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On a bulk gap strategy for quantum lattice models

Amanda Young

TL;DR

This work presents a bulk gap strategy to rigorously bound the spectral gap of quantum lattice models in the thermodynamic limit when edge excitations lie below the bulk gap. It recasts standard spectral-gap techniques within an invariant-subspace framework, enabling ground-state and edge-state sectors to be treated separately and avoiding edge-mode contamination in gap bounds. The approach is instantiated for a truncated $1/3$-filled Haldane pseudopotential on the cylinder, where ground-state tilings (and edge tilings) furnish invariant subspaces, and explicit finite-size and martingale-based bounds yield a size-independent lower bound on the bulk gap under realistic parameter regimes. The results illuminate the structure of edge states, establish explicit subspace gap bounds, and demonstrate robust bulk-gap behavior despite edge excitations, with potential applicability to other models with boundary-localized low-energy modes.

Abstract

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [Warzel, Young '22] and [Warzel, Young '23] while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.

On a bulk gap strategy for quantum lattice models

TL;DR

This work presents a bulk gap strategy to rigorously bound the spectral gap of quantum lattice models in the thermodynamic limit when edge excitations lie below the bulk gap. It recasts standard spectral-gap techniques within an invariant-subspace framework, enabling ground-state and edge-state sectors to be treated separately and avoiding edge-mode contamination in gap bounds. The approach is instantiated for a truncated -filled Haldane pseudopotential on the cylinder, where ground-state tilings (and edge tilings) furnish invariant subspaces, and explicit finite-size and martingale-based bounds yield a size-independent lower bound on the bulk gap under realistic parameter regimes. The results illuminate the structure of edge states, establish explicit subspace gap bounds, and demonstrate robust bulk-gap behavior despite edge excitations, with potential applicability to other models with boundary-localized low-energy modes.

Abstract

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [Warzel, Young '22] and [Warzel, Young '23] while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.
Paper Structure (17 sections, 14 theorems, 120 equations, 4 figures)

This paper contains 17 sections, 14 theorems, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The truncated $1/3$-filled Haldane pseudopotential
  3. Spectral gap techniques and the bulk gap strategy
  4. The invariant subspace approach
  5. Spectral gap methods
  6. A coarse-grained finite size criterion
  7. The martingale method
  8. Invariant subspaces of the truncated Haldane pseudopotential
  9. Ground state tilings for the periodic model
  10. Ground state tilings for the open boundary model
  11. Edge state tilings
  12. Tiling space framentation and isospectral properties
  13. Applying the bulk gap strategy to the Haldane pseudopotential
  14. A lower bound on $E_1({\mathcal{V}}_\Lambda)$
  15. A lower bound on $E_0({\mathcal{V}}_\Lambda^\perp)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Fix any nonzero $\lambda \in {\mathbb C}$ and $\kappa>0.$ There exists a monotone increasing function $f:[0,\infty)\to[0,\infty)$ such that if $f(|\lambda|^2)<1/3$, then where

Figures (4)

  • Figure 1: An equivalence class of VMD tilings generated by the replacement rule. We use the void on the second site to unravel the periodic tiling to an interval.
  • Figure 2: A BMVD tiling equivalence class generated by the replacement rules. Boundary tiles can only be placed on their designated boundary, but are not required to be used.
  • Figure 3: An edge tiling equivalence class generated by the replacement rules. Note that removing the boundary site of the edge tile produces a BVMD tiling.
  • Figure 4: A truncation of a BVMD equivalence class. The right boundary of $\Lambda'$ separates the particles of a pair of neighboring monomers, but the left boundary does not. Hence, $n_R(\Lambda')=1$. The bottom two tilings produces the two distinct root tilings on $\Lambda'$ from obtained from truncating any $T\leftrightarrow R.$

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 2.1: Generalized Knabe Bound Knabe1988
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.4: The Martingale Method Nachtergaele1996Nachtergaele2018
  • proof
  • Corollary 2.5
  • proof
  • Proposition 3.1
  • ...and 12 more