Infrared properties of boundaries in 1-d quantum systems

TL;DR

This paper explores the infrared behavior of boundary entropy in 1-d quantum systems, examining conditions for boundary criticality and boundedness as temperature approaches zero, using real-time analytical methods.

Contribution

It demonstrates that boundary entropy decreases with temperature and links boundary beta-functions to the gradient of boundary entropy, providing new insights into boundary criticality.

Findings

Boundary entropy s(T) decreases with T.

Failure of boundary properties relates to pathological behaviors.

Boundary beta-function is the gradient of boundary entropy.

Abstract

We present some partial results on the general infrared behavior of bulk-critical 1-d quantum systems with boundary. We investigate whether the boundary entropy, s(T), is always bounded below as the temperature T decreases towards 0, and whether the boundary always becomes critical in the IR limit. We show that failure of these properties is equivalent to certain seemingly pathological behaviors far from the boundary. One of our approaches uses real time methods, in which locality at the boundary is expressed by analyticity in the frequency. As a preliminary, we use real time methods to prove again that the boundary beta-function is the gradient of the boundary entropy, which implies that s(T) decreases with T. The metric on the space of boundary couplings is interpreted as the renormalized susceptibility matrix of the boundary, made finite by a natural subtraction.