The $t\ensuremath{-}{J}_{z}$ model is the strongly anisotropic limit of the $t\ensuremath{-}J$ model which captures some general properties of doped antiferromagnets (AF's). The absence of spin fluctuations simplifies the analytical treatment of hole motion in an AF background, and allows us to calculate single- and two-hole spectra with a high accuracy using a regular diagram technique combined with a real-space approach. At the same time, numerical studies of this model via exact diagonalization on small clusters show negligible finite-size effects for a number of quantities, thus allowing a direct comparison between analytical and numerical results. Both approaches demonstrate that the holes have a tendency to pair in p- and d-wave channels at realistic values of $t/J.$ Interactions leading to pairing and effects selecting p and d waves are thoroughly investigated. The role of transverse spin fluctuations is considered using perturbation theory. Based on the results of the present study, we discuss the pairing problem in the realistic $t\ensuremath{-}J$-like model. Possible implications for preformed pairs formation and phase separation are drawn.
The t\ensuremath−Jz model is the strongly anisotropic limit of the t\ensuremath−J model which captures some general properties of doped antiferromagnets (AF's). The absence of spin fluctuations simplifies the analytical treatment of hole motion in an AF background, and allows us to calculate single- and two-hole spectra with a high accuracy using a regular diagram technique combined with a real-space approach. At the same time, numerical studies of this model via exact diagonalization on small clusters show negligible finite-size effects for a number of quantities, thus allowing a direct comparison between analytical and numerical results. Both approaches demonstrate that the holes have a tendency to pair in p- and d-wave channels at realistic values of t/J. Interactions leading to pairing and effects selecting p and d waves are thoroughly investigated. The role of transverse spin fluctuations is considered using perturbation theory. Based on the results of the present study, we discuss the pairing problem in the realistic t\ensuremath−J-like model. Possible implications for preformed pairs formation and phase separation are drawn.
The phase diagram of the planar t-J model at small hole doping is investigated by finite-size scaling of exact diagonalization data of \ensuremath{\surd}N \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}N clusters (N\ensuremath{\le}26). Hole-droplet binding energies, compressibility, and static spin and charge correlations are calculated. Short-range antiferromagnetic correlations can produce attractive forces between holes leading to a very rich phase diagram including a liquid of d-wave hole pairs (for J/t\ensuremath{\gtrsim}0.2), a liquid of hole droplets (quartets) for larger J/t ratios (J/t\ensuremath{\gtrsim}0.5), and, at even larger coupling J/t, an instability towards phase separation.
We analytically calculate the ground state pairing symmetry and excitation spectra of two holes doped into the half-filled $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}$ model in the strong-coupling limit $({J}_{z}⪢\ensuremath{\mid}t\ensuremath{\mid},\ensuremath{\mid}{t}^{\ensuremath{'}}\ensuremath{\mid},\ensuremath{\mid}{t}^{\ensuremath{''}}\ensuremath{\mid})$. In leading order, this reduces to the ${t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}$ model, where there are regions of d-wave, s-wave, and (degenerate) p-wave symmetry. We find that the $t\ensuremath{-}{J}_{z}$ model maps in lowest order onto the ${t}^{\ensuremath{'}}\ensuremath{-}{t}^{\ensuremath{''}}\ensuremath{-}{J}_{z}$ model on the boundary between d and p symmetry, with a flat lower band in the pair excitation spectrum. In higher order, d-wave symmetry is selected from the lower pair band. However, we observe that the addition of the appropriate ${t}^{\ensuremath{'}}<0$ and/or ${t}^{\ensuremath{''}}>0$, the signs of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$ found in the hole-doped cuprates, could drive the hole-pair symmetry to p-wave symmetry, implying the possibility of competition between p-wave and d-wave pair ground states. (An added ${t}^{\ensuremath{'}}>0$ and/or ${t}^{\ensuremath{''}}<0$ generally tend to promote d-wave symmetry.) We perturbatively construct an extended quasipair for the $t\ensuremath{-}{J}_{z}$ model. In leading order, there are contributions from sites at a distance of 2 lattice spacings apart; however, contributions from sites two lattice spacings apart, also of the same order, vanish identically. Finally, we compare our approach with analytic calculations for a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette and with existing numerical work, and discuss possible relevance to the physical parameter regime.
We calculated the binding energy of two holes with antiparallel spin in the t-J model using a variational wave function based on the string or spin-bag picture. The symmetry of the bound state is dx2-y2. Although the size of matrices to be diagonalized is only 3\ifmmode\times\else\texttimes\fi{}3 we could reproduce the results of various extensive numerical calculations with reasonable accuracy. We show that a sizeable contribution of the binding energy is due to the quantum fluctuations in the ground state of the Heisenberg antiferromagnet. In contrast to exact diagonalization results on small clusters in the thermodynamical limit the binding energy decreases rapidly with increasing t/J. A detailed discussion of the various finite-size effects in a small cluster is also included.
The formation of bound states of holes in an antiferromagnetic spin-1/2 background is studied using numerical techniques applied to the t-J Hamiltonian on clusters with up to 26 sites. An analysis of the binding energy as a function of cluster size suggests that a two-hole bound state is formed for couplings larger than a ``critical'' value J/t${\mathrm{\ensuremath{\Vert}}}_{\mathit{c}}$ The symmetry of the bound state is dx2-y2. We also observed that its ``quasiparticle'' weight Z2h (defined in the text), is finite for coupling J/t\ensuremath{\ge}J/t${\mathrm{\ensuremath{\Vert}}}_{\mathit{c}}$. Thus, in the region J/t\ensuremath{\ge}J/t${\mathrm{\ensuremath{\Vert}}}_{\mathit{c}}$ the bound state of two holes behaves like a quasiparticle with charge Q=2e, spin S=0, and dx2-y2 internal symmetry. The relation with recent ideas that have suggested the possibility of d-wave pairing in theories of the high-Tc cuprates is briefly discussed.
Abstract. Within the recently proposed doped-carrier representation of the projected lattice electron operators we derive a full Ising version of the t–J model. This model possesses the global discrete Z2 symmetry as a maximal spin symmetry of the Hamiltonian at any values of the coupling constants, t and J. In contrast, in the spin anisotropic limit of the t–J model, usually referred to as the t–Jz model, the global SU(2) invariance is fully restored at Jz = 0, so that only the spin-spin interaction has in that model the true Ising form. We discuss a relationship between those two models and the standard isotropic t–J model. We show that the low-energy quasiparticles in all three models share the qualitatively similar properties at low doping and small values of J/t. The main advantage of the proposed Ising t–J model over the t–Jz one is that the former allows for the unbiased Monte Carlo calculations on large clusters of up to 10 3 sites. Within this model we discuss in detail the destruction of the antiferromagnetic (AF) order by doping as well as the interplay between the AF order and hole mobility. We also discuss the effect of the exchange interaction and that of the next nearest neighbour hoppings on the destruction of the AF order at finite doping. We show that the short-range AF order is observed in a wide range of temperatures and dopings, much beyond the boundaries of the AF phase. We explicitly demonstrate that the local no double occupancy constraint plays the dominant role in destroying the magnetic order at finite doping. Finally, a role of inhomogeneities is discussed. PACS numbers: 71.10.Fd, 71.10.Pm, 74.25.JbIsing t–J model close to half filling: A Monte Carlo study 2 1.
Numerical studies, from variational calculation to exact diagonalization, all indicate that the quasiparticle generated by introducing one hole into a two-dimensional quantum antiferromagnet has the same nature as a string state in the t-Jz model. Based on this observation, we attempt to visualize the quasiparticle formation and subsequent coherent propagation at low energy by studying the generalized t-Jz-${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ model in which we first diagonalize the t-Jz model and then perform a degenerate perturbation in ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$. We construct the quasiparticle state and derive an effective Hamiltonian describing the coherent propagation of the quasiparticle and its interaction with the spin wave excitations in the presence of the N\'eel order. We expect that qualitative properties of the quasiparticle remain intact when analytically continuing ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ from the anisotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$Jz to the isotropic ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=Jz limit, despite the fact that the spin wave excitations change from gapful to gapless. Extrapolating to ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=Jz, our quasiparticle dispersion and spectral weight compare well with the exact numerical results for small clusters. \textcopyright{} 1996 The American Physical Society.