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arXiv:2305.18959

Bose-Einstein condensation of interacting bosons: A two-step proof

András Sütö

Published May 30, 2023• 1 citation
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Abstract

We prove two equilibrium properties of a system of interacting atoms in three or higher dimensional continuous space. (i) If the particles interact via pair potentials of a nonnegative Fourier transform, their self-organization into infinite permutation cycles is simultaneous with off-diagonal long-range order. If the cycle lengths tend to infinity not slower than the square of the linear extension of the system, there is also Bose-Einstein condensation. (ii) If the pair potential is also nonnegative, cycles composed of a nonzero fraction of the total number of particles do appear if the density exceeds a temperature-dependent threshold value. The two together constitute the proof that in such a system Bose-Einstein condensation takes place at high enough densities.

DOI
10.48550/arxiv.2305.18959

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We prove two equilibrium properties of a system of interacting atoms in three or higher dimensional continuous space. (i) If the particles interact via pair potentials of a nonnegative Fourier transform, their self-organization into infinite permutation cycles is simultaneous with off-diagonal long-range order. If the cycle lengths tend to infinity not slower than the square of the linear extension of the system, there is also Bose-Einstein condensation. (ii) If the pair potential is also nonnegative, cycles composed of a nonzero fraction of the total number of particles do appear if the density exceeds a temperature-dependent threshold value. The two together constitute the proof that in such a system Bose-Einstein condensation takes place at high enough densities.
Based on the paper "Fourier formula for quantum partition functions", arXiv:2106.10032 [math-ph], we show that in an infinite system of identical bosons interacting via a positive-type pair potential there is off-diagonal long-range order if and only if a nonzero fraction of the particles form infinite permutation cycles. In particular, there is Bose-Einstein condensation if and only if the diverging cycle lengths increase at least as fast with NNN, the number of particles, as N2/dN^{2/d}N2/d in d≥3d\geq 3d≥3 dimensions. This extends a similar result known for the ideal Bose gas.
In this third paper of a series that started with arXiv:2106.10032 [math-ph] and continued with arXiv:2108.02659 [math-ph] we show that in d≥3d\geq 3d≥3 dimensions at low temperatures or high densities bosons interacting via pair potentials that are both positive and positive type form permutation cycles whose length diverges proportionally with the number of particles. Based on the second-cited paper, this implies Bose-Einstein condensation.
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.
The proof of Jasnow and Fisher that no spontaneous ordering can occur in one- and two-dimensional systems with continuous symmetry is shown to fail for nonlocal potentials of sufficiently long range. The occurrence of Bose-Einstein condensation for finite temperature (in the infinite-volume limit) is, therefore, not forbidden for such potentials. This result also follows from the postulate of the completeness of the noninteracting equilibrium states. Thus, the completeness postulate represents a weaker assumption on the nature of an interacting equilibrium state (or, equivalently, on the basic interactions between particles) than is usually found in the literature.
Using the rigorous path integral formalism of Feynman and Kac we prove London's eighty years old conjecture that during the superfluid transition in liquid helium Bose-Einstein condensation (BEC) takes place. The result is obtained by proving first that at low enough temperatures macroscopic permutation cycles appear in the system, and then showing that this implies BEC. We find also that in the limit of zero temperature the infinite cycles cover the whole system, while BEC remains partial. For the Bose-condensed fluid at rest we define a macroscopic wave function. Via the equivalence of 1/2 spins and hard-core bosons the method extends to lattice models. We show that at low enough temperatures the spin-1/2 axially anisotropic Heisenberg models, including the isotropic ferro- and antiferromagnet and the XY model, undergo magnetic ordering.
The existence of a Bose-Einstein condensation in an interacting many-boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair-Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose-field operatorφ(x), whereφ(x) has nonvanishing ground state expectation value, in accordance with the usual c-number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low-density limit. Bose-Einstein condensation exists when the over-all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operatorφ. Yet the c-number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.