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arXiv:2303.04925 · Physical review. B./Physical review. B

Exploring spin antisymmetrized geminal power Ansätze for strongly correlated spin systems

Z.L. Liu, Fei Gao, Guo P. Chen, Thomas M. Henderson, J. Dukelsky, Gustavo E. Scuseria

Published Aug 25, 2023• 11 citations
arXivHTMLPDF

Abstract

The antisymmetrized geminal power (AGP), a wave function equivalent to number-projected Hartree-Fock-Bogoliubov (HFB), and number-projected Bardeen-Cooper-Schrieffer (BCS) when working in the paired (natural orbitals) basis, has proven to be an excellent reference for strong pairing interactions. Several correlation methods have also been applied on top of AGP. In this work, we show how AGP can also be applied to spin systems by simply basing its formulation on a spin $su(2)$ algebra. We here implement spin AGP and spin AGP-based correlation techniques and benchmark them on the XXZ and ${\mathrm{J}}_{1}\text{\ensuremath{-}}{\mathrm{J}}_{2}$ Heisenberg models, both in one and two dimensions. Our results indicate that spin AGP is a promising starting point for modeling spin systems.

DOI
10.1103/physrevb.108.085136

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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
Symmetry-projected wave function methods capture static correlation by breaking and restoring the symmetries of a system. In this article, we present the symmetry-projected spin antisymmetrized geminal power (spin-AGP) state projected onto space group symmetry as well as complex conjugation, spin-flip, and time-reversal symmetries. The method is benchmarked on the 1D XXZ model and the 2D J1 - J2 model with square and triangular lattices. Our results indicate that symmetry projection methods provide a powerful tool for frustrated spin systems.
Single-reference methods such as Hartree-Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential Ansätze in this paper. We investigate two approaches in particular. Similar to Wahlen-Strothman et al. [Phys. Rev. B 91, 041114(R) (2015)], we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting the Schrödinger equation. The second approach is based on approximating the unitary pair-hopper Ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches.
The antisymmetrized geminal power (AGP) wave function has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP-based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J -CI provides excellent accuracy in energies when applied to the pairing Hamiltonian.
Abstract For variational algorithms on the near term quantum computing hardware, it is highly desirable to use very accurate ansatze with low implementation cost. Recent studies have shown that the antisymmetrized geminal power (AGP) wavefunction can be an excellent starting point for ansatze describing systems with strong pairing correlations, as those occurring in superconductors. In this work, we show how AGP can be efficiently implemented on a quantum computer with circuit depth, number of CNOTs, and number of measurements being linear in system size. Using AGP as the initial reference, we propose and implement a unitary correlator on AGP and benchmark it on the ground state of the pairing Hamiltonian. The results show highly accurate ground state energies in all correlation regimes of this model Hamiltonian.
The antisymmetrized geminal power (AGP) wave function has a long history and considerable conceptual appeal, but in many situations its accuracy is wanting. Here, we consider a form of configuration interaction (CI) based upon the AGP wave function and take advantage of its killing operators to construct an excitation manifold. Our geminal CI reduces to standard single-determinant-based CI in the limit in which AGP reduces to a single determinant. It substantially improves upon AGP in the reduced BCS Hamiltonian, which serves as a prototype for the kinds of strong pairing correlations relevant in Bardeen-Cooper-Schrieffer-style superconductivity. Moreover, our geminal CI naturally generalizes to add correlation to more general geminal-based wave functions than AGP.