Single hole in a quantum antiferromagnet: Finite-size-scaling approach

Exact diagonalizations of small clusters up to 26 sites are used to extrapolate the ground-state energy of a single hole in the Néel state of the t-${\mathit{J}}_{\mathit{z}}$ model or in the quantum antiferromagnetic (t-J model). A surprisingly rapid convergence with system size is observed for a wide range of parameters and for the largest clusters considered. In the Ising limit the energy dependence with ${\mathit{J}}_{\mathit{z}}$ is remarkably close to a ${\mathit{J}}_{\mathit{z}}^{2/3}$ law for intermediate coupling in agreement with Brinkman-Rice type of calculations. Nevertheless a small but significant bandwidth indicates that the hole can propagate coherently in this limit. Clearly the polaronic behavior ∼${\mathit{J}}_{\mathit{z}}^{1/2}$, for very small ${\mathit{J}}_{\mathit{z}}$, is out of the range of the method. When quantum fluctuations are introduced (isotropic case) the ground-state energy behavior still remains close to that of the Ising limit although an overall increase of the energy by a factor 0.3t is observed. This suggests that the ‘‘string scenario’’ might have some relevance there.