The variational two-electron reduced density matrix (v2RDM) method is generalized for the description of total angular momentum ( J ) and projection of total angular momentum ( M J ) states in atomic systems described by nonrelativistic Hamiltonians, and it is shown that the approach exhibits serious deficiencies. Under ensemble N -representability constraints, v2RDM theory fails to retain the appropriate degeneracies among various J states for fixed spin ( S ) and orbital angular momentum ( L ), and for fixed L, S, and J, the manifold of M J states is not necessarily degenerate. Moreover, a substantial energy error is observed for a system for which the two-electron reduced density matrix is exactly ensemble N -representable; in this case, the error stems from violations in pure-state N -representability conditions. Unfortunately, such violations do not appear to be good indicators of the reliability of energies from v2RDM theory in general. Several states are identified for which energy errors are near zero and yet pure-state conditions are clearly violated.
The direct variational optimization of the ground-state two-electron reduced-density matrix (2-RDM) is typically performed under ensemble N-representability conditions. Accordingly, variationally obtained 2-RDMs for degenerate ground states may not represent a pure state. When considering only ground-state energetics, the ensemble nature of the 2-RDM is of little consequence. However, the use of ensemble densities within an extended random phase approximation (ERPA) yields astonishingly poor estimates of excitation energies, even for simple atomic systems [H. van Aggelen et al., Comput. Theor. Chem. 1003, 50-54 (2013)]. Here, we outline an approach for the direct variational optimization of ground-state 2-RDMs that satisfy pure-state N-representability known as generalized Pauli constraints. Within the ERPA, 2-RDMs that satisfy both ensemble conditions and the generalized Pauli constraints yield much more reliable estimates of excitation energies than those that satisfy only ensemble conditions.
Despite its fundamental importance in the theory of strongly correlated electrons, the nature of the ground state of the two-dimensional doped Hubbard model remains intensely debated. Variational approaches provide a powerful route to this problem, but their conclusions can depend sensitively on the chosen wave-function parameterization, the mean-field initialization, or the pinning fields used to guide the optimization, as well as on boundary conditions. This can favor one type of symmetry breaking over another, making it difficult to distinguish the genuine interplay of intertwined or competing orders from biases induced by the variational parameterization. Here, we introduce the Symmetry-Preserving Backflow Pairing (SBP) ansatz, a neural-network wave function that respects translational symmetry by construction and thereby avoids these broken-symmetry minima. The SBP ansatz reaches state-of-the-art variational energies for the
t-
t′ Hubbard model on lattices up to
24×24 with
504 electrons, below those of competing pure stripe solutions. By extrapolating to the thermodynamic limit, we find robust evidence for
d-wave superconducting order, resolving a long-standing question about the
1/8-doped model at
t′/t=−0.2 and
U/t=8.0. Built on general principles of symmetry and locality, the SBP wave function provides a broadly applicable variational representation for challenging interacting fermionic systems.
This paper is concerned with the pure-state
N-representability problem for systems under a magnetic field. Necessary and sufficient conditions are given for a spin-density
2×2 matrix
R to be representable by a Slater determinant. We also provide sufficient conditions on the paramagnetic current
j for the pair
(R,j) to be Slater-representable in the case where the number of electrons
N is greater than 12. The case
N<12 is left open.