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International Journal of Modern Physics B

RELATION BETWEEN MICROSCOPIC AND MACROSCOPIC CHARACTERISTICS OF STATISTICAL SYSTEMS

A.J. COLEMAN, Vyacheslav I. Yukalov

Published Nov 15, 1996• 13 citations
Publisher

Abstract

Order parameters of physical systems and order indices of their density matrices are, respectively, macroscopic and microscopic characterisics of a statistical system. We claim that there are situations in which the index provides more information than the parameter. This is proved by the exhibition of an exactly solvable model of a Bose system with mid-range order for which the order parameter is zero but for which an order index changes continuously as the Hamiltonian is varied.

DOI
10.1142/s0217979296001872

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Order parameters of physical systems and order indices of their density matrices are, respectively, macroscopic and microscopic characterisics of a statistical system. We claim that there are situations in which the index provides more information than the parameter. This is proved by the exhibition of an exactly solvable model of a Bose system with mid-range order for which the order parameter is zero but for which an order index changes continuously as the Hamiltonian is varied.
We introduce a novel notion, order indices of reduced density matrices, characterizing different types of order, long-range as well as mid-range, occurring in statistical systems. The order indices define the behaviour of the largest eigenvalues of density matrices in the thermodynamic limit. These indices are especially useful for describing mid-range order, where they are directly connected with the large-distance asymptotic forms of off-diagonal density matrices. Several examples illustrate the idea.
A new notion is introduced of matrix order indices which relate the matrix norm and its trace. These indices can be defined for any given matrix. They are especially important for matrices describing many-body systems, equilibrium as well as nonequilibrium, for which the indices present a quantitative measure of the level of ordering. They characterize not only the long-range order, but also midrange order. In the latter case, when order parameters do not exist, the matrix indices are well defined, providing an explicit classification of various mid-range orders. The matrix order indices are suitable for describing phase transitions with both off-diagonal and diagonal order. Contrary to order parameters whose correct definition requires the thermodynamic limit, the matrix indices do not necessarily need the latter. Because of this, such indices can distinguish between different phases of finite systems, thus, allowing for the classification of crossover phase transitions. PACS: 02.10.Yn; 05.70.Fh; 64.60.Cn
The review is devoted to two important quantities characterizing many-body systems, order indices and the measure of entanglement production. Order indices describe the type of order distinguishing statistical systems. Contrary to the order parameters characterizing systems in the thermodynamic limit and describing long-range order, the order indices are applicable to finite systems and classify all types of orders, including long-range, mid-range, and short-range orders. The measure of entanglement production quantifies the amount of entanglement produced in a many-partite system by a quantum operation. Despite that the notions of order indices and entanglement production seem to be quite different, there is an intimate relation between them, which is emphasized in the review.
A measure of correlations for statistical systems is advanced, called correlation index. The measure is general, being applicable to quantum and classical, equilibrium and nonequilibrium statistical systems. After formulating the general notion of correlation indices for arbitrary trace-class operators, the calculation of the correlation indices are illustrated by several examples of density operators and reduced density matrices. The computation of the correlation indices for equilibrium systems is exemplified for bosonic systems with the Bose-Einstein condensate, superconducting systems, and for spin systems. Correlation indices for nonequilibrium systems are considered for nonlinear dynamics of spin and pseudospin systems.
The survey discusses the problem of measuring correlations in composite quantum and statistical systems. After briefly recalling the standard methods of describing correlations, the emphasis is on the method of correlation indices. The latter method for quantifying the strength of correlations in composite physical systems is sufficiently general allowing for measuring in a unique way the correlation strength in both quantum and statistical systems, for static as well as for dynamic processes. The correlation indices, can be defined for correlation operators, reduced density matrices, and other operators containing information on correlations in the studied composite systems. The correlation indices measure all types of correlations, quantum correlations, such as entanglement, as well as classical correlations. They are applicable for characterizing static as well as dynamical processes. Examples are given of correlation indices for several quantum states and for spin and pseudospin systems. The quantification of correlations in a nonequilibrium system is exemplified by calculating the correlation index for a trapped Bose-Einstein condensate subject to the action of an alternating field.