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Disordered Ground States of Ergodic Quantum Spin Systems

Eric B. Roon, Jeffrey H. Schenker

Abstract

In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction $\{\mathfrak{h}(Z)\}_{Z\Subset \mathbb{Z}^ν}$ satisfying a statistical version of translation invariance. We show such systems always have disordered ground states in the thermodynamic limit with the same symmetry. A key tool we use is a disordered version of the Lieb-Robinson bounds, which hold almost surely under mild conditions on $\mathfrak{h}$. Along the way, we formalize the notion of a random state on a $C^*$-algebra and prove a weak-$\ast$ version of the Riesz-Markov-Kakutani theorem, which seems not to have been recorded in the vector measures literature. As a consequence of the existence of the aforementioned disordered ground states, we show that the spectrum of the GNS Hamiltonain associated to the bulk dynamics is deterministic with respect to the disorder.

Disordered Ground States of Ergodic Quantum Spin Systems

Abstract

In this letter, we fill a hole in the existing literature about disordered quantum spin systems generated by a random local interaction satisfying a statistical version of translation invariance. We show such systems always have disordered ground states in the thermodynamic limit with the same symmetry. A key tool we use is a disordered version of the Lieb-Robinson bounds, which hold almost surely under mild conditions on . Along the way, we formalize the notion of a random state on a -algebra and prove a weak- version of the Riesz-Markov-Kakutani theorem, which seems not to have been recorded in the vector measures literature. As a consequence of the existence of the aforementioned disordered ground states, we show that the spectrum of the GNS Hamiltonain associated to the bulk dynamics is deterministic with respect to the disorder.
Paper Structure (12 sections, 19 theorems, 67 equations)

This paper contains 12 sections, 19 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Spin Systems
  4. Random states on Quasilocal Algebras.
  5. Quasilocality Estimates for Disordered Interactions
  6. Main Results
  7. Ergodic Local Hamiltonians
  8. Covariant Disordered Ground States
  9. The Spectrum of a Bulk Hamiltonian in an Ergodic Ground State
  10. An Application
  11. Measurability Concerns
  12. Weak-$\ast$ Measurability and Duality

Key Result

Theorem A

If $\mathbb{Z}^\nu$ is equipped with a translation-invariant metric, and $\{h^{\Lambda}_\omega\}$ is an ergodic interaction as in (eqn:translation-covariance) above, then there exist ergodic disordered bulk states $\psi_\omega$ satisfying

Theorems & Definitions (40)

  • Theorem A: Theorem \ref{['thm:exist']} Informal
  • Theorem B: Corollary \ref{['thm:Bulk_gap_const']} Informal
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Remark 3.1
  • Proposition 3.2: Lieb-Robinson Bound
  • Corollary 3.3
  • proof
  • ...and 30 more