Exploring Novel Quantum Embedding Methods with a Non-orthogonal Decomposition of Slater Determinants

The Schmidt decomposition of quantum many-body states, which is the basis of many powerful quantum many-body techniques, relies on a partition of the whole system in terms of orthogonal local basis functions. By carefully investigating the spectrum of the truncated density matrices in a non-orthogonal basis, we propose in this work a non-orthogonal decomposition of Slater determinants, which reduces to the conventional Schmidt decomposition in the orthogonal limit. The new decomposition provides a natural tool for building local correlation spaces when the orbitals are overlapping, and we are able to extend some existing embedding methods based on the Schmidt decomposition like density matrix embedding theory (DMET) to overlapping subsystem partitions to achieve greater flexibility and efficiency. We also propose a new quantum embedding strategy as another utility of the non-orthogonal decomposition that bridges the \textit{ab initio} model potential (AIMP) theory and DMET, which does not require a mean-field calculation of the whole system like AIMP, and is able to capture quantum entanglement like DMET.