# Rigorous proof for the non-local correlation functions in the antiferromagnetic seamed transverse Ising ring

## TL;DR

This paper rigorously proves that all states in the lowest gapless spectrum of a frustrated transverse Ising ring share the same non-local correlation function, indicating temperature invariance within this energy range.

## Contribution

It provides a rigorous proof that the asymptotic correlation function is identical for all states in the lowest gapless spectrum of the model.

## Key findings

- All lowest gapless states exhibit the same asymptotic correlation function.
- Thermal correlation functions are temperature-invariant within the lowest gapless spectrum.
- The correlation function is non-local and persists in the thermodynamic limit.

## Abstract

An unusual correlation function is conjectured by M. Campostrini et al. (Phys. Rev. E 91, 042123 (2015)) for the ground state of a transverse Ising chain with geometrical frustration in one of the translationally invariant cases. Later, we demonstrated the correlation function and showed its non-local nature in the thermodynamic limit based on the rigorous evaluation of a Toeplitz determinant (J. Stat. Mech. 113102 (2016)). In this paper, we prove rigorously that all the states that forming the lowest gapless spectrum (including the ground state) in the kink phase exhibit the same asymptotic correlation function. So, in a point of view of cannonical ensemble, the thermal correlation function is inert to temperature within the energy range of the lowest gapless spectrum.