Effects of interactions on the topological classification of free fermion systems

Fidkowski, Lukasz; Kitaev, Alexei

doi:10.1103/physrevb.81.134509

Cited by 581 publications

(828 citation statements)

References 23 publications

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“…Z T 2 represents time-reversal symmetry, U (1) represents U (1) symmetry, etc. As a comparison, the results for noninteracting fermionic gapped/SPT phases [48][49][50], as well as the interacting symmetric phases in 1D [44,[51][52][53], are also listed. Note that the symmetric interacting and noninteracting fermionic gapped phases can be the SPT phases or intrinsically topologically ordered phases.…”

Section: A Summary Of Main Resultsmentioning

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 Summary Of Main Resultsmentioning

confidence: 99%

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

2014

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“…First, with relatively few wires (N 8) one can experimentally explore the Z → Z 8 reduction of the BDI classification by interactions [46,47], very similar to Refs. [58,59].…”

Section: Let Us Now Examine a General T -Invariant Four-fermion Intermentioning

confidence: 99%

“…The full architecture continues to approximately preserve T provided (i) the dot carries negligible spin-orbit coupling and (ii) the B field orients in the plane of the dot so that orbital effects are absent. Here the setup falls into class BDI, which in the free-fermion limit admits an integer topological invariant ν ∈ Z [44,45] that counts the number of Majorana zero modes at each end; interactions collapse the classification to Z 8 [46,47]. In essence our device leverages nanowires to construct a topological phase with a free-fermion invariant ν = N: All bilinear couplings iM jk γ j γ k are forbidden by T and thus cannot be generated by the dot under the conditions specified above.…”

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confidence: 99%

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

2017

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The Sachdev-Ye-Kitaev (SYK) model describes a collection of randomly interacting Majorana fermions that exhibits profound connections to quantum chaos and black holes. We propose a solid-state implementation based on a quantum dot coupled to an array of topological superconducting wires hosting Majorana zero modes. Interactions and disorder intrinsic to the dot mediate the desired random Majorana couplings, while an approximate symmetry suppresses additional unwanted terms. We use random-matrix theory and numerics to show that our setup emulates the SYK model (up to corrections that we quantify) and discuss experimental signatures. DOI: 10.1103/PhysRevB.96.121119 Introduction. Majorana fermions provide building blocks for many novel phenomena. As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.The SYK Hamiltonian reads

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“…17,60,61 In the latter case, instead of imposing a sublattice symmetry, we would impose a reality condition (originating from the fact that we are dealing with Majorana fermions) combined with time-reversal symmetry. The reality condition is equivalent to a charge-conjugation symmetry and when combined with time-reversal plays a role similar to sublattice symmetry.…”

Section: B Generalitiesmentioning

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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Effects of interactions on the topological classification of free fermion systems

Cited by 581 publications

References 23 publications

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

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

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

Effective field theories for topological insulators by functional bosonization

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