arXiv 0810.06112008-10-03hep-thcond-mat.stat-mechquant-ph

1+1维超对称系统边界重整化群流的一般性质

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

Daniel Friedan, Anatoly Konechny

备注：: 21 pages, Latex
发表信息：: Adv. Theor. Math. Phys. 13 (2009) 1847-1874

摘要：

考虑一般的超对称一维有边界量子系统，在体中是临界的，而在边界处不是。边界条件空间上的重整化群流由边界耦合常数λ^{a}的边界β函数β^{a}（λ）生成。证明了一个梯度公式\partial \ln z/ \partial λ^{a} =- g_{ab} ^{S}β^{b}其中z（λ）是给定温度T=1/β时的边界配分函数，g_{ab} ^{S}（λ）是超对称边界条件空间上的某个正定度量.证明依赖于边界处的正则紫外行为。任何短距离行为由固定点控制的系统都满足这一要求。梯度公式意味着边界能- \partial \ln z/ \partial β= -Tβ^{a} \partial_{a} \ln z是非负的。等价地，在重整化群流下，\ln z（λ）量减小。

英文摘要：

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 β^{a}(λ) for the boundary coupling constants λ^{a}. We prove a gradient formula \partial\ln z/\partialλ^{a} =-g_{ab}^{S}β^{b} where z(λ) is the boundary partition function at given temperature T=1/β, and g_{ab}^{S}(λ) 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/\partialβ= -Tβ^{a}\partial_{a}\ln z, is nonnegative. Equivalently, the quantity \ln z(λ) decreases under the renormalization group flow.

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