Abstract We explore the suitability of a feed-forward neural network (FNN) to represent the ground state of the fermionic Hubbard model in one dimension (1D). We consider the model at half-filling, represent the ground state wave function in terms of an FNN and optimize it using the variational Monte Carlo (VMC) method. The results are compared with the exact Bethe Ansatz solution. We find that for lattice sizes which give a ‘filled-shell’ condition for the non-interacting Fermi sea wave function, a simple FNN performs very well at all values of Hubbard interaction U. For lattice sizes where this condition is not obtained, the simple FNN fails and we find a modified network with a ‘sign’ component (sFNN) to work in such cases. On the flip side, though we find the FNN to be successful in providing an unbiased variational wave function for the fermionic many-body system in 1D, the computational cost for the wave function scales up rapidly with lattice size which limits its applicability.
We investigate the dynamics of two bosons trapped in an infinite one-dimensional optical lattice potential within the framework of the Bose-Hubbard model and derive an exact expression for the wave function at finite time. As initial condition we chose localized atoms that are separated by a distance of
d lattice sites and carry a center-of-mass quasimomentum. An initially localized pair (
d=0) is found to be more stable as quantified by the pair probability (probability to find two atoms at the same lattice site) when the interaction and/or the center-of-mass quasimomentum is increased. For initially separated atoms ($d\ensuremath{\ne}0$) there exists an optimal interaction strength for pair formation. Simple expressions for the wave function, the pair probability, and the optimal interaction strength for pair formation are computed in the limit of infinite time. Whereas the time-dependent wave function differs for values of the interaction strength that differ only by the sign, important observables such as the density and the pair probability do not. With a symmetry analysis this behavior is shown to extend to the
N-particle level and to fermionic systems. Our results provide a complementary understanding of the recently observed [Winkler et al., Nature (London) 441, 853 (2006)] dynamical stability of atom pairs in a repulsively interacting lattice gas.