The modern polarization theory[1,2] casts the bulk macroscopic polarization of crystalline materials as a geometric phase [3].  This enables the calculation of the bulk polarization as well as its cumulants[4,5,6,7,8] (variance, skew, kurtosis) in crystalline materials.   In addition, even some non-crystalline materials are often simulated under periodic boundary conditions (for example, disordered or quasi-periodic systems), meaning that the tools of the modern polarization theory are useful and applicable[9], in particular in the characterization of localization transitions.  Recently the modern polarization analog of the Binder cumulant was developed[8], allowing for the location of phase transition points, and for finite size scaling around those transition points.  Auxiliary tools to support Binder cumulant based finite size scaling calculations were also developed, for example, a renormalization based method was used [9] in the analysis of Anderson localization.

Although not immediately obvious, there is a deep connection between quasiperiodic models and models describing the quantum Hall effect.  The canonical model for the former, the Aubry-André model (AAM), is equivalent to the Harper-Hofstadter model, used to explain the integer quantum Hall effect.  The AAMl, consists of a tight-binding term and a potential modulation characterized by an irrational wave number.  This model exhibits a localization transition at finite potential strength, and it was the first simple model to exhibit such behavior.  Since its original appearance extensions of this model include irrationally modulated hopping or superconducting pairing terms, leading to a real plethera of interesting localization phenomena characterized by fractal order parameters.  Quasi-periodic models have also been applied to study twisted bi-layer graphene.  Here the full two-dimensional models are difficult to calculate, but dimensional reduction leads to a one-dimensional model which is an extended Aubry-André model.

The purpose of this research project is to apply the modern polarization theory, including its recent developments[7,8,9], to study extended quasi-periodic models.  As a first step, the well-known phase diagram of the Aubry-André model will be verified.  Subsequently the Aubry-André model with second nearenst neighbor hoppings will be studied.  Although this is a simple extension of the original model, its phase diagram has not been established precisely, previous works show considerable error bars.  The combination of the modern polarization theory and finite size scaling methods á la Binder allow for an accurate location of the phase transition points, so we can map the entire phase diagram.   In addition, we are also able to characterize individual eigenstates and their localization properties, which can not be done with the original Resta-Sorella formulation.  In addition to second nearest neighbor hopping, we will also study other Hermitian extensions, such as superconducting pairing.  The known phenomena of interest in this case is the so called critical phase, a phase which falls in between the localized and delocalized phases.

A further point of interest is the study another set of extensions, that of non-Hermitian systems[11].  Non-Hermitian AAM have been studied already, but not using the methods of the modern polarization theory.  This part of the project would involve developing the modern theory of polarization (Zak phase and cumulants) for non-Hermitian models.

References:

[1]. R. D. King-Smith and David Vanderbilt, Phys. Rev. B, 47 1651R (1993).

[2]. R. Resta, Rev. Mod. Phys., 66 899 (1994).

[3]  M. V. Berry, Proc. R. Soc. Lond. A 392 45 (1984).

[4]. R. Resta, Phys. Rev. Lett. 80 1800 (1998).

[5]  R. Resta and S. Sorella, Phys. Rev. Lett. 82 370 (1999).

[6] I. Souza, T. Wilkins, and R. M. Martin, Phys. Rev. B 62 1666 (2000).

[7] B. Hetényi and B. Dóra, Phys. Rev. B 99 085126 (2019).

[8] B. Hetényi and S. Cengiz, Phys. Rev. B, 106 195151 (2022).

[9] B. Hetényi. S. Parlak, and M. Yahyavi, Phys. Rev. B 104 214207 (2021).

[10] S. Aubry and G. André, Ann. Israel Phys. Soc. Vol. 3 133 18 (1980).

[11] E. J. Bergholtz, J. C. Budich, F. K. Kunst, Rev. Mod. Phys. 93 015005 (2021).