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Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model

Angelo Lucia, Alvin Moon, Amanda Young

Abstract

We use cluster expansions to establish local indistiguishability of the finite-volume ground states for the AKLT model on decorated hexagonal lattices with decoration parameter at least 5. Our estimates imply that the model satisfies local topological quantum order (LTQO), and so the spectral gap above the ground state is stable against local perturbations.

Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model

Abstract

We use cluster expansions to establish local indistiguishability of the finite-volume ground states for the AKLT model on decorated hexagonal lattices with decoration parameter at least 5. Our estimates imply that the model satisfies local topological quantum order (LTQO), and so the spectral gap above the ground state is stable against local perturbations.
Paper Structure (15 sections, 18 theorems, 164 equations, 5 figures)

This paper contains 15 sections, 18 theorems, 164 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Stability of the frustration-free ground state
  4. Characterizations of ground states
  5. The ground states and bulk state of the decorated AKLT Hamiltonian
  6. Graphs and weights for the hard core polymer representation
  7. The hard-core polymer representation of $Z_N$
  8. The hard-core loop representation of $\mathring{Z}_{N}$
  9. A comparison lemma
  10. Indistinguishability of ground states
  11. Cluster expansion preliminaries
  12. Indistinguishability of the finite volume ground states
  13. Appendix
  14. Modifications to the stability argument
  15. Cluster expansion preliminaries

Key Result

Theorem 2.1

For the decorated AKLT model with $d\geq 5$, there is a frustration-free state $\omega^{(d)}:{\mathcal{A}}_{\Gamma^{(d)}}\to{\mathbb C}$ so that for any normalized $\psi_n\in \ker(H_{\Lambda_n^{(d)}})$ and observable $A\in{\mathcal{A}}_{\mathring{\Lambda}_k^{(d)}}$ with $1\leq k < n$, where $F_\alpha(n,k)=102ke^{-2\alpha(n-k)}$ and, with respect to $f(x) = \frac{x+1-\sqrt{x^2+1}}{x}$,

Figures (5)

  • Figure 1: The decorated hexagonal lattice for $d=2$.
  • Figure 2: Illustration of $\Lambda_2^{(1)}$ and $\Gamma_2^{(1)}$. The latter is used to verify Assumption \ref{['assump:localgap']} for the spectral gap stability argument. The red vertices comprise $\partial \Lambda_2^{(1)}$, and $\mathring{\Lambda}_2^{(1)}$ corresponds to the black vertices and edges between them. We suppress the dependence of $\tilde{x}\in\tilde{\Gamma}_0$ for simplicity.
  • Figure 3: Illustration of the separating partition. The part ${\mathcal{T}}_2^{\tilde{0}}$ is the set of the dual lattice points where two dotted lines intersect. The index set ${\mathcal{I}}_2\subseteq \tilde{\Gamma}^{(0)}$ is set of points contained in the fundamental cell outlined in red.
  • Figure 4: Example of polymers from ${\mathcal{S}}_{5,2}^{(1)}$. The decoration is suppressed outside of $\Lambda_2^{(1)}$ for clarity.
  • Figure 5: All the possible types of polymers that contribute for cases 1-3 that pass through the vertices $a$-$d$. The contributing endpoints and edges to these polymers are colored. For case 1, odd length polymers must pass through a "corner" as illustrated by the polymers passing through vertex $b$. For case 3a, all polymers of length four contribute. For case 3b, the figure shows all length four polymers passing through vertices $a$ and $c$, all length three polymers passing through $b$, and all length 2 polymers passing through $d$.

Theorems & Definitions (36)

  • Theorem 2.1: Ground State Indistinguishability
  • Theorem 2.2: Spectral Gap Stability
  • Corollary 3.5: LTQO
  • proof
  • proof : Proof of Theorem \ref{['thm:stability']}
  • Theorem 4.1: Kennedy1988KK89
  • Lemma 4.2: Bulk-boundary map
  • proof
  • Definition 4.3
  • Definition 4.4
  • ...and 26 more