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Modern Physics Letters B

ORDER INDICES AND MID-RANGE ORDER

Alan Coleman, Vyacheslav I. Yukalov

Published Oct 1, 1991• 24 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984991002021

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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.
Abstract Finite Bose systems cannot display a genuine Bose–Einstein condensate with infinite long-range order. But, if the number of trapped atoms is sufficiently large, a kind of Bose–Einstein condensation does occur, with the properties of the arising quasi-condensate being very close to the genuine condensate. Although the quasi-condensate does not enjoy long-range order, it has mid-range order. This paper shows that the level of mid-range order in finite Bose systems can be characterized by order indices of density matrices.
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.