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arXiv:cond-mat/9510121 · Physical Review Letters

Effective σ Model Formulation for Two Interacting Electrons in a Disordered Metal

Klaus M. Frahm, Axel Müller–Groeling, Jean‐Louis Pichard

Published Feb 26, 1996• 36 citations
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Abstract

We derive an analytical theory for two interacting electrons in a $d$-dimensional random potential. Our treatment is based on an effective random matrix Hamiltonian. After mapping the problem on a nonlinear $\ensuremath{\sigma}$ model, we exploit similarities with the theory of disordered metals to identify a scaling parameter, investigate the level correlation function, and study the transport properties of the system. In agreement with recent numerical work we find that pair propagation is subdiffusive and that the pair size grows logarithmically with time.

DOI
10.1103/physrevlett.76.1509

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We derive an analytical theory for two interacting electrons in a ddd-dimensional random potential. Our treatment is based on an effective random matrix Hamiltonian. After mapping the problem on a nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model, we exploit similarities with the theory of disordered metals to identify a scaling parameter, investigate the level correlation function, and study the transport properties of the system. In agreement with recent numerical work we find that pair propagation is subdiffusive and that the pair size grows logarithmically with time.
We numerically investigate the transport properties of disordered interacting electrons in three dimensions in the metallic as well as in the insulating phases. The disordered many-particle problem is modeled by the quantum Coulomb glass which contains a random potential, long-range unscreened Coulomb interactions and quantum hopping between different sites. We have recently developed the Hartree-Fock based diagonalization (HFD) method which amounts to diagonalizing the Hamiltonian in a suitably chosen energetically truncated basis. This method allows us to investigate comparatively large systems. Here we calculate the combined effect of disorder and interactions on the dissipative conductance. We find that the qualitative influence of the interactions on the conductance depends on the relative disorder strength. For strong disorder interactions can significantly enhance the transport while they suppress the conductance for weak disorder.
We derive and study an effective spin model that explains the anomalous spin dynamics in the one-dimensional Hubbard model with strong potential disorder. Assuming that charges are localized, we show that spins are delocalized and their subdiffusive transport originates from a singular random distribution of spin exchange interactions. The exponent relevant for the subdiffusion is determined by the Anderson localization length and the density of the electrons. Although the analytical derivations are valid for low particle density, numerical results for the full model reveal a qualitative agreement up to half filling.
The nonlinear sigma model is a well-established theoretical tool for studies of transport and thermodynamics in disordered electronic systems. The conventional sigma model approach for interacting systems does not account for particle-hole asymmetry. It is therefore not suited for studying quantities that are sensitive to this effect such as the thermoelectric transport coefficient. Here, we derive a minimal extension of the Keldysh nonlinear sigma model tailored for two-dimensional interacting systems. We argue that this model can be used to systematically study the combined effect of interactions and disorder on thermoelectric transport. As a first step in this direction, we use the model to analyze the structure of the heat density-density correlation function and calculate interaction corrections to its static part. The calculation of interaction corrections to the dynamical part of the correlation function and the thermodynamic transport coefficient is left for future work.
Abstract We numerically investigate the transport properties of disordered interacting electrons in three dimensions in the metallic as well as in the insulating phases. The disordered many‐particle problem is modeled by the quantum Coulomb glass which contains a random potential, long‐range unscreened Coulomb interactions and quantum hopping between different sites. We have recently developed the Hartree‐Fock based diagonalization (HFD) method which amounts to diagonalizing the Hamiltonian in a suitably chosen energetically truncated basis. This method allows us to investigate comparatively large systems. Here we calculate the combined effect of disorder and interactions on the dissipative conductance. We find that the qualitative influence of the interactions on the conductance depends on the relative disorder strength. For strong disorder interactions can significantly enhance the transport while they suppress the conductance for weak disorder.
We studied several aspects of the Mott metal-insulator transition in the disordered case. The model on which we based our analysis is the disordered Hubbard model, which is the simplest model capable of capturing the Mott metal-insulator transition. We investigated this model through statistical dynamical mean-field theory (statDMFT). This theory is a natural extension of dynamical mean-field theory (DMFT), which has been used with relative success in the past several years with the purpose of describing the Mott transition in the clean case. As is the case for the latter theory, statDMFT incorporates the electronic correlation effects only in their local manifestations. Disorder, on the other hand, is treated in such a way as to incorporate Anderson localization effects. With this technique, we analyzed the disordered two-dimensional Mott transition, using the quantum Monte Carlo algorithm to solve the associated single-impurity problems. We found spinodal lines at which the metal and insulator ceased to be metastable. We also studied spatial fluctuations of local quantities, such as self-energy and local Green's function, and showed the appearance of metallic regions within the insulator and vice versa. We carried out an analysis of finite-size effects and showed that, in agreement with the theorems of Imry and Ma [Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975).], the first-order transition is smeared in the thermodynamic limit. We analyzed transport properties by means of a mapping to a random classical resistor network and calculated both the average current and its distribution across the metal-insulator transition.
The interacting disordered electron problem is reviewed with emphasis on the quantum phase transitions that occur in a model system and on the field-theoretic methods used to describe them. An elementary discussion of conservation laws and diffusive dynamics is followed by a detailed derivation of the extended nonlinear sigma model, which serves as an effective field theory for the problem. A general scaling theory of metal-insulator and related transitions is developed, and explicit renormalization-group calculations for the various universality classes are reviewed and compared with experimental results. A discussion of pertinent physical ideas and phenomenological approaches to the metal-insulator transition not contained in the sigma-model approach is given, and phase-transition aspects of related problems, like disordered superconductors and the quantum Hall effect, are discussed. The review concludes with a list of open problems.