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International Journal of Modern Physics B

DISORDERED ELECTRON LIQUID WITH INTERACTIONS

Alexander M. Finkel’stein

Published May 20, 2010• 30 citations

Abstract

The metal–insulator transition (MIT) observed in a two-dimensional dilute electron liquid raises the question about the applicability of the scaling theory of disordered electrons, the approach pioneered by Phil Anderson and his collaborators,8 for the description of this transition. In this context, we review here the scaling theory of disordered electrons with electron–electron interactions. We start with the disordered Fermi liquid, and show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions. This model has a direct relation with the disordered Fermi liquid, but can be more generally applicable, since it is a minimal model for disordered interacting electrons. The discussion is mostly about the general structure of the theory emphasizing the connection of the scaling parameters entering the NLSM with conservation laws. Next, we show that the MIT, as described by the NLSM with interactions, is a quantum phase transition and identify the parameters needed for the description of the kinetics and thermodynamics of the interacting liquid in the critical region of the transition. Finally, we discuss the MIT observed in Si -MOSFETs. We consider it as an example of the Anderson transition in the presence of the electron interactions. We demonstrate that the two-parameter RG equations, which treat disorder in the one-loop approximation but incorporate the full dependence on the interaction amplitudes, describe accurately the experimental data in Si -MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The fact that this drop can be reproduced theoretically, together with the argument that Anderson localization should occur at strong disorder, justified the existence of the MIT within the scaling theory.

DOI: 10.1142/s0217979210064642

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The metal–insulator transition (MIT) observed in a two-dimensional dilute electron liquid raises the question about the applicability of the scaling theory of disordered electrons, the approach pioneered by Phil Anderson and his collaborators,8 for the description of this transition. In this context, we review here the scaling theory of disordered electrons with electron–electron interactions. We start with the disordered Fermi liquid, and show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions. This model has a direct relation with the disordered Fermi liquid, but can be more generally applicable, since it is a minimal model for disordered interacting electrons. The discussion is mostly about the general structure of the theory emphasizing the connection of the scaling parameters entering the NLSM with conservation laws. Next, we show that the MIT, as described by the NLSM with interactions, is a quantum phase transition and identify the parameters needed for the description of the kinetics and thermodynamics of the interacting liquid in the critical region of the transition. Finally, we discuss the MIT observed in Si -MOSFETs. We consider it as an example of the Anderson transition in the presence of the electron interactions. We demonstrate that the two-parameter RG equations, which treat disorder in the one-loop approximation but incorporate the full dependence on the interaction amplitudes, describe accurately the experimental data in Si -MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The fact that this drop can be reproduced theoretically, together with the argument that Anderson localization should occur at strong disorder, justified the existence of the MIT within the scaling theory.

Quantum materials exhibit a rich dynamic of physical parameters, which, when combined, can lead to entirely different behaviors. These parameters constantly compete with each other, with the most influential parameters determining the state of the system. For example, in the case of metal–insulator transitions, electron–electron interactions compete with the kinetic energy of the electrons and disorder. Understanding these complex dynamics is crucial for both fundamental physics and the development of novel technological applications, particularly given the role of disorder in tuning critical temperatures, a property with significant potential benefit in the fabrication of new devices where temperature requirements are still the bottleneck. In this article, properties of the Mott metal–insulator transition within disordered electron systems are explored using the disordered Hubbard model, the simplest Hamiltonian for capturing the metal–insulator transition. The model solutions are obtained using the self-consistent statistical dynamical mean-field theory (statDMFT). statDMFT incorporates local electronic correlation effects while allowing for Anderson localization due to disorder.

We uncover a disorder-driven instability in the diffusive Fermi liquid phase of a class of many-fermion systems, indicative of a metal-insulator transition of first-order type, which arises solely from the competition between quenched disorder and interparticle interactions. Our result is expected to be relevant for sufficiently strong disorder in

d=3

spatial dimensions. Specifically, we study a class of half-filled, Hubbard-like models for spinless fermions with (complex) random hopping and short-ranged interactions on bipartite lattices in $d\ensuremath{\geqslant}2$. In a given realization, the hopping disorder breaks time-reversal invariance but preserves the special ``nesting'' symmetry responsible for the charge density wave instability of the ballistic Fermi liquid. This disorder may arise, e.g., from the application of a random magnetic field to the otherwise clean model. We derive a low-energy effective field theory description for this class of disordered, interacting fermion systems, which takes the form of a Finkel'stein nonlinear sigma model $(\mathrm{FNL}\ensuremath{\sigma}\mathrm{M})$ (A. M. Finkel'stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]). We analyze the $\mathrm{FNL}\ensuremath{\sigma}\mathrm{M}$ using a perturbative, one-loop renormalization group analysis controlled via an

ϵ

expansion in

d=2+ϵ

dimensions. We find that in

d=2

dimensions, the interactions destabilize the conducting phase known to exist in the disordered, noninteracting system. The metal-insulator transition that we identify in

d>2

dimensions

(ϵ>0)

occurs for disorder strengths of order

ϵ

, and is therefore perturbatively accessible for

ϵ⪡1

. We emphasize that the disordered system has no localized phase in the absence of interactions, so that a localized phase and the transition into it can only appear due to the presence of the interactions.

This chapter focuses on experimental results obtained in low-disordered strongly interacting two-dimensional (2D) electron systems; in particular in (100)-silicon metal-oxide-semiconductor field-effect transistors (MOSFET). It describes the additional confirmation in favor of zero-temperature zero-field metal-insulator transition is provided by magnetic and thermopower measurements. The chapter argues that the metal-insulator transition in silicon samples with very low level of disorder is driven by interactions. In a low-disordered 2D electron system in silicon MOSFETs, the resistivity at a certain "critical" electron density shows virtually no temperature dependence over a wide range of temperatures. In 2000, it was experimentally found that the ratio between the spin and the cyclotron splittings in silicon MOSFETs strongly increases at low electron densities. The strong enhancement of the spin susceptibility has indicated that at low electron densities, the system behaves well beyond the weakly interacting Fermi liquid.

This chapter focuses on studies of the critical behavior of conductivity in two-dimensional electron systems (2DES) with different amounts and types of disorder as well as different ranges of Coulomb interactions, in order to identify possible universality classes of the 2D metal-insulator transition (MIT). It discusses the manifestations of glassiness in both low- and high-disorder samples. The chapter examines the critical region near the 2D MIT in particular on describing conductivity measurements that have demonstrated dynamical scaling, the main signature of the MIT as a quantum phase transition. It also focuses on experiments that probe the nature of the insulating state, as well as charge dynamics across the MIT, in both high- and low-disorder 2DESs. There is substantial evidence that in many materials near the MIT both strong electronic correlations and disorder play an important role, and thus their competition is expected to lead to glassy behavior of electrons, in analogy with other frustrated systems.

We consider the Finkelstein action describing a system of spin-polarized or spinless electrons in $2+2\ensuremath{\epsilon}$ dimensions, in the presence of disorder as well as the Coulomb interactions. We extend the renormalization-group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an $\ensuremath{\epsilon}$ expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature, and/or electron density. We extend the results for the interacting electron gas in $2+2\ensuremath{\epsilon}$ dimensions to include the quantum critical behavior of the plateau transitions in the quantum Hall regime. Although these transitions have a very different microscopic origin and are controlled by a topological term in the action $(\ensuremath{\theta}$ term), the quantum critical behavior is in many ways the same in both cases. We show that the two independent critical exponents of the quantum Hall plateau transitions, previously denoted as $\ensuremath{\nu}$ and p, control not only the scaling behavior of the conductances ${\ensuremath{\sigma}}_{\mathrm{xx}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ at finite temperatures T, but also the non-Fermi-liquid behavior of the specific heat ${(c}_{v}\ensuremath{\propto}{T}^{p}).$ To extract the numerical values of $\ensuremath{\nu}$ and p it is necessary to extend the experiments on transport to include the specific heat of the electron gas.

We study the temperature-dependent conductivity $\ensuremath{\sigma}(T)$ and spin susceptibility $\ensuremath{\chi}(T)$ of the two-dimensional disordered Hubbard model. Calculations of the current-current correlation function using a quantum Monte Carlo method show that repulsion between electrons can significantly enhance the conductivity, and at low temperatures change the sign of $d\ensuremath{\sigma}/\mathrm{dT}$ from positive (insulating behavior) to negative (conducting behavior). This result suggests the possibility of a metallic phase, and consequently a metal-insulator transition, in a two-dimensional microscopic model containing both interactions and disorder. The metallic phase is a non-Fermi liquid with local moments as deduced from $\ensuremath{\chi}(T)$.