General properties of the boundary renormalization group flow for supersymmetric systems in 1+1 dimensions

We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but not at the boundary. The renormalization group flow on the space of boundary conditions is generated by the boundary beta functions \beta^{a}(\lambda) for the boundary coupling constants \lambda^{a}. We prove a gradient formula \partial\ln z/\partial\lambda^{a} =-g_{ab}^{S}\beta^{b} where z(\lambda) is the boundary partition function at given temperature T=1/\beta, and g_{ab}^{S}(\lambda) is a certain positive-definite metric on the space of supersymmetric boundary conditions. The proof depends on canonical ultraviolet behavior at the boundary. Any system whose short distance behavior is governed by a fixed point satisfies this requirement. The gradient formula implies that the boundary energy, -\partial\ln z/\partial\beta = -T\beta^{a}\partial_{a}\ln z, is nonnegative. Equivalently, the quantity \ln z(\lambda) decreases under the renormalization group flow.