Comparative study of the density matrix embedding theory for Hubbard models

We examine the performance of the density matrix embedding theory (DMET) recently proposed in Knizia and Chan [Phys. Rev. Lett. 109, 186404 (2012)]. The core of this method is to find a proper one-body potential that generates a good trial wave function for projecting a large-scale original Hamiltonian to a local subsystem with a small number of bases. The resultant ground state of the projected Hamiltonian can locally approximate the true ground state. However, the lack of the variational principle makes it difficult to judge the quality of the choice of the potential. Here we focus on the entanglement spectrum (ES) as a judging criterion; accurate evaluation of the ES guarantees that the corresponding reduced density matrix well reproduces all physical quantities on the local subsystem. We apply the DMET to the Hubbard model on the one-dimensional chain, zigzag chain, and triangular lattice, and test several variants of potentials and cost functions. It turns out that ES serves as a more sensitive quantity than the energy and double occupancy to probe the quality of the DMET outcomes. A symmetric potential reproduces the ES of the phase that continues from a noninteracting limit. The Mott transition as well as symmetry-breaking transitions can be detected by the singularities in the ES. However, the details of the ES in the strongly interacting parameter region depends much on these variants, meaning that the present DMET algorithm allowing for numerous variant is insufficient to fully characterize the particular phases that require characterization by the ES.