Infinite Cycles in Boson Lattice Models

Abstract

We study the relationship between long cycles and Bose-Einstein condensation (BEC) in the case of several models. A convenient expression for the density of particles on cycles of length $q$ is obtained, in terms of $q$ unsymmetrised particles coupled with a boson field. Using this formulation we reproduce known results on the Ideal Bose Gas, Mean-Field and Perturbed Mean-Field Models, where the condensate density exactly equals the long cycle density. Then we consider the Infinite-Range-Hopping Bose-Hubbard model, with and without hard-cores. For the hard-core case, we find we can disregard the hopping contribution of the q unsymmetrised particles to obtain an exact expression for the density of particles on long cycles. It is shown that only the cycle of length one contributes to the cycle density. We conclude that while the existence of a non-zero long cycle density coincides with the occurrence of Bose-Einstein condensation, the respective densities are not necessarily equal. For the case of a finite on-site repulsion, we obtain an expression for the cycle density by neglecting the $q$ unsymmetrised hop. Inspired by the Approximating Hamiltonian method we conjecture a simplified expression for the short cycle density as a ratio of single-site partition functions. In the absence of condensation we prove that this simplification is exact and find that in this case the long-cycle density vanishes. In the presence of condensation we justify this simplification when a gauge-symmetry breaking term is introduced in the Hamiltonian. Assuming our conjecture is correct, we compare numerically the long-cycle density with the condensate and again find that though they coexist, in general they are not equal.

arXiv commentPh.D. thesis

References in corpus (11)

• Proof of Bose-Einstein Condensation for Dilute Trapped Gases
• Dynamics and Thermodynamics of the Bose-Hubbard model
• Justification of c-Number Substitutions in Bosonic Hamiltonians
• Percolation transition in the Bose gas II
• Percolation, Bose-Einstein Condensation, and String Proliferation
• Long Cycles in a Perturbed Mean Field Model of a Boson Gas
• The relation between Feynman cycles and off-diagonal long-range order
• What do emulsification failure and Bose-Einstein condensation have in common?
• The Approximating Hamiltonian Method for the Imperfect Boson Gas
• Condensation in a Disordered Infinite-Range Hopping Bose-Hubbard Model
• C^*-algebraic approach to the Bose-Hubbard model