  We study analytically the finite-temperature phase diagram of proton ordering
of a quasi-two dimensional hydrogen-bonded system, namely the squaric acid
crystal($\text{H}_2\text{C}_4\text{O}_4$). We take into account the four-spin
interaction model at the zeroth order. Using an improvised loop algorithm
within the Stochastic series expansion quantum monte carlo method, we find two
distinct phases as we increase the temperature and magnetic-field. One of the
phase is the $\Pi_f$, the phase with long range ferroelectric order and the
other being an intermediate state with strong local correlations, i.e, a
quantum liquid-like state $\Pi_{ql}$. The transition to $\Pi_{f}$ shows a very
small anomalous peak in the specific heat with strong dependence of critical
temperature on the strength of dipole-dipole interaction. The presence of the
small peak is attributed to the absence of macroscopic degeneracy in the
presence of dipole-dipole interaction and re-entrance of such degeneracy to
some extent at small temperature. Though the degenerate ground state manifold
is identical for a four-spin interaction or appropriate two-spin interaction
model at zeroth order, we find that for the former case, the required strength
of dipole-dipole interaction is quite larger to induce a ferroelectric phase.
The work also presents an intricate connection of quantum fluctuation and
thermal fluctuation in the presence of competing interaction with entropic
effects.
