Topological Orders in Rigid States

Wen, Xiao-Gang

doi:10.1142/s0217979290000139

Cited by 1,053 publications

(1,515 citation statements)

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“…Recently, it was realized that highly entangled quantum states can give rise to new kind of quantum phases beyond Landau symmetry breaking [1][2][3], which include topologically ordered phases [4][5][6], and symmetry-protected topological (SPT) phases [7,8] (see Fig. 1).…”

Section: A Short-and Long-range Entangled Statesmentioning

confidence: 99%

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T 2 = 1 time-reversal Z T 2 and fermion-number-parity Z f 2 symmetries described by a full symmetry group Z T 2 × Z f 2 ). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z 2 × Z f 2 . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.

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Section: A Short-and Long-range Entangled Statesmentioning

confidence: 99%

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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“…When applied to elections in the (lowest) Landau level, this approach is equivalent to the composite particle (composite boson and composite fermion) theories. 23,[28][29][30][31][32][89][90][91][92][93][94] The functional bosonization is also readily applicable to fractional quantum Hall states formed on a lattice, i.e., "fractional Chern insulators". See Refs.…”

Section: Dimensional Reductionmentioning

confidence: 99%

Effective field theories for topological insulators by functional bosonization

et al. 2013

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Effective field theories that describes the dynamics of a conserved U(1) current in terms of "hydrodynamic" degrees of freedom of topological phases in condensed matter are discussed in general dimension D = d+1 using the functional bosonization technique. For non-interacting topological insulators (superconductors) with a conserved U(1) charge and characterized by an integer topological invariant [more specifically, they are topological insulators in the complex symmetry classes (class A and AIII) and in the "primary series" of topological insulators in the eight real symmetry classes], we derive the BF-type topological field theories supplemented with the Chern-Simons (when D is odd) or the θ-term (when D is even). For topological insulators characterized by a Z2 topological invariant (the first and second descendants of the primary series), their topological field theories are obtained by dimensional reduction. Building on this effective field theory description for noninteracting topological phases, we also discuss, following the spirit of the parton construction of the fractional quantum Hall effect by Block and Wen, the putative "fractional" topological insulators and their possible effective field theories, and use them to determine the physical properties of these non-trivial quantum phases.

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“…Much of this progress has occurred in three areas of research: (1) the study of topological phases in condensed matter systems such as FQH systems [Wen and Niu (1990); Blok and Wen (1990); Read (1990); Fröhlich and Kerler (1991)], quantum dimer models [Rokhsar and Kivelson (1988); Read and Chakraborty (1989); Moessner and Sondhi (2001); Ardonne et al (2004)], quantum spin models [Kalmeyer and Laughlin (1987) ;Wen et al (1989); Wen (1990); Read and Sachdev (1991); Wen (1991a); Senthil and Fisher (2000); Wen (2002b); Sachdev and Park (2002) ;Balents et al (2002)], or even superconducting states [Wen (1991b); Hansson et al (2004)], (2) the study of lattice gauge theory [Wegner (1971); Banks et al (1977); Kogut and Susskind (1975); Kogut (1979)], and (3) the study of quantum computing by anyons [Kitaev (2003); Ioffe et al (2002); Freedman et al (2002)]. The phenomenon of string condensation is important in all of these fields, though the string picture is often de-emphasized.…”

mentioning

confidence: 99%

“…Ref. [Wen (1990)] used ground state degeneracy, particle statistics, and edge excitations to partially characterize topologically ordered states. Later, Ref.…”

mentioning

confidence: 99%