Subsystem distance after a local operator quench

TL;DR

This paper studies how the trace and Schatten distances between reduced density matrices evolve over time after local operator quenches in 2D conformal field theory and 1D quantum spin chains, revealing orthogonality properties and matching numerical results with CFT predictions.

Contribution

It provides a detailed analysis of subsystem distances after local quenches, including numerical verification in spin chains and analytical predictions from conformal field theory, highlighting orthogonality of RDMs.

Findings

RDMs of intervals with quasiparticles are orthogonal to those without quasiparticles.

RDMs of intervals with quasiparticles at different positions are orthogonal.

Numerical results in spin chains match CFT predictions in the appropriate limit.

Abstract

We investigate the time evolution of the subsystem trace distance and Schatten distances after local operator quenches in two-dimensional conformal field theory (CFT) and in one-dimensional quantum spin chains. We focus on the case of a subsystem being an interval embedded in the infinite line. The initial state is prepared by inserting a local operator in the ground state of the theory. We only consider the cases in which the inserted local operator is a primary field or a sum of several primaries. While a nonchiral primary operator can excite both left-moving and right-moving quasiparticles, a holomorphic primary operator only excites a right-moving quasiparticle and an anti-holomorphic primary operator only excites a left-moving one. The reduced density matrix (RDM) of an interval hosting a quasiparticle is orthogonal to the RDM of the interval without any quasiparticles. Moreover, the RDMs of two intervals hosting quasiparticles at different positions are also orthogonal to each other. We calculate numerically the entanglement entropy, R\'enyi entropy, trace distance, and Schatten distances in time-dependent states excited by different local operators in the critical Ising and XX spin chains. These results match the CFT predictions in the proper limit.