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

We study the finite-temperature phase diagram of proton ordering of a quasi-two dimensional hydrogen-bonded system, namely the squaric acid crystal (H₂C₄O₄) using quantum Monte Carlo. We take into account the four-spin plaquette interaction at the zeroth order followed by next nearest neighbor Ising interaction within a plaquette, dipole-dipole interaction and an external transverse magnetic field respectively. Using an improvised loop algorithm within the stochastic series expansion (SSE) quantum Monte Carlo method, we find two distinct phases as we increase the temperature and magnetic-field. One of the phase is the Π_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 Π_ql. The transition to Π_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. The work also discusses an intricate connection of quantum fluctuation and thermal fluctuation in the presence of competing interaction with entropic effects.