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.
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.
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.
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. 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/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/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.
The two-leg $t\ensuremath{-}J$ ladder forms a spin liquid at half filling that evolves to a Luther-Emery liquid upon doping. Our aim is to obtain a complete phase diagram for isotropic coupling (i.e., rungs and legs equal) as a function of electron density n and the ratio
J/t. Two known limiting cases are
n<21, which is a single band Luttinger liquid, and small hole doping $\ensuremath{\delta}\ensuremath{\ll}1$ for $J/\stackrel{\ensuremath{\rightarrow}}{t}0,$ which is a Nagaoka ferromagnet. Using Lanczos techniques we examine the region between the Nagaoka and Luther-Emery phases for
1>n>21. We find evidences for gapless behavior in both spin and charge channels for
J/t<0.3, consistent with Luttinger liquids in both bonding and antibonding bands (i.e.,
C2S2). This proposal is based on the behavior of spin and charge correlation functions. For example, the hole-hole correlation function that displays hole pairing at larger
J/t shows hole-hole repulsion in this region. As a further test, we examined the dependence of the energy on a relative phase shift between bonding and antibonding bands. For
J/t<0.3 this is very weak, indicating a lack of pairing between these channels.
Two-leg $t\ensuremath{-}J$ ladders are investigated in the framework of a combination of the phase string formulation and bond-operator representation. We develop a mean-field theory in the strong rung interaction regime, i.e., ${J}_{\ensuremath{\perp}}\ensuremath{\gg}J,t,$ which provides a unified description of the undoped insulating phase and the low-doping phase---the so-called
C1S0 phase. Both of them are characterized by the resonating-valence-bond (RVB) order parameter, with gap opened up in all spin excitations. The ground state of the doped phase is intrinsically a superconductor with a d-wave symmetry, which is driven by the RVB correlations. The ground-state energy is in good agreement with numerical results. Phase separation is shown to occur beyond some critical value of
J/t for given doping concentration. The local structure of hole pairs as well as the spectra of various spin and charge modes are analyzed in comparison with other approaches.