Representation of single-reference expansions of ground states of small two-dimensional cluster treated with the Hubbard model using superpositions of tree-tensor networks

In a previous work (Phys. Rev. B 101, 045109 (2020)), we have made a proposal to extend the coupled-cluster (CC) method to the strong correlations regime using tensor networks to represent the wave function, resulting in a CC method in which it is entanglement that is truncated instead or the cluster operator T. To do so, we have developed a new type of tensor representation for single-reference expansions of the wave function which consists in superpositions of tree-tensor-networks (STTN). One important property of the STTN is that some tensors are shared amongst different terms in the expansion, reducing the total number of parameters for a given accuracy. In addition, the terms in the superpositions correspond to different possible types of correlations between excited particles and holes, bearing some similarity with the different channels in diagrammatic expansions of correlations functions. In this work we use the STTN to represent the coefficients and cluster operator amplitudes of exact single-reference ground states of small two-dimensional clusters treated with the Hubbard model, using natural orbitals. First, we find that the STTN are suitable to represent the wave function coefficients or T-amplitudes at strong coupling. Second, using superpositions instead of a single tree for a given term in the expansion substantially improve the accuracy for a given number of parameters. Finally, we find that, at strong coupling, a direct representation of the wave function coefficients using STTN is more accurate than using STTN for the cluster operator amplitudes.