Cumulants associated with geometric phases and their implementation in modern theory of crystalline polarization

Abstract

Many fields have been influenced by Berry's geometric phase because of its physical meaning and observable effects. One of the breakthroughs that stem from geometric phases is the modern theory of polarization. The expectation value of the position was not possible to calculate for crystalline structures because of ill-defined position operator. The modern theory of polarization showed that the geometric phase obtained by Zak phase, integral across the Brillouin zone, gives thefirst cumulants so that polarization is obtainable by the geometric phase. This indicates that cumulants are essential for studies such as polarization, charge transport, and electron localization. In the context of the modern theory of polarization, gauge-invariant cumulants are derived but they are not geometric even though they are physically well defined. In order to deal with this issue, a Binder cumulant associated with the adiabatic cycle is introduced, so called geometric Binder cumulant. Since the definition of Binder cumulants is based on a ratio of two cumulants, it is possible to eliminate factors that prevent the quantity to become geometric. An alternative way to extract cumulants associated with the adiabatic cycle is proposed as well. Error terms of the Cumulants are improved when they are extracted in an alternative way. Distortion around the transition points which modern theory of polarization has been reduced significantly. Geometric Binder cumulant is implemented to observe the difference between gapped and gapless band structures. One-dimensional and two-dimensional models are investigated and phase transition between metallic and insulating states is clearly observed. SSH model is investigated to make a comparison with the modern theory of polarization and development in the formalism is shown. Geometric Binder cumulant also lets us observe the correlated model and a method based on renormalization group theory is used to locate transition points in the correlated model. Results are in good agreement with each other. An alternative way to extract cumulants is also extended to two-dimensional systems and phase transition is observed in two-dimensional systems with the usage of geometric Binder cumulant. Regardless of whether the two-dimensional system has a zero-dimensional or one-dimensional Fermi surface, Geometric Binder cumulant is a quantity that is sensitive for the metallic and insulating cases. For the open gap case, geometric Binder cumulant is affected by the system size, and the effect of the system size is distinct. An increase in the system size improves the quantity.