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Stability of ground state degeneracy to long-range interactions

Matthew F. Lapa, Michael Levin

Abstract

We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g., power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (1) the stability of the gap, and (2) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.

Stability of ground state degeneracy to long-range interactions

Abstract

We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g., power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small in the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (1) the stability of the gap, and (2) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.

Paper Structure

This paper contains 37 sections, 6 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stability of the transverse field Ising model to long-range interactions
  3. Description of the model and its symmetry
  4. Main result
  5. Applications to other models
  6. Instability for interactions that violate the summability condition
  7. Proof of the main result
  8. Outline of the proof
  9. Step 1: A formula for $Z_{\pm}$
  10. Setting up the expansion of $Z_{\pm}$
  11. Definition of the support sets $X$ and factorization of the weights
  12. The bound on the weights
  13. Review of the polymer expansion
  14. Step 2: Convergent polymer expansion and its consequences
  15. Mapping our model to a polymer model
  16. ...and 22 more sections

Key Result

Theorem 1

There exists a $L$-independent constant $\lambda_0>0$ such that, if $|\lambda| < \lambda_0$, then (1) $H$ has a unique ground state and a finite energy gap in each sector $\mathcal{H}_{\pm}$ of the Hilbert space with fixed $\mathcal{S}$ eigenvalue, and (2) the ground state energy splitting $|E_{+}(\ where $c_1$ and $c_2$ are positive constants that depend on $\Delta$, $h$, and $\lambda$, but not o

Figures (2)

  • Figure 1: A microscopic domain wall worldline configuration (the red lines in the left panel) and its corresponding support set (shown in the right panel). The support set is made up of boxes, plaquettes, and dashed lines, and the plaquettes and boxes are shaded in light blue. The red crosses indicate the action of a $\sigma^x$ operator at the given position in spacetime.
  • Figure 2: We endow $\Lambda$ with a graph structure such that each vertex is connected by edges to 14 of its neighbors. This figure shows the 14 edges that emanate from the central red vertex (which could represent a box or a plaquette on the blocked spacetime lattice). The edges in $\Lambda$ that do not connect to the red vertex are not shown in this figure.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Claim 1
  • Claim 2
  • Theorem 2: Simplified convergence criterion
  • Theorem 3
  • Theorem : Theorem 3.4 of Ref. brydges