  Variational methods are of fundamental importance and widely used in
theoretical physics, especially for strongly interacting systems. In this work,
we present a set of variational equations of state (VES) for pure states of an
interacting quantum Hamiltonian. The VES can be expressed in terms of the
variation of the density operators or static correlation functions. We derive
the algebraic relationship between a known pure state density matrix and its
variation, and obtain the VES by applying this relation to the averaged
Heisenberg-equations-of-motion for the exact density matrix. Additionally, we
provide a direct expression of the VES in terms of correlation functions to
make it computable. We present three nontrivial applications of the VES: a
perturbation calculation of correlation functions of the transverse field Ising
model in arbitrary spatial dimensions, a study of a longitudinal field
perturbation to the one-dimensional transverse field Ising model at the
critical point and variational calculation of magnetization and ground state
energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice.
For the second one, our results not only recover the scaling limit, but also
indicate the possibility of continuous tuning of the critical exponents by
adjusting the longitudinal fields differently from the scaling limit. For the
Heisenberg model, we obtained results numerically comparable to established
results with simple calculations. The VES approach provides a powerful and
versatile tool for studying interacting quantum systems.
