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arXiv:cond-mat/9509123 · Physical review. B, Condensed matter

Luttinger-liquid behavior and superconducting correlations int-Jladders

C A Hayward, Didier Poilblanc

Published May 1, 1996• 81 citations
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Abstract

The low-energy behavior of the isotropic t-J ladder system is investigated using exact diagonalization techniques, specifically finding the Drude weight, the charge velocity, and the compressibility. By applying the ideas of Luttinger-liquid theory, we determine the correlation exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ which defines the behavior of the long-range correlations in the system. The boundary to phase separation is determined and a phase diagram is presented. At low electron density, a Tomonaga-Luttinger-like phase is stabilized while at higher electron densities a gapped phase with power law pairing correlations is stabilized: A large region of this gapped phase is found to exhibit dominant superconducting correlations. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.11721

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The low-energy behavior of the isotropic t-J ladder system is investigated using exact diagonalization techniques, specifically finding the Drude weight, the charge velocity, and the compressibility. By applying the ideas of Luttinger-liquid theory, we determine the correlation exponent K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​ which defines the behavior of the long-range correlations in the system. The boundary to phase separation is determined and a phase diagram is presented. At low electron density, a Tomonaga-Luttinger-like phase is stabilized while at higher electron densities a gapped phase with power law pairing correlations is stabilized: A large region of this gapped phase is found to exhibit dominant superconducting correlations. \textcopyright{} 1996 The American Physical Society.
The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around Jc{\mathit{J}}_{\mathit{c}}Jc​/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.
We investigate a t-J ladder model by numerical diagonalization method. By calculating correlation functions and assuming the Luttinger liquid relation, we obtained a possible phase diagram of the ground state as a function of J/t and electron density nnn. We also found that behavior of correlation functions seems to consist with the prediction of Luttinger liquid relation. The result suggests that the superconducting phase appear in the region of J/t>∼0.5J/t \displaystyle{ \mathop{>}_{\sim}} 0.5J/t>∼​0.5 for high electron density and J/t>∼2.0J/t \displaystyle{ \mathop{>}_{\sim}} 2.0J/t>∼​2.0 for low electron density.
We examine the ground-state phase diagram of the ttt-JJJ model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction JJJ and the density nnn, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as JJJ is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger JJJ, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value ( 0.1t~0.1t 0.1t) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.
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.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.
We investigate numerically the ground state phase diagram of the one-dimensional extended Hubbard model, including an on-site interaction U and a nearest-neighbor interaction V. We focus on the ground state phases of the model in the $V\ensuremath{\gg}U$ region, where previous studies have suggested the possibility of dominant superconducting pairing fluctuations before the system phase separates at a critical value V=VPS.{V=V}_{\mathrm{PS}}.V=VPS​. Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For V<VPS,V<{V}_{\mathrm{PS}},V<VPS​, our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.