The modern theory of polarization casts the dielectric polarization as a geometric phase (Zak-Berry phase).   Due to the fact that this quantity is not a simple operator expectation value, traditional finite size scaling approaches are not applicable to it.  In systems where a local order parameter exists, the Binder cumulant (a ratio of statistical cumulants) is guaranteed to locate classical and quantum phase transition points via the finite size scaling hypothesis.  In this talk, it will be shown that the so-called gauge invariant cumulants associated with the geometric phase can be used to construct the analog of the Binder cumulant for adiabatic cycles.  The formalism is general, in the sense that such "Berry-Binder cumulants" can be constructed for any adiabatic cycle with isolated degeneracy points, and they take particular finite values at gap closure.   We apply the formalism to the location of gap closure points in a variety of systems in one and two dimensions, including topological, disordered, and correlated systems.  Our approach is sensitive to gap closure, even in cases where the Fermi surface is down by two dimensions compared to the dimension of the system (Dirac points in graphene or the topological Haldane model).  We also develop a renormalization scheme based on the modern polarization theory, and apply it to disordered systems in one, two, and three dimensions.  In one and three dimensions our approach concurs with the famous "gang-of-four" results, in two dimensions we run into system size limitations, but our preliminary results are not inconsistent with the scaling theory of localization.

 

[1]: B. Hetényi and B. Dóra, "Quantum phase transitions from analysis of the polarization amplitude", Phys. Rev. B 99 085126 (2018).

[2]: B. Hetényi, "Interaction-driven polarization shift in the t-V-V' lattice fermion model at half filling: emergent Haldane phase", Phys. Rev. Research, 2 023266 (2020).

[3]: B. Hetényi, S. Parlak, and M. Yahyavi, "Scaling and renormalization in the modern theory of polarization: application to disordered systems", Phys. Rev. B 104 214207 (2021).

[4]: B. Hetényi and S. Cengiz, "Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems" 106 195151 (2022).