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We study analytically the finite-temperature phase diagram of proton orderingof a quasi-two dimensional hydrogen-bonded system, namely the squaric acidcrystal($\text{H}_2\text{C}_4\text{O}_4$). We take into account the four-spininteraction model at the zeroth order. Using an improvised loop algorithmwithin the Stochastic series expansion quantum monte carlo method, we find twodistinct phases as we increase the temperature and magnetic-field. One of thephase is the $\Pi_f$, the phase with long range ferroelectric order and theother being an intermediate state with strong local correlations, i.e, aquantum liquid-like state $\Pi_{ql}$. The transition to $\Pi_{f}$ shows a verysmall anomalous peak in the specific heat with strong dependence of criticaltemperature on the strength of dipole-dipole interaction. The presence of thesmall peak is attributed to the absence of macroscopic degeneracy in thepresence of dipole-dipole interaction and re-entrance of such degeneracy tosome extent at small temperature. Though the degenerate ground state manifoldis identical for a four-spin interaction or appropriate two-spin interactionmodel at zeroth order, we find that for the former case, the required strengthof dipole-dipole interaction is quite larger to induce a ferroelectric phase.The work also presents an intricate connection of quantum fluctuation andthermal fluctuation in the presence of competing interaction with entropiceffects.

Dipole-dipole interaction induced phases in Hydrogen-bonded squaric acid crystal