Crosscap States in Integrable Field Theories and Spin Chains

TL;DR

This paper develops exact formulas for crosscap state overlaps in integrable field theories and spin chains, revealing their entanglement properties and RG flow behavior, with implications for holography and quantum quenches.

Contribution

It introduces new exact overlap formulas for crosscap states in integrable models and explores their entanglement and RG flow characteristics, connecting to holography and minimal models.

Findings

Overlap decreases monotonically along RG flow in most cases.

Crosscap states exhibit linear entanglement entropy growth up to half-system size.

Proposes generalization of staircase model to D-series minimal models.

Abstract

We study crosscap states in integrable field theories and spin chains in 1+1 dimensions. We derive an exact formula for overlaps between the crosscap state and any excited state in integrable field theories with diagonal scattering. We then compute the crosscap entropy, i.e. the overlap for the ground state, in some examples. In the examples we analyzed, the result turns out to decrease monotonically along the renormalization group flow except in cases where the discrete symmetry is spontaneously broken in the infrared. We next introduce crosscap states in integrable spin chains, and obtain determinant expressions for the overlaps with energy eigenstates. These states are long-range entangled and their entanglement entropy grows linearly until the size of the subregion reaches half the system size. This property is reminiscent of pure-state black holes in holography and makes them interesting for use as initial conditions of quantum quench. As side results, we propose a generalization of Zamolodchikov's staircase model to flows between D-series minimal models, and discuss the relation to fermionic minimal models and the GSO projection.