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Reviews of Modern Physics

Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors

Chen Ning Yang

Published Oct 1, 1962• 1,438 citations
Publisher

Abstract

This review is concerned with the concept that in a many-body system of bosons or fermions, it is possible to have an off-diagonai long-range order of the reduced density matrices in the coordinate space representation. The onset of such an order ieads to a new thermodynamic phase of the system. It is reasonabie to assume that superfluid He II and the superconductors are phases characterized by the existence of such an order. (W.D.M.)

DOI
10.1103/revmodphys.34.694

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This review is concerned with the concept that in a many-body system of bosons or fermions, it is possible to have an off-diagonai long-range order of the reduced density matrices in the coordinate space representation. The onset of such an order ieads to a new thermodynamic phase of the system. It is reasonabie to assume that superfluid He II and the superconductors are phases characterized by the existence of such an order. (W.D.M.)
On the basis of gauge invariance, it is proven in an elementary and straightforward manner, but without invoking any ad hoc assumption, that the existence of off-diagonal long-range order in one-particle reduced density matrix in Bose liquids implies both the irrotational flow in a simply connected region and the quantization of circulation in a multiply connected region, the two fundamental properties of a Bose superfluid. The origin for both is the phase coherence of condensate wave functions. Some relevant issues are also addressed.
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.
For a strongly interacting boson system, it has been proposed that the concept of off-diagonal long-range ordering (ODLRO) is equivalent to Bose-Einstein condensation for a free boson system. Therefore, the existence of ODLRO in the ground state implies superfluidity. The author checks the validity of this proposition from another point of view. He rigorously shows that, for a hard-core lattice-boson system with a short-ranged interaction ODLRO is suppressed when a charged excitation gap develops and, hence, the boson system becomes a Mott-insulator. Finally, by using his theorem, he shows some interesting properties of the antiferromagnetic Heisenberg model, which can be taken as a hard-core lattice-boson system.
Relations between the existence of ODLRO in the reduced density matrixρ n (for alln≧1) and that of superconducting currents in a hollow cylinder are discussed. Some concrete behaviours ofρ n , including the power law decaying behaviour, in the coordinate space representation are considered and the corresponding effective superconducting particle densities are given. Some restraint conditions caused by the existence of the ODLRO (including quasi-ODLRO) have been found and discussions on the minimum unit of flux quantization are also included.
This paper deals with ordinary material systems whose elementary constitutents are fermions. It is pointed out that in such systems there can occur two kinds of bosons with quite different physical and mathematical characteristics. Type I bosons are bound complexes of an even number of fermions (such as 4He^{4}\mathrm{He}4He); and type II bosons are elementary excitations which are bound complexes of fermions and their holes (such as excitons). When the first type condenses, a superfluid state results with so-called off-diagonal, long-range order; while when the second type condenses, there is no superfluidity, but a change in spatial order. Thus both kinds of long-range order are related to Bose condensation.
The thermodynamic properties of the hard-sphere Bose gas are calculated from the one- and two-particle density matrices. The one- and two-particle density matrices are evaluated in the Bogoliubov approximation. The off-diagonal elements of the one- and two-particle density matrices are studied for the presence or absence of off-diagonal long-range order in various temperature regions. Off-diagonal long-range order was found in the single-particle density matrix below the Bose-Einstein transition temperature. The two-particle density matrix also exhibited off-diagonal long-range order whenever the single-particle density matrix did.