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

TL;DR

Addresses how boundary degrees of freedom in 1D quantum critical systems behave under renormalization at finite temperature. Introduces a physically-defined boundary metric and proves the gradient formula $g_{ab}(\lambda)\beta^{b}(\lambda)=-\partial_{a}s$, implying monotonic decrease of the boundary entropy and boundary energy with RG scale and with temperature away from fixed points. Demonstrates that the Affleck–Ludwig g-decrease conjecture follows as a corollary and clarifies the relationship to a string-theory gradient formula. Notes that the question of a universal lower bound for $s$ remains open, with implications for the IR behavior of boundary modes.

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.

Figures (1)

• Figure 1: The space-time is periodic in imaginary time $\tau$, with period $\beta$. The boundary is at $x=0$.