Existence of a phase with finite localization length in the double scaling limit of N-orbital models

doi:10.1016/j.aop.2021.168484

Annals of Physics

Volume 435, Part 1, December 2021, 168484

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Abstract

Among the models of disordered conduction and localization, models with $N$ orbitals per site are attractive both for their mathematical tractability and for their physical realization in coupled disordered grains. However Wegner proved that there is no Anderson transition and no localized phase in the $N \to \infty$ limit, if the hopping constant $K$ is kept fixed (Wegner, 1979; Khorunzhy and Pastur, 1993). Here we show that the localized phase is preserved in a different limit where $N$ is taken to infinity and the hopping $K$ is simultaneously adjusted to keep $N K$ constant. We support this conclusion with two arguments. The first is numerical computations of the localization length showing that in the $N \to \infty$ limit the site-diagonal-disorder model possesses a localized phase if $N K$ is kept constant, but does not possess that phase if $K$ is fixed. The second argument is a detailed analysis of the energy and length scales in a functional integral representation of the gauge invariant model. The analysis shows that in the $K$ fixed limit the functional integral’s spins do not exhibit long distance fluctuations, i.e. such fluctuations are massive and therefore decay exponentially, which signals conduction. In contrast the $N K$ fixed limit preserves the massless character of certain spin fluctuations, allowing them to fluctuate over long distance scales and cause Anderson localization.

Section snippets

The site-diagonal-disorder model

In the first sections of this article we study Wegner’s site-diagonal-disorder model with $N$ orbitals on each site, which at $N = 1$ reduces to the Anderson model [1], [2]. The on-site Hamiltonian $H_{i}$ for site $i$ is an $N \times N$ random matrix which describes interactions between the $N$ orbitals at site $i$ . It is the natural $N \times N$ generalization of the Anderson model’s disorder potential. The mean squared value of $H_{i}$ ’s matrix elements is proportional to $N^{- 1}$ , so that when a single site is taken in isolation, its

Numerical calculation of the localization length when the hopping strength $K$ is kept fixed

In Fig. 1 we present the localization length $ξ$ , which we calculate using the transfer matrix method [15]. The transfer matrix method for calculating localization lengths operates by building a long chain with length $L$ and measuring the average decay within the chain. We keep the energy fixed at $E = 2.8$ , fairly close to the band edge which is near $E = 3.5$ in the 1-D chain, in order to reduce the scattering length and the computational cost.

We calculate $ξ$ with the hopping strength $K$ fixed at three

Numerical calculation of the localization length in the double scaling limit where $N K$ is kept fixed

Now we turn to the more interesting case where the product $N K$ is kept fixed while taking the $N \to \infty$ limit. Fig. 2 shows the localization length $ξ$ with $N K$ fixed at $N K = 1, 2, 3$ for both the unitary and orthogonal ensembles. When $N K$ is fixed to these values the scattering length always remains small and does not significantly affect the computational cost of calculating the localization length. Therefore we fix the energy at the band center $E = 0$ and calculate $ξ$ at large values of $N$ up to $N = 192$ .

Fig. 2

Discussion of the numerical results

These results are a numerical proof that the one-dimensional site-diagonal-disorder model is localized in the double scaling limit where $N K$ is kept fixed and $N$ is taken to infinity. This localized property is expected of one-dimensional wires with short-range uncorrelated disorder and short-range hopping. Wires in this class have been shown both numerically and analytically to be localized, except in the special case of certain ensembles with perfectly conducting channels [4], [5].

These

The gauge invariant model

In order to go beyond one dimension, and to obtain non-numerical insight, we will now analyze both the $K$ fixed limit and the $N K$ fixed limit using analytical analysis of a functional integral. In order to ease our mathematical analysis, we will switch from Wegner’s site-diagonal-disorder model to Wegner’s gauge invariant model, both of which were introduced in the same paper [1]. The only difference between the two models is that in the gauge invariant model the hopping between sites is mediated

Functional integral formulation of the gauge invariant model

The conversion of disordered models to functional integrals is a well developed topic that originated with Schafer and Wegner in 1980 [19], [20]. The main point of these functional integrals is that their spin variables obey a continuous symmetry and therefore are capable of a transition from a spontaneously-broken-symmetry phase to a symmetric phase. In the symmetry-broken phase the spins are correlated over long distances, which corresponds to electronic conduction over long distances. In the

Two energy and length scales

In this section we will use the hybrid functional integral in Eq. (6) to show that the gauge invariant model has two natural energy scales. The first is the scattering energy scale $\tilde{ε} N$ . The second is the much smaller energy scale $\tilde{ε} N K$ which controls conduction and localization. Corresponding to these two energy scales are two length scales: the scattering length which is equal to 1 in the gauge invariant model because scattering occurs on site, and the localization length which can be much

Summary

In summary, we have presented evidences that two disordered models, each with $N$ orbitals per site, possess a localized phase with finite localization length in the double scaling limit where $N \to \infty$ and $N K$ is kept fixed. In the site-diagonal-disorder model we directly computed the localization length and showed that it remains finite in one dimension when $N K$ is kept fixed. In the gauge invariant model, in any dimension or geometry, we showed that the $N K$ fixed limit preserves dynamic spin variables

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We especially thank Università degli Studi di Roma “La Sapienza”, Italy, the Asia Pacific Center for Theoretical Physics, South Korea, Sophia University, and the Zhejiang Institute of Modern Physics, where much of this work was done. We thank T. Ohtsuki for help during the early stages of this work. We also acknowledge support from the Institute of Physics of the Chinese Academy of Sciences, Nanyang Technological University, Singapore, Royal Holloway University of London, United Kingdom, and

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Special Issue “Localisation 2020”: Editorial Summary
2021, Annals of Physics

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Existence of a phase with finite localization length in the double scaling limit of N-orbital models

Abstract

Section snippets

The site-diagonal-disorder model

Numerical calculation of the localization length when the hopping strength K is kept fixed

Numerical calculation of the localization length in the double scaling limit where NK is kept fixed

Discussion of the numerical results

The gauge invariant model

Functional integral formulation of the gauge invariant model

Two energy and length scales

Summary

Declaration of Competing Interest

Acknowledgments

Ann. Phys.

Nuclear Phys. B

Nuclear Phys. B

Nuclear Phys. B

Phys. Rev. B

Phys. Rev.

Comm. Math. Phys.

Progr. Theoret. Phys. Suppl.

Rep. Progr. Phys.

Phys. Rev. Lett.

J. Phys. Soc. Japan

J. Phys. Soc. Japan

Phys. Rev. Lett.

Phys. Rev. Lett.

Eigenvalue Statistics and Localization for Random Band Matrices with Fixed Width and Wegner Orbital Model

Int. Math. Res. Not.

Phys. Rev. Lett.

Z. Phys. B

Phys. Rev. Lett.

J. Phys. A: Math. Gen.

Sov. Phys.—JETP

Supersymmetry in Disorder and Chaos

Numerical calculation of the localization length when the hopping strength $K$ is kept fixed

Numerical calculation of the localization length in the double scaling limit where $N K$ is kept fixed