Inhomogeneous solutions to thermal Hartree–Fock equations in first-order phase transitions

It is shown that the damping algorithm for the density matrix in self-consistent-field iterations, originally proposed for quantum chemistry of molecules in the ground state, can yield convergent solutions to thermal Hartree–Fock equations representing inhomogeneous domain structures in first-order phase transitions. A lattice gas of spinless Fermions with nearest-neighbour attractive interaction is analysed as an illustration. In the metastable or unstable region of the phase diagram, numerical solutions converge to phase-separated states consisting of gas and liquid domains, characterised by a double-peak structure in the single-particle energy spectrum. Broadening of the energy spectrum due to quantum-mechanical tunnelling effect is also demonstrated. The present method would facilitate direct numerical access to phase-transition regions of various quantum/classical systems without causing breakdown