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arXiv:cond-mat/9507029 · Physical review. B, Condensed matter

Effects of weak disorder on two coupled Hubbard chains

E. Orignac, Thierry Giamarchi

Published Apr 15, 1996• 27 citations
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Abstract

We consider the effect of weak nonmagnetic disorder on two chains of interacting fermions (with and without spins) coupled by interchain hopping. For the spinless case, interchain hopping increases localization for repulsive interactions but stabilizes the $s$-wave superconducting phase for attractive interactions. For the case with spin, the $d$-wave phase arising from purely repulsive interactions in the clean system is destroyed by an infinitesimal disorder, while for attractive interactions, the $s$-wave superconductivity is more resistant to disorder than in the one-chain case. In each case we compute the localization length and the conductivity.

DOI
10.1103/physrevb.53.r10453

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Physical review. B, Condensed matter•Aug 1, 1992•89 citations
We consider the effect of weak nonmagnetic disorder on two chains of interacting fermions (with and without spins) coupled by interchain hopping. For the spinless case, interchain hopping increases localization for repulsive interactions but stabilizes the sss-wave superconducting phase for attractive interactions. For the case with spin, the ddd-wave phase arising from purely repulsive interactions in the clean system is destroyed by an infinitesimal disorder, while for attractive interactions, the sss-wave superconductivity is more resistant to disorder than in the one-chain case. In each case we compute the localization length and the conductivity.
We study the effects of disorder on a system of two coupled chains of strongly correlated fermions (the ladder system), using a renormalization-group technique. The stability of the phases of the pure system has been investigated as a function of interactions both for fermions with spin and spinless fermions. For spinless fermions the repulsive side is strongly localized whereas the system with attractive interactions is stable with respect to disorder, at variance with the single-chain case. For fermions with spins, the repulsive side is also localized, and in particular the ddd-wave superconducting phase found for the pure system is totally destroyed by an arbitrarily small amount of disorder. On the other hand, the attractive side is again remarkably stable with respect to localization. We have also computed the charge stiffness, the localization length, and the temperature dependence of the conductivity for the various phases. In the parameter range where ddd-wave superconductivity would occur for the pure system the conductivity is found to decrease monotonically with temperature, even at high temperature, and we discuss this surprising result. For a model with one-site repulsion and nearest-neighbor attraction, the most stable phase is an orbital antiferromagnet. Although this phase has no divergent superconducting fluctuation it can have a divergent conductivity at low temperature. Finally, to make a comparison of our results with experimental ladder systems, we treated the interladder coupling in a mean-field approximation. We argue based on our results that the superconductivity observed in some of these compounds cannot be a simple stabilization of the ddd-wave phase found for a pure single ladder. The application of our results to systems such as quantum wires is also discussed. In particular, the corrections to conductance in a two-channel quantum wire have been obtained as a function of system length, temperature, and interactions.
The effect of interchain hopping and interaction on Anderson localization has been investigated for two coupled chains of spinless fermions with nearest neighbor interaction. Using the renormalization group method, we find that interchain hopping plays a significant role when the correlation gap induced by the interplay of interchain hopping and intrachain interaction becomes larger than the Anderson localization energy scale induced by impurities. The strong competition between localization and delocalization is examined for attractive interaction.
The effect of interchain hopping and interaction on Anderson localization has been investigated for two coupled chains of spinless fermions with nearest neighbor interaction. Using the renormalization group method, we find that interchain hopping plays a significant role when the correlation gap induced by the interplay of interchain hopping and intrachain interaction becomes larger than the Anderson localization energy scale induced by impurities. The strong competition between localization and delocalization is examined for attractive interaction.
This work analyzes the behavior of coupled disordered one-dimensional systems as modelled by identical fermionic Hubbard chains with the on-site potential disorder and coupling emerging through the interchain hopping ${t}^{\ensuremath{'}}$. The study is motivated by the experiment on fermionic cold atoms on a disordered lattice, where a decay rate of the quenched density wave was measured. We present a derivation of the decay rate $\mathrm{\ensuremath{\Gamma}}$ within perturbation theory and show that, even at large disorder along the chains, the interaction leads to finite $\mathrm{\ensuremath{\Gamma}}>0$, the mechanism being the interaction-induced coupling of in-chain localized and interchain extended single-fermion states. Explicit expressions for $\mathrm{\ensuremath{\Gamma}}$ are presented for a weak interaction $U<t,{t}^{\ensuremath{'}}$, but extended also to the regime $t>U>{t}^{\ensuremath{'}}$. It is shown that, in both regimes, $\mathrm{\ensuremath{\Gamma}}$ increases with the interchain hopping ${t}^{\ensuremath{'}}$, as well as decreases with increasing disorder.
The effect of a weak random potential on two-leg Hubbard ladders is investigated. The random potential is shown to induce Anderson localization except for attractive enough interactions, supressing completely d-wave superconductivity. These localization effects remain very strong even for many ladders coupled by Josephson coupling. Both dc and ac conductivities and localization lengths are obtained. Consequences for the superconducting ladder compound SrxCa14−xCu24O41+δ are discussed. The two-chain Hubbard ladder is a toy model of a metal with a spin-gap, reminiscent of the metallic phase of underdoped cuprates. Recent experiments[1] on the superconducting transition in Sr14−xCaxCu24O41 have been interpreted in the framework of this model. However, impurities are only considered as a source of holes and not a source of random potential although it is known that an infinitesimal disorder localizes all electronic states in one dimension. It is therefore important to investigate the disordered Hubbard ladder. The disordered Hubbard ladder is defined by the Hamiltonian: H = −t ‖ (c † i+1,p,σci,p,σ + c † i,p,σci+1,p,σ) − t⊥ i,p,σ i,σ
The T=0 phase diagram for two Hubbard chains with interchain hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ and on-site repulsion U is derived within a renormalization-group (RG) approach. We explicitly allow for renormalization of the Fermi momenta. A nonzero ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ drives the chains towards strong coupling, and only for large ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ is a Luttinger liquid recovered. By use of weak-coupling RG, a qualitative picture of the phase diagram can be obtained. Our calculations show two different strong-coupling regimes: one with nonvanishing effective interchain hopping (〈${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$〉\ensuremath{\ne}0), not confining, for small U, and another with 〈${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$〉=0, confining, for large U. Numerical diagonalizations for a small cluster support the RG results.