Local Commuting Projector Hamiltonians and the Quantum Hall Effect

Kapustin, Anton; Fidkowski, Lukasz

doi:10.1007/s00220-019-03444-1

Cited by 45 publications

(53 citation statements)

References 15 publications

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“…Therefore such systems cannot have edge modes with a nonzero chiral central charge. This is an energy counterpart of the recently proved result that in such systems the zero-temperature electric Hall conductance vanishes [13]. In the concluding section we compare our approach to thermal Hall conductance with the results in the literature.…”

Section: Introduction and Overviewsupporting

confidence: 56%

“…4 This implies that the chiral central charge of the edge modes must vanish for such a Hamiltonian. One can also show that the zero-temperature electric Hall conductance vanishes for such systems, but the proof is very different [13].…”

Section: A Relative Invariant Of Gapped 2d Lattice Systemsmentioning

confidence: 99%

“…The chain ∂A is called the boundary of the chain A. Using this notation, the conservation equation (13) can be written as…”

Section: B Chains and Cochainsmentioning

confidence: 99%

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Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems

Kapustin

Spodyneiko

2020

Phys. Rev. B

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We show that derivatives of thermal Hall conductance of a 2d lattice quantum system with respect to parameters of the Hamiltonian are well-defined bulk quantities provided correlators of local observables are short-range. This is despite the fact that thermal Hall conductance itself has no meaning as a bulk transport coefficient. We use this to define a relative topological invariant for gapped 2d lattice quantum systems at zero temperature. Up to a numerical factor, it can be identified with the difference of chiral central charges for the corresponding edge modes. This establishes bulk-boundary correspondence for the chiral central charge. We also show that for Local Commuting Projector Hamiltonians relative thermal Hall conductance vanishes identically, while for free fermionic systems it is related to the electric Hall conductance via the Wiedemann-Franz

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Section: Introduction and Overviewsupporting

confidence: 56%

Section: A Relative Invariant Of Gapped 2d Lattice Systemsmentioning

confidence: 99%

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Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems

Kapustin

Spodyneiko

2020

Phys. Rev. B

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“…On the plane dual to each link, we put a 2D p þ ip chiral superconductor. Different from other layers, the p þ ip superconductor layer does not have a fixed-point wave function on a discrete lattice [80]. So we do not discuss this layer decoration until the end of this section.…”

Section: ð166þmentioning

confidence: 99%

Construction and Classification of Symmetry-Protected Topological Phases in Interacting Fermion Systems

Wang

2020

Phys. Rev. X

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The classification and lattice model construction of symmetry-protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixedpoint wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group G f ¼ Z f 2 × ω 2 G b which is a central extension of bosonic symmetry group G b (may contain time-reversal symmetry) by the fermion parity symmetry group Z f 2 ¼ f1; P f g. Our construction is based on the concept of an equivalence class of finite-depth fermionic symmetric local unitary transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We also discuss the systematical construction and classification of boundary anomalous SPT states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point and space group symmetry as well as continuum Lie group symmetry.

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“…Despite this progress, the construction of tractable microscopic models for topological states of matter starting from local spin or electronic degrees of freedom remains challenging. Especially challenging are chiral (i.e., time-reversalbreaking) topological phases, which cannot be represented by exactly solvable lattice models whose Hamiltonians consist of local commuting projectors [19] (in contrast to, e.g., Kitaev's toric code and quantum double models [20]). There is, however, an approach that allows for the development of tractable models even in the case of chiral phases: the coupledwire construction.…”

Section: A Motivationmentioning

confidence: 99%

Ground-state degeneracy of non-Abelian topological phases from coupled wires

et al. 2019

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We construct a family of two-dimensional non-Abelian topological phases from coupled wires using a non-Abelian bosonization approach. We then demonstrate how to determine the nature of the non-Abelian topological order (in particular, the anyonic excitations and the topological degeneracy on the torus) realized in the resulting gapped phases of matter. This paper focuses on the detailed case study of a coupled-wire realization of the bosonic su(2) 2 Moore-Read state, but the approach we outline here can be extended to general bosonic su(2) k topological phases described by non-Abelian Chern-Simons theories. We also discuss possible generalizations of this approach to the construction of three-dimensional non-Abelian topological phases. IADECOLA, NEUPERT, CHAMON, AND MUDRY PHYSICAL REVIEW B 99, 245138 (2019) 245138-2 GROUND-STATE DEGENERACY OF NON-ABELIAN … PHYSICAL REVIEW B 99, 245138 (2019)(2.6e)With these definitions, it follows that the Hamiltonian density associated with the free Lagrangian density (2.1) is given by(2.7)Rewriting the free theory (2.1) in terms of the currents (2.4) amounts to performing a non-Abelian bosonization of the free theory. This rewriting highlights the fact that a theory of multiple flavors of free fermions can be broken up into independent charge [u(1)], spin [su(2) N c ], and color (orbital) [su(N c ) 2 ] sectors. B. Intrawire interactionsHaving rewritten the free theory (2.1) in terms of the non-Abelian currents (2.4), we now wish to isolate the su(2) N c spin degrees of freedom by removing the u(1) charge and su(N c ) 2 color (orbital) degrees of freedom from the low-energy sector of the theory. We accomplish this by adding interactions that gap out the latter pair of degrees of freedom. 245138-3 IADECOLA, NEUPERT, CHAMON, AND MUDRY PHYSICAL REVIEW B 99, 245138 (2019) The interaction (3.2) can be better understood by rewriting the su(2) k currents in terms of auxiliary degrees of freedom. This rewriting must preserve the su (2) k current algebra, which is encoded in the operator product expansion (OPE) [66] J a L,y (v) J˜a L,ỹ (w) ∼ δ y,ỹ (k/2) δ aã v 2 − w 2 + i aãb J b L,y (w) v − w , (3.3) 245138-4 GROUND-STATE DEGENERACY OF NON-ABELIAN … PHYSICAL REVIEW B 99, 245138 (2019)

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Local Commuting Projector Hamiltonians and the Quantum Hall Effect

Abstract: We prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians. 1 arXiv:1810.07756v1 [cond-mat.str-el]

Cited by 45 publications

References 15 publications

Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems

Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems

Construction and Classification of Symmetry-Protected Topological Phases in Interacting Fermion Systems

Ground-state degeneracy of non-Abelian topological phases from coupled wires

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