We consider two-dimensional Hamiltonians on a torus with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap, and a conserved local charge, as defined precisely in the text. Using the local charge operators, we introduce a boundary magnetic flux in the horizontal and vertical direction and evolve the ground state quasi-adiabatically around a square of size one magnetic flux, in flux space. At the end of the evolution we obtain a trivial Berry phase, which we compare, via a method reminiscent of Stokes' Theorem, to the Berry phase obtained from an evolution around a small loop near the origin. As a result, we prove, without any averaging assumption, that the Hall conductance for interacting electron systems is quantized in integer multiples of e^2/h up to small corrections bounded by a function that decays as a stretched exponential in the linear size L. Finally, we discuss extensions to the fractional case under an additional topological order assumption to describe the multiple degenerate ground states.
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of
G-crossed braided tensor categories (
G×BTCs) for SETs. We consider systems with
U(1) charge conservation, magnetic translational, and spatial rotational symmetries, in the continuum and for all
5 orientation-preserving crystalline space groups in two dimensions, allowing arbitrary rational magnetic flux per unit cell, and assuming that symmetries do not permute anyons. In the crystalline setting, applicable to fractional Chern insulators and spin liquids, symmetry fractionalization is fully characterized by a generalization to non-Abelian states of the charge, spin, discrete torsion, and area vectors, which specify fractional charge, angular momentum, linear momentum, and fractionalization of the translation algebra for each anyon. The topological response theory contains
9 terms, which attach charge, linear momentum, and angular momentum to magnetic flux, lattice dislocations, disclinations, corners, and units of area. Using the
G×BTC formalism, we derive the formula relating charge filling to the Hall conductivity and flux per unit cell; in the continuum this relates the filling fraction and the Hall conductivity without assuming Galilean invariance. We provide systematic formulas for topological invariants within the
G×BTC framework; this gives, for example, a new categorical definition of the Hall conductivity.
We study charge transport for zero-temperature infinite-volume gapped lattice systems in two dimensions with short-range interactions. We show that the Hall conductance is locally computable and is the same for all systems that are in the same gapped phase. We provide a rigorous version of Laughlin’s flux-insertion argument, which shows that for short-range entangled systems, the Hall conductance is an integer multiple of e2/h. We show that the Hall conductance determines the statistics of flux insertions. For bosonic short-range entangled systems, this implies that the Hall conductance is an even multiple of e2/h. Finally, we adapt a proof of quantization of the Thouless charge pump to the case of infinite-volume gapped lattice systems in one dimension.
For T - L x L a finite subset of Z{sup 2}, let H{sub o} denote a Hamiltonian on T with periodic boundary conditions and finite range, finite strength intetactions and a unique ground state with a nonvanishing spectral gap. For S {element_of} T, let q{sub s} denote the charge at site s and assume that the total charge Q = {Sigma}{sub s {element_of} T} q{sub s} is conserved. Using the local charge operators q{sub s}, we introduce a boundary magnetic flux in the horizontal and vertical direction and allow the ground state to evolve quasiadiabatically around a square of size one magnetic flux, in flux space. At the end of the evolution we obtain a trivial Berry phase, which we compare, via a method reminiscent of Stokes Theorem. to the Berry phase obtained from an evolution around an exponentially small loop near the origin. As a result, we show, without any averaging assumption, that the Hall conductance is quantized in integer multiples of e{sup 2}/h up to exponentially small corrections of order e{sup -L/{zeta}}, where {zeta}, is a correlation length that depends only on the gap and the range and strength of the interactions.
We consider the edge transport properties of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. With respect to previous work, our result allows to consider a generic class of quantum Hall systems, displaying arbitrarily many edge modes. Our proof quantifies the validity and the limitations of the Luttinger liquid effective description for the edge currents. In particular, due to edge states scattering, the effective description alone is not able to predict the universality of the edge conductance. The exact quantization follows after fully taking into account the bulk degrees of freedom, whose precise contribution to the edge transport is determined thanks to lattice conservation laws.
Abstract We study Kitaev’s quantum double model for arbitrary finite gauge group in infinite volume, using an operator-algebraic approach. The quantum double model hosts anyonic excitations which can be identified with equivalence classes of ‘localized and transportable endomorphisms’, which produce anyonic excitations from the ground state. Following the Doplicher–Haag–Roberts (DHR) sector theory from AQFT, we organize these endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons. We show that this category is equivalent to
RepfD(G) Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.