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Physical Review Letters

Crossover and scaling phenomena in a disordered Fermi liquid

D. Belitz, T. R. Kirkpatrick

Published Sep 18, 1989• 28 citations
Publisher

Abstract

We consider Finkelshtein's model for a disordered Fermi liquid. We show that logarithmic terms at two-loop order suppress the fixed point proposed by Castellani et al. Spin transport slows down dramatically, and in d=2+\ensuremath{\epsilon} a sizable scaling region exists. We show that in the scaling region the apparent conductivity exponent is zero to all orders in a loop expansion. This implies that the charge transport is unaffected by the crossover in the spin system. Experiments compare favorably with this scenario. Further measurements of the Wilson ratio are proposed.

DOI
10.1103/physrevlett.63.1296

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We consider Finkelshtein's model for a disordered Fermi liquid. We show that logarithmic terms at two-loop order suppress the fixed point proposed by Castellani et al. Spin transport slows down dramatically, and in d=2+\ensuremath{\epsilon} a sizable scaling region exists. We show that in the scaling region the apparent conductivity exponent is zero to all orders in a loop expansion. This implies that the charge transport is unaffected by the crossover in the spin system. Experiments compare favorably with this scenario. Further measurements of the Wilson ratio are proposed.
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.
The authors consider a generic model for a disordered Fermi liquid. By means of an infinite resummation of the loop expansion they show that there is a phase transition involving the spin degrees of freedom which can be identified with a new fixed point in the renormalisation group flow equations. The critical exponents obtained for 2<d<4 are apparently exact. The large-disorder phase is described as an incompletely frozen spin phase with no long-range order and subdiffusive spin transport. The results are in good agreement with experiments on Si:P.
Disorder operators are a type of non-local observables for quantum many-body systems, measuring the fluctuations of symmetry charges inside a region. It has been shown that disorder operators can reveal global aspects of many-body states that are otherwise difficult to access through local measurements. We study the disorder operator for U(1) (charge or spin) symmetry in two-dimensional Fermi and non-Fermi liquid states, using the multidimensional bosonization formalism. For a region AAA, the logarithm of the charge disorder parameter in a Fermi liquid with isotropic interactions scales asymptotically as lAln⁡lAl_A\ln l_AlA​lnlA​, with lAl_AlA​ being the linear size of the region AAA. We calculate the proportionality coefficient in terms of Landau parameters of the Fermi liquid theory. We then study models of the Fermi surface coupled to gapless bosonic fields realizing non-Fermi liquid states. In a simple spinless model, where the fermion density is coupled to a critical scalar, we find that at the quantum critical point the scaling behavior of the charge disorder operators is drastically modified to lAln⁡2lAl_A \ln^2 l_AlA​ln2lA​. We also consider the composite Fermi liquid state and argue that the charge disorder operator scales as lAl_AlA​.
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.
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)\mathcal{R}(T)R(T), even at small R(T)\mathcal{R}(T)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)\mathcal{R}(T)R(T) has only been evaluated to lowest order, this should be sufficient to describe experiments in the region of very small R\mathcal{R}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\mathcal{R}⪡1R⪡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)\mathcal{R}(T)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.