Friedan Daniel, Konechny Anatoly
Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854-8019, USA.
Phys Rev Lett. 2004 Jul 16;93(3):030402. doi: 10.1103/PhysRevLett.93.030402. Epub 2004 Jul 12.
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((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. It remains open whether the boundary entropy is always bounded below.
边界β函数生成了重整化群，该重整化群作用于具有边界的一维量子系统的普适类，这些系统在体相中是临界的，但在边界处不是临界的。我们证明了边界β函数的梯度公式，将其表示为在固定非零温度下边界熵s的梯度。该梯度公式意味着，除了在临界点（此时它保持不变）外，s在重整化下会减小。在临界点处，数量exp(s)是阿弗莱克（Affleck）和路德维希（Ludwig）的“基态简并度”g，因此我们证明了他们长期以来的猜想，即g在重整化下从一个临界点到另一个临界点会减小。该梯度公式还意味着，除了在临界点（此时它与温度无关）外，s随温度降低。边界熵是否总是有下界仍然未知。
