Linear Combinations of Cluster Mean-Field States Applied to Spin Systems.

We present an innovative cluster-based method employing linear combinations of diverse cluster mean-field states and apply it to describe the ground state of strongly correlated spin systems. In cluster mean-field theory, the ground state wave function is expressed as a factorized tensor product of optimized cluster states. While our prior work concentrated on a single cMF tiling, this study removes that constraint by combining different tilings of cMF states. Selection criteria, including translational symmetry and spatial proximity, guide this process. We present benchmark calculations for the one- and two-dimensional J1 - J2 and XXZ Heisenberg models. Our findings highlight two key aspects. First, the method offers a semiquantitative description of the 0.4 ≲ J2/J1 ≲ 0.6 regime of the J1 - J2 model─a particularly challenging regime for existing methods. Second, our results demonstrate the capability of our method to provide qualitative descriptions for all the models and regimes considered, establishing it as a valuable reference. However, the inclusion of additional (weak) correlations is necessary for quantitative agreement, and we explore methods to incorporate these extra correlations.

Hartree-Fock-Bogoliubov theory for number-parity-violating fermionic Hamiltonians.

It is usually asserted that physical Hamiltonians for fermions must contain an even number of fermion operators. This is indeed true in electronic structure theory. However, when the Jordan-Wigner (JW) transformation is used to map physical spin Hamiltonians to Hamiltonians of spinless fermions, terms that contain an odd number of fermion operators may appear. The resulting fermionic Hamiltonian thus does not have number parity symmetry and requires wave functions that do not have this symmetry either. In this work, we discuss the extension of standard Hartree-Fock-Bogoliubov (HFB) theory to the number-parity-nonconserving case. These ideas had appeared in the literature before but, perhaps for lack of practical applications, had, to the best of our knowledge, never been employed. We here present a useful application for this more general HFB theory based on coherent states of the SO(2M + 1) Lie group, where M is the number of orbitals. We also show how using these unusual mean-field states can provide significant improvements when studying the JW transformation of chemically relevant spin Hamiltonians.

Correlated pair ansatz with a binary tree structure.

We develop an efficient algorithm to implement the recently introduced binary tree state (BTS) ansatz on a classical computer. BTS allows a simple approximation to permanents arising from the computationally intractable antisymmetric product of interacting geminals and respects size-consistency. We show how to compute BTS overlap and reduced density matrices efficiently. We also explore two routes for developing correlated BTS approaches: Jastrow coupled cluster on BTS and linear combinations of BT states. The resulting methods show great promise in benchmark applications to the reduced Bardeen-Cooper-Schrieffer Hamiltonian and the one-dimensional XXZ Heisenberg Hamiltonian.

Symmetry-projected cluster mean-field theory applied to spin systems.

We introduce Sz spin-projection based on cluster mean-field theory and apply it to the ground state of strongly correlated spin systems. In cluster mean-fields, the ground state wavefunction is written as a factorized tensor product of optimized cluster states. In previous work, we have focused on unrestricted cluster mean-field, where each cluster is Sz symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all Sz sectors, breaking Sz symmetry. In addition, a projection scheme is used to restore global Sz, which gives rise to the Sz spin-projected generalized cluster mean-field (SzGcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D J1 - J2 and XXZ Heisenberg models. Our results indicate that the new methods (GcMF and SzGcMF) provide a qualitative and semi-quantitative description of the Heisenberg lattices in the regimes considered, suggesting them as useful references for further inter-cluster correlations, which are discussed in this work.

Thermofield Theory for Finite-Temperature Electronic Structure.

Wave function methods have offered a robust, systematically improvable means to study ground-state properties in quantum many-body systems. Theories like coupled cluster and their derivatives provide highly accurate approximations to the energy landscape at a reasonable computational cost. Analogues of such methods to study thermal properties, though highly desirable, have been lacking because evaluating thermal properties involve a trace over the entire Hilbert space, which is a formidable task. Besides, excited-state theories are generally not as well studied as ground-state ones. In this mini-review, we present an overview of a finite-temperature wave function formalism based on thermofield dynamics to overcome these difficulties. Thermofield dynamics allows us to map the equilibrium thermal density matrix to a pure state, i.e., a single wave function, albeit in an expanded Hilbert space. Ensemble averages become expectation values over this so-called thermal state. Around this thermal state, we have developed a procedure to generalize ground-state wave function theories to finite temperatures. As explicit examples, we highlight formulations of mean-field, configuration interaction, and coupled cluster theories for thermal properties of Fermions in the grand-canonical ensemble. To assess the quality of these approximations, we also show benchmark studies for the one-dimensional Hubbard model, while comparing against exact results. We will see that the thermal methods perform similarly to their ground-state counterparts, while merely adding a prefactor to the asymptotic computational cost. They also inherit all the properties, good or bad, from the ground-state methods, signifying the robustness of our formalism and the scope for future development.

Reprogramming cell fates towards novel cancer immunotherapies.

Recent advances in our understanding of host immune and cancer cells interactions have made immunotherapy a prominent choice in cancer treatment. Despite such promise, cell-based immunotherapies remain inapplicable to many patients due to severe limitations in the availability and quality of immune cells isolated from donors. Reprogramming technologies that facilitate the engineering of cell types of interest, are emerging as a putative solution to such challenges. Here we focus on the recent progress being made in reprogramming technologies with respect to the immune system and their potential for clinical applications.

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

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.

Coupled Cluster and Perturbation Theories Based on a Cluster Mean-Field Reference Applied to Strongly Correlated Spin Systems.

We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly correlated spin systems. In cluster mean-field, the ground state wave function is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allow for a straightforward improvement which uses perturbation theory and coupled-cluster to account for intercluster correlations. We present benchmark calculations on the 1D chain and 2D square J1-J2 Heisenberg model, using cluster mean-field, perturbation theory, and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4, and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly correlated systems. Some ways to improve upon the methods presented in this work are discussed.

A power series approximation in symmetry projected coupled cluster theory.

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.

Construction of linearly independent non-orthogonal AGP states.

We show how to construct a linearly independent set of antisymmetrized geminal power (AGP) states, which allows us to rewrite our recently introduced geminal replacement models as linear combinations of non-orthogonal AGPs. This greatly simplifies the evaluation of matrix elements and permits us to introduce an AGP-based selective configuration interaction method, which can reach arbitrary excitation levels relative to a reference AGP, balancing accuracy and cost as we see fit.

Exploring non-linear correlators on AGP.

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.

Direct aerobic NSZD of a basalt vadose zone LNAPL source in Hawaii.

In recent years, a number of methods have been used to measure the biodegradation of petroleum light non-aqueous phase liquids (LNAPL) at petroleum release sites, a process known as natural source zone depletion (NSZD). Most commonly, NSZD rates have been measured at sites with unconsolidated geology and relatively shallow groundwater (<50 ft. bgs, <15 m bgs). For this study, we have used two methods (1. carbon dioxide flux measured using carbon traps and 2. heat flux based on subsurface temperature gradients) to measure NSZD rates at a petroleum release site in Hawaii with basalt geology and deep groundwater (>300 ft. bgs, >100 m bgs). Both methods documented the occurrence of NSZD at the facility and the two methods yield estimates of the NSZD rate that agreed within a factor of 2 (4600 to 7400 gal/yr; 17,000 to 28,000 L/yr for the flux method and 8600 to 13,000 gal/yr; 33,000 to 49,000 L/yr for the temperature method). Soil gas samples collected directly above the water table and at shallower depths within the vadose zone indicated aerobic conditions throughout the vadose zone (oxygen >13%) and no detectable methane. These results indicate that NSZD occurs at this site through the direct aerobic biodegradation of LNAPL rather than the two-step process of anaerobic methanogenesis followed by methane oxidation at a shallow depth interval documented at other sites.

Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems.

We present a wave function representation for the canonical ensemble thermal density matrix by projecting the thermofield double state against the desired number of particles. The resulting canonical thermal state obeys an imaginary-time evolution equation. Starting with the mean-field approximation, where the canonical thermal state becomes an antisymmetrized geminal power (AGP) wave function, we explore two different schemes to add correlation: by number-projecting a correlated grand-canonical thermal state and by adding correlation to the number-projected mean-field state. As benchmark examples, we use number-projected configuration interaction and an AGP-based perturbation theory to study the hydrogen molecule in a minimal basis and the six-site Hubbard model.

Visual Function and Macular Carotenoid Changes in Eyes with Retinal Drusen-An Open Label Randomized Controlled Trial to Compare a Micronized Lipid-Based Carotenoid Liquid Supplementation and AREDS-2 Formula.

PURPOSE: To compare the changes in visual and ocular parameters in individuals with retinal drusen who were treated with two commercially available nutritional supplements. METHODS: An open-label, single-center, randomized, parallel-treatment with an observational control group design was utilized. The treatment groups included individuals with fine retinal drusen sub-clinical age-related macular degeneration (AMD), while the control group consisted of ocular normal individuals. The treatment groups were randomly assigned to the micronized lipid-based carotenoid supplement, Lumega-Z (LM), or the PreserVision Age-Related Eye Disease Study 2 (AREDS-2) soft gel (PV). Visual performance was evaluated using the techniques of visual acuity, dark adaptation recovery and contrast sensitivity, at baseline, three months, and six months. Additionally, the macular pigment optical density (MPOD) was measured. The control group was not assigned any carotenoid supplement. The right eye and left eye results were analyzed separately. RESULTS: Seventy-nine participants were recruited for this study, of which 68 qualified and 56 participants had useable reliable data. Of the individuals who completed this study, 25 participants belonged to the LM group, 16 belonged to the PV group, and 15 to the control group. The LM group demonstrated statistically significant improvements in contrast sensitivity function (CSF) in both eyes at six months (p < 0.001). The LM group displayed a positive linear trend with treatment time in CSF (p < 0.001), with benefits visible after just three months of supplementation. Although there was a trend showing improvement in CSF in the PV group, the change was not significant after a Bonferroni-corrected p-value of p < 0.00625. Visual acuity, dark adaptation recovery and MPOD did not significantly improve in either treatment groups. CONCLUSION: The LM group demonstrated greater and faster benefits in visual performance as measured by CSF when compared to the PV group. This trial has been registered at clinicaltrials.gov (NCT03946085).

Correlating the antisymmetrized geminal power wave function.

Strong pairing correlations are responsible for superconductivity and off-diagonal long-range order in the two-particle density matrix. The antisymmetrized geminal power wave function was championed many years ago as the simplest model that can provide a reasonable qualitative description for these correlations without breaking number symmetry. The fact remains, however, that the antisymmetrized geminal power is not generally quantitatively accurate in all correlation regimes. In this work, we discuss how we might use this wave function as a reference state for a more sophisticated correlation technique such as configuration interaction, coupled cluster theory, or the random phase approximation.

Geminal Replacement Models Based on AGP.

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.

The effect of acute exercise on environmentally induced symptoms of dry eye.

The purpose of this study was to investigate the effects of acute exercise on environmentally induced symptoms of dry eye. Twelve participants without dry eye disease volunteered to complete three experimental visits in a randomized order; (1) control condition seated for 1 h at a relative humidity (RH) of 40% (CONT), (2) dry condition seated for 1 h at a RH of 20% (DRY), and (3) exercise condition seated for 40 min followed by 20 min of cycling exercise at a RH of 20% (EXER). Tear volume, tear matrix metalloproteinase 9 (MMP-9), perception of dry eye symptoms (frequency and severity), core temperature, and ocular surface temperature (OST) were measured at the end of each exposure. The perception of dry eye frequency and MMP-9 concentration were significantly higher in DRY compared to CONT (P < 0.012), with no differences in EXER compared to CONT. The results suggest that an acute bout of exercise may attenuate symptoms of environmentally induced dry eye, and warrant further research.

Efficient evaluation of AGP reduced density matrices.

We propose and implement an algorithm to calculate the norm and reduced density matrices (RDMs) of the antisymmetrized geminal power of any rank with polynomial cost. Our method scales quadratically per element of the RDMs. Numerical tests indicate that our method is very fast and capable of treating systems with a few thousand orbitals and hundreds of electrons reliably in double-precision. In addition, we present reconstruction formulas that allow one to decompose higher order RDMs in terms of linear combinations of lower order ones and geminal coefficients, thereby reducing the computational cost significantly.

Thermofield Theory for Finite-Temperature Coupled Cluster.

We present a coupled cluster and linear response theory to compute properties of many-electron systems at nonzero temperatures. For this purpose, we make use of the thermofield dynamics, which allows for a compact wave function representation of the thermal density matrix, and extend our recently developed framework ( J. Chem. Phys. 2019 , 150 , 154109 , DOI: 10.1063/1.5089560 ) to parametrize the so-called thermal state using an exponential ansatz with cluster operators that create thermal quasiparticle excitations on a mean-field reference. As benchmark examples, we apply this method to both model (one-dimensional Hubbard and Pairing) and ab initio (atomic Beryllium and molecular Hydrogen) systems, while comparing with exact results.