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

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 respect to excitation number. We show that, if the coefficients of Slater determinants (SD) with $l\leq k$ excitations can be defined with a number of free parameters $N_{\leq k}$ 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 $N_{\leq k}$ 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 $N_{\leq k}$. First, one can use LRTD to represent the amplitudes of the CC cluster operator $T$ which, unlike in the CC case, is not truncated with respect to excitation number, and the energy and tensor parameters are given by a LRTD-adapted version of standard CC equations. Second, the LRTD can also be used to directly parametrize the CI coefficients, which involves different equations of the CC form. We derive those equations for up to quadruple-excitation coefficients, using a different type of excitation operator in the CC wave function ansatz, and a Hamiltonian representation in terms of excited particle and hole operators. We complete the proposed CC extensions by constructing 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 extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.