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Journal of Physics C Solid State Physics

Effects of weak, short-ranged interactions on the Anderson transition

Abhishek Singh, W. L. McMillan

Published Apr 10, 1985• 3 citations
Publisher

Abstract

The effects of weak, short-ranged interactions on the non-interacting Anderson transition in three dimensions are calculated using a macroscopic renormalisation-group method. The on-site Hubbard (singlet) interaction is found to be relevant at the noninteracting Anderson fixed point; the nearest-neighbour triplet interaction is found to be irrelevant.

DOI
10.1088/0022-3719/18/10/016

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The effects of weak, short-ranged interactions on the non-interacting Anderson transition in three dimensions are calculated using a macroscopic renormalisation-group method. The on-site Hubbard (singlet) interaction is found to be relevant at the noninteracting Anderson fixed point; the nearest-neighbour triplet interaction is found to be irrelevant.
We study the influence of short-range electron-electron interactions on scaling behavior near the integer quantum Hall plateau transitions. Short-range interactions are known to be irrelevant at the renormalization group fixed point which represents the transition in the noninteracting system. We find, nevertheless, that transport properties change discontinuously when interactions are introduced. Most importantly, in the thermodynamic limit the conductivity at finite temperature is zero without interactions, but nonzero in the presence of arbitrarily weak interactions. In addition, scaling as a function of frequency $\ensuremath{\omega}$ and temperature T is determined by the scaling variable $\ensuremath{\omega}{/T}^{p}$ (where p is the exponent for the temperature dependence of the inelastic scattering rate) and not by $\ensuremath{\omega}/T,$ as it would be at a conventional quantum phase transition described by an interacting fixed point. We express the inelastic exponent p and the thermal exponent zT{z}_{T}zT​ in terms of the scaling dimension $\ensuremath{-}\ensuremath{\alpha}<0$ of the interaction strength and the dynamical exponent z (which has the value z=2),z=2),z=2), obtaining $p=1+2\ensuremath{\alpha}/z$ and zT=2/p.{z}_{T}=2/p.zT​=2/p.
We study the effects of the electron-electron interaction on the critical properties of the plateau transitions in the integer quantum Hall effect. We find the renormalization group dimension associated with short-range interactions to be $\ensuremath{-}0.66\ifmmode\pm\else\textpm\fi{}0.04$. Thus the noninteracting fixed point (characterized z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2 and $\ensuremath{\nu}\ensuremath{\approx}2.3$) is stable. For the Coulomb interaction, we find the correlation effect is a marginal perturbation at a Hartree-Fock fixed point ( z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1, $\ensuremath{\nu}\ensuremath{\approx}2.3$) by dimension counting. Further calculations are needed to determine its stability upon loop corrections.
A new real-space renormalisation group method is developed and tested for the case of the Anderson-Hubbard model. The development is concentrated on obtaining and subsequent analysis of one-dimensional recurrence relations which take into account the effects of interaction and temperature. The results are presented in a form that naturally permits the extension of the given treatment to the case of higher dimensions and off-diagonal bond percolation type of disorder. Although the method itself is not limited to low temperatures, actual computations were made for the case of low temperatures and weak-electron-electron coupling. For this case, the scaling expressions obtained for the ground-state energy and the energy gap are in satisfactory agreement with available exact results.
The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At 2 /U 2 ) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t) alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t) 1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.
The periodic Anderson model (PAM) is studied within the framework of dynamical mean-field theory, with particular emphasis on the interaction-driven Mott transition it contains, and on resultant Mott insulators of both Mott-Hubbard and charge-transfer type. The form of the PAM phase diagram is first deduced on general grounds using two exact results, over the full range of model parameters and including metallic, Mott, Kondo and band insulator phases. The effective low-energy model which describes the PAM in the vicinity of a Mott transition is then shown to be a one-band Hubbard model, with effective hoppings that are not in general solely nearest neighbour, but decay exponentially with distance. This mapping is shown to have a range of implications for the physics of the problem, from phase boundaries to single-particle dynamics; all of which are confirmed and supplemented by NRG calculations. Finally we consider the locally degenerate, non-Fermi liquid Mott insulator, to describe which requires a two-self-energy description. This is shown to yield a number of exact results for the associated local moment, charge, and interaction-renormalised levels, together with a generalisation of Luttinger's theorem to the Mott insulator.