Holes in the $t-J_z$ model: A diagrammatic study

The $t-J_z$ model is the strongly anisotropic limit of the $t-J$ 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 $t/J.\mathrm{}$ 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 $t-J$-like model. Possible implications for preformed pairs formation and phase separation are drawn.