Approximating strongly correlated spin and fermion wavefunctions with correlator product states

We explore correlator product states for the approximation of correlated wavefunctions in arbitrary dimensions. We show that they encompass many interesting states including Laughlin's quantum Hall wavefunction, Huse and Elser's frustrated spin states, and Kitaev's toric code. We further establish their relation to common families of variational wavefunctions, such as matrix and tensor product states and resonating valence bond states. Calculations on the Heisenberg and spinless Hubbard models show that correlator product states capture both two-dimensional correlations (independent of system width) as well as non-trivial fermionic correlations (without sign problems). In one-dimensional simulations, correlator product states appear competitive with matrix product states with a comparable number of variational parameters, suggesting they may eventually provide a route to practically generalise the density matrix renormalisation group to higher dimensions.