Fermionic Orbital Optimization in Tensor Network States

Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such nonlocal fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this Letter, we propose a way to overcome this challenge. We suggest a method intertwining the optimization over matrix product states with suitable fermionic Gaussian mode transformations. The described algorithm generalizes basis changes in the spirit of the Hartree-Fock method to matrix-product states, and provides a black box tool for basis optimization in tensor network methods.

Figure 1

Illustration of the general ansatz class of an MPS with varying physical basis, where $g(U)$ is a Gaussian transformation defined by a mode transformation $U\in U(np)$ as described in the main text.

Figure 2

Numerical results for the Be ring of 6 Be atoms with interatomic distance of 3.3 Å. All calculations have been performed with a $U(1)\times U(1)$ symmetric open boundary MPS and local mode transformations which keep the $SU(2)$ symmetry of the Hamiltonian. Both diagrams show results of the described optimization for the physical basis. In the left panel we show the bond dimension needed for a bounded truncation error ${\epsilon }_{\text{trc}}\le {10}^{-6}$ and $D_{\min }=64$ when starting in the HF basis. The dark blue dashed line corresponds to a calculation in the HF basis, the blue dotted and light blue line correspond to the first and the 10th iteration of the calculation with basis optimization. The right panel compares the relative error in energy $(⟨\psi |H|\psi ⟩-E_0)/E_0$ obtained by calculations with $D_{\max }=256$, where the reference value for the ground state energy $E_0$ has been obtained from a calculation with $D_{\max }=2048$ in the localized basis. The dark blue dashed and red dashed-dotted lines show the results for a calculation in the HF and localized basis, respectively. In light blue and orange we plot the relative error of the 15th and 10th iteration of the calculation with basis optimization starting in the HF and localized basis, respectively.