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Physical review. B, Condensed matter

Anderson localization and interactions in one-dimensional metals

Thierry Giamarchi, Heinz J. Schulz

Published Jan 1, 1988• 854 citations
Publisher

Abstract

A one-dimensional interacting electron gas in a random potential exhibits a localized-delocalized transition for increasingly attractive interactions. We develop here a renormalization-group approach to study this transition. Our treatment allows us to obtain the phase diagram and the exponents of the correlation functions in the delocalized regime. The boundary between the two regimes is found to depend both on disorder and the strength of the interactions. For (nearly) spin-isotropic interactions the delocalized phase is dominated by superconducting fluctuations of either singlet or triplet type. The temperature dependence of the conductivity in the delocalized phase is also obtained and a nonuniversal power-law behavior is found. A description of the crossover towards the localized phase is given and the localization length is computed. An analogous description is developed for the localized-superfluid transition of a one-dimensional boson gas. In this case the transition to the localized regime occurs for increasingly repulsive interactions. We suggest a phase diagram with two different localized phases. Finally, we discuss some possible implications of our model for real quasi-one-dimensional metals.

DOI
10.1103/physrevb.37.325

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We describe a renormalization group method to study the localized-delocalized transition of a one-dimensional interacting electron gas in a random potential. We obtain the phase diagram and the exponents of the correlation functions in the delocalized regime. The boundary between the two regimes is found to depend both on disorder and on the strength of the interactions. The delocalized phase is dominated by superconducting fluctuations. We find the asymptotic behaviour of the localization length and the temperature dependence of the conductivity. An analogous description is developed for the localized-superfluid transition of a one-dimensional boson gas. In this case the transition to the localized regime occurs for increasingly repulsive interactions.
We consider a one-dimensional system of interacting bosons in a random potential. At zero temperature, it can be either in the superfluid or in the insulating phase. We study the transition at weak disorder and moderate interaction. Using a systematic approach, we derive the renormalization group equations at two-loop order and discuss the phase diagram. We find the universal form of the correlation functions at the transitions and compute the logarithmic corrections to the main universal power-law behavior. In order to mimic large density fluctuations on a single site, we study a simplified model of disordered two-leg bosonic ladders with correlated disorder across the rung. Contrarily to the single-chain case, the latter system exhibits a transition between a superfluid and a localized phase where the exponents of the correlation functions at the transition do not take universal values.
By use of the method in the previous paper [J. Phys. Soc. Jpn. 53 (1984) 3918], we study the Anderson localization in a one-dimensional half-filled extended Hubbard model with randomness. A transition from localization into delocalization takes place as the attractive interaction for superconductivity increases. It is shown that the effect of localization on singlet superconductivity (SS) is stronger than that on triplet superconductivity (TS) due to the large quantum fluctuation of TS compared with that of SS.
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Electrons in one-dimensional disordered systems exhibit Anderson localization in the absence of mutual interactions. However the transition is possible between localized and delocalized state as the interaction is turned on. In this paper such interplay between Anderson localization and interactions has been examined for spin dependent Tomonaga model based on the phase Hamiltonian derived by the bosonization of the fermion field. The problem is the extention of the classical impurity pinning of the charge density wave to that of quantum systems with phase variables of not only charge but also spin. The critical value of the interaction constant has been derived by the use of selfconsistent-harmonic approximation to the nonlinearity.
Using the density matrix renormalization group algorithm, we investigate the lattice model for spinless fermions in one dimension in the presence of a strong interaction and disorder. The phase sensitivity of the ground state energy is determined with high accuracy for systems up to a size of 60 lattice constants. This quantity is found to be log normally distributed. The fluctuations grow algebraically with system size with a universal exponent of $\ensuremath{\approx}2/3$ in the localized region of the phase diagram. Surprisingly we find, for an attractive interaction, a delocalized phase of finite extension. The boundary of this delocalized phase is determined.