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arXiv:2209.04563 · The Journal of Chemical Physics

Strong–weak duality via Jordan–Wigner transformation: Using fermionic methods for strongly correlated su(2) spin systems

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

Published Nov 1, 2022• 16 citations
arXivHTMLPDF

Abstract

The Jordan–Wigner transformation establishes a duality between su(2) and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree–Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Our proof of principle results for XXZ and J1-J2 Heisenberg (in 1D and 2D) indicates that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams and provides a better starting point for addressing challenging spin problems.

DOI
10.1063/5.0125124

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The Jordan–Wigner transformation establishes a duality between su(2) and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree–Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Our proof of principle results for XXZ and J1-J2 Heisenberg (in 1D and 2D) indicates that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams and provides a better starting point for addressing challenging spin problems.
High Resolution Image Download MS PowerPoint Slide Recent work has highlighted that the strong correlation inherent in spin Hamiltonians can be effectively reduced by mapping spins to Fermions via the Jordan–Wigner transformation (JW). The Hartree–Fock method is straightforward in the Fermionic domain and may provide a reasonable approximation to the ground state. Correlation with respect to the Fermionic mean field can be recovered based on Lie-algebraic similarity transformation (LAST) with two-body correlators. Specifically, a unitary LAST variant eliminates the dependence on site ordering, while a nonunitary LAST yields size-extensive correlation energies. Whereas the first recent demonstration of such methods was restricted to small spin systems, we present efficient implementations using analytical gradients for the optimization with respect to the mean-field reference and the LAST parameters, thereby enabling the treatment of larger clusters, including systems with local spins s > 1 2 .
The Jordan-Wigner transformation permits one to convert spin 1/2 operators into spinless fermion ones, or vice versa. In some cases, it transforms an interacting spin Hamiltonian into a noninteracting fermionic one, which is exactly solved at the mean-field level. Even when the resulting fermionic Hamiltonian is interacting, its mean-field solution can provide surprisingly accurate energies and correlation functions. Jordan-Wigner is, however, only one possible means of interconverting spin and fermionic degrees of freedom. Here, we apply several such techniques to the XXZ and J1-J2 Heisenberg models, as well as to the pairing or reduced Bardeen-Cooper-Schrieffer Hamiltonian, with the aim of discovering which of these mappings is most useful in applying fermionic mean-field theory to the study of spin Hamiltonians.
The Jordan–Wigner transformation is a powerful tool for converting systems of spins into systems of fermions, or vice versa. While this mapping is exact, the transformation itself depends on the labelling of the spins. One consequence of this dependence is that approximate solutions of a Jordan–Wigner-transformed Hamiltonian may depend on the (physically inconsequential) labelling of the spins. In this work, we turn to an extended Jordan–Wigner transformation which remedies this problem and which may also introduce some correlation atop the Hartree–Fock solution of a transformed spin Hamiltonian. We demonstrate that this extended Jordan–Wigner transformation can be thought of as arising from a unitary version of the Lie algebraic similarity transformation (LAST) theory. We show how these ideas, particularly in combination with the standard (non-unitary) version of LAST, can provide a potentially powerful tool for the treatment of the XXZ and J1−J2 Heisenberg Hamiltonians.
Quantum chemistry aims to understand and predict molecular structure and properties by solving the time-independent Schr\"odinger equation within fixed nuclei (Born-Oppenheimer) approximation. The Hamiltonian may have continuous symmetries (e.g., spin) or discrete symmetries (for example, point group). Mean-field wave functions may break some or all of these symmetries. By taking the symmetry-adapted pieces of these broken symmetry states, one can restore the physical character of the wave function while also treating relevant strong correlations. On the other hand, mean-field methods lack dynamic correlation arising from short-range interactions. Still, single-reference methods such as configuration interaction or coupled cluster theory can handle these correlations efficiently. Ideally, we would like to combine these approaches: performing configuration interaction or coupled cluster theory on a broken symmetry reference while restoring the desired symmetries. This work introduces a new approach to integrating symmetry projection and coupled cluster theory. By borrowing techniques from previous research, the new method can restore continuous symmetries like UUU(1) and SUSUSU(2) and discrete symmetries like point group and time reversal. This method achieves accurate results in challenging systems and has the same computational scaling as traditional coupled cluster theory. Jordan-Wigner transformation is a potent tool in spin models, but its application is challenging due to the inherent strings. The same techniques we use to combine symmetry projection and coupled cluster theory can be used to combine Jordan-Wigner transformation and coupled cluster. The strings are the ' symmetry projection ' operator when constructing the Hartree-Fock and coupled cluster wave functions for a Jordan-Wigner transformed Hamiltonian. Leveraging the method of symmetry projection, we obtain excellent results for both 1-D and 2-D spin models.
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 a combinatorial computational cost. In many cases, pair coupled cluster doubles provide 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 transformation to turn the seniority zero problem back into a Fermionic one, we can provide mean-field 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. This success is rooted in the proof we provide, showing that the Hartree-Fock wave function on the Jordan-Wigner-transformed Hamiltonian transforms back to variational coupled cluster doubles in the seniority zero representation, but restricted to have determinant rather than permanent amplitude coefficients, without compromising its overall accuracy.
The antisymmetrized geminal power (AGP) wavefunction has proven to be an excellent reference for strong interactions in paired electron systems. In this thesis, we extend the application of AGP methods to spin systems by utilizing the spin to pairing mapping, owing to the su(2) algebra present in both paired electron and spin systems. We have investigated the spin AGP method applied to the XXZ model and J1 − J2 model, where we have identified what we refer to as the multimodal extreme AGP as the mean-field optimized solution. We have also developed correlated methods such as configuration interaction (CI) and coupled cluster (CC) atop spin AGP for these systems. In addition, we have shown that space group symmetry plays a significant role in 2D spin lattices, and have developed space group symmetry projection methods for spin AGP. While the spin AGP method associates a spin with a pair of fermions, Jordan- Wigner Hartree–Fock (JWHF) maps a spin into a single fermion and uses a string to bridge between the spin su(2) and the fermion anticommutation algebras. We have tested JWHF on various spin systems and have additionally discussed the potential application of space group symmetry projection methods to JWHF