We study the metal-insulator transition in the highly correlated electron gas, described by the Hubbard Hamiltonian, within a renormalization group approach. The scaling theory of critical phenomena and results associated with the existence of a zero-temperature fixed point are used to derive expressions for the susceptibility and specific heat at very low temperatures. We discuss the enhancement of these quantities in the Fermi liquid due to the proximity of the transition to an insulating state. Finally we compare our results with those expected for the case of a magnetic instability.
We study the metal-insulator transition in the highly correlated electron gas, described by the Hubbard Hamiltonian, within a renormalization group approach. The scaling theory of critical phenomena and results associated with the existence of a zero-temperature fixed point are used to derive expressions for the susceptibility and specific heat at very low temperatures. We discuss the enhancement of these quantities in the Fermi liquid due to the proximity of the transition to an insulating state. Finally we compare our results with those expected for the case of a magnetic instability.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.
Introduction In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V 2 O 3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible , i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T , of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.
The interpretation of the metal-insulator transition phenomena in disordered two-dimensional electron systems in terms of density-dependent scaling variables suggests the existence of a quantum critical point at some critical electron density. However a first principles scaling theory based on renormalization group (RG) methods predicts a strong temperature dependence of the dimensionless resistivity R(T), even at small R(T), that is not observed. The observed properties are in fact consistent with a weakly disordered Fermi liquid, and there are no indications of strong temperature dependence induced by scaling. While the RG expansion in a power series in R(T) has only been evaluated to lowest order, this should be sufficient to describe experiments in the region of very small R. A further apparent anomaly is a return from metal-like to insulating-like behavior for increasing density. We explain these fundamental discrepancies between the first principles theory and experiment. We find that the R⪡1 data in the currently attainable temperature range are in a weak scaling regime described by the logarithmic approximation. We independently determine the density dependent prefactor of the logarithm using data for the spin susceptibility and effective mass. We find good agreement between theory and experiment for R(T) in the diffusive regime. We point out that there are corrections to the leading logarithm approximation that should be observable at still lower temperatures.
The problem of disordered interacting electrons is considered. We study a model for a generic disordered Fermi liquid without cooper pairs and in the absence of any spin-flip mechanisms. We prove to within logarithmic accuracy that there is a stable renormalization-group fixed point to all orders in a loop expansion. We prove that the conductivity exponent at this fixed point is identically equal to zero, while the spin-diffusion constant scales to zero with an exponent \ensuremath{\gamma}=4+O(d-2). By an explicit two-loop calculation we then show that this fixed point is suppressed by logarithmic terms, but a sizeable scaling region persists. The main conclusion is that the metal-insulator transition is preceded on the metallic side by a near instability in the spin system. There is a scaling region leading from a spin-diffusion phase to a region of very slow spin transport. Throughout this scaling region the charge transport is unaffected. Experiments are discussed in the light of these results, and further experiments to test the theory are proposed.
We use scaling and renormalization-group techniques to analyze the leading nonanalyticities in a Fermi liquid. We show that a physically motivated scaling hypothesis reproduces the results known from perturbation theory for the density of states, the density-of-states fluctuations, the specific heat, the spin susceptibility, and the nematic magnetic susceptibility. We also discuss the absence of nonanalytic terms in the density susceptibility. We then use a recent effective field theory for clean electron systems to derive the scaling hypothesis by means of renormalization-group techniques. This shows that the exponents (although not the prefactors) of the nonanalyticities that were previously derived by means of perturbative techniques are indeed exact, and can be understood as the leading corrections to scaling at the stable Fermi-liquid fixed point.