Statistical ensembles for phase coexistence states specified by noncommutative additive observables

Abstract: A phase coexistence state cannot be specified uniquely by any intensive parameters, such as the temperature and the magnetic field, because they take the same values over all coexisting phases. It can be specified uniquely only by an appropriate set of additive observables. Hence, to analyze phase coexistence states the statistical ensembles that are specified by additive observables have been employed, such as the microcanonical and restricted ensembles. However, such ensembles are ill-defined or ill-behaved when some of the additive observables do not commute with each other. Here, we solve this fundamental problem by extending a generalized ensemble in such a way that it is applicable to phase coexistence states which are specified by noncommutative additive observables. We prove that this ensemble correctly gives the density matrix corresponding to phase coexistence states of general quantum systems as well as the thermodynamic functions. Furthermore, these ensembles are convenient for practical calculations because of good analytic properties and useful formulas by which temperature and other intensive parameters are directly obtained from the expectation values of the additive observables. As a demonstration, we apply our formulation to a two-dimensional system whose phase coexistence states are specified by an additive observable (order parameter) that does not commute with the Hamiltonian.