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Scaling of the formation probabilities and universal boundary entropies in the quantum XY spin chain
Journal of Statistical Mechanics: Theory and Experiment ( IF 2.4 ) Pub Date : 2020-08-25 , DOI: 10.1088/1742-5468/aba9d4
F Ares 1 , M A Rajabpour 2 , J Viti 1, 3, 4
Affiliation  

We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse $\sigma^z$-basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal Boundary Entropy at the critical point. The latter is a benchmark of the underlying Boundary Conformal Field Theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the Boundary Entropies are computed by exploiting a generalization of the Euler-MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.

中文翻译:

量子XY自旋链中形成概率和通用边界熵的缩放

我们准确地计算了在横向 $\sigma^z$ 基础上在给定自旋配置中找到 XY 链基态的概率。通过确定对各种配置概率的有限体积修正,我们获得了临界点处的通用边界熵。后者是表征每个量子态的基本边界共形场理论的基准。为了确定概率的标度,我们证明了一个定理,该定理以分解形式将循环矩阵的子矩阵的特征值表示为原始矩阵的特征值的函数。最后,通过将 Euler-MacLaurin 公式推广到不可微函数来计算边界熵。结果表明,在某些情况下,自旋构型可以流向 Cardy 状态的线性叠加。我们的方法和工具相当通用,可以应用于所有映射到自由费米子哈密顿量的周期性量子链。
更新日期:2020-08-25
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