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arXiv:cond-mat/9401050 · Physical review. B, Condensed matter

Pairing and excitation spectrum in dopedt-Jladders

Hirokazu Tsunetsugu, Matthias Troyer, Thomas Maurice Rice

Published Jun 1, 1994• 140 citations
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Abstract

Exact-diagonalization studies for a doped t-J ladder (or double chain) show hole pairing in the ground state. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet mode. At half-filling the former vanish but the latter evolves continuously into the triplet band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large.

DOI
10.1103/physrevb.49.16078

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arXiv:cond-mat/9606144•Physical review. B, Condensed matter•Aug 1, 1996•20 citations
Exact-diagonalization studies for a doped t-J ladder (or double chain) show hole pairing in the ground state. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet mode. At half-filling the former vanish but the latter evolves continuously into the triplet band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large.
We have numerically investigated the doped t-J ladder using exact diagonalization. We have studied both the limit of strong interchain coupling and isotropic coupling. The ladder scales to the Luther-Emery liquid regime in the strong interchain coupling limit. In this strong coupling limit there is a simple picture of the excitation spectrum that can be continued to explain the behavior at isotropic coupling. At J=0 we have indications of a ferromagnetic ground state. At a large J/t the ladder is phase separated into holes and a Heisenberg ladder. At intermediate coupling the ground state shows hole pairing with a modified d-wave symmetry. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet magnon mode. At half filling the former vanish but the latter evolves continuously into the magnon band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large. The dynamical structure factors are calculated and are found to be very similar to calculations on two-dimensional clusters. \textcopyright{} 1996 The American Physical Society.
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 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.
A two-chain ladder model is considered that is described by the strong-coupling t - t ´- J - J ´ Hamiltonian. For the case of two holes moving in a background of antiferromagnetically interacting spins, exact, analytical results are derived for the ground-state energy and low-lying excitation spectrum. The ground state is a bound state of two holes with total spin S = 0. The charge excitation is gapless and the spin excitation has a gap. The corresponding wave functions are also exactly determined. The bound hole pair is found to have symmetry of the d-wave type. In the limit of strong rung coupling, the model maps onto an effective hard-core-boson model which exhibits dominant superconducting pairing correlations.
Results for a doped three-leg ttt-JJJ ladder obtained using the density-matrix renormalization group are reported. At low hole doping, the holes form a dilute gas with a uniform density. The momentum occupation of the odd band shows a sharp decrease at a large value of kF{k}_{F}kF​ similar to the behavior of a lightly doped ttt-JJJ chain, while the even modes appear gapped. The spin-spin correlations decay as a power law consistent with the absence of a spin gap, but the pair-field correlations are negligible. At larger doping we find evidence for a spin gap and as xxx increases further we find three-hole diagonal domain walls. In this regime there are significant pair-field correlations and the internal pair orbital has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-like symmetry.
Photoemission spectra for the isotropic two-leg $t\ensuremath{-}J$ ladder are calculated at various hole-doping levels using exact diagonalization techniques. Low-energy sharp features caused by short-range antiferromagnetic correlations are observed at finite doping levels close to half-filling, above the naive Fermi momentum. These features should be observable in angle-resolved photoemission experiments. In addition, the formation of a ddd-wave pairing condensate as the ratio Jt\frac{J}{t}tJ​ is increased leads to dynamically generated spectral weight for momenta close to kF{k}_{F}kF​ where the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ order parameter is large.