一维量子多体系统中极化振幅的标度性
Scaling of polarization amplitude in quantum many-body systems in one dimension
摘要：
Resta根据大规范变换算符的基态期望值提出了一维系统中电极化的强度的定义。在热力学极限中期望值的消失意味着系统是导体。我们研究了无间隙导电Tomonaga-Luttinger液相中$S=1/2$ XXZ链中Resta的极化振幅（期望值）及其几种推广。我们得到了自由费米子点（XY链）的最低阶微扰理论解析表达式，以及具有长程相互作用的Haldane-Shastry模型的精确结果。我们也得到了数值结果，主要是使用精确对角化方法。我们发现，振幅表现出幂律缩放的系统大小（链长）和消失的热力学极限。另一方面，指数依赖于模型，即使当低能量极限是由Tomonaga-Luttinger液体与相同的Luttinger参数。我们发现，当Umklapp项被消除时，指数发生变化，这表明Umklapp项的重要性。
英文摘要：
Resta proposed a definition of the electric polarization in one-dimensional systems in terms of the ground-state expectation value of the large gauge transformation operator. Vanishing of the expectation value in the thermodynamic limit implies that the system is a conductor. We study Resta's polarization amplitude (expectation value) in the $S=1/2$ XXZ chain and its several generalizations, in the gapless conducting Tomonaga-Luttinger Liquid phase. We obtain an analytical expression in the lowest-order perturbation theory about the free fermion point (XY chain), and an exact result for the Haldane-Shastry model with long-range interactions. We also obtain numerical results, mostly using the exact diagonalization method. We find that the amplitude exhibits a power-law scaling in the system size (chain length) and vanishes in the thermodynamic limit. On the other hand, the exponent depends on the model even when the low-energy limit is described by the Tomonaga-Luttinger Liquid with the same Luttinger parameter. We find that a change in the exponent occurs when the Umklapp term(s) are eliminated, suggesting the importance of the Umklapp terms.
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