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arXiv:1810.10878 · Physical review. B./Physical review. B

Quantum phase transitions from analysis of the polarization amplitude

Balázs Hetényi, Balázs Dóra

Published Feb 15, 2019• 22 citations
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Abstract

In the modern theory of polarization, polarization itself is given by a geometric phase. In calculations for interacting systems the polarization and its variance are obtained from the polarization amplitude. We interpret this quantity as a discretized characteristic function and derive formulas for its cumulants and moments. In the case of a noninteracting system, our scheme leads to the gauge-invariant cumulants known from polarization theory. We study the behavior of such cumulants for several interacting models. In a one-dimensional system of spinless fermions with nearest neighbor interaction the transition at which gap closure occurs can be clearly identified from the finite size scaling exponent of the variance. When next nearest neighbor interactions are turned on a model with a richer phase diagram emerges, but the finite size scaling exponent is still an effective way to identify the localization transition.

DOI
10.1103/physrevb.99.085126

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In the modern theory of polarization, polarization itself is given by a geometric phase. In calculations for interacting systems the polarization and its variance are obtained from the polarization amplitude. We interpret this quantity as a discretized characteristic function and derive formulas for its cumulants and moments. In the case of a noninteracting system, our scheme leads to the gauge-invariant cumulants known from polarization theory. We study the behavior of such cumulants for several interacting models. In a one-dimensional system of spinless fermions with nearest neighbor interaction the transition at which gap closure occurs can be clearly identified from the finite size scaling exponent of the variance. When next nearest neighbor interactions are turned on a model with a richer phase diagram emerges, but the finite size scaling exponent is still an effective way to identify the localization transition.
In this work, we focus on two questions. One, we complement the machinery to calculate geometric phases along adiabatic cycles as follows. The geometric phase is a line integral along an adiabatic cycle and if the cycle encircles a degeneracy point, the phase becomes nontrivial. If the cycle crosses the degeneracy point, the phase diverges. We construct quantities which are well defined when the path crosses the degeneracy point. We do this by constructing a generalized Bargmann invariant and noting that it can be interpreted as a cumulant generating function, with the geometric phase being the first cumulant. We show that particular ratios of cumulants remain finite for cycles crossing a set of isolated degeneracy points. The cumulant ratios take the form of the Binder cumulants known from the theory of finite size scaling in statistical mechanics (we name them geometric Binder cumulants). Two, we show that the developed machinery can be applied to perform finite size scaling in the context of the modern theory of polarization. The geometric Binder cumulants are size independent at gap-closure points or regions with closed gap (Luttinger liquid). We demonstrate this by model calculations for a one-dimensional topological model, several two-dimensional models, and a one-dimensional correlated model. In the case of two dimensions, we analyze to different situations, one in which the Fermi surface is one dimensional (a line) and two cases in which it is zero dimensional (Dirac points). For the geometric Binder cumulants, the gap-closure points can be found by one-dimensional scaling even in two dimensions. As a technical point, we stress that only certain finite difference approximations for the cumulants are applicable since not all approximation schemes are capable of extracting the size scaling information in the case of a closed-gap system.
We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and cumulants, quantities which characterize the fluctuations of an underlying probability distribution. In all of the cases we review, the fluctuations are those of a quantum system. We show that the original formalism for geometric phases, in which a quantum system is taken around an adiabatic cycle, can be extended to the case when degeneracy points are encountered along the cycle (quasiadiabatic cycles). The essential tool for this extension is a generalized Bargmann invariant which plays the role of a generating function. From the cumulants generated this way one can form ratios according to the Binder cumulant scheme in statistical mechanics. Such geometric Binder cumulants are sensitive to gap closure, as such, they are useful in identifying metal-insulator transitions, localization, and quantum phase transitions. We present example calculations on simple model systems, whose localization properties are well known, to validate to approach. We also complement our geometric Binder cumulant calculations with results for the fidelity susceptibility, a quantity directly related to the quantum geometry of the parameter space.
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.
The order parameter cumulants of infinite matrix product ground states are evaluated across a quantum phase transition. A scheme using the Binder cumulant, finite-entanglement scaling, and scaling functions to obtain the critical point and exponents of the correlation length and cumulants is presented. Analogous to the scaling relations that relate the exponents of various thermodynamic quantities, a cumulant exponent relation is derived and used to check the consistency and relationship between the cumulant exponents. This scheme gives a numerically economical way of accurately obtaining the critical exponents. Examples of this scheme are shown for four one-dimensional models---the transverse field Ising model, the topological Kondo insulator, the S=1S=1S=1 Heisenberg chain with single-ion anisotropy and the Bose-Hubbard model. A two-dimensional model is also exemplified in the square lattice transverse field Ising model on an infinite cylinder. These exemplary systems portray a variety of local and string order parameters as well as phase transition classes that can be studied with the scaling functions and infinite matrix product states.
We discuss the polarization amplitude of quantum spin systems in one dimension. In particular, we closely investigate it in gapless phases of those systems based on the two-dimensional conformal field theory. The polarization amplitude is defined as the ground-state average of a twist operator which induces a large gauge transformation attaching the unit amount of the U(1)U(1)U(1) flux to the system. We show that the polarization amplitude under the periodic boundary condition is sensitive to perturbations around the fixed point of the renormalization-group flow rather than the fixed point itself even when the perturbation is irrelevant. This dependence is encoded into the scaling law with respect to the system size. In this paper, we show how and why the scaling law of the polarization amplitude encodes the information of the renormalization-group flow. In addition, we show that the polarization amplitude under the antiperiodic boundary condition is determined fully by the fixed point in contrast to that under the periodic one and that it visualizes clearly the nontriviality of spin systems in the sense of the Lieb-Schultz-Mattis theorem.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.