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Progress of Theoretical Physics

Renormalized Inter-Chain Hopping vs Charge Gap in Two Coupled Chains

Masahisa Tsuchiizu, Yoshikazu Suzumura, Thierry Giamarchi

Published Mar 1, 1999• 15 citations
Publisher

Abstract

The role of intra-chain interactions, which renormalize inter-chain hopping, has been examined for two coupled chains with umklapp scattering by use of the renormalization group method.It is clarified that the confinement-deconfinement transition occurs when the charge gap becomes larger than the renormalized inter-chain hopping.
DOI
10.1143/ptp.101.763

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A role of umklapp scattering has been examined for two-coupled chains with both forward and backward scattering by applying the renormalization group method to the bosonized Hamiltonian. It has been found that a state with relevant interchain hopping changes into a state with irrelevant (confined) one when the magnitude of umklapp scattering becomes larger than that of interchain hopping. The critical value of umklapp scattering for such a confinement-deconfinement transition is calculated as the function of interchain hopping and intrachain interactions. A crossover from one-dimensional regime into that of coupled chains is also shown with decreasing temperature.
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.
Using a nonperturbative functional renormalization group approach involving both fermionic and bosonic fields we calculate the interaction-induced change of the Fermi surface of spinless fermions moving on two chains connected by weak interchain hopping ${t}_{\ensuremath{\perp}}$. For a model containing interband backward scattering only we show that the distance $\ensuremath{\Delta}$ between the Fermi momenta associated with the bonding and the antibonding band can be strongly reduced, corresponding to a large reduction of the effective interchain hopping ${t}_{\ensuremath{\perp}}^{*}\ensuremath{\propto}\ensuremath{\Delta}$. A self-consistent one-loop approximation neglecting marginal vertex corrections and wave-function renormalizations predicts a confinement transition for sufficiently large interchain backscattering, where the renormalized ${t}_{\ensuremath{\perp}}^{*}$ vanishes. However, a more accurate calculation taking vertex corrections and wave-function renormalizations into account predicts only weak confinement in the sense that $0<\ensuremath{\mid}{t}_{\ensuremath{\perp}}^{*}\ensuremath{\mid}⪡\ensuremath{\mid}{t}_{\ensuremath{\perp}}\ensuremath{\mid}$. Our method can be applied to other strong-coupling problems where the dominant scattering channel is known.
We study dimensionality effects on the charge gap in the dimerized Hubbard model at quarter filling, with two approaches. First, we examine three chains coupled via the interchain one-particle hopping integral t b , by the density-matrix renormalization-group (DMRG) method. Next, we consider the d =1+ε dimensional model, using the perturbative renormalization-group (PRG) method. The dimensionality is controlled through t b and ε, respectively. Both approaches lead to the conclusion that, for a finite dimerization ratio, the charge gap decreases as the dimensionality increases.
We study two (very) weakly-coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale mmm is much larger than the bare single-electron interchain hopping t⊥t_{\perp}t⊥​. First, we find that the divergence of the intrachain Umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: This is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-pairs produces a magnetic interchain exchange J⊥=t⊥2/m>0J_{\perp}={t_{\perp}}^2/m>0J⊥​=t⊥​2/m>0. The result is an insulating state with spin-gapped excitations as in the delocalized limit (i.e. for rather large interchain hoppings), where the two-leg ladder is in the well-known D-Mott phase. Unlike for Bechgaard salts, the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4... (but not too large).