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arXiv:cond-mat/0204514 · Physical review. B, Condensed matter

Two-hole and four-hole bound states in at-J ladder at half-filling

Weihong Zheng, Chris J. Hamer

Published Aug 15, 2002• 3 citations
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Abstract

The two-hole excitation spectrum of the $t\ensuremath{-}J$ ladder at half-filling is studied using linked-cluster series expansion methods. A rich spectrum of bound states emerges, particularly at small $t/J.$ Their dispersion relations and coherence lengths are computed, along with the threshold behavior as the bound states merge into the continuum. A class of four-hole bound states is also studied, leading to the conclusion that phase separation occurs for $t/J\ensuremath{\lesssim}0.5,$ in agreement with other studies.

DOI
10.1103/physrevb.66.085112

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The two-hole excitation spectrum of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J ladder at half-filling is studied using linked-cluster series expansion methods. A rich spectrum of bound states emerges, particularly at small t/J.t/J.t/J. Their dispersion relations and coherence lengths are computed, along with the threshold behavior as the bound states merge into the continuum. A class of four-hole bound states is also studied, leading to the conclusion that phase separation occurs for t/J\ensuremath≲0.5,t/J\ensuremath{\lesssim}0.5,t/J\ensuremath≲0.5, in agreement with other studies.
Series expansions at T=0T=0T=0 are used to study properties of the half-filled $t\ensuremath{-}J$ ladder doped with one or two holes, and at quarter filling. Dispersion curves are obtained for one-hole symmetric and antisymmetric (bonding and antibonding) excitations and for the two-hole bound state. The line in the phase diagram that separates bound and unbound states is determined. For quarter filling we compute the ground-state energy and estimate the location of the phase separation line. Comparisons with other numerical and analytical results are presented.
We study one- and two-hole properties of the $t\ensuremath{-}J$ model at half-filling on the square lattice using series expansion methods at T=0.T=0.T=0. The dispersion curve for one-hole excitations is calculated and found to be qualitatively similar to that obtained by other methods, but the bandwidth for small t/Jt/Jt/J is some 20% larger than given previously. We also obtain the binding energy and dispersion relation for two-hole bound states. The lowest bound state as t/Jt/Jt/J increases is found to be first d wave, and then p wave, in accordance with predictions based upon the Kohn-Luttinger effect. We also carry out a similar study for the $t\ensuremath{-}{J}_{z}$ model.
Spin gaps in coupled t-J ladders are investigated by exact diagonalization of small clusters up to 4\ifmmode\times\else\texttimes\fi{}8 sites. At half-filling, the numerical results for the triplet excitation spectrum are in very good agreement with a second-order perturbation expansion in terms of small interladder and intraladder exchange couplings between rungs (J/J'0.25). The band of local triplet excitations moving coherently along the ladder (with momenta close to \ensuremath{\pi}) is split by the interladder coupling. For intermediate couplings, finite-size scaling is used to estimate the spin gap. In the isotropic infinite four-chain system (two coupled ladders) we find a spin gap of 0.245J, roughly half of the single-ladder spin gap. When the system is hole doped, bonding and antibonding bound pairs of holes can propagate coherently along the chains and the spin gap remains finite.
The Hubbard model on a two-leg ladder structure has been studied by a combination of series expansions at T = 0 and the density-matrix renormalization group. We report results for the ground-state energy E 0 and spin gap Δ s at half-filling, as well as dispersion curves for one- and two-hole excitations. For small U both E 0 and Δ s show a dramatic drop near t / t ⊥ ~0.5, which becomes more gradual for larger U . This represents a crossover from a `band insulator' phase to a strongly correlated spin liquid. The various features are collected in a `phase diagram' for the model.
The phase diagram of the two-leg $t\text{\ensuremath{-}}{J}_{z}$ ladder is explored, using the density-matrix renormalization group method. Results are obtained for energy gaps, electron density profiles, and correlation functions for the half filled and quarter filled cases. The effective Lagrangian velocity parameter ${v}_{\ensuremath{\rho}}$ is shown to vanish at half filling. The behavior of the one-hole gap in the Nagaoka limit is investigated, and found to disagree with theoretical predictions. A tentative phase diagram is presented, which is quite similar to the full $t\text{\ensuremath{-}}J$ ladder, but scaled up by a factor of about 2 in coupling. Near half filling a Luther-Emery phase is found, which may be expected to show superconducting correlations, while near quarter filling the system appears to be in a Tomonaga-Luttinger phase.
Holes weakly doped into a four-leg t-JJJ ladder bind in pairs. At dopings exceeding a critical doping of ${\ensuremath{\delta}}_{c}\ensuremath{\simeq}\frac{1}{8}$ four-hole clusters are observed to form in density-matrix renormalization-group calculations. The symmetry of the ground-state wave function does not change and we are able to reproduce this behavior qualitatively with an effective bosonic model in which the four-leg ladder is represented as two coupled two-leg ladders and hole pairs are mapped on hard-core bosons moving along and between these ladders. At lower dopings, $\ensuremath{\delta}<{\ensuremath{\delta}}_{c},$ a one-dimensional bosonic representation for hole pairs works and allows us to calculate accurately the Luttinger liquid parameter ${K}_{\ensuremath{\rho}}$ that takes the universal value ${K}_{\ensuremath{\rho}}=1$ as half-filling is approached.