After a thorough introduction of Berry’s geometric phase, starting from scratch, the modern theory of polarization will be developed. It will be shown that polarization in crystalline systems is not given by a usual operator expectation value, but a geometric phase of a particular kind, one which results from integration over an open path with symmetry related endpoints (Zak phase). Given this, it is not obvious that the usual procedures of calculations based on operators are applicable, for example, finite-size scaling calculations are based on calculating the moments (cumulants) of operators. It will be argued that the key difference between the case of polarization and that of usual observables is that the characteristic function involved only exists on a discrete set of points, and that the derivatives of such a characteristic function are by definition finite difference derivatives. Based on this observation a formalism will be developed which allows for accurate finite-size scaling using the Binder cumulant, a well-known tool in the analysis of critical points for classical or quantum systems. The results of example calculations will be provided.
Reference: B. Hetényi and S. Cengiz, Phys. Rev. B, 106 195151 (2022).