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Physical review. B, Condensed matter

Strong-coupling phases of two Hubbard chains with interchain hopping

Michele Fabrizio, Alberto Parola, Erio Tosatti

Published Aug 1, 1992• 89 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.46.3159

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The T=0 phase diagram for two Hubbard chains with interchain hopping t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ and on-site repulsion U is derived within a renormalization-group (RG) approach. We explicitly allow for renormalization of the Fermi momenta. A nonzero t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ drives the chains towards strong coupling, and only for large t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​ 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 (〈t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​〉\ensuremath{\ne}0), not confining, for small U, and another with 〈t\ensuremath⊥{\mathit{t}}_{\mathrm{\ensuremath{\perp}}}t\ensuremath⊥​〉=0, confining, for large U. Numerical diagonalizations for a small cluster support the RG results.
We have calculated the charge gap and spin gap for the two-chain Hubbard model as a function of the on-site Coulomb interaction and the interchain hopping amplitude. We used the density matrix renormalization group method and developed a method to calculate separately the gaps numerically for the symmetric and antisymmetric modes with respect to the exchange of the chain indices. We have found very different behaviors for the weak and strong interaction cases. Our calculated phase diagram is compared to the one obtained by Balents and Fisher using the weak coupling renormalization group technique.
Using functional renormalization group methods, we present a self-consistent calculation of the true Fermi momenta kFa{k}_{F}^{a}kFa​ (antibonding band) and kFb{k}_{F}^{b}kFb​ (bonding band) of two spinless interacting metallic chains coupled by small interchain hopping. In the regime where the system is a Luttinger liquid, we find that $\ensuremath{\Delta}={k}_{F}^{b}\ensuremath{-}{k}_{F}^{a}$ is self-consistently determined by $\ensuremath{\Delta}={\ensuremath{\Delta}}_{1}{[1+{g}_{0}^{2}\mathrm{ln}{({\ensuremath{\Lambda}}_{0}∕\ensuremath{\Delta})}^{2}]}^{\ensuremath{-}1}$, where g0{g}_{0}g0​ is the dimensionless interchain backscattering interaction, ${\ensuremath{\Delta}}_{1}$ is the Hartree-Fock result for ${k}_{F}^{b}\ensuremath{-}{k}_{F}^{a}$, and ${\ensuremath{\Lambda}}_{0}⪢\ensuremath{\Delta}$ is an ultraviolet cutoff. For ${g}_{0}^{2}\mathrm{ln}{({\ensuremath{\Lambda}}_{0}∕{\ensuremath{\Delta}}_{1})}^{2}⪢1$ even weak interachain backscattering leads to a strong reduction of the distance between the Fermi momenta.
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.
Interchain hopping in systems of coupled chains of correlated electrons is investigated by exact diagonalization and quantum Monte Carlo methods. For two weakly coupled Hubbard chains at commensurate densities (e.g., n=1/3), the splitting at the Fermi level between bonding and antibonding bands is strongly reduced (but not suppressed) by repulsive interactions extending to a few lattice spacings. The magnitude of this reduction is directly connected to the exponent \ensuremath{\alpha} of the one-dimensional Luttinger liquid. However, we show that the incoherent part of the single-particle spectral function is much less affected by the interchain coupling. This suggests that incoherent interchain hopping could occur for intermediate \ensuremath{\alpha} values. \textcopyright{} 1996 The American Physical Society.
We study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.
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.