Title
"Ground-state Energies of Spinless Free Fermions and Hard-core Bosons"
Nie Wenxing (ISSP, The University of Tokyo)
Abstract
Statistics of identical particles is one of the most fundamental concepts in quantum physics. A consequence of the particle statistics appears in the groundstate energy.
We compare the groundstate energies of bosons and fermions with the same form of the Hamiltonian. If both are noninteracting, the groundstate energy of bosons is always lower (`natural' inequality E_0^B<= E_0^F), owing to Bose-Einstein Condensation (BEC). However, the comparison is nontrivial when bosons do interact and thus BEC is no longer perfect. We examine how general is the `natural' inequality and when it can be violated.
We first present a sufficient condition for the `natural' inequality E_0^B <= E_0^F, and moreover its strict version E_0^B <E_0^F to hold, using Perron-Frobenius theorem. We find when the hopping is unfrustrated (all the hopping amplitudes are non-negative), hard-core bosons must have a lower groundstate energy than fermions. Our proof reveals how the quantum statistics affects the groundstate energy in general interacting systems. When the hopping is frustrated, bosons can have higher groundstate energy than fermions.
In two-dimensional system with flux, we find the ground-state energy of hard-core bosons is higher for a range of flux and particle density by exact diagonalization of small clusters. Furthermore, we also prove rigorously that this inversion indeed occurs in several examples in the thermodynamic limit, by various techniques [1].
[1] W.X. Nie, H. Katsura and M. Oshikawa, arXiv:1302.0013.
