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

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=(2²/m)|ln(a²)|^–1, to leading order, with a relative error at most O(|ln(a²)|^–1/5). Here N is the number of particles, =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.