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Jordan-Wigner Transformation for the Description of Strong Correlation in Fermionic Systems

Thomas M. Henderson, Guo P. Chen, Gustavo E. Scuseria

TL;DR

This work introduces an extended Jordan-Wigner (EJW) framework to recast the zero-seniority (DOCI) sector of fermionic Hamiltonians into a fermionic form amenable to standard electronic-structure methods. By applying an EJW transformation to the locally seniority-conserving Hamiltonian and optimizing the EJW strings jointly with Hartree-Fock, the EJW-HF approach achieves DOCI-quality energies with polynomial scaling, while producing more accurate density matrices than pCCD. Benchmarks on the 1D Hubbard model (and doped/attractive variants) and small molecules (H$_8$ and N$_2$) show EJW-HF often surpasses pCCD in both energies and density-matrix fidelity, though higher-seniority effects remain important for dynamic correlation. The method provides a Hermitian, variational alternative for strongly correlated regimes and suggests avenues for refinement, such as incorporating Hartree-Fock–Bogoliubov treatments or number-projection to further improve robustness and applicability.

Abstract

Seniority is a useful way of organizing Hilbert space for strongly correlated systems. The exact zero-seniority wave function, doubly-occupied configuration interaction (DOCI), provides accurate results (given the right orbitals) for many strongly-correlated electronic systems, but has combinatorial computational cost. In many cases, pair coupled cluster doubles provides a polynomial-cost approximation that closely reproduces the energies of DOCI, but it breaks down in some cases and, as shown herein, it does not provide particularly good density matrices. In this work, we demonstrate that by using the Jordan-Wigner approximation to turn the seniority zero problem back into a fermionic one, we can provide variational results of DOCI quality for the Hubbard model and a few small molecular dissociation examples, with polynomial cost, both for the energies and for density matrices, all while being protected from collapse.

Jordan-Wigner Transformation for the Description of Strong Correlation in Fermionic Systems

TL;DR

This work introduces an extended Jordan-Wigner (EJW) framework to recast the zero-seniority (DOCI) sector of fermionic Hamiltonians into a fermionic form amenable to standard electronic-structure methods. By applying an EJW transformation to the locally seniority-conserving Hamiltonian and optimizing the EJW strings jointly with Hartree-Fock, the EJW-HF approach achieves DOCI-quality energies with polynomial scaling, while producing more accurate density matrices than pCCD. Benchmarks on the 1D Hubbard model (and doped/attractive variants) and small molecules (H and N) show EJW-HF often surpasses pCCD in both energies and density-matrix fidelity, though higher-seniority effects remain important for dynamic correlation. The method provides a Hermitian, variational alternative for strongly correlated regimes and suggests avenues for refinement, such as incorporating Hartree-Fock–Bogoliubov treatments or number-projection to further improve robustness and applicability.

Abstract

Seniority is a useful way of organizing Hilbert space for strongly correlated systems. The exact zero-seniority wave function, doubly-occupied configuration interaction (DOCI), provides accurate results (given the right orbitals) for many strongly-correlated electronic systems, but has combinatorial computational cost. In many cases, pair coupled cluster doubles provides a polynomial-cost approximation that closely reproduces the energies of DOCI, but it breaks down in some cases and, as shown herein, it does not provide particularly good density matrices. In this work, we demonstrate that by using the Jordan-Wigner approximation to turn the seniority zero problem back into a fermionic one, we can provide variational results of DOCI quality for the Hubbard model and a few small molecular dissociation examples, with polynomial cost, both for the energies and for density matrices, all while being protected from collapse.
Paper Structure (8 sections, 40 equations, 7 figures)

This paper contains 8 sections, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The (Local) Seniority-Conserving Hamiltonian
  3. Jordan-Wigner Transformation
  4. The JW Transformation of the JW-HF Wave Function
  5. Results
  6. The Hubbard Model
  7. Molecular Examples
  8. Discussion

Figures (7)

  • Figure 1: Errors per electron with respect to DOCI in the 1D half-filled repulsive Hubbard model. Left panel: OBC. Right panel: PBC. Note that we have had numerical difficulties with 14 sites with OBC so have truncated the plot at $U/t = 6$.
  • Figure 2: Errors per electron with respect to DOCI in the 1D attractive Hubbard model, in PBC. The discontinuities reflect the fact that there are two different DOCI curves that cross around $U/t \sim -4.5$ (for 6 sites) and $U/t \sim -3.5$ (for 10 sites).
  • Figure 3: Errors per electron with respect to DOCI in the 1D repulsive Hubbard model with 10 sites and 6 electrons, in PBC. The discontinuities reflect the fact that there are two different DOCI curves that cross around $U/t \sim 8$.
  • Figure 4: Errors per electron with respect to DOCI in the $2 \times 4$ Hubbard model with 6 electrons, in PBC.
  • Figure 5: Fractional errors in density matrix elements for the 10-site half-filled Hubbard model in PBC, at $U/t = 3.5$. Left panel: errors in $Z^{(010)}$ (occupation numbers of DOCI orbitals). Middle panel: Errors in $Z^{(020)}.$ Right panel: Errors in $Z^{(101)}$.
  • ...and 2 more figures