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Novel statistical ensembles for phase coexistence states specified by noncommutative additive observables.

Yasushi Yoneta, Akira Shimizu

Published Nov 20, 2021• 0 citations
OpenAlex

Abstract

Coexisting phases in a phase coexistence state cannot be distinguished by thermodynamic forces, such as temperature and chemical potential, because the forces take the same values over all coexisting phases. Therefore, to investigate a phase coexistence state, it is necessary to employ an ensemble in which all additive observables that distinguish the coexisting phases have macroscopically definite values. Although the microcanonical ensemble is conventionally employed as such an ensemble, it becomes ill-defined when some of the additive observables do not commute with each other, and a new ensemble has been craved. We propose a novel class of ensembles such that the additive observables, which are generally noncommutative, always have macroscopically definite values even in a first-order phase transition region. Using these ensembles, we propose a concrete method to construct phase coexistence states of general quantum systems. Furthermore, these ensembles are convenient for practical calculations because of good analytic properties. To demonstrate that our formulation successfully gives phase coexistence states of quantum systems, we apply it to a two-dimensional system whose coexisting phases are distinguished by an additive observable (order parameter) that does not commute with the Hamiltonian. To the author's best knowledge, this is the first work that obtains phase coexistence states separated by phase interfaces at finite temperature in such a quantum system.

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Coexisting phases in a phase coexistence state cannot be distinguished by thermodynamic forces, such as temperature and chemical potential, because the forces take the same values over all coexisting phases. Therefore, to investigate a phase coexistence state, it is necessary to employ an ensemble in which all additive observables that distinguish the coexisting phases have macroscopically definite values. Although the microcanonical ensemble is conventionally employed as such an ensemble, it becomes ill-defined when some of the additive observables do not commute with each other, and a new ensemble has been craved. We propose a novel class of ensembles such that the additive observables, which are generally noncommutative, always have macroscopically definite values even in a first-order phase transition region. Using these ensembles, we propose a concrete method to construct phase coexistence states of general quantum systems. Furthermore, these ensembles are convenient for practical calculations because of good analytic properties. To demonstrate that our formulation successfully gives phase coexistence states of quantum systems, we apply it to a two-dimensional system whose coexisting phases are distinguished by an additive observable (order parameter) that does not commute with the Hamiltonian. To the author's best knowledge, this is the first work that obtains phase coexistence states separated by phase interfaces at finite temperature in such a quantum system.
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 that 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 their 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.
Abstract We study the dynamics of quantum statistical ensembles at first-order phase transition points of finite macroscopic systems. First, we show that at the first-order phase transition point of the system, with an order parameter that does not commute with the Hamiltonian, any quantum state with an order parameter that has a nonzero value always evolves towards a macroscopically distinct state after a sufficiently long time. From this result, we argue that the stationarity required for statistical ensembles should be interpreted as stationarity on a sufficiently long but finite timescale. Finally, we prove that the density matrix of the squeezed ensemble, a class of generalized statistical ensembles proposed as the only concrete method of constructing phase coexistence states applicable to general quantum systems, is locally stationary on timescales diverging in the thermodynamic limit. Our results support the validity of the squeezed ensemble from a dynamical point of view and open the door to nonequilibrium statistical physics at the first-order phase transition point.
All ensembles of statistical mechanics are equivalent in the sense that they give the equivalent thermodynamic functions in the thermodynamic limit. However, when investigating microscopic structures in the first-order phase transition region, one must choose an appropriate statistical ensemble. The appropriate choice is particularly important when one investigates finite systems, for which even the equivalence of ensembles does not hold. We propose a class of statistical ensembles, which always give the correct equilibrium state even in the first-order phase transition region. We derive various formulas for this class of ensembles, including the one by which temperature is obtained directly from energy without knowing entropy. Moreover, these ensembles are convenient for practical calculations because of good analytic properties. We also derive formulas which relate statistical-mechanical quantities of different ensembles, including the conventional ones, for finite systems. The formulas are useful for obtaining results with smaller finite-size effects, and for improving the computational efficiency. The advantages of the squeezed ensembles are confirmed by applying them to the Heisenberg model and the frustrated Ising model.
A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterized by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures.
A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterised by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order (i.e. continuous) phase transitions at finite temperatures.
Stationary states of quantum many-body Hamiltonians are invariant under the Hamiltonian evolution. Besides ground and thermal states, this class includes microcanonical ensembles that are of fundamental importance in statistical physics. We consider the preparation of general stationary states by leveraging recent advances in the field of open-system dynamics. In particular, constructions based on exact KMS-detailed balance with respect to Gibbs states of noncommuting Hamiltonians have only recently been proposed as a tool for their efficient preparation and, by extension to small temperatures, for ground state preparation. We extend these constructions to the problem of stationary state preparation, providing general criteria that characterize when such states have efficient implementations, along with specific results on the approximation of microcanonical ensembles. An interesting application of our work are tests of conjectured ensemble equivalences for local observables between microcanonical and Gibbs ensembles.