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 nonzero temperature. The gradient formula implies that $s$ decreases under renormalization, except at critical points (where it stays constant). At a critical point, the number $\exp \left(s\right)$ 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. It remains open whether the boundary entropy is always bounded below.

• Received 22 December 2003

DOI:https://doi.org/10.1103/PhysRevLett.93.030402