A quantum quasicrystal was proposed to exist by mean-field, variational arguments in two-dimensional Rashba spin-orbit coupled BECs with dipolar interactions. Despite this remarkable prediction, there is little known about the superfluid character or stability of this quasicrystalline state against thermal and quantum fluctuations, whose importance is implied by the massive single-particle degeneracy of the Rashba Hamiltonian. Here, we apply field-theoretic numerical simulations based on a boson coherent state path integral representation to simulate the finite-temperature behavior of a periodic approximant of an octagonal quasicrystal. We find that the quasicrystal state maintains a 50% superfluid fraction down to low temperature, suggesting supersolid character. We compute a phase diagram that depicts a region of thermodynamic stability for the octagonal quasicrystal phase at low temperature. At intermediate temperature, the quasicrystal undergoes a first-order transition into a crystalline BEC with hexagonal symmetry. Our findings support a possible experimental realization of the octagonal quasicrystal in ${}^{164}\mathrm{Dy}$ quantum gases below a temperature of 75 nK.

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• Finite temperature field theory
• Phase transitions
• Spin-orbit coupling
• Quasicrystals
• Spinor Bose-Einstein condensates
• Supersolids
• Langevin algorithm
• Path-integral methods