In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (i.e. a topological order) can have a thermal Hall conductance
κxy=c3hπ2kB2T, where the dimensionless
c is called chiral central charge. If there is a
U1 symmetry, a gapped quantum phase can also have a Hall conductance
σxy=νhe2, where the dimensionless
ν is called filling fraction. In this paper, we derive some quantization conditions of
c and
ν, via a cobordism approach to define Chern--Simons topological invariants which are associated with
c and
ν. In particular, we obtain quantization conditions that depend on the ground state degeneracies on Riemannian surfaces, and quantization conditions that depend on the type of spacetime manifolds where the topological partition function is non-zero.
A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real valued in any local product basis.
Abstract We consider an infinitely extended system of fermions on a d-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength
ε ε is of order
O(ε∞) O ( ε ∞ ) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in
ε ε up to terms of order
O(ε∞) O ( ε ∞ ) . The proportionality factor
σH σ H is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for
d=2 d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with
σH σ H in expectation up to
O(ε∞) O ( ε ∞ ) , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.
This thesis studies the mathematical structure of quantum Hall transport in lattice systems, with an emphasis on relating mutiple definitions of Hall conductance in different mathematical branches: Geometrically as a bulk Chern number for a Bloch bundle; analytically as an edge spectral flow for a half-space Hamiltonian; and in operator algebra language as a many body charge transport index generated by magnetic flux insertion. The main results of the thesis reveal restrictions imposed on these indices by frustration-free structure, and improved locality estimates for interacting many body systems.The first result concerns frustration free free-Fermion (F 4 ) Hamiltonians. For translation-invariant, strictly local, gapped F4 Hamiltonians on Z 2 , we prove that the bulk index vanishes. More precisely, the two Bloch bundle associated with the image/kernel of the momentum space Hamiltonian is shown to be trivial by an algebraic argument using polynomial modules and a free resolution. Thus a frustration-free free-Fermion model cannot realize a nontrivial integer quantum Hall phase at the level of the bulk Chern index.The second new result is the corresponding edge statement. For a translation-invariant F4 Hamiltonian restricted to a half-plane, we prove that the edge index vanishes. We first identified that frustration free is equivalent to flat ground state band. Detailed analysis of spectral flow of the edge Hamiltonian shows that the edge mode is an analytic function and can be absorbed by the flat band. Consequently, the edge spectral flow across a regular energy in the bulk gap is constant and must equal its trivial value below the spectrum. This establishes, in this setting, that the frustration-free structure rules out nonzero edge transport as well as nonzero bulk transport. The third new result is an exponential clustering theorem for finite-volume quantum spin systems with a possibly degenerate and split ground-state sector. Assuming only a uniform spectral gap above a low-energy spectral patch of sufficiently small width, we prove exponential decay of correlations in the form|⟨Ω, ABΩ⟩ − ⟨Ω, AP BΩ⟩| ≤ C∥P∥1∥A∥∥B∥e −cd(X,Y ),for local observables A and B with disjoint supports. This improves the locality control needed in many body Hall conductance arguments, where previous estimates under comparable hypotheses were superpolynomial. Together, these results clarify how locality and spectral gaps constrain both free-Fermion and interacting formulations of topological transport.
In the Kitaev honeycomb model, a weak magnetic field gaps the gapless Majorana cone, driving the system into a gapped non-Abelian phase with Ising topological order. The edge supports a single chiral Majorana mode with chiral central charge
c−=1/2, yielding a half-quantized thermal Hall conductance
κxy/T=π2kB2/6h. We rigorously investigate whether observing this half-quantized value uniquely identifies the non-Abelian Ising anyon sector, or whether Abelian topological orders or extrinsic mechanisms can produce the same signature. By combining the edge-bulk correspondence with the classification of strictly two-dimensional bosonic topological orders, we show that a chiral central charge
c−=1/2 (mod 8) is the unique fingerprint of the Ising non-Abelian anyon theory among minimal bosonic topological orders. Abelian bosonic orders necessarily have integer
c−, with the minimal nonzero value being
c−=1. Thus, a robustly half-quantized
κxy/T uniquely distinguishes the Majorana sector from Abelian alternatives, provided the system is a genuine 2D bosonic spin system free of invertible
E8 stacking or phonon contamination. ## 1. Introduction The Kitaev honeycomb model is an exactly solvable model of a quantum spin liquid, characterized by fractionalized excitations known as anyons. In the absence of a magnetic field, the model hosts a gapless Majorana cone. A weak magnetic field in the [111] direction gaps this cone and induces a non-Abelian phase with Ising topological order. This phase supports a single chiral Majorana edge mode, which contributes to a quantized thermal Hall conductance. The central question addressed in this preprint is whether the observation of a half-quantized thermal Hall conductance, corresponding to a chiral central charge
c−=1/2, uniquely identifies the non-Abelian Ising anyon sector, or if Abelian topological orders can mimic this signature. ## 2. Background The thermal Hall conductance $\kappa_{x