On the Boundary Entropy of One-dimensional Quantum Systems at Low Temperature

TL;DR

This paper proves a gradient formula for boundary entropy in one-dimensional quantum systems, showing it decreases under renormalization and temperature changes, confirming the conjecture that ground-state degeneracy decreases along renormalization flows.

Contribution

The paper establishes a gradient formula linking boundary entropy to the boundary beta-function, confirming the decrease of ground-state degeneracy under renormalization in 1D quantum systems.

Findings

Boundary entropy decreases under renormalization except at critical points.

Ground-state degeneracy g decreases under renormalization, confirming a long-standing conjecture.

Boundary entropy remains constant at critical points regardless of temperature.

Abstract

The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta-function, expressing it as the gradient of the boundary entropy s at fixed non-zero temperature. The gradient formula implies that s decreases under renormalization except at critical points (where it stays constant). At a critical point, the number exp(s) is the ``ground-state degeneracy,'' g, of Affleck and Ludwig, so we have proved their long-standing conjecture that g decreases under renormalization, from critical point to critical point. The gradient formula also implies that s decreases with temperature except at critical points, where it is independent of temperature. The boundary thermodynamic energy u then also decreases with temperature. It remains open whether the boundary entropy of a 1-d quantum system is always bounded below. If s is bounded below, then u is also bounded below.