Feynman cycles in the Bose gas

Daniel Ueltschi

Published Jan 1, 2006• 14 citations

Abstract

Abstract. We study the lengths of the cycles formed by trajectories in the Feynman-Kac representation of the Bose gas. We discuss the occurrence of infinite cycles and their relation to Bose-Einstein condensation.

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Feynman cycles in the Bose gas

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Feynman cycles in the Bose gas

Daniel Ueltschi

Published Jan 1, 2006• 14 citations

OpenAlex

Abstract

Related papers

Feynman cycles in the Bose gas

Daniel Ueltschi

arXiv:math-ph/0605002Journal of Mathematical PhysicsDec 1, 200624 citations

Infinite Cycles in Boson Lattice Models

G. Boland

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.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:1002.3103Irish Mathematical Society BulletinJan 1, 20090 citations

Relation between Feynman Cycles and Off-Diagonal Long-Range Order

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arXiv:cond-mat/0604005Physical Review LettersOct 26, 200622 citations

Infinite cycles of interacting bosons

András Sütö

Abstract In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity in the thermodynamic quantities which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail.Abstract In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity in the thermodynamic quantities which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail.

arXiv:2409.19695Physica ScriptaOct 11, 20240 citations

Permutation cycles in the Bose–Einstein condensation of a trapped ideal gas

William J. Mullin

Physica B Condensed MatterJul 1, 20003 citations

Length scales for BEC in the dilute Bose gas

Søren Fournais

We give a short proof of Bose Einstein Condensation of dilute Bose gases on length scales much longer than the Gross-Pitaevskii scale.

arXiv:2011.00309EMS series of congress reportsMay 18, 202118 citations

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.