The Embedded Density Matrix Renormalization Group: Size-Extensive and Quasi-Exact for Nonlinear Quantum Chemistry

Tensor networks (TNs) and the breadth of algorithms acting
on them
have seen astounding success in simulating quantum many-body systems
in the strongly interacting regime with both accuracy and efficiency.
In the context of quantum chemistry, Steven White’s density
matrix renormalization group (DMRG) continues to take center stage
as the TN method of choice, seeing countless theoretical and computational
breakthroughs in recent decades yet remaining fettered by a few persistent
shortcomings, notably, a lack of size-extensivity and quasi-exactness. Here, we present a simple
yet versatile framework for circumventing these issues: the bootstrap
embedded density matrix renormalization group (BE-DMRG), and numerically
validate its size-extensive and quasi-exact ground-state properties
for a test bed of strongly correlated molecular systems (linear H-chains; quasi-linear E-polyacetylene; 2D and 3D H-lattices;
2D arene flakes). Spanning a breadth of system sizes
(10 to 200 orbitals) and entanglement topologies (linear to highly
nonlinear), we demonstrate the robustness of the BE-DMRG for problems
far beyond the reach of conventional DMRG implementations. Furthermore,
by detailing BE-DMRG convergence behavior with respect to exact diagonalization,
we find the rate of convergence with bond dimension to be significantly
faster than, yet just as reliable as, that of conventional DMRG. Ultimately,
we find that the embedded DMRG might serve as a natural extension
of White’s original formulation to higher dimensions without
the need for higher-order tensor networks. The coupling of tensor
network theories to the framework of quantum embedding, more broadly,
may become an incomparably powerful tool for the study of strongly
correlated molecules and materials.