Scaling and renormalization in the modern theory of polarization: application to disordered systems

Abstract

We develop a scaling theory and a renormalization technique in the context of the modern theory of polarization. The central idea is to use the characteristic function (also known as the polarization amplitude) in place of the free energy in the scaling theory and in place of the Boltzmann probability in a position-space renormalization scheme. We derive a scaling relation between critical exponents which we test in a variety of models in one and two dimensions. We then apply the renormalization to disordered systems. In one dimension the renormalized disorder strength tends to infinity indicating the entire absence of extended states. Zero(infinite) disorder is a repulsive(attractive) fixed point. In two and three dimensions, at small system sizes, two additional fixed points appear, both at finite disorder, $W_a$($W_r$) is attractive(repulsive) such that $W_a<W_r$. In three dimensions $W_a$ tends to zero, $W_r$ remains finite, indicating metal-insulator transition at finite disorder. In two dimensions we are limited by system size, but we find that both $W_a$ and $W_r$ decrease significantly as system size is increased.

References in corpus (8)

• Anderson Transitions
• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Time Reversal Polarization and a Z_2 Adiabatic Spin Pump
• Coherent Backscattering of Ultracold Atoms
• Anisotropic 2D diffusive expansion of ultra-cold atoms in a disordered potential
• Reconstruction of the polarization distribution of the Rice-Mele model
• Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization
• Polarization amplitude near quantum critical points

Cited by in corpus (2)

• Scaling of the bulk polarization in extended and localized phases of a quasiperiodic model
• Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems