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The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 4^2 lattice in the region of J/t0. 2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.
Liu et al. (Sat,) studied this question.