横场伊辛链自旋模型费米子表象的问题

Pathology of the Fermionic Representation of Transverse Ising Model

摘要:

人们在研究一维横场伊辛模型时，通常将模型中的自旋算符转换成费米子算符，一般是通过Jordan-Wigner变换来进行.但是变换前后的两个模型，即原自旋算符表达的模型与费米子算符表达的模型，其等价性需要伴随一些条件，如不细致处理，会造成后一模型的自由度冗余.本文中，我们通过对表象变换前后的两个模型进行详细对比，找出导致这两个模型等价的各个因素，包括边界条件和格点数目奇偶等，揭示在不同要素和条件下两个模型等价的内涵，并通过详细的实例来说明如何通过投影法筛除费米子表象模型的冗余自由度.

Abstract:

When studying the quantum Ising chain in a transverse field, we usually transform the spin operators to fermion operators by applying the Jordan-Wigner transformation,i.e.we take it for granted that the spin model and corresponding fermionic model are equivalent naively. But deliberate treatment could reveal that the equivalence accompanies some delicate conditions.Redundant degrees of freedom should be projected out carefully, especially for finite length of chain. In this paper, we compare the two models in detail so as to find the conditions of equivalence of them,including the interaction, the boundary conditions, and odevity of the number of lattice sites.We also exemplify in detail how to project out the redundant degrees of freedom for the fermionic representation.