We study a system of doped antiferromagnet in three dimensions at finite temperatures using the $t$-$J$ model, a canonical model of strongly correlated electrons. We employ the slave-fermion representation of electrons, in which an electron is described as a composite of a charged spinless holon and a chargeless spinon. We introduce two kinds of U(1) gauge fields on links as auxiliary fields, one describing resonating valence bonds of antiferromagnetic nearest-neighbor spin pairs and the other for nearest-neighbor hopping amplitudes of holons and spinons in the ferromagnetic channel. To perform a numerical study of the system, we integrate out the fermionic holon field by using the hopping expansion in powers of the hopping amplitude, which is legitimate for the region in and near the insulating phase. The resultant effective model is described in terms of bosonic spinons, two U(1) gauge fields, and a collective field for hole pairs. We study this model by means of Monte Carlo simulations, calculating the specific heat, spin correlation functions, and instanton densities. We obtain a phase diagram in the hole concentration-temperature plane, which is in good agreement with that observed recently for clean and homogeneous underdoped samples.

• Received 23 July 2010

DOI:https://doi.org/10.1103/PhysRevB.83.064502