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arXiv:cond-mat/9608126 · Physical review. B, Condensed matter

Correlation functions in the three-chain Hubbard ladder

T. Kimura, Kazuhiko Kuroki, Hideo Aoki

Published Oct 1, 1996• 32 citations
arXivHTMLPDF

Abstract

In order to check whether odd-numbered Tomonaga-Luttinger ladders are dominated by antiferromagnetic correlations associated with gapless spin excitations, correlation functions of the doped three-chain Hubbard model are obtained with the bosonization at the renormalization-group fixed point. The correlation of the singlet superconducting pairing across the central and edge chains is found to be dominant, reflecting two gapful spin modes, while the intraedge spin-density-wave correlation, reflecting the gapless mode, is only subdominant. This implies that, when there are multiple spin modes, a dominant superconductivity can arise from the presence of some spin gap(s) despite the coexistence of power-law correlated spins.

DOI
10.1103/physrevb.54.r9608

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In order to check whether odd-numbered Tomonaga-Luttinger ladders are dominated by antiferromagnetic correlations associated with gapless spin excitations, correlation functions of the doped three-chain Hubbard model are obtained with the bosonization at the renormalization-group fixed point. The correlation of the singlet superconducting pairing across the central and edge chains is found to be dominant, reflecting two gapful spin modes, while the intraedge spin-density-wave correlation, reflecting the gapless mode, is only subdominant. This implies that, when there are multiple spin modes, a dominant superconductivity can arise from the presence of some spin gap(s) despite the coexistence of power-law correlated spins.
In order to shed light whether the `even-odd conjecture' (even numbers of legs will superconduct accompanied by a spin gap while odd ones do not) for correlated electrons in ladder systems, the pairing correlation is studied for the Hubbard model on a two- and three-leg ladders. We have employed both the weak-coupling renormalization group and the quantum Monte Carlo (QMC) method for strong interactions. For the two-leg Hubbard ladder, a systematic QMC (with a controlled level spacings) has detected an enhanced pairing correlation, which is consistent with the weak-coupling prediction. We also calculate the correlation functions in the three-leg Hubbard ladder and show that the weak-coupling study predicts the dominant superconductivity, which refutes the naive even-odd conjecture. A crucial point is a spin gap for only some of the multiple spin modes is enough to make the ladder superconduct with a pairing symmetry (d-like here) compatible with the gapped mode. A QMC study for the three-leg ladder endorses the enhanced pairing correlation.
Abstract We study the ground state properties of the Hubbard model on three-leg triangular cylinders using large-scale density-matrix renormalization group simulations. At half-filling, we identify an intermediate gapless spin liquid phase, which has one gapless spin mode and algebraic spin–spin correlations but exponential decay scalar chiral–chiral correlations, between a metallic phase at weak coupling and Mott insulating dimer phase at strong interaction. Upon light doping the gapless spin liquid, the system exhibits power-law charge-density-wave (CDW) correlations but short-range single-particle, spin–spin, and chiral–chiral correlations. Similar to CDW correlations, the superconducting correlations also decay in power-law but oscillate in sign as a function of distance, which is consistent with the striped pair-density wave. When further doping the gapless spin liquid phase or doping the dimer order phase, another phase takes over, which has similar CDW correlations but all other correlations decay exponentially.
The spin and charge excitations of the t-J ladder are studied by exact diagonalization techniques for several electron densities. The various modes are classified according to their spin (singlet or triplet excitations) and their parity under a reflection with respect to the symmetry axis along the chains and a finite-size scaling of the related gaps is performed. At low doping, formation of hole pairs leads to a spin gap for all J/t ratios. This phase is characterized by (at least) one vanishing energy mode only in the charge bonding channel when Kx{\mathit{K}}_{\mathit{x}}Kx​\ensuremath{\rightarrow}0 consistent with the existence of superconducting pairing correlations. At larger doping the spin gap disappears. Although the antibonding modes remain gapped, low-energy Kx{\mathit{K}}_{\mathit{x}}Kx​\ensuremath{\sim}0 and Kx{\mathit{K}}_{\mathit{x}}Kx​\ensuremath{\sim}2kF{\mathit{k}}_{\mathit{F}}kF​ spin and charge bonding modes are found consistent with a single-band Luttinger scenario. At sufficient low electron density and above a critical value of J/t we also expect another phase of electron pairs with gapped spin excitations.
We study the ground-state spin correlations in the gapless incommensurate regime of a S=1/2S=1/2S=1/2 XXZ\mathrm{XXZ}XXZ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open XXZ\mathrm{XXZ}XXZ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter K and the amplitudes of the correlation functions. The value of K estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of K on the magnetization M. The $K\ensuremath{-}M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate 1/T1{1/T}_{1}1/T1​ and dynamical spin structure factors.