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 $E_0/N = (2π\hbar^2ρ/m){|\ln (ρa^2)|^{-1}}$, to leading order, with a relative error at most ${\rm O} (|\ln (ρa^2)|^{-1/5})$. 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, $E_0$ 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.