All thermodynamic properties can be computed with these imaginary time ... of the Enrico Fermi Summer School on Non-Linear Molecular. Dynamics (1985).
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Numerical Simulations in Superfluid Helium
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1990 Phys. Scr. 1990 11 (http://iopscience.iop.org/1402-4896/1990/T33/002) View the table of contents for this issue, or go to the journal homepage for more
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Physica Scripta. Vol. T33, 11, 1990.
Numerical Simulations in Superfluid Helium David Ceperley National Center for Supercomputing Applications and Department of Physics, University of Illinois, Urbana, Illinois, 61801, USA
Rewived December 8 , 1989: accepted December 15, 1989
In 1953, Richard Feynman [l] showed how superfluidity can be understood from the point of view of imaginary-time path integrals. In this picture, each atom becomes a cyclic polymer and the superfluid transition occurs when the atomic paths join up to make larger units which stretch a macroscopic distance. Boson statistics imply that all possible ways of connecting or “permuting” the “polymers” have equal weight. Feynman’s path integrals are an exact isomorphism between quantum statistical mechanics of bosons and the classical statistical mechanics of these special “polymers”. We [2-41 have implemented Feynman’s path integral method on a computer. Efficient methods, based on the Metropolis algorithm have been developed to move the paths quickly through path configuration space and permutation space. In addition, accurate approximations to the many-body offdiagonal density matrix involving pairs and triplets of atoms have been calculated. All thermodynamic properties can be computed with these imaginary time path integrals. Among the properties of bulk liquid and solid helium that have been computed are the internal energy, specific heat, the superfluid density [7], the momentum distribution [5, 91, and the static structure factor. All of the calculated properties are in close agreement with the measured quantities. The existence of a superfluid, i.e., when the interior of a liquid becomes decoupled from the boundaries, is revealed by paths of “polymers” which wrap around the periodic boundaries of the simulation cell. Recently, we have completed studies of superfluid 4He droplets [lo] and films 181. Small droplets of 64-128 atoms show
almost complete superfluid behavior. The phase transition in superfluid He films seems to follow the Kosterlitz-Thouless behaviour. The method has also been extended to calculate tunneling frequencies in crystal 3He.The tunneling frequency is shown to be proportional [6] to the free energy needed to cross-link several “polymers” in a polymeric crystal. It is only possible to treat this fermion system because in the crystal phase exchange is a rare event. We find the computed frequencies explain the frustrated nuclear spin ordering of 3He at millikelvin temperatures and are in approximate agreement with the multiple-exchange model.
References 1. Feynman, R . P., Statistical Mechanics, Benjamin, Reading, Mass.
(1972). 2. Pollock, E. L. and Ceperley, D. M., Phys. Rev. B30, 2555 (1984). 3. Ceperley. D. M. and Pollock, E. L., Phys. Rev. Lett. 56, 351 (1986). 4. Ceperley, D. M., “The Simulation of Liquid Helium”, in Proceedings of the Enrico Fermi Summer School on Non-Linear Molecular Dynamics (1985). 5. Ceperley, D. M . and Pollock, E. L., Can. J. Physics, 65, 1416 (1987). 6. Ceperley, D. M. and Jacucci, G . , Phys. Rev. Lett. 58, 1648 (1987). 7. Pollock, E. L. and Ceperley, D. M., Phys. Rev. B36, 8343 (1987). 8. Ceperley. D. M. and Pollock, E. L., Phys. Rev. B39. 2084 (1989). 9. Ceperley, D. M., “The Momentum Distribution of 4He at Non-zero Temperature”, to appear in the proceedings of the Argonne Conference on Momentum Distributions (1989). 10. Sindzingre, P., Klein, M . L. and Ceperley. D. M., Phys. Rev. Lett. 63. 15 (1989).
