PHYSICAL REVIEW 8 VOLUME 38, NUMBER 7 1 SEPTEMBER 1988 Multiple-spin-exchange calculation of the T =0 properties of solid He H. Godfrin A T& T Bell Laboratories, Murray Hill, New Jersey 07974 and Institute Laue-Langeuin, 156X 38042 Grenoble Cedex, France D. D. Osheroff A T& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received 24 August 1987; revised manuscript received 17 February 19&8) The multiple-exchange Hamiltonian used to describe spin-spin interactions in solid He is developed in terms of Pauli spin operators for an arbitrary number of particles in the exchange cy- cle. The T =0 mean-field properties of the "up-up-down-down" (U2D2) and canted normal antifer- romagnetic magnetic phases are then calculated including two-, three-, four-, five-, and six-particle exchange processes. These properties agree well with experimentally determined melting-curve values when evaluated using the exchange frequencies calculated by Ceperley and Jacucci. INTRODUCTION Nuclear magnetic order in solid He is believed to be a consequence of the Fermi statistics and exchange of spin- —, ' He atoms. ' The multiple-spin-exchange model, ' based on the work by Dirac and Thouless, has been suc- cessful in providing a good description of the experimen- tal properties of this system. It is based on the observa- tion that the steric hinderance of the hard cores of the interatomic potentials can favor the coherent exchange of more than two particles. The richness of the observed magnetic phase diagram is understood as the result of competing ferromagnetic and antiferromagnetic interac- tions, the former being due to exchange of an odd num- ber of particles, and the latter to that of an even number of particles. Considering only a small number of exchange process- es (three- and four-particle exchange), Roger, Delrieu, and Hetherington' were able to give a convincing theoretical model for solid He. Later work by Roger demonstrated that two-particle exchange was probably as important as three- and planar four-particle exchange. This motivated the development of three-parameter mod- els which allowed an estimation of the exchange frequen- cies by fitting the experimental database. Very recently Ceperley and Jacucci determined the frequencies of several exchange processes (up to six spins) using path-integral Monte Carlo techniques. This work confirmed that the most important types of exchange were indeed those considered previously. ' However, the frequencies of a large number of other exchange pro- cesses were found not to be negligible. It was therefore desirable to calculate the properties of the ordered phases taking into account all these exchange processes. They are calculated at zero temperature in this paper, for the U2D2 and the canted normal antiferromagnetic (CNAF) phases. The ordered phases are described in Sec. I. The multiple-spin-exchange Hamiltonian is given in terms of Pauli operators in Sec. II. The magnetic properties of the U2D2 and CNAF phases are calculated in Sec. III. Sec- tion IV is devoted to a comparison with experimental re- sults. I. THE ORDERED PHASES OF SOLID He CNAE 0 0 2 2 I /' LI i I T(rnK j PARA FIG. 1. Experimental phase diagram of melting solid 'He, as a function of temperature and magnetic field, according to Refs. 8, 11, 12, 14, and 15. The phase diagram of solid He on the melting curve, as a function of temperature and magnetic field, is shown in Fig. 1. Three phases have been observed: a paramag- netic phase, the low-field ordered phase, and the high- field ordered phase. These have been described in detail in recent review articles. ' The structure of the low-field phase is believed to be of the U2D2 type, illustrated in Fig. 2. This has been 38 4492 1988 The American Physical Society