  In the present paper, we study a system of doped antiferromagnet in three
dimensions at finite temperatures by 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. In order to perform 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 and the 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.
