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Edge Hamiltonian for Free Fermion Quantum Hall Models

Simon Du, Martin Fraas, Abi Gopal, Nathan Singh

TL;DR

This work validates Kitaev's microscopic edge Hamiltonian construction for translation-invariant free-fermion quantum Hall models, showing that the edge-only operator $\widetilde{H}_{edge}$ supports discrete edge modes with correct chirality and a spectral flow equal to the bulk Hall conductance. By analyzing the Fourier transform and large-energy asymptotics, the authors derive an explicit asymptotic spectrum $E(m,j,k_y)$ whose edge-subspace contributions capture the topological winding encoded by Berry phases. They prove that the edge spectrum’s winding number equals the Hall conductance (the sum of occupied bands’ Chern numbers), and illustrate these results with Harper–Hofstadter numerics that align with theory. The findings provide a rigorous bridge between microscopic edge constructions and macroscopic topological invariants, with potential implications for extracting edge dynamics from bulk data in interacting or lattice systems.

Abstract

We investigate a proposal of Kitaev for a microscopic construction of a Hamiltonian intended to describe the edge dynamics of a quantum Hall system. We show that the construction works in the setting of translation-invariant free-fermion Hamiltonians. In this case, the resulting edge Hamiltonian exhibits only edge modes, and these modes have the correct chirality.

Edge Hamiltonian for Free Fermion Quantum Hall Models

TL;DR

This work validates Kitaev's microscopic edge Hamiltonian construction for translation-invariant free-fermion quantum Hall models, showing that the edge-only operator supports discrete edge modes with correct chirality and a spectral flow equal to the bulk Hall conductance. By analyzing the Fourier transform and large-energy asymptotics, the authors derive an explicit asymptotic spectrum whose edge-subspace contributions capture the topological winding encoded by Berry phases. They prove that the edge spectrum’s winding number equals the Hall conductance (the sum of occupied bands’ Chern numbers), and illustrate these results with Harper–Hofstadter numerics that align with theory. The findings provide a rigorous bridge between microscopic edge constructions and macroscopic topological invariants, with potential implications for extracting edge dynamics from bulk data in interacting or lattice systems.

Abstract

We investigate a proposal of Kitaev for a microscopic construction of a Hamiltonian intended to describe the edge dynamics of a quantum Hall system. We show that the construction works in the setting of translation-invariant free-fermion Hamiltonians. In this case, the resulting edge Hamiltonian exhibits only edge modes, and these modes have the correct chirality.

Paper Structure

This paper contains 5 sections, 15 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setup and Results
  3. Fourier Transform of the Operator $\widetilde{H}$
  4. Large Energy Asymptotics
  5. Numerical Experiments

Key Result

Theorem 2.2

The spectrum of $\widetilde{H}_{\mathrm{edge}}(k_y)$ is discrete. The spectral flow of the spectrum of $\widetilde{H}_{\text{edge}}(k_y)$ is equal to the Hall conductance.

Figures (5)

  • Figure 1: The spectrum of the Harper-Hofstadter model on a half plane with $\alpha = 1/3$. The model has three bands. The lines connecting the bands are the edge modes. The Chern numbers of these bands are $-1, 2, -1$.
  • Figure 2: The spectrum of $\widetilde{H}$ at $E_F = 1.5$, right figure, and $E_F = -1.5$, left figure, in blue, compared to \ref{['approxSpectrum']} in orange.
  • Figure 3: The spectrum of $\widetilde{H}$ at $E_F = 1.5$, sampled at 500 sites. The blue is the spectrum of $\widetilde{H}_{\text{edge}}$, the green the spectrum of $\widetilde{H}_{\text{edge}}^{-}$
  • Figure 4: The spectrum of $\widetilde{H}$ at $E_F = -1.5$, sampled at 500 sites. The blue is the spectrum of $\widetilde{H}_{\text{edge}}$, the green the spectrum of $\widetilde{H}_{\text{edge}}^{-}$
  • Figure 5: Parallel transport ignoring the dotted lines

Theorems & Definitions (29)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 19 more