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A short proof of stability of topological order under local perturbations

S. Bravyi, M. B. Hastings

TL;DR

The article presents a short, alternate proof of spectral-gap stability for topologically ordered quantum Hamiltonians where $H_0$ is a sum of local commuting projectors. Using Lieb-Robinson bounds and exact quasi-adiabatic continuation, it shows that for sufficiently small local perturbations $V$ with exponentially decaying interactions, the spectrum of $H=H_0+V$ remains organized into bands derived from the low-energy spectrum of $H_0$, separated by a constant gap, with the lowest-band width decaying faster than any power of the system size. A key contribution is reducing global block-diagonal perturbations to locally block-diagonal ones and proving stability under locally block-diagonal perturbations via relative boundedness, then extending to general local perturbations through exact continuation. This yields a concise, complementary proof of gap stability for topological order that complements previous longer approaches and tightens the understanding of perturbative robustness in topologically ordered phases.

Abstract

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian $H_0$ can be written as a sum of local pairwise commuting projectors on a $D$-dimensional lattice. We consider a perturbed Hamiltonian $H=H_0+V$ involving a generic perturbation $V$ that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of $V$ is below a constant threshold value then $H$ has well-defined spectral bands originating from the low-lying eigenvalues of $H_0$. These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of $H_0$ decays faster than any power of the lattice size.

A short proof of stability of topological order under local perturbations

TL;DR

The article presents a short, alternate proof of spectral-gap stability for topologically ordered quantum Hamiltonians where is a sum of local commuting projectors. Using Lieb-Robinson bounds and exact quasi-adiabatic continuation, it shows that for sufficiently small local perturbations with exponentially decaying interactions, the spectrum of remains organized into bands derived from the low-energy spectrum of , separated by a constant gap, with the lowest-band width decaying faster than any power of the system size. A key contribution is reducing global block-diagonal perturbations to locally block-diagonal ones and proving stability under locally block-diagonal perturbations via relative boundedness, then extending to general local perturbations through exact continuation. This yields a concise, complementary proof of gap stability for topological order that complements previous longer approaches and tightens the understanding of perturbative robustness in topologically ordered phases.

Abstract

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological quantum order for which the unperturbed Hamiltonian can be written as a sum of local pairwise commuting projectors on a -dimensional lattice. We consider a perturbed Hamiltonian involving a generic perturbation that can be written as a sum of short-range bounded-norm interactions. We prove that if the strength of is below a constant threshold value then has well-defined spectral bands originating from the low-lying eigenvalues of . These bands are separated from the rest of the spectrum and from each other by a constant gap. The width of the band originating from the smallest eigenvalue of decays faster than any power of the lattice size.

Paper Structure

This paper contains 7 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Hamiltonians describing TQO
  3. Local Hamiltonians and the Lieb-Robinson bounds
  4. Reduction from global to local block-diagonality
  5. Stability under locally block-diagonal perturbation
  6. Exact quasi-adiabatic continuation
  7. Stability proof

Key Result

Theorem 1

There exist constants $J_0,c_1>0$ depending only on $\mu$ and the spatial dimension $D$ such that for all $J\le J_0$ the spectrum of $H_0+V$ is contained (up to an overall energy shift) in the union of intervals $\bigcup_{k\ge 0} I_k$, where $k$ runs over the spectrum of $H_0$ and $I_k$ is the close for some $\delta$ bounded by $J$ times a quantity decaying faster than any power of $L$.

Theorems & Definitions (21)

  • Theorem 1
  • Corollary 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Definition 4
  • Lemma 2
  • Lemma 3
  • ...and 11 more