The Ground State Energy of a Dilute Two-dimensional Bose Gas

Abstract

The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be $E_0/N=(2\pi {\hslash }^2\rho /m)|\ln (\rho a^2){|}^{-1}$, to leading order, with a relative error at most $\mathrm{O}(|\ln (\rho a^2){|}^{-1/5})$. Here $N$ is the number of particles, $\rho =N/V$ 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, $E_0$ 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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