Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological quantum order: stability under local perturbations

Sergey Bravyi, Matthew Hastings, Spyridon Michalakis

TL;DR

The paper proves that topologically ordered phases described by frustration-free commuting Hamiltonians on a D-dimensional lattice retain a stable spectral gap under generic, exponentially decaying local perturbations, provided two topological-quantum-order conditions (TQO-1 and TQO-2) hold. It develops a discrete Hamiltonian-flow framework to convert arbitrary local perturbations into a block-diagonal form while preserving locality via Lieb-Robinson bounds, achieving an exponentially small perturbative width for the lowest energy band and protected gaps between bands. A linearized block-diagonalization scheme and quasi-adiabatic continuation are then used to show that topological invariants and logical operator structures survive under perturbation, enabling robust adiabatic manipulation of topological quantum information. The results apply to stabilizer codes and models like the toric code and Levin-Wen string-nets, and they provide a rigorous foundation for the stability and controllability of topological quantum order in realistic, perturbed settings.

Abstract

We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H_0 we prove that there exists a constant threshold ε>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions the perturbed Hamiltonian H=H_0+εV has well-defined spectral bands originating from O(1) smallest eigenvalues of H_0. These bands are separated from the rest of the spectrum and from each other by a constant gap. The band originating from the smallest eigenvalue of H_0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb-Robinson bound.

Topological quantum order: stability under local perturbations

TL;DR

The paper proves that topologically ordered phases described by frustration-free commuting Hamiltonians on a D-dimensional lattice retain a stable spectral gap under generic, exponentially decaying local perturbations, provided two topological-quantum-order conditions (TQO-1 and TQO-2) hold. It develops a discrete Hamiltonian-flow framework to convert arbitrary local perturbations into a block-diagonal form while preserving locality via Lieb-Robinson bounds, achieving an exponentially small perturbative width for the lowest energy band and protected gaps between bands. A linearized block-diagonalization scheme and quasi-adiabatic continuation are then used to show that topological invariants and logical operator structures survive under perturbation, enabling robust adiabatic manipulation of topological quantum information. The results apply to stabilizer codes and models like the toric code and Levin-Wen string-nets, and they provide a rigorous foundation for the stability and controllability of topological quantum order in realistic, perturbed settings.

Abstract

We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H_0 we prove that there exists a constant threshold ε>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions the perturbed Hamiltonian H=H_0+εV has well-defined spectral bands originating from O(1) smallest eigenvalues of H_0. These bands are separated from the rest of the spectrum and from each other by a constant gap. The band originating from the smallest eigenvalue of H_0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb-Robinson bound.

Paper Structure

This paper contains 22 sections, 20 theorems, 193 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Sketch of the stability proof
  4. Hamiltonians describing TQO
  5. Frustration-free commuting Hamiltonians
  6. Formal definition of TQO
  7. Verification of TQO conditions for stabilizer Hamiltonians
  8. Unstable version of the toric code model
  9. Relatively bounded perturbations
  10. Definition and basic properties
  11. Stability of TQO under block-diagonal perturbations
  12. Hamiltonian flow equations
  13. Outline of the method
  14. Local decompositions of Hamiltonians
  15. Proof of the main theorem
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose $H_0$ obeys TQO-1,2. Then there exist constants $J_0,c_1,c_2>0$ 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 sp and

Figures (1)

  • Figure 1: Energy bands $I_k$ describing the spectrum of a perturbed Hamiltonian $H_0+V$.

Theorems & Definitions (39)

  • Theorem 1
  • Corollary 2.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.1
  • ...and 29 more