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Reviews in Mathematical Physics

ANOMALOUS SUPERCONDUCTIVITY IN COUPLED LUTTINGER LIQUIDS

Vieri Mastropietro

Published Dec 1, 2000• 1 citation
Publisher

Abstract

According to Anderson ([1], p. 54) "the equivalent of BCS theory for a fully spin-charge separated Luttinger liquid has not been formally worked out". We consider a model for two one-dimensional spinning Luttinger liquids coupled via a Cooper tunnelling Hamiltonian. We show that the partition function is the four-dimensional integral of an exponential whose exponent has an extremal point obtained solving an anomalous non-BCS self-consistence equation. If the extremal point is a global minimum the model is completely solved by the saddle point theorem and the anomalous gap generation is proved. We find that the Luttinger interaction enhances strongly T c if the intrachain Luttinger interaction is much bigger than the interchain interaction.

DOI
10.1142/s0129055x0000054x

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According to Anderson ([1], p. 54) "the equivalent of BCS theory for a fully spin-charge separated Luttinger liquid has not been formally worked out". We consider a model for two one-dimensional spinning Luttinger liquids coupled via a Cooper tunnelling Hamiltonian. We show that the partition function is the four-dimensional integral of an exponential whose exponent has an extremal point obtained solving an anomalous non-BCS self-consistence equation. If the extremal point is a global minimum the model is completely solved by the saddle point theorem and the anomalous gap generation is proved. We find that the Luttinger interaction enhances strongly T c if the intrachain Luttinger interaction is much bigger than the interchain interaction.
A self-consistence equation describing a superconductor whose normal state is a Luttinger liquid is computed in a rigorous way by renormalization group techniques. The gap and the critical temperatures are quite different with respect to the ones of normal superconductors (i.e. superconductors whose normal state is a Fermi liquid) and in particular, the critical temperature can be quite high for suitable values of the parameters. Such self-consistence equation can be obtained in a BCS mean field theory for a model of two Mattis models exchanging Cooper pairs. The connection of this model with recent theories of high T c superconductivity is briefly discussed.
In the context of the Anderson theory of high T c cuprates, we develop a BCS theory for Luttinger liquids. If the Luttinger interaction is much stronger than the BCS potential we find that the BCS equation is quite modified compared to usual BCS equation for Fermi liquids. In particular T c predicted by the BCS equation for Luttinger liquids is quite higher than the usual T c for Fermi liquids. 1 The anomalous BCS equation The equivalent of BCS theory for a Luttinger liquid has not formally worked out, despite the relevance of such theory for the problem of superconductivity in the high-T c cuprates, see [1] or the discussion in the following section. Such theory should describe superconductors which in their normal state are Luttinger liquids. In this paper we develop such theory using constructive quantum field theory techniques, applied in many other Luttinger liquid problems, see [2], [3]. In particular, we compute in a rigorous way the BCS self-consistence equation for a spinni...
We consider the problem of tunneling between spinless one-dimensional (1D) Luttinger-liquid chains. We show how to map the problem onto the four-state chiral clock model together with a free boson for the total charge mode. We use a variety of results, some of them exact, from the integrable chiral Potts model and the study of commensurate/incommensurate transitions to deduce the physics of coupled Luttinger liquids. For those intrachain interaction strengths for which interchain tunneling is relevant, we find that it can lead to the formation of symmetric and antisymmetric bands with split Fermi surfaces, depending on the relative strengths of interchain tunneling and interchain interactions. With split Fermi surfaces, interchain transport is coherent. It is not possible to have two gapless Fermi surfaces with the same Fermi momentum when interchain tunneling is relevant. However, interchain interactions can drive the formation of a gap, in which case interchain transport will be incoherent. We comment on the possible relevance of our results to c-axis transport in high-Tc{T}_{c}Tc​ superconductors.
We consider a one-dimensional system consisting of electrons with short-ranged repulsive interactions and coupled to small-momentum transfer acoustic phonons. The interacting electrons are bosonized and described in terms of a Luttinger liquid which allows us to calculate exactly the one- and two-electron Green function. For non-interacting electrons, the coupling to phonons alone induces a singularity at the Fermi surface which is analogous to that encountered for electrons with an instantaneous attractive interaction. The exponents which determine the presence of singlet/triplet superconducting pairing fluctuations, and spin/charge density wave fluctuations are strongly affected by the presence of the Wentzel-Bardeen singularity, resulting in the favoring of superconducting fluctuations. For the Hubbard model the equivalent of a phase diagram is established, as a function of: the electron-phonon coupling, the electron filling factor, and the on-site repulsion between electrons. The Wentzel-Bardeen singularity can be reached for arbitrary values of the electron-phonon coupling constant by varying the filling factor. This provides an effective mechanism to push the system from the antiferromagnetic into the metallic phase, and finally into the superconducting phase as the electron filling factor is increased towards half-filling.
One-particle interchain hopping in a system of coupled Luttinger liquids is investigated using exact diagonalizations techniques. First, the two-chain problem of spinless fermions is studied in order to see the behavior of the band splitting as a function of the exponent $\ensuremath{\alpha}$ which characterizes the one-dimensional (1D) Luttinger liquid. Moderate intrachain interactions can lead to a strong reduction of this splitting. The onset of the confinement within the individual chains (defined by a vanishing splitting) seems to be governed by $\ensuremath{\alpha}$. We give numerical evidence that interchain coherent hopping can be totally suppressed for $\ensuremath{\alpha}\ensuremath{\sim}0.4$ or even smaller $\ensuremath{\alpha}$ values. The transverse conductivity is shown to exhibit a strong incoherent part. Even when coherent interchain hopping is believed to occur (at small $\ensuremath{\alpha}$ values), it is shown that the coherent Drude weight is always significantly smaller than the incoherent weight. Implications for the optical experiments in quasi-1D organic or high-Tc{T}_{c}Tc​ superconductors are outlined.
We consider two one-dimensional interacting electron systems that are coupled via interchain tunneling to calculate transport properties in the presence of impurities or a single barrier by using the bosonization formalisms for the non-Luttinger-liquid phases on the phase diagram obtained by Fabrizio [Phys. Rev. B 48, 15 838 (1993)]. We find for the weak- and strong-interaction phases, where the superconducting correlation dominates in the ground state, that the interchain transfer enhances the conductivity and the Anderson localization is suppressed. This shows that multichain systems can have unique transport properties.