Energy of fermionic ground states with low-entanglement single-reference expansions, and tensor-based strong-coupling extensions of the coupled-cluster method

Abstract

We consider a fermionic system for which there exist a single-reference Configuration-Interaction (CI) expansion of the ground state wave function that converges, albeit not necessarily rapidly, with the increasing number of particle-hole excitations. For such systems, we show that, whenever the coefficients of Slater determinants (SD) with l ≤ k excitations can be defined with a number of free parameters N ≤k bounded polynomially in k, the ground state energy E only depends on a small fraction of all the wave function parameters, and is the solution of equations of the Coupled-Cluster (CC) form. This generalizes the standard CC method, for which N ≤k is bounded by a constant. Based on that result and low-rank tensor decompositions (LRTD), we discuss two possible extensions of the CC approach for wave functions with general polynomial bound for N ≤k . The most straightforward of those extensions uses the LRTD to represent the amplitudes of the CC cluster operator T which, unlike in the CC case, is not truncated with respect to number of excitations, and the tensor parameters are given by a LRTD-adapted version of standard CC equations. The LRTD can also be used to directly parametrize the wave function coefficients, which involves different equations of the CC form. We derive those equations for the coefficients of SD's with l ≤ 4 excitations, using the CC exponential wave function ansätz with a different type of excitation operator, and a representation of the Hamiltonian in terms of excited particle and hole operators. To complete the proposed computational methods, we construct compact tensor representations of coefficients, or T -amplitudes, using superpositions of tree tensor networks which take into account different possible types of entanglement between excited particles and holes. Finally, we discuss why the proposed CC extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.