Excited state R\'enyi entropy and subsystem distance in two-dimensional non-compact bosonic theory -- I. Single-particle states

TL;DR

This paper analytically calculates the second R'enyi entropy and Schatten distance for excited states in a 2D free massless non-compact bosonic theory, linking conformal field theory results with harmonic chain models and exploring universal behaviors.

Contribution

It provides the first analytical computation of R'enyi entropy and Schatten distance for excited states in this theory and confirms the universality in large momentum limits.

Findings

Second R'enyi entropy matches between field theory and harmonic chain models.

Universal form of R'enyi entropy in large momentum limit.

Subsystem Schatten distance exhibits universal behavior with momentum difference.

Abstract

We investigate the R\'enyi entropy of the excited states produced by the current and its derivatives in the two-dimensional free massless non-compact bosonic theory, which is a two-dimensional conformal field theory. We also study the subsystem Schatten distance between these states. The two-dimensional free massless non-compact bosonic theory is the continuum limit of the finite periodic gapless harmonic chains with the local interactions. We identify the excited states produced by current and its derivatives in the massless bosonic theory as the single-particle excited states in the gapless harmonic chain. We calculate analytically the second R\'enyi entropy and the second Schatten distance in the massless bosonic theory. We then use the wave functions of the excited states and calculate the second R\'enyi entropy and the second Schatten distance in the gapless limit of the harmonic chain, which match perfectly with the analytical results in the massless bosonic theory. We verify that in the large momentum limit the single-particle state R\'enyi entropy takes a universal form. We also show that in the limit of large momenta and large momentum difference the subsystem Schatten distance takes a universal form but it is replaced by a corrected form when the momentum difference is small. Finally we also comment on the mutual R\'enyi entropy of two disjoint intervals in the excited states of the two-dimensional free non-compact bosonic theory.