2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/225723In this note, we discuss the quantum version of the melting crystal corner in one, two, and three dimensions, generalizing the treatment for the quantum dimer model. Using a mapping to spin chains we find that the two--dimensional case (growth of random partitions) is integrable and leads directly to the Hamiltonian of the Heisenberg XXZ ferromagnet. The three--dimensional case of the melting crystal corner is described in terms of a system of coupled XXZ spin chains. We give a conjecture for its mass gap and analyze the system numerically.34 pages, 26 picturesStatistical MechanicsHigh Energy Physics - TheoryMathematical Physicstext