  We present a path integral formalism for expressing matrix elements of the
density matrix of a quantum many-body system between any two coherent states in
terms of standard Matsubara action with periodic(anti-periodic) boundary
conditions on bosonic(fermionic) fields. We show that this enables us to
express several entanglement measures for bosonic/fermionic many-body systems
described by a Gaussian action in terms of the Matsubara Green function. We
apply this formalism to compute various entanglement measures for the
two-dimensional Bose-Hubbard model in the strong-coupling regime, both in the
presence and absence of Abelian and non-Abelian synthetic gauge fields, within
a strong coupling mean-field theory. In addition, our method provides an
alternative formalism for addressing time evolution of quantum-many body
systems, with Gaussian actions, driven out of equilibrium without the use of
Keldysh technique. We demonstrate this by deriving analytical expressions of
the return probability and the counting statistics of several operators for a
class of integrable models represented by free Dirac fermions subjected to a
periodic drive in terms of the elements of their Floquet Hamiltonians. We
provide a detailed comparison of our method with the earlier, related,
techniques used for similar computations, discuss the significance of our
results, and chart out other systems where our formalism can be used.

Strongly Correlated Electrons