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Symmetry projection methods in strongly correlated systems

Ruiheng Song

Published Sep 30, 2024• 0 citations
OpenAlex

Abstract

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 $U$(1) and $SU$(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.

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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.
Coupled cluster theory is the method of choice for weakly correlated systems. But in the strongly correlated regime, it faces a symmetry dilemma, where it either completely fails to describe the system or has to artificially break certain symmetries. On the other hand, projected Hartree-Fock theory captures the essential physics of many kinds of strong correlations via symmetry breaking and restoration. In this work, we combine and try to retain the merits of these two methods by applying symmetry projection to broken symmetry coupled cluster wave functions. The non-orthogonal nature of states resulting from the application of symmetry projection operators furnishes particle-hole excitations to all orders, thus creating an obstacle for the exact evaluation of overlaps. Here we provide a solution via a disentanglement framework theory that can be approximated rigorously and systematically. Results of projected coupled cluster theory are presented for molecules and the Hubbard model, showing that spin projection significantly improves unrestricted coupled cluster theory while restoring good quantum numbers. The energy of projected coupled cluster theory reduces to the unprojected one in the thermodynamic limit, albeit at a much slower rate than projected Hartree-Fock.
The goal of electronic structure theory is to solve the time-independent Schrodinger equation with a method that scales polynomially with the system size. The problem has been solved in the weakly-correlated regime, where coupled cluster theory (CC) with singles, doubles and perturbative triples (CCSD(T)) can achieve chemical accuracy. However, in the strongly-correlated regime, where the two-body interaction dominates, there are still no methods that can achieve a comparable accomplishment. Strong correlation always comes hand in hand with spontaneous symmetry breaking in mean-field theory. For example, coupled cluster theory faces a symmetry dilemma in the strongly correlated regime, where it either completely fails to describe the system if symmetry adapted, or has to artificially break certain symmetries in order to produce reasonable energies. It is plausible that the problem of strong correlation can be solved by curing the symptoms, i.e., fixing the symmetries. Symmetry projection has been carried out at the mean-field level, but it was unclear until this work how to incorporate correlation on top of it because of the language barrier between symmetry projection and coupled cluster theory. This thesis is devoted to combining CC and symmetry projection, focusing on spin and number symmetry. Different flavours of theories are obtained depending on whether a restricted or unrestricted reference is used. On the restricted side, the spin projected unrestricted Hartree Fock (SUHF) state can be rewritten as a polynomial of particle-hole excitations, acting on the restricted reference. Thus, this formalism opens up many ways to combine SUHF with CC. One the unrestricted side, the concept of disentangled cluster is introduced to describe the CC wavefunction transformed by symmetry rotation. A scheme basing on differential equations is devised to approximate these disentangled clusters, which scales polynomially and is systematically improvable. By combining symmetry projection and coupled cluster theory, the symmetry dilemma can be solved to a good extent.
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.
Projected Hartree-Fock theory provides an accurate description of many kinds of strong correlations but does not properly describe weakly correlated systems. On the other hand, single-reference methods, such as configuration interaction or coupled cluster theory, can handle weakly correlated problems but cannot properly account for strong correlations. Ideally, we would like to combine these techniques in a symmetry-projected coupled cluster approach, but this is far from straightforward. In this work, we provide an alternative formulation to identify the so-called disentangled cluster operators, which arise when we combine these two methodological strands. Our formulation shows promising results for model systems and small molecules.
The failures of single-reference coupled-cluster theory for strongly correlated many-body systems is flagged at the mean-field level by the spontaneous breaking of one or more physical symmetries of the Hamiltonian. Restoring the symmetry of the mean-field determinant by projection reveals that coupled-cluster theory fails because it factorizes high-order excitation amplitudes incorrectly. However, symmetry-projected mean-field wave functions do not account sufficiently for dynamic (or weak) correlation. Here we pursue a merger of symmetry projection and coupled-cluster theory, following previous work along these lines that utilized the simple Lipkin model system as a test bed [J. Chem. Phys. 146, 054110 (2017)]. We generalize the concept of a symmetry-projected mean-field wave function to the concept of a symmetry projected state, in which the factorization of high-order excitation amplitudes in terms of low-order ones is guided by symmetry projection and is not exponential, and combine them with coupled-cluster theory in order to model the ground state of the Agassi Hamiltonian. This model has two separate channels of correlation and two separate physical symmetries which are broken under strong correlation. We show how the combination of symmetry collective states and coupled-cluster theory is effective in obtaining correlation energies and order parameters of the Agassi model throughout its phase diagram.
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