# Constraints on order and disorder parameters in quantum spin chains

## TL;DR

This paper establishes fundamental constraints on order and disorder parameters in gapped, translationally invariant Ising symmetric quantum spin chains, revealing conditions for their coexistence and implications for self-dual chains.

## Contribution

It proves that such spin chains must have either a nonzero order or disorder parameter, and that self-dual chains are either gapless or degenerate, extending these results beyond translational symmetry.

## Key findings

- Gapped, translationally invariant, Ising symmetric chains have either a nonzero order or disorder parameter.
- A chain cannot have both a nonzero order and a nonzero disorder parameter simultaneously.
- Self-dual chains are either gapless or have degenerate ground states.

## 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.