Determination of electronic excitation energies within the doubly occupied configuration interaction space by means of the Hermitian operator method.

In this work, we formulate the equations of motion corresponding to the Hermitian operator method in the framework of the doubly occupied configuration interaction space. The resulting algorithms turn out to be considerably simpler than the equations provided by that method in more conventional spaces, enabling the determination of excitation energies in N-electron systems under an affordable polynomial computational cost. The implementation of this technique only requires to know the elements of low-order reduced density matrices of an N-electron reference state, which can be obtained from any approximate method. We contrast our procedure against the reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev integrable models, pointing out the reliability of our proposal.

Variational determination of ground and excited-state two-electron reduced density matrices in the doubly occupied configuration space: A dispersion operator approach.

This work implements a variational determination of the elements of two-electron reduced density matrices corresponding to the ground and excited states of N-electron interacting systems based on the dispersion operator technique. The procedure extends the previously reported proposal [Nakata et al., J. Chem. Phys. 125, 244109 (2006)] to two-particle interaction Hamiltonians and N-representability conditions for the two-, three-, and four-particle reduced density matrices in the doubly occupied configuration interaction space. The treatment has been applied to describe electronic spectra using two benchmark exactly solvable pairing models: reduced Bardeen-Cooper-Schrieffer and Richardson-Gaudin-Kitaev Hamiltonians. The dispersion operator combined with N-representability conditions up to the four-particle reduced density matrices provides excellent results.

Variational reduced density matrix method in the doubly occupied configuration interaction space using three-particle N-representability conditions.

Ground-state energies and two-particle reduced density matrices (2-RDMs) corresponding to N-particle systems are computed variationally within the doubly occupied configuration interaction (DOCI) space by constraining the 2-RDM to satisfy a complete set of three-particle N-representability conditions known as three-positivity conditions. These conditions are derived and implemented in the variational calculation of the 2-RDM with standard semidefinite programming algorithms. Ground state energies and 2-RDMs are computed for N2, CO, CN-, and NO+ molecules at both equilibrium and nonequilibrium geometries as well as for pairing models at different repulsive interaction strengths. The results from the full three-positivity conditions are compared with those from the exact DOCI method and with approximated 2-RDM variational ones obtained within two-positivity and two-positivity plus a subset of three-positivity conditions, as recently reported [D. R. Alcoba et al., J. Chem. Phys. 148, 024105 (2018) and A. Rubio-García et al., J. Chem. Theory Comput. 14, 4183 (2018)]. The accuracy of these numerical determinations and their low computational cost demonstrate the usefulness of the three-particle variational constraints within the DOCI framework.

Merging symmetry projection methods with coupled cluster theory: Lessons from the Lipkin model Hamiltonian.

Coupled cluster and symmetry projected Hartree-Fock are two central paradigms in electronic structure theory. However, they are very different. Single reference coupled cluster is highly successful for treating weakly correlated systems but fails under strong correlation unless one sacrifices good quantum numbers and works with broken-symmetry wave functions, which is unphysical for finite systems. Symmetry projection is effective for the treatment of strong correlation at the mean-field level through multireference non-orthogonal configuration interaction wavefunctions, but unlike coupled cluster, it is neither size extensive nor ideal for treating dynamic correlation. We here examine different scenarios for merging these two dissimilar theories. We carry out this exercise over the integrable Lipkin model Hamiltonian, which despite its simplicity, encompasses non-trivial physics for degenerate systems and can be solved via diagonalization for a very large number of particles. We show how symmetry projection and coupled cluster doubles individually fail in different correlation limits, whereas models that merge these two theories are highly successful over the entire phase diagram. Despite the simplicity of the Lipkin Hamiltonian, the lessons learned in this work will be useful for building an ab initio symmetry projected coupled cluster theory that we expect to be accurate in the weakly and strongly correlated limits, as well as the recoupling regime.

Excited-state quantum phase transitions in the two-spin elliptic Gaudin model.

We study the integrability of the two-spin elliptic Gaudin model for arbitrary values of the Hamiltonian parameters. The limit of a very large spin coupled to a small one is well described by a semiclassical approximation with just one degree of freedom. Its spectrum is divided into bands that do not overlap if certain conditions are fulfilled. In spite of the fact that there are no quantum phase transitions in each of the band heads, the bands show excited-state quantum phase transitions separating a region in which the parity symmetry is broken from another region in which time-reversal symmetry is broken. We derive analytical expressions for the critical energies in the semiclassical approximation, and confirm the results by means of exact diagonalizations for large systems.

Many-body characterization of particle-conserving topological superfluids.

What distinguishes trivial superfluids from topological superfluids in interacting many-body systems where the number of particles is conserved? Building on a class of integrable pairing Hamiltonians, we present a number-conserving, interacting variation of the Kitaev model, the Richardson-Gaudin-Kitaev chain, that remains exactly solvable for periodic and antiperiodic boundary conditions. Our model allows us to identify fermion parity switches that distinctively characterize topological superconductivity (fermion superfluidity) in generic interacting many-body systems. Although the Majorana zero modes in this model have only a power-law confinement, we may still define many-body Majorana operators by tuning the flux to a fermion parity switch. We derive a closed-form expression for an interacting topological invariant and show that the transition away from the topological phase is of third order.

Composite boson mapping for lattice boson systems.

We present a canonical mapping transforming physical boson operators into quadratic products of cluster composite bosons that preserves matrix elements of operators when a physical constraint is enforced. We map the 2D lattice Bose-Hubbard Hamiltonian into 2×2 composite bosons and solve it within a generalized Hartree-Bogoliubov approximation. The resulting Mott insulator-superfluid phase diagram reproduces well quantum Monte Carlo results. The Higgs boson behavior in the superfluid phase along the unit density line is unraveled and in remarkable agreement with experiments. Results for the properties of the ground and excited states are competitive with other state-of-the-art approaches, but at a fraction of their computational cost. The composite boson mapping here introduced can be readily applied to frustrated many-body systems where most methodologies face significant hurdles.

Commensurability effects for fermionic atoms trapped in 1D optical lattices.

Fermionic atoms in two different hyperfine states confined in optical lattices show strong commensurability effects due to the interplay between the atomic density wave ordering and the lattice potential. We show that spatially separated regions of commensurable and incommensurable phases can coexist. The commensurability between the harmonic trap and the lattice sites can be used to control the amplitude of the atomic density waves in the central region of the trap.