fermionic symmetryprotected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the Z8 and Z16 classification of topological superconductors in one and three dimensions.The Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula [1, 2] is the prototype for topological characterization of phases of matter. It relates the quantized Hall conductance to the (first) Chern number defined for Bloch wave functions. At the level of many-body physics, the quantized Hall conductance can also be formulated in terms of ground state wave functions in the presence of twisted boundary conditions ("the many-body Chern number") [3]. In contrast to local order parameters, the TKNN integer distinguishes different quantum phases of matter by focusing on their global topological properties.More recently, the discovery of topological insulators and superconductors [4, 5] has led to a new research frontier, generally referred to as symmetry protected topological (SPT) phases. These phases are adiabatically connected to topologically trivial states, i.e., atomic insulators which can be represented as simple product states without any entanglement. Nevertheless, they are topologically distinct once a symmetry condition, e.g., timereversal symmetry, is imposed. A complete classification of the noninteracting fermionic SPT phases protected by non-spatial discrete symmetries [6][7][8], as well as crystalline SPT phases protected by spatial symmetries [9][10][11][12], were achieved.However, it was later discovered that the noninteracting topological classification is not the full story, and can be dramatically altered once interaction effects are taken into account [13]. Since then, there have been several works which discuss the breakdown of the noninteracting classification in the presence of interactions [14][15][16][17][18][19][20][21][22][23][24][25].There are various topological invariants for noninteracting fermionic SPT phases using single-particle states (e.g., Bloch wave functions). For example, the Z 2 -valued topological invariants have been introduced for topological insulators both in two and three spatial dimensions [26][27][28][29]. For topological superconductors protected by time-reversal symmetry, the integer-valued topological invariants ("the winding number") have been introduced [6]. However, the discovered breakdown of non-interacting classification clearly indicates that the situation at the interacting level is more intricate, and a general framework to distinguish interacting fermionic SPT phases is lacking. This should be contrasted from the quantized Hall conductance, which can be formulate...

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Symmetry-protected topological phases, generalized Laughlin argument, and orientifolds

Cited by 60 publications

References 70 publications

Symmetry Protection of Critical Phases and a Global Anomaly in1+1Dimensions

Symmetry Protection of Critical Phases and a Global Anomaly in1+1Dimensions

Field-Theory Representation of Gauge-Gravity Symmetry-Protected Topological Invariants, Group Cohomology, and Beyond

Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

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