Abstract. We consider the strictly correlated electron (SCE) limit of the fermionic quantum many-body problem in the second-quantized formalism. This limit gives rise to a multi-marginal optimal transport (MMOT) problem. Here the marginal state space for our MMOT problem is the binary set {0, 1}, and the number of marginals is the number L of sites in the model. The costs of storing and computing the exact solution of the MMOT problem both scale exponentially with respect to L. We propose an efficient convex relaxation which can be solved by semidefinite programming (SDP). In particular, the semidefinite constraint is only of size 2L × 2L. Moreover, the SDP-based method yields an approximation of the dual potential needed to the perform selfconsistent field iteration in the so-called Kohn-Sham SCE framework, which, once converged, yields a lower bound for the total energy of the system. We demonstrate the effectiveness of our methods on spinless and spinful Hubbard-type models. Numerical results indicate that our relaxation methods yield tight lower bounds for the optimal cost, in the sense that the error due to the semidefinite relaxation is much smaller than the intrinsic modeling error of the Kohn-Sham SCE method. We also describe how our relaxation methods generalize to arbitrary MMOT problems with pairwise cost functions. 1. Introduction. For ground-state electronic structure calculations, the success of the widely-used Kohn-Sham density functional theory (DFT) [13, 15] hinges on the accuracy of the approximate exchange-correlation functionals. Although tremendous progress has been made in the construction of approximate functionals [33, 2, 17, 32] , these approximations are mostly derived by fitting known results for the uniform electron gas, single atoms, small molecules, and perfect crystal systems. Such functionals often perform well when the underlying quantum systems are 'weakly correlated,' i.e., when the single-particle energy is significantly more important than the electron-electron interaction energy. In order to extend the capability of DFT to the treatment of strongly correlated quantum systems, one recent direction of functional development considers the limit in which the electron-electron interaction energy is infinitely large compared to other components of the total energy. The resulting limit is known as strictly correlated electron (SCE) [38, 37, 5, 22, 10] limit. The SCE limit provides an alternative route to derive exchange-correlation energy functionals. The study of Kohn-Sham DFT with SCE-based functionals is still in its infancy, but such approaches have already been used to treat strongly correlated model systems and simple chemical systems (see e.g. [29, 7, 12]). A system of N interacting electrons in a d-dimensional space can be described using either the first-quantized or the second-quantized representation. In the firstquantized representation, the number of electrons N is fixed, and the electronic wavefunction is an anti-symmetric function in N L 2 (R d ; C 2 ), which is a subset of the tensor product space N L 2 (R d ; C 2 ). Here C 2 corresponds to the spin degree of free-
The variational embedding method can also be understood as a way to tighten the variational lower bound obtained in REF for fermionic many-body problems based on the strictly correlated electron (SCE) formulation.
Semidefinite relaxation of multi-marginal optimal transport for strictly correlated electrons in second quantization
arXiv: Optimization and Control
Mathematics Physics