Density Matrix Research Articles

An Ab Initio Journey toward the Molecular‐Level Understanding and Predictability of Subnanometric Metal Clusters

Current advances in synthesizing and characterizing atomically precise monodisperse metal clusters (AMCs) at the subnanometer scale have opened up new possibilities in quantum materials research. Their quantizied “molecule‐like” electronic structure showcases unique stability, and physical and chemical properties differentiate them from larger nanoparticles. When integrated into inorganic materials that interact with the environment and sunlight, AMCs serve to enhance their (photo)catalytic activity and optoelectronic properties. Their tiny size makes AMCs isolated in the gas phase amenable to atom‐scale modeling using either density functional theory (DFT) or methods at a high level of ab initio theory, even addressing nonadiabatic (e.g., Jahn–Teller) effects. Surface‐supported AMCs can be routinely modeled using DFT, enabling long real‐time molecular dynamics simulations. Their optical properties can also be addressed using time‐dependent DFT or reduced density matrix (RDM) theory. These theoretical–computational efforts aim to achieve predictability and molecular‐level understanding of the stability and properties of AMCs as function of their composition, size, and structural fluxionality in different thermodynamical conditions (temperature and pressure). In this perspective, the potential of ab initio and DFT‐based modeling is illustrated through recent studies of unsupported and surface‐supported AMCs. Future directions of research are also discussed, including applications and methodological enhancements beyond the state‐of‐the‐art.

The Clifford Algebra of the Density Matrix: An Elementary Approach

The Clifford Algebra of the Density Matrix: An Elementary Approach

Vibrational Entanglement through the Lens of Quantum Information Measures.

We introduce a quantum information analysis of vibrational wave functions to understand complex vibrational spectra of molecules with strong anharmonic couplings and vibrational resonances. For this purpose, we define one- and two-modal entropies to guide the identification of strongly coupled vibrational modes and to characterize correlations within modal basis sets. We evaluate these descriptors for multiconfigurational vibrational wave functions which we calculate with the n-mode vibrational density matrix renormalization group algorithm. Based on the quantum information measures, we present a vibrational entanglement analysis of the vibrational ground and excited states of CO2, which display strong anharmonic effects due to the symmetry-induced and accidental (near-) degeneracies. We investigate the entanglement signature of the Fermi resonance and discuss the maximally entangled state arising from the two degenerate bending modes.

Real-time dynamics of the Schwinger model as an open quantum system with Neural Density Operators

Ab-initio simulations of multiple heavy quarks propagating in a Quark-Gluon Plasma are computationally difficult to perform due to the large dimension of the space of density matrices. This work develops machine learning algorithms to overcome this difficulty by approximating exact quantum states with neural network parametrisations, specifically Neural Density Operators. As a proof of principle demonstration in a QCD-like theory, the approach is applied to solve the Lindblad master equation in the 1 + 1d lattice Schwinger Model as an open quantum system. Neural Density Operators enable the study of in-medium dynamics on large lattice volumes, where multiple-string interactions and their effects on string-breaking and recombination phenomena can be studied. Thermal properties of the system at equilibrium can also be probed with these methods by variationally constructing the steady state of the Lindblad master equation. Scaling of this approach with system size is studied, and numerical demonstrations on up to 32 spatial lattice sites and with up to 3 interacting strings are performed.

Direct electron transfer mediated electrochemical activation of persulfates by reticulated vitreous carbon (RVC) cathode

The electrochemical activation of peroxymonosulfate (PMS) and peroxydisulfate (PDS) for pollutant degradation has gained increasing attention. In this work, reticulated vitreous carbon (RVC) electrodes were prepared at a low cost and investigated for the cathodic activation of PMS/PDS for Acid Orange 7 degradation. The RVC was characterized by SEM, TGA, XRD, Raman spectroscopy, XPS, EIS, and compression test. The superior activity outperformed previous reports in terms of low current density operation (1–10 mA/cm2), high degradation rates (0.13–0.49 min−1), low energy consumption (0.03–0.23 kWh/m3/order) without involvement of catalysts. The process was optimized for current density, PMS dosage, pollutant concentration, pH, water matrix, and supporting electrolyte. Under the optimal current density (3 mA/cm2), the addition of 0.5 mM of PMS boosted the degradation rate by 9.6 times to 0.4470 min−1 and reduced the electrical energy consumption by 90.1 % to 0.0243 kWh/m3/order in 50 mM of Na2SO4. Moreover, up to 117 mg/L of H2O2 was produced in 1 hour by the reduction of oxygen, with current efficiencies between 78–24 %. The process showed little deterioration in 10 cycles, worked for a variety of pollutants, and was not affected by the water matrix. The H-cell experiments confirmed that 90 % degradation took place on the cathode. Scavenging experiments and electrochemical tests suggested that the transitional state of PMS* and O2•− mediated the direct electron transfer mechanism for AO7 degradation. Eighteen degradation products were identified by HPLC-MS and categorized into 4 degradation pathways. Finally, their toxicity was assessed by the ECOSAR programme.

Information structure of a polarization state: the concept of metaspin

A complete description of the polarization properties of a random, stationary, generally three-dimensional (3D) optical field in a point is provided by the 3×3 polarization matrix. We show that its nine degrees of freedom can be represented by nine independent and decoupled parameters with simple and significant physical meanings. These cover the three orientation angles determining the intrinsic reference frame with respect to an arbitrary one, the three principal intensities representing the strengths of the components of the electric field along the respective intrinsic reference axes, and a real-valued vector, which we term metaspin, whose three components are given by the intrinsic correlations of the field components. Consequently, any given polarization state has an associated intensity-isotropic state, called the metaspin state, whose spin vector is fully determined by the metaspin vector. We also show that the concept of metaspin provides an illustrative synthesis procedure for 3D polarization states. The results can straightforwardly be applied to any 3×3 density matrix.

Numerical design of non-toxic high-efficiency tandem solar cell with distinct hole transport materials

The metal-halide perovskite is at the forefront of photovoltaic research and has the potential to combine low manufacturing costs with excellent power conversion efficiency. In this study, a solar cell capacitance simulator was used to analyze the performance of a tandem solar cell in which MASnI3 was used as the lower cell active layer, and NaZn0.7Cu0.3Br3 was used as the upper cell active layer. The primary objective of this research is to search for a device structure with enhanced efficiency. Here, it is evaluated how absorber thickness, temperature, hole transport layer, defect density, and metal work function affect solar cell performance to reach this aim. After looking at the data for several different solar cell configurations, it is seen that indium tin oxide (ITO) / titanium dioxide (TiO2)/sodium zinc copper bromide (NaZn0.7Cu0.3Br3)/ methylammonium tin iodide (MASnI3) / copper oxide (CuO)/gold (Au) offers the best efficiency at 32.58 %, fill factor at 82.19 %, short circuit current density at 34.8389 mA/cm2, and open circuit voltage at 1.1375 Vs. Device simulations showed an optimal MASnI3 absorber thickness of about 1 µm. Finally, the simulation results reveal that the device's efficiency declines when the absorber's defect density and cell operation temperature rise. The device topologies are stable at around 300 Kelvin. Last but not least, any conductive material can be used as an anode if it has a work function that is more than or roughly equivalent to 5.10 eV.

Unveiling the influence of end-capped acceptors modification on photovoltaic properties of non-fullerene fused ring compounds: a DFT/TD-DFT study.

Herein, unique A-D-A configuration-based molecules (NBD1-NBD7) were designed from the reference compound (NBR) by utilizing the end-capped acceptor modification approach. Various electron-withdrawing units -F, -Cl, -CN, -NO2, -CF3, -HSO3, and -COOCH3, were incorporated into terminals of reference compound to designed NBD1-NBD7, respectively. A theoretical investigation employing the density functional theory (DFT) and time-dependent DFT (TD-DFT) was performed at B3LYP/6-311G(d,p) level. To reveal diverse opto-electronic and photovoltaic properties, the frontier molecular orbitals (FMOs), absorption maxima (λ max), density of states (DOS), exciton binding energy (E b), open-circuit voltage (V oc) and transition density matrix (TDM) analyses were executed at the same functional. Moreover, the global reactivity parameters (GRPs) were calculated using the HOMO-LUMO energy gaps from the FMOs. Significant results were obtained for the designed molecules (NBD1-NBD7) as compared to NBR. They showed lesser energy band gaps (2.024-2.157 eV) as compared to the NBR reference (2.147 eV). The tailored molecules also demonstrated bathochromic shifts in the chloroform (671.087-717.164 nm) and gas phases (623.251-653.404 nm) as compared to NBR compound (674.189 and 626.178 nm, respectively). From the photovoltaic perspectives, they showed promising results (2.024-2.157 V). Furthermore, the existence of intramolecular charge transfer (ICT) in the designed compounds was depicted via their DOS and TDM graphical plots. Among all the investigated molecules, NBD4 was disclosed as the excellent candidate for solar cell applications owing to its favorable properties such as the least band gap (2.024 eV), red-shifted λ max in the chloroform (717.164 nm) and gas (653.404 nm) phases as well as the minimal E b (0.126 eV). This is due to the presence of highly electronegative -NO2 unit at the terminal of electron withdrawing acceptor moiety, which leads to increased conjugation and enhanced the intramolecular charge transfer (ICT) rate. The obtained insights suggested that the designed molecules could be considered as promising materials for potential applications in the realm of OSCs.

Elucidating the potential of nitrogen-containing heterocyclic core-based non-fullerene acceptors for organic photovoltaic applications

Intramolecular noncovalent interactions are recognized as an efficient route to construct non-fused-core-based non-fullerene acceptors (NFAs) for efficient and cost-effective organic solar cells (OSCs). Herein, we engineered ten distinct structured NFAs molecules based on nitrogen-containing six-membered heterocycle cores. We incorporated various efficient functional groups into the synthetic reference molecule (RM) to design a new efficient series of NFAs (RM1-RM10). These materials are intensively investigated by performing advanced quantum chemical simulations, and the results are compared with those of synthetic RM molecules. We revealed the potential of these materials by investigating their optical, electronic, optoelectronic, and photovoltaic characteristics. By implementing various theoretical models, we explore the inner hidden potential of this designed series (RM1-RM10), such as density of state, transition density matrix, and electrostatic potential studies have been performed. Our applied theoretical modeling approach enables these photovoltaic materials (RM1-RM10) to have customized attributes by tuning their energy levels, binding energy, optical features, and also the photovoltaic characteristics of these materials. Newly developed molecules (RM1-RM10) show promising optoelectronic properties, including a lower energy gap (ranging from 1.81 eV to 1.63 eV) and an absorbed maximum absorption wavelength (760.36 nm), compared to the reference values, which have an energy gap of 1.81 eV and a wavelength of 684.16 nm. Moreover, to demonstrate the charge transfer behavior of these materials, a donor: acceptor (PTB7-Th:RM9) complex study is also performed to reveal the charge transfer phenomenon at the donor: acceptor interface. Therefore, our presented molecular modeling strategy holds significant potential for manufacturing materials with tailored qualities, which is crucial in advancing the organic photovoltaics field.

Computer Simulation of Charge Transfer In a DNA Molecule within a Simple Model of an Open Quantum System

The quantum-statistical model proposed earlier by Skourtis and Nitzan to describe a charge (hole) propagation in a fragment of artificial DNA molecule has been generalized and further investigated. A numerical simulation of a charge transport was carried out taking into account two different dissipative processes, including the capture of the charged carrier by the environment and the decoherence of its quantum wave properties due to the influence of stochastic fields of the environment. The interaction of the carrier with the environment is regulated by a small set of phenomenological parameters whose values vary in the process of simulation. Within this model, the propagation of a hole carrier in DNA is described by solving the quantum Franke-Kossakovski–Lindblad–Glauber–Sudarshan equation (hereinafter referred to as the Lindblad equation) for the carrier density matrix using the Lindblad MPO Solver program. Results of numerical analysis of the model are in a good agreement with experimental observations and demonstrate two different types of the charged carrier motion, presumably tunneling and incoherent hopping. The model is put in more general context and non-unitary dynamics of the hole carrier is treated within the framework of a theory of continuous quantum measurements by the environment in an open quantum system. The main concepts of the theory of decoherence and superselection for open quantum systems and the prospects for their application for further study of various mechanisms of motion of a charged carrier in DNA are briefly discussed.

Entanglement and pseudo entanglement dynamics versus fusion in CFT

The fusion rules and operator product expansion (OPE) serve as crucial tools in the study of operator algebras within conformal field theory (CFT). Building upon the vision of using entanglement to explore the connections between fusion coefficients and OPE coefficients, we employ the replica method and Schmidt decomposition method to investigate the time evolution of entanglement entropy (EE) and pseudo entropy (PE) for linear combinations of operators in rational conformal field theory (RCFT). We obtain a formula that links fusion coefficients, quantum dimensions, and OPE coefficients. We also identify two definition schemes for linear combination operators. Under one scheme, the EE captures information solely for the heaviest operators, while the PE retains information for all operators, reflecting the phenomenon of pseudo entropy amplification. Irrespective of the scheme employed, the EE demonstrates a step-like evolution, illustrating the effectiveness of the quasiparticle propagation picture for the general superposition of locally excited states in RCFT. From the perspective of quasiparticle propagation, we observe spontaneous block-diagonalization of the reduced density matrix of a subsystem when quasiparticles enter the subsystem.

Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches.

Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations.

On the future sustainable ultra-high-speed maglev: A superconductor magnet technology enabling high energy efficiency and robustness

The blooming ultrahigh-speed SC maglev (superconducting magnetically levitated train) prompts green travel worldwide. We utilized high-temperature superconductor (HTS) magnets and have developed an energy-economical SC maglev prototype (Energ. Convers. Manage. (291)117247). In this paper, as a continuation, we further improve and maximize the performance of its key part – the HTS magnets, especially in compact high magnetic field and energy density operating conditions, by innovating an HTS magnet technology named the Shunted Extreme-NI Concept. The operational energy-robustness, -density, and -economy of HTS magnets, compared to the conventional type, are improved by 96.8%, 66.7%, and 52.5%, respectively. For a full-scale maglev equipped with this technology, the estimated energy consumption per passenger kilometer can be further reduced to below 20% of that of airplanes under the same cruising speed of 1000 km/h. This work is dedicated to advancing the energy performances of HTS magnets, heading to the future sustainable maglev transportation.

Entanglement and Bell inequalities with boosted tt¯

The Large Hadron Collider provides a unique opportunity to study quantum entanglement and violation of Bell inequalities at the highest energy available today. In this paper, we will investigate these quantum correlations with top quark pair production, which represents a system of two-qubits. The semileptonic top pair channel has a factor of 6 increase in statistics and easier reconstruction with respect to the dileptonic channel. Although measuring the spin polarization of the hadronic top quark is known to be challenging, our study indicates that it is feasible to reconstruct the spin density matrix of the two-qubit system using an optimal hadronic polarimeter. This is achieved with the aid of jet substructure techniques and NN-inspired reconstruction methods, which improve the mapping between subjets and quarks. We find that entanglement can already be observed at more than the 5σ level with existing data, and violation of Bell inequalities may be probed above the 4σ level at the HL-LHC with 3 ab−1 of data. Published by the American Physical Society 2024

Managing temperature in open quantum systems strongly coupled with structured environments.

In non-perturbative non-Markovian open quantum systems, reaching either low temperatures with the hierarchical equations of motion (HEOM) or high temperatures with the Thermalized Time Evolving Density Operator with Orthogonal Polynomials Algorithm (T-TEDOPA) formalism in Hilbert space remains challenging. We compare different ways of modeling the environment. Sampling the Fourier transform of the bath correlation function, also called temperature dependent spectral density, proves to be very effective. T-TEDOPA [Tamascelli et al., Phys. Rev. Lett. 123, 090402 (2019)] uses a linear chain of oscillators with positive and negative frequencies, while HEOM is based on the complex poles of an optimized rational decomposition of the temperature dependent spectral density [Xu et al., Phys. Rev. Lett. 129, 230601 (2022)]. Resorting to the poles of the temperature independent spectral density and of the Bose function separately is an alternative when the problem due to the huge number of Bose poles at low temperatures is circumvented. Two examples illustrate the effectiveness of the HEOM and T-TEDOPA approaches: a benchmark pure dephasing case and a two-bath model simulating the dynamics of excited electronic states coupled through a conical intersection. We show the efficiency of T-TEDOPA to simulate dynamics at a finite temperature by using either continuous spectral densities or only all the intramolecular oscillators of a linear vibronic model calibrated from abinitio data of a phenylene ethynylene dimer.

Tripartite entanglement in H→ZZ,WW decays

The decays of the Higgs boson produce a state in which the spins of the decay products and the orbital angular momentum (L) are highly entangled. We obtain the tripartite density operator, as well as reduced operators, for the decay into two weak bosons. For H→ZZ in the four-lepton final state we also estimate the statistical sensitivity at the Large Hadron Collider and future upgrades, using a binned method to reconstruct the density operators from distributions. With the expected Run 3 data, establishing genuine tripartite entanglement would be possible beyond the 5σ level. The violation of Bell inequalities involving the spins of the two weak bosons could also be established beyond the 5σ level. Published by the American Physical Society 2024

Simulating multipulse NMR spectra of polycrystalline solids in the frequency domain.

An approach is presented for simulating multipulse nuclear magnetic resonance (NMR) spectra of polycrystalline solids directly in the frequency domain. The approach integrates the symmetry pathway concept for multipulse NMR with efficient algorithms for calculating spinning sideband amplitudes and performing interpolated finite-element numerical integration over all crystallite orientations in a polycrystalline sample. The numerical efficiency is achieved through a set of assumptions used to approximate the evolution of a sparse density matrix through a pulse sequence as a set of individual transition pathway signals. The utility of this approach for simulating the spectra of complex materials, such as glasses and other structurally disordered materials, is demonstrated.

Symmetry-preserving quadratic Lindbladian and dissipation driven topological transitions in Gaussian states

The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U(1) symmetry and the DIII class without U(1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.

Occupied-Virtual Orbitals for Chemical Valence with Applications to Charge Transfer in Energy Decomposition Analysis.

In this article, we introduce the occupied-virtual orbitals for chemical valence (OVOCV). The OVOCVs can replace or complement the closely related idea of the natural orbitals for chemical valence (NOCV). The input is a difference density matrix connecting any initial single determinant to any final determinant, at a given molecular geometry, and a given one-particle basis. This arises in problems such as orbital rearrangement or charge transfer (CT) in energy decomposition analysis (EDA). The OVOCVs block-diagonalize the density difference operator into 2 × 2 blocks, which are spanned by one level that is filled in the initial state (the occupied OVOCV) and one that is empty (the virtual OVOCV). By contrast, the NOCVs fully diagonalize the density difference matrix and therefore are orbitals with mixed occupied-virtual character. Use of the OVOCVs makes it much easier to identify the donor and acceptor orbitals. We also introduce two different types of EDA methods with the OVOCVs and, most importantly, a charge decomposition analysis method that fixes the unreasonably large CT amount obtained directly from NOCV analysis. The square of the CT amount associated with each NOCV pair emerges as the appropriate value from the OVOCV analysis. When connecting the same initial and final states, this value is identical to the CT amount obtained from the independent absolutely localized molecular orbital (ALMO) complementary occupied-virtual orbital pair (COVP) analysis. The total, summed over all pairs, is also exactly the same as the independently suggested excitation number, as proved herein. Several examples are presented to compare NOCVs and OVOCVs: stretched H2+, a strong halogen bond between tetramethylthiourea and iodine, coordination of ethene in Zeise's salt, and binding in the Cp3La···C≡NCy complex.

Proximate Tomonaga-Luttinger liquid in an anisotropic Kitaev-Gamma model

The characterization of quantum spin liquid phases in Kitaev materials has been a subject of intensive studies over the recent years, both theoretically and experimentally. Most theoretical studies have focused on an isotropically interacting model with its coupling strength being equivalent on each bond in an attempt to simplify the problem. In this Letter, we study an extended spin-1/2 Kitaev-Γ model on a honeycomb lattice with an additional tuning parameter that controls the coupling strength on one of the bonds: We connect the limit of isolated Kitaev-Γ chains, which is known to exhibit an emergent SU(2)1 Tomonaga-Luttinger liquid phase [Yang , ], to the two-dimensional model. We report on an instance, in which properties of the Tomonaga-Luttinger liquid persist for finite interchain coupling forming a phase we call the proximate Tomonaga-Luttinger liquid (pTLL). This pTLL is strongly anisotropic in character analogous to , and features spinon-like excitations similar to those of the antiferromagnetic Heisenberg chain. We use numerical exact diagonalization and density matrix renormalization group on various cluster geometries in a complementary way to overcome finite-size limitations. Published by the American Physical Society 2024