Luttinger液体Rényi-Shannon熵的相变
Phase transition in the Rényi-Shannon entropy of Luttinger liquids
摘要：
研究了中心电荷为$c=1$的临界量子自旋链的Rényi-Shannon熵在Rényi参数$n$的某一值$n_c$处发生相变，该值依赖于Luttinger参数（或紧化半径R）.使用一个新的复制自由制定，熵表示为在$n-$依赖值的刚度评估的单片配分函数的组合。当顶点操作符在边界处变得相关时，转换发生。我们的数值结果（XXZ和$J_1-J_2$模型的精确对角化）与分析预测一致：在$n_c=4/R^2$之上，对熵的次主导和普适贡献对于开链是$\ln(L)(R^2-1)/(4n-4)$，对于周期链是$\ln(R)/(1-n)$（在自由费米子点R=1）。在以前的作品中使用的复制品的方法无法预测这种转变，原来是正确的，只有$n<n_c$。从二维Rokhsar-Kivelson态的角度来看，该跃迁在纠缠谱中显示出丰富的结构。
英文摘要：
The Rényi-Shannon entropy associated to critical quantum spins chain with central charge $c=1$ is shown to have a phase transition at some value $n_c$ of the Rényi parameter $n$ which depends on the Luttinger parameter (or compactification radius R). Using a new replica-free formulation, the entropy is expressed as a combination of single-sheet partition functions evaluated at $n-$ dependent values of the stiffness. The transition occurs when a vertex operator becomes relevant at the boundary. Our numerical results (exact diagonalizations for the XXZ and $J_1-J_2$ models) are in agreement with the analytical predictions: above $n_c=4/R^2$ the subleading and universal contribution to the entropy is $\ln(L)(R^2-1)/(4n-4)$ for open chains, and $\ln(R)/(1-n)$ for periodic ones (R=1 at the free fermion point). The replica approach used in previous works fails to predict this transition and turns out to be correct only for $n<n_c$. From the point of view of two-dimensional Rokhsar-Kivelson states, the transition reveals a rich structure in the entanglement spectra.
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