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arXiv:2112.02042 · Physical review. B./Physical review. B

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

Balázs Hetényi, Selçuk Parlak, Mohammad Yahyavi

Published Dec 15, 2021• 25 citations
arXivHTMLPDF

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 and ${W}_{r}$ remains finite, indicating a 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.

DOI
10.1103/physrevb.104.214207

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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: Wa{W}_{a}Wa​ (Wr)({W}_{r})(Wr​) is attractive (repulsive) such that Wa<Wr{W}_{a}<{W}_{r}Wa​<Wr​. In three dimensions, Wa{W}_{a}Wa​ tends to zero and Wr{W}_{r}Wr​ remains finite, indicating a metal-insulator transition at finite disorder. In two dimensions, we are limited by system size, but we find that both Wa{W}_{a}Wa​ and Wr{W}_{r}Wr​ decrease significantly as system size is increased.
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Based on finite-size scaling ideas a new approach to analyse the singularities associated with second-order phase transitions is presented. Under certain requirements, the method allows to derive an upper limit for the critical exponent from finite-size data that describe the system far away from the transition. By extrapolation the critical exponent ν is determined. As an example, the approach is applied to the metal-insulator transition in disordered systems. The result ν = 1.35 ± 0.15 is independent of the employed random distribution. With field theoretical arguments we discuss and generalize this result. Our results are consistent with the assumption that one-parameter scaling and universality are valid in the 3D Anderson model.
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Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.
Abstract We have seen that disordered systems give rise to a host of interesting effects related to the localization of states. Thus, from the outset we have explicitly to take account of the localization in the theoretical framework, which makes computer experiments on finite systems an important first-principles method. This situation immediately poses several essential questions, such as: (i) how can the properties of macroscopic systems be deduced from the results for finite systems; and (ii) how can the concept of Anderson localization be extended to disordered systems with electron-electron interactions in a numerical model in which disorder and electron correlation are treated on an equal footing? In this chapter we address these questions. In this section we introduce the concept of renormalization for disordered systems. Specifically, we examine real-space renormalization schemes.
Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.