Phase transitions for an open system consisting of an ensemble of interacting quantum subsystems with discrete spectrum are studied in the mean-field approximation. In the considered model, the change of an internal symmetry of a thermodynamic system upon the second-order phase transition is due to changing symmetry of distribution of charge/spin density inside each quantum subsystem. The latter can be caused by either splitting of one of lowest degenerated energy level or closing a gap between the levels and appearance of avoided crossing. The effect of external parameters (pressure, field, composition, etc.) results in direct change of internal control parameters: level spacing and/or the strength of interaction between adjacent quantum subsystems. Considering a simplest case of the two-level quantum subsystems, expressions for the free energy as a function of the internal control parameters were obtained in analytical form. The behavior of the heat capacity and susceptibility for different regions of the low-temperature phase diagram including the area of quantum fluctuations was determined.
Phase transitions for an open system consisting of an ensemble of interacting quantum subsystems with discrete spectrum are studied in the mean-field approximation. In the considered model, the change of an internal symmetry of a thermodynamic system upon the second-order phase transition is due to changing symmetry of distribution of charge/spin density inside each quantum subsystem. The latter can be caused by either splitting of one of lowest degenerated energy level or closing a gap between the levels and appearance of avoided crossing. The effect of external parameters (pressure, field, composition, etc.) results in direct change of internal control parameters: level spacing and/or the strength of interaction between adjacent quantum subsystems. Considering a simplest case of the two-level quantum subsystems, expressions for the free energy as a function of the internal control parameters were obtained in analytical form. The behavior of the heat capacity and susceptibility for different regions of the low-temperature phase diagram including the area of quantum fluctuations was determined.
A general spin system with a nonaxially symmetric Hamiltonian containing Jx, Jz-linear and Jz-quadratic terms, widely used in many-body fermionic and bosonic systems and in molecular magnetism, is considered for the variations of general parameters describing intensity interaction changes of each of its terms. For this model Hamiltonian, a semiclassical energy surface (ES) is obtained by means of the coherent-state formalism. An analysis of this ES function, based on catastrophe theory, determines the separatrix in the control parameter space of the system Hamiltonian: the loci of singularities representing semiclassical phase transitions. Here we show that distinct regions of qualitatively different spectrum structures, as well as a singular behavior of quantum states, are ruled by this separatrix: here we show that the separatrix not only describes ground-state singularities, which have been associated with quantum phase transitions, but also reveals the structure of the excited spectrum, distinguishing different quantum phases within the parameter space. Finally, we consider magnetic susceptibility and heat capacity of the system at finite temperature, in order to study thermal properties and thermodynamical phase transitions in the perspective of the separatrix of this Hamiltonian system.
We study the thermodynamic phase diagram of a one-dimensional quantum spin chain subjected to both mean-field and nearest-neighbor interactions, and to a transverse magnetic field h. The purpose is to determine the effect of the quantum fluctuations, due to the transverse field, on the phase diagram, in particular with respect to the occurrence of ensemble inequivalence. We denote our model as a quantum Nagle-Kardar model. To perform the calculation of the canonical partition function, we show that, due to the presence of the mean-field term, in the thermodynamic limit one can use the Hubbard-Stratonovich transformation in spite of the non-commutativity of the different operators appearing in the Hamiltonian, and we adopt a procedure of successive approximations that lead to the determination of the phase diagram thanks to a scaling property of the phase transition lines. The results show that the ensemble inequivalence, present in the classical Nagle-Kardar model, is removed above a threshold value hc for the transverse field. For h larger than hc the phase diagram exhibits only second-order phase transition lines, implying therefore restoration of ensemble equivalence.
The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.
It is investigated how the order of phase transition in a system of many identical interacting subsystems depends on the degeneracies of the states of each subsystem and on the interaction between subsystems; in particular, on the symmetry and uniformity of the degeneracy and on the symmetry of the interaction. By denoting the state of the i-th subsystem by a spin-like variable σi (i = 1,2,…,N), the degeneracy is expressed as D(σi) as a function of σi and the interaction between the i-th and j-th subsystems as V(σi,σj). (In the text use is made of such a normalized value as ΣσiD(σi) = 1). Then the symmetry means the equalities D(σi) = D(-σi) and V(σi,σj) = V(-σi,-σj) and the uniformity of degeneracy means the independency of D(σi) upon σi. The system of magnetic spins with exchange interactions belongs to the uniform symmetric category, which is used as a reference standard system in the present paper. The thermostatistical average of σi which is denoted by σ represents the order parameter in the system. Dependence of σ upon temperature and some applied field are investigated to see the features of the phase transition. In the non-uniform symmetric case it is found that the phase transition of first order can occur for an adequate form of D(σi), and that it is necessarily of second order in the uniform symmetric case. The induced transition by the applied field is also possible in the former case. In the asymmetric case the phase transition of a peculiar type as well as the induced transition may occur. Under some respective conditions on the degeneracy and interaction parameters, transitions of first and second orders can occur. Analytic properties of the partition functions are also discussed. An exact problem is proposed for the simplest model which is expected to exhibit the first-order phase transition.
Based on the fact that most of low temperature experiments of quantum systems are explored in an external field on condition that the particle numbers, volumes and energies of systems may be changed, the thermodynamic properties of weakly interacting Fermi systems in weak magnetic field are studied by using the statistical distribution of the completely open system with variable particle number, volume, and energy (N-E-V distribution). Firstly, the analytical expressions of internal energy and heat capacity, which are in the Fermi integral form, are obtained in the general case, and the analytical expressions and numerical results of energy and heat capacity are given under the extreme condition of supper-low temperature. The calculation results by the N-E-V distribution (with particle number density being variable) are compared with those by a pseudopotential method (with particle number density being unchanged). It can be found that the deviations of the internal energy and heat capacity calculated by the two different methods are very small, and the N-E-V distribution method can partially compensate for the error caused by the pseudo potential method. The most interesting point of the results obtained by the N-E-V distribution method is that there is a phase transition temperature in the weakly interacting Fermi system in weak magnetic field under the low temperature condition. The phase transition temperature is just in the range where occur the Fermi systems, Bose-Einstein condensation (BEC), Bardeen-Cooper-Schrieffer (BCS) phase transition, and BEC-BCS crossover, and does not vary with strength nor characteristic (attraction or repulsion) of the weak interaction, but it decreases with the strengthening of the external magnetic. When there is no external magnetic, the phase transition temperature is highest (more than 0.184 times Fermi temperature).
Several basic problems of the theory of quantum phase transitions are reviewed. The effect of the quantum correlations on the phase transition properties is considered with the help of basic models of statistical physics. The effect of quenched disorder on the quantum phase transitions is also discussed. The review is performed within the framework of the thermodynamic scaling theory and by the most general methods of statistical physics for the treatment of phase transitions: general lengthscale arguments, exact solutions, mean field approximation, Hubbard-Stratonovich transformation, Feynman path integral approach, and renormalization group in the field theoretical variant. Some new ideas and results are presented. Outstanding theoretical problems are mentioned. 1.