Two-hole problem in the $t$-$J$ model: A canonical transformation approach

The $t$-$J$ model in the spinless-fermion representation is studied. An effective Hamiltonian for the quasiparticles is derived using a canonical transformation approach. It is shown that the rather simple form of the transformation generator allows one to take into account the effect of hole interactions with the short-range spin waves and to describe the single-hole ground state. Obtained results are very close to ones of the self-consistent Born approximation. Further accounting of the long-range spin-wave interaction is possible on a perturbative basis. Spin-wave exchange and an effective interaction due to minimization of the number of broken antiferromagnetic bonds are included in the effective quasiparticle Hamiltonian. The two-hole bound state problem is solved using a Bethe-Salpeter equation. The only bound state found to exist in the region of $1<\mathrm{}(t/J)<5\mathrm{}$ is the $d$ wave. Both types of the hole-hole interaction are important for its formation. A discussion of the possible relation of the obtained results to the problem of superconductivity in real systems is presented.