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Journal of Physics Condensed Matter

The spin freezing transition in the disordered electron problem

T. R. Kirkpatrick, D. Belitz

Published Jun 11, 1990• 14 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/2/23/017

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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.
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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 phase transition between a Fermi liquid and a pseudomagnetic phase, which has recently been predicted to occur in disordered electron systems, is considered. We determine the exact asymptotic critical behavior at the transition, establish the connection between the integral-equation approach used earlier and a renormalization-group analysis, and determine the qualitative phase diagram. The critical behavior is found to be governed in part by log-log normal terms rather than by power laws. However, the critical region is extremely small, and on realistic scales transport and thermodynamic quantities obey preasymptotic, nonuniversal power laws. We present an extensive discussion of the properties of the renormalization group for this model, and of the physical and experimental implications of our results.
We study critical behavior of the diluted two-dimensional Ising model in the presence of disorder correlations which decay algebraically with distance as $\ensuremath{\sim}{r}^{\ensuremath{-}a}$. Mapping the problem onto two-dimensional Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a nontrivial fixed point which is stable for 0.995<a<20.995<a<20.995<a<2 and is characterized by the correlation length exponent $\ensuremath{\nu}=2/a+O({(2\ensuremath{-}a)}^{3})$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent ${\ensuremath{\eta}}_{2}=1/2\ensuremath{-}(2\ensuremath{-}a)/4+O({(2\ensuremath{-}a)}^{2})$.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.