  The quantum Heisenberg model is studied in the geometrically frustrated
body-centered tetragonal lattice(BCT lattice) with antiferromagnetic interlayer
coupling J1 and intralayer first and second neighbor coupling J2 and J3. We
introduce a variational method: each interaction term can be decoupled
partially in the purely magnetic Weiss and in the spin-liquid (SL) mean-field
channels. We find that the most stable variational solutions correspond to the
three different possible long range magnetic orders that are respectively
governed by J1, J2, and J3. We characterize three different purely SL
non-magnetic solutions that are variationally the second most stable states
after the purely magnetic ones. This suggests that quantum fluctuations induced
by the frustration between J1-J2-J3 coupling should destroy magnetic orders and
stabilize the formation of SL in large areas of parameters. The SL solution
governed by J1 breaks the lattice translation symmetry. This Modulated SL is
associated to a commensurate ordering wave vector (1,1,1). We discuss the
relevance of our results for heavy fermion Uru2Si2 and cuprate superconductors
that have a BCT lattice structure. Also, the general variational method
introduced here can be applied to any other system where interaction terms can
be decoupled in two different mean-field channels.
