Abstract
The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be E0/N=(2πℏ2ρ/m)|ln(ρa2)|−1, to leading order, with a relative error at most O(|ln(ρa2)|−1/5). Here N is the number of particles, ρ=N/V is the particle density and a is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, E0 is not simply N(N−1)/2 times the energy of two particles in a large box of volume (area, really) V. It is much larger.
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Lieb, E.H., Yngvason, J. The Ground State Energy of a Dilute Two-Dimensional Bose Gas. Journal of Statistical Physics 103, 509–526 (2001). https://doi.org/10.1023/A:1010337215241
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DOI: https://doi.org/10.1023/A:1010337215241