We discuss the characterization of the polarization for insulators under the periodic boundary condition in terms of the Berry phase, clarifying confusing subtleties. For band insulators, the Berry phase can be formulated in terms of the Bloch function in momentum space. More generally, in the presence of interactions or disorders, one can instead use the many-body ground state as a function of the flux piercing the ring. However, the definition of the Bloch function and the way of describing the flux are not unique. As a result, the value of the Berry phase and its behavior depend on how precisely it is defined. In particular, identifying the Berry phase as a polarization, its change represents a polarization current, which also depends on the definition. We demonstrate this by elucidating mutual relations among different definitions of the Berry phase and showing that they correspond to currents measured differently in real space. Despite the nonuniqueness of the polarization current, the total charge transported during a Thouless pumping process is independent of the definition, reflecting its topological nature.

中文翻译：

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体极化的不等价贝里相

我们讨论了周期性边界条件下的绝缘子极化的贝里相表征，以澄清令人困惑的细微之处。对于带状绝缘子，可以根据动量空间中的Bloch函数来公式化Berry相。更一般地，在存在相互作用或紊乱的情况下，可以替代地使用多体基态作为穿透环的通量的函数。但是，Bloch函数的定义和描述通量的方式并不是唯一的。结果，Berry阶段的值及其行为取决于其定义的精确程度。特别地，将贝里相识别为极化，其变化表示极化电流，这也取决于定义。我们通过阐明Berry相的不同定义之间的相互关系并证明它们对应于在现实空间中以不同方式测量的电流来证明这一点。尽管极化电流存在非唯一性，但在Thouless抽运过程中传输的总电荷与定义无关，反映了其拓扑性质。