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Constraints on order and disorder parameters in quantum spin chains

Michael Levin

TL;DR

This work establishes precise constraints linking order and disorder parameters in Ising-symmetric spin chains. Using quantum-information tools (notably the Fuchs–van de Graaf inequality) together with Hastings-area-law-type entanglement bounds and AKLV entanglement results, the authors prove that any gapped, Ising-symmetric 1D chain must host either a nonzero order parameter or a nonzero disorder parameter, with explicit bounds: a $(\delta,\ell)$ parameter where $\delta=1/72$ and $\ell\lesssim \tilde{\mathcal{O}}((\log d)^3/\epsilon^2)$. They further show that a chain cannot support both an order parameter and a disorder parameter globally, nor a global odd-disorder parameter if the gap remains finite; these constraints extend to inhomogeneous chains and imply that self-dual chains are either gapless or have degenerate ground states. The paper also presents a three-region decomposition in non-translationally invariant settings and provides rigorous bounds via Hastings-type projections and AKLV entanglement, with implications for gapped boundaries of topological phases. Overall, the results give a near-complete set of constraints on order/disorder probes in 1D Ising-symmetric systems and suggest avenues for generalization to other symmetry classes and higher dimensions.

Abstract

We derive general constraints on order and disorder parameters in Ising symmetric spin chains. Our main result is a theorem showing that every gapped, translationally invariant, Ising symmetric spin chain has either a nonzero order parameter or a nonzero disorder parameter. We also prove two more constraints on order and disorder parameters: (i) it is not possible for a gapped, Ising symmetric spin chain to have both a nonzero order parameter and a nonzero disorder parameter; and (ii) it is not possible for a spin chain of this kind to have a nonzero disorder parameter that is odd under the symmetry. These constraints have an interesting implication for self-dual Ising symmetric spin chains: every self-dual spin chain is either gapless or has a degenerate ground state in the thermodynamic limit. All of these constraints generalize to spin chains without translational symmetry. Our proofs rely on previously known bounds on entanglement and correlations in one dimensional systems, as well as the Fuchs-van de Graaf inequality from quantum information theory.

Constraints on order and disorder parameters in quantum spin chains

TL;DR

This work establishes precise constraints linking order and disorder parameters in Ising-symmetric spin chains. Using quantum-information tools (notably the Fuchs–van de Graaf inequality) together with Hastings-area-law-type entanglement bounds and AKLV entanglement results, the authors prove that any gapped, Ising-symmetric 1D chain must host either a nonzero order parameter or a nonzero disorder parameter, with explicit bounds: a parameter where and . They further show that a chain cannot support both an order parameter and a disorder parameter globally, nor a global odd-disorder parameter if the gap remains finite; these constraints extend to inhomogeneous chains and imply that self-dual chains are either gapless or have degenerate ground states. The paper also presents a three-region decomposition in non-translationally invariant settings and provides rigorous bounds via Hastings-type projections and AKLV entanglement, with implications for gapped boundaries of topological phases. Overall, the results give a near-complete set of constraints on order/disorder probes in 1D Ising-symmetric systems and suggest avenues for generalization to other symmetry classes and higher dimensions.

Abstract

We derive general constraints on order and disorder parameters in Ising symmetric spin chains. Our main result is a theorem showing that every gapped, translationally invariant, Ising symmetric spin chain has either a nonzero order parameter or a nonzero disorder parameter. We also prove two more constraints on order and disorder parameters: (i) it is not possible for a gapped, Ising symmetric spin chain to have both a nonzero order parameter and a nonzero disorder parameter; and (ii) it is not possible for a spin chain of this kind to have a nonzero disorder parameter that is odd under the symmetry. These constraints have an interesting implication for self-dual Ising symmetric spin chains: every self-dual spin chain is either gapless or has a degenerate ground state in the thermodynamic limit. All of these constraints generalize to spin chains without translational symmetry. Our proofs rely on previously known bounds on entanglement and correlations in one dimensional systems, as well as the Fuchs-van de Graaf inequality from quantum information theory.

Paper Structure

This paper contains 13 sections, 13 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Two more constraints on order and disorder parameters
  4. Application to self-dual spin chains
  5. Proof of main result
  6. Definitions
  7. Main argument
  8. The Fuchs-van de Graaf inequality and the proof of Lemma \ref{['orddis']}
  9. Some bounds on correlations and the proof of Lemma \ref{['ordparam']}
  10. A bound on entanglement and the proof of Lemma \ref{['weakimpstrong']}
  11. Proofs of additional constraints
  12. Discussion
  13. Acknowledgments

Key Result

Theorem 1

If $H$ is translationally invariant, then $|\Omega\rangle$ has either a $(\delta, \ell)$ order parameter or a $(\delta, \ell)$ disorder parameter defined on the whole spin chain (i.e. $X = \{1,...,L\}$) with

Figures (2)

  • Figure 1: (a) A spin chain made up of $L$ spins in a ring geometry. (b) A $(\delta, \ell)$ order parameter is a collection of operators $\{O_i\}$, supported on the intervals $[i-\ell, i+\ell]$, obeying certain conditions.
  • Figure 2: Example of $I^\pm$ notation: the spins in the interval $I$ are shown in black; the spins in $I^-$ and $I^+$ are shown in red and yellow; and the spins in $I' \equiv (I^- \cup I \cup I^+)^c$ are shown in blue.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • proof
  • Definition 3
  • Definition 4
  • ...and 23 more