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arXiv:cond-mat/0002106 · Physical review. B, Condensed matter

Quantum criticality at the metal-insulator transition

D. Schmeltzer

Published Jan 24, 2001• 1 citation
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Abstract

We introduce an alternative method to analyze the many-body problem with disorder. The method is an extension of the real space renormalization group based on the operator product expansion. We consider the problem in the presence of interactions, a large elastic mean free path, and finite temperatures. As a result scaling is stopped either by temperature or the length scale set by the diverging many-body length scale (superconductivity). Due to disorder a superconducting instability might take place at ${T}_{\mathrm{SC}}\ensuremath{\rightarrow}0,$ giving rise to a metallic phase or $T>{T}_{\mathrm{SC}}.$ For repulsive interactions at $\stackrel{\ensuremath{\rightarrow}}{T}0$ we flow towards the localized phase, which is analyzed within the diffusive Finkelstein theory. For strong repulsive backward interactions and nonspherical Fermi surfaces characterized by $|d\mathrm{ln}N(b)/\mathrm{ln}b|\ensuremath{\ll}1$ one finds a fixed point ${(D}^{*},{\ensuremath{\Gamma}}_{2}^{*})$ in the plane $(D,{\ensuremath{\Gamma}}_{2}^{(s)}).$ $[D\ensuremath{\propto}{(K}_{F}{l)}^{\ensuremath{-}1}$ is the disorder coupling constant, ${\ensuremath{\Gamma}}_{2}^{(s)}$ is the particle-hole triplet interaction, b is the length scale, and $N(b)$ is the number of channels.] For weak disorder, $D<{D}^{*},$ one obtains a metallic behavior with the resistance $\ensuremath{\rho}(D,{\ensuremath{\Gamma}}_{2}^{(s)},T)=\ensuremath{\rho}(D,{\ensuremath{\Gamma}}_{2}^{(s)},T)\ensuremath{\simeq}{\ensuremath{\rho}}^{*}f((D\ensuremath{-}{D}^{*}{)/D}^{*}{(1/T}^{z{\ensuremath{\nu}}_{1}}))$ $[{\ensuremath{\rho}}^{*}=\ensuremath{\rho}{(D}^{*},{\ensuremath{\Gamma}}_{2}^{*},1),$ $z=1,$ and ${\ensuremath{\nu}}_{1}>1],$ and large ferromagnetic fluctuations caused by the stable fixed point ${\ensuremath{\Gamma}}_{2}^{*}.$

DOI
10.1103/physrevb.63.075105

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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.
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