Density Matrix Research Articles

Reduced form of statistical equilibrium equations and radiative transfer equations

Abstract An extension of the vector algebra for irreducible spherical tensor operators (ITOs) is proposed, which involves coupling two ITOs into a third one (V (kv)= T (kt)× U (ku)*), with the additional condition that one of the operators is a complex conjugate and therefore is not an ITO. The correctness condition is considered for this extension. The new coupling rule treats the density matrices ρ(Kl) and all tensors related to light polarization (J (Kr) or Τ (K)) equally.
This approach has been applied to rewrite the fundamental statistical equilibrium equations and radiation transfer equations relevant to astrophysical polarimetry. The revised equations are more concise and eliminate the need for the nj symbols. Instead, they utilize vector and scalar products of ITOs. All rate coefficients in the equations consist of two independent factors, the reduced matrix element from dipole moments and the product (vector or scalar) of the tensors ρ(Kl) and J (Kr) (or Τ (K)).

Optimizing non-fullerene acceptor molecules constituting fluorene core for enhanced performance in organic solar cells: a theoretical methodology.

Looking for novel outstanding performance materials suitable for organic solar cells, we constructed a range of non-fullerene acceptors (NFAs) evolved from the recently synthesized acceptor molecule identified as DICTIF, structured around fluorene core where 2-(2,3-dihydro-3-oxo-1H-inden-1-ylidene) propanedinitrile presented the terminals end-groups. Employing density functional theory (DFT) and time dependent-DFT (TD-DFT) simulations, we have simulated the impact of altering the end groups of DICTIF molecule by five assorted acceptors molecules, for the purpose of exploring their opto-electronic properties and their performance in organic solar cell (OSC) applications. We proved that the designed non-fullerene acceptors provide enhanced efficiency compared to the synthesized molecule, such as planar geometries and narrower energy gap ranging from 1.51 to 1.95 eV. A red shift in absorption was observed for all tailored molecules (λmax = 583.5-711.4 nm) as compared to the reference molecule (λmax = 578 nm).Various decisive factors such as frontier molecular orbitals (FMOs), exciton binding energy (EB), absorption maximum (λmax), open circuit voltage (VOC), reorganization energies (RE), transition density matrix (TDM), reduced density gradient (RDG), and electron-hole overlap have also been computed for analyzing the performance of NFAs. Low reorganizational energy values facilitate charge mobility which improves the conductivity of all the designed molecules. This study showed that our novel tailored molecules might be suitable candidates for the fabrication of highly efficient photovoltaic materials. After testing various hybrid functionals, optimized geometries were assigned using DFT HSEH1PBE/6-31G(d) level of theory. Electronic excitations and absorption spectra were investigated using the TD-DFT MPW1PW91/6-31G(d) level of theory. We ascertained that HSEH1PBE/6-31G(d) level of theory yield the closest calculated highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy levels of the DICTIF to the corresponding experimental ones and that TD-MPW1PW91//6-31G(d) was the most suitable level of theory for exploring electronic excitations and finding the maximum of absorption (λmax).

Parallel Implementation of the Density Matrix Renormalization Group Method Achieving a Quarter petaFLOPS Performance on a Single DGX-H100 GPU Node.

We report cutting edge performance results on a single node hybrid CPU-multi-GPU implementation of the spin adapted ab initio Density Matrix Renormalization Group (DMRG) method on current state-of-the-art NVIDIA DGX-H100 architectures. We evaluate the performance of the DMRG electronic structure calculations for the active compounds of the FeMoco, the primary cofactor of nitrogenase, and cytochrome P450 (CYP) enzymes with complete active space (CAS) sizes of up to 113 electrons in 76 orbitals [CAS(113, 76)] and 63 electrons in 58 orbitals [CAS(63, 58)], respectively. We achieve 246 teraFLOPS of sustained performance, an improvement of more than 2.5× compared to the performance achieved on the DGX-A100 architectures and an 80× acceleration compared to an OpenMP parallelized implementation on a 128-core CPU architecture. Our work highlights the ability of tensor network algorithms to efficiently utilize high-performance multi-GPU hardware and shows that the combination of tensor networks with modern large-scale GPU accelerators can pave the way toward solving some of the most challenging problems in quantum chemistry and beyond.

Theoretical investigation of benzodithiophene-based donor molecules in organic solar cells: from structural optimization to performance metrics.

Achieving high power conversion efficiency (PCE) remains a significant challenge in the advancement of organic solar cells (OSCs). In the field of organic photovoltaics (OPVs), considerable progress has been made in optimizing molecular structures to improve the PCE. However, innovative material design strategies specifically aimed at enhancing PCE are still needed. Here, we have designed BDTS-2DPP-based molecules and propose a molecular design approach to develop donor materials that can significantly improve the PCE of OSCs. Density functional theory (DFT) and time-dependent DFT (TD-DFT) methods have been adopted in both gas and solvent phases. Our newly designed molecule M1 shows the highest absorption value (λ max = 846 nm), highest electron reorganization energy (λ e = 0.18 eV), and the lowest energy gap (E g = 1.81 eV) among all the designed molecules. M1 molecule also exhibits the highest dipole moment in both gas (10.62 D) and solvent phase (13.62 D), and their ground and excited state dipole moment difference is also higher (μ e - μ g = 2.99 D), which enhances its separation to make it a suitable candidate for charge transfer between HOMO-LUMO (97%). Newly designed molecule M3 is observed to have the highest voltage when the current is zero (V oc = 1.15 V) highest PCE value (21.90%) and highest fill factor (FF) value (89.42%). The lowest excitation binding energy is estimated by newly designed molecule M2 (E b = 0.30 eV), which indicates a higher rate of dissociation during the excitation as observed in transition density matrix (TDM) plots. Utilizing electron density difference maps, the newly designed molecules in dichloromethane solvent exhibited consistent intramolecular charge transfer (ICT). The designed molecules were evaluated against reference molecule R to determine if they exhibit superior optoelectronic capabilities. It is found that all designed molecules (M1-M5) exhibit reduced band gaps, are red-shifted in wavelength in comparison to a reference molecule R, and have remarkable charge motilities in terms of reorganisation energies.

Kinetics in the Two-Level System with Strong Time-De-pendent Coupling of Its States to the Phonon Bath: Spin-Boson Description

Using the methods of nonequilibrium statistical mechanics, the master equation for the density matrix of an open dissipative quantum system is obtained under conditions, when the coupling between the electronic states of the system and the nuclear displacements in it is controlled by the alternating field. A time-dependent polaron transformation is proposed, which made it possible to solve kinetic equations using an expansion in a parameter characterizing transitions between “phonon-dressed” electronic states of the system. As an example, a mechanism is shown that can control the kinetics in a two-level system by applying a periodic force field to electron-phonon coupling.

Signatures of the attractive interaction in spin spectra of one-dimensional cuprate chains

Identifying the minimal model for cuprates is crucial for explaining the high-Tc pairing mechanism. Recent photoemission experiments have suggested a significant near-neighbor attractive interaction V in cuprate chains, favoring pairing instability. To determine its strength, we systematically investigate the dynamical spin structure factors S(q,ω) using the density matrix renormalization group. Our analysis quantitatively reveals a notable softening in the two-spinon continuum, particularly evident in the intense spectrum at large momentum. This softening is primarily driven by the renormalization of the superexchange interaction, as determined by a comparison with the slave-boson theory. We also demonstrate the feasibility of detecting this spectral shift in thin-film samples using resonant inelastic x-ray scattering. Therefore, this provides a distinctive fingerprint for the attractive interaction, motivating future experiments to unveil essential ingredients in cuprates. Published by the American Physical Society 2024

Tailoring Diversified Peripheral Anchor Groups in Spirofluorene‐Dithiolane‐Based Hole Transporting Materials for Efficient Organic and Perovskite Solar Cells from First‐Principle

AbstractThis quantum mechanical approach recommends push–pull molecular engineering to fabricate hole‐transporting materials (HTMs) for photovoltaic cells. It integrates acceptor moieties via thiophene to fluorene core, resulting in five novel HTMs (SFD‐1 to SFD‐5). The results exhibit that derivative HTMs show excellent coherence in excitation, dispersion, and transportation of charge carriers, ensuring robust hole mobility. The anchor moieties functionalized HTMs unveil excellent band alignment with perovskite with fitting HOMO energy levels (−4.93–−5.35 eV), less optical absorption in visible portion ( < 520). This acceptor integration has improved the hole mobility in derivatives, accredited to the smaller hole reorganization energy (0.14–0.68 eV), and greater hole transfer integral (0.22–0.33 eV). The transition density matrix analysis exhibited robust electronic coupling, subtler charge carrier overlapping and greater charge transfer length (7.48–13.73 Å). This resulted in an excellent upsurge in intrinsic charge transference (70.75–92.70%) and smaller exciton binding energy, leading to easier exciton dissociation, and fewer recombination fatalities. However, an adequate variation in dipole moment (4.04 D to 16.34 D) and Gibbs solvation‐free energy (−18.06 to −21.89 kcal mol−1) ensures facile film formation and processability. In conclusion, this approach recommends these flourene‐based HTMs are highly desireable for forthcoming solar cell technology.

Strong Pairing Originated from an Emergent Z_{2} Berry Phase in La_{3}Ni_{2}O_{7}.

The recent discovery of high-temperature superconductivity in La_{3}Ni_{2}O_{7} offers a fresh platform for exploring unconventional pairing mechanisms. Starting with the basic argument that the electrons in d_{z^{2}} orbitals nearly form local moments, we examine the effect of the Hubbard interaction U on the binding strength of Cooper pairs based on a single-orbital bilayer model with intralayer hopping t_{∥} and interlayer superexchange J_{⊥}. By extensive density matrix renormalization group calculations, we observe a remarkable enhancement in binding energy as much as 10-20 times larger with U/t_{∥} increasing from 0 to 12 at J_{⊥}/t_{∥}∼1. We demonstrate that such a substantial enhancement stems from a kinetic-energy-driven mechanism. Specifically, a Z_{2} Berry phase will emerge at large U due to the Hilbert space restriction (Mottness), which strongly suppresses the mobility of single particle propagation as compared to U=0. However, the kinetic energy of the electrons (holes) can be greatly restored by forming an interlayer spin-singlet pairing, which naturally results in a superconducting state even for relatively small J_{⊥}. An effective hard-core bosonic model is further proposed to estimate the superconducting transition temperature at the mean-field level.

Phases and coherence of strongly interacting finite bosonic systems in shallow optical lattice

We explore the ground states of strongly interacting bosons in the vanishingly small and weak lattices using the multiconfiguration time-dependent Hartree method for bosons (MCTDHB) which calculate numerically exact many-body wave function. Two new many-body phases: fragmented or quasi superfluid (QSF) and incomplete fragmented Mott or quasi Mott insulator (QMI) are emerged due to the strong interplay between short-range contact interaction and lattice depth. Fragmentation is utilized as a figure of merit to distinguish these two new phases. We utilize the eigenvalues of the reduced one-body density matrix and define an order parameter that characterizes the pathway from a very weak lattice to a deep lattice. We provide a detailed investigation through the measures of one- and two-body correlations and information entropy. We find that the structures in one- and two-body coherence are good markers to understand the gradual built-up of intra-well correlation and decay of inter-well correlation with increase in lattice depth. For the dipolar interaction, the many-body features become more distinct and true Mott state can appear even in a shallow lattice. Whereas, for incommensurate fraction of particles, incomplete localization happens that exhibits distinct features in the measure of two-body coherence.

Quantum embedding method with transformer neural network quantum states for strongly correlated materials

The neural-network quantum states (NNQS) method is rapidly emerging as a powerful tool in quantum mechanisms. While significant advancements have been achieved in simulating simple molecules using NNQS, the ab initio simulation of complex solid-state materials remains challenging. Here in this work, we have adopted the periodic density matrix embedding theory to extend the NNQS method to deal with complex solid-state systems. Our approach notably reduces the computational problem size while maintaining high accuracy. We have validated the accuracy and efficiency of our method against traditional methodologies and experimental data in extended systems, and have investigated the magnetic ordering and charge density wave state in transition metal compounds. The findings from our research indicate that the integration of quantum embedding with intuitive chemical fragmentation can significantly enhance the NNQS simulation of realistic materials.

Abstract B081: In situ multi-modal characterization of pancreatic ductal adenocarcinoma reveals tumor cell identity as a defining factor of the surrounding microenvironment

Abstract Pancreatic Ductal Adenocarcinoma (PDAC) is heterogeneous with low tumor purity, prominent microenvironment and complex architecture, which preclude identification of shared tumor intrinsic biology within and across patients. We overcame these challenges by applying complementary spatial omics approaches – providing necessary resolution and context – to primary untreated PDAC tumors from 39 patients capturing 341,949 low-bulk and 531,718 single-cell spatial transcriptomes. Of these, 59,403 low-bulk profiles and 205,665 single-cells represented tumor cells. We leveraged this data to build on prior reports of tumor cell subtypes in PDAC. We identify 5 distinct malignant subtypes, excluding ductal-like and intraepithelial neoplasia cells, that span the classical-to-basal spectrum. One tumor subtype is highly proliferative and enriched for the DNA synthesis-condensation-mitosis transcriptional network suggesting this subset disproportionately contribute to tumor growth. Strikingly, the spectrum of tumor cell subtypes was present within all patients, suggesting that subtype-based treatment stratification may need refinement. We find that each subtype has its own distinct regulators and histology. The classical subtype has extensive cell intrinsic regulation, higher cell density and lower extracellular matrix content. In contrast, the basal subtype phenotype results from a combination of cell intrinsic and tumor microenvironment (TME) interactions, wherein the basal subtype has a lower cell density, is surrounded by activated stroma and dense collagen matrix, and is poised for hypoxic adaptation. We derive the composition of cellular neighborhoods providing an explanation for the complex architecture seen in PDAC. Tumor cell neighborhoods stratified based on which side of the classical-to-basal spectrum they fell. For example, cells towards the classical-side of the spectrum had more homogenous tumor neighborhoods that generally lacked intermediate and basal tumor cells. In contrast, cells towards the basal-side of the spectrum were likely to contain classical neighbors. More broadly, neighborhoods along this continuum also contained different proportions of immune and stromal cells, cell densities, collagen deposition, and TME adaptation. These observations support a model where tumor subtypes exist within disparate niches. Lastly, our atlas unifies the catalog of previously reported stromal cells (fibroblasts and stellate cells versus myofibroblastic and inflammatory cancer associated fibroblasts (CAF)) and describes a novel interferon stimulated gene (ISG) resistance signature (ISG.RS) fibroblast. We map the location of these various stromal cells relative to tumor subtypes. For example, myofibroblast CAFs and ISG.RS fibroblasts are enriched near the basal tumor subtype whereas inflammatory CAF abundance increases with distance away from tumor cells. In sum, our atlas provides a lens for stratifying tumor complexity in a cancer cell centric manner and provides a path for testing therapeutic hypothesis in the right cell subtype and context. Citation Format: Brian Rabe, Anna Lyubetskaya, Andrew Kavran, Yulong Bai, Andrew Fisher, Hannah Pliner, Alba Font-Tello, Anne Lewin, Ruifeng Hu, Alexandre P Alloy, Yelena Cheng, Chao Dai, Yunfan Fan, Constance Brett, Todd Brett, Lauren Giampapa, Soeren Stahlschmidt, Fayaz Seifuddin, Steven Vasquez Grinnell, Daniel Carrera, Carlos Rios, Tom Lila, Pradeep Kar, Abhishek Shukla, Rachael Bashford Rogers, Lara Heij, Mike Mason, Ryan Golhar, Isaac Neuhaus, Enas Abu Shah, Shivan Sivakumar, Jimena Trillo Tinoco, Benjamin J Chen, Konstantinos J Mavrakis, Eugene Drokhlyansky. In situ multi-modal characterization of pancreatic ductal adenocarcinoma reveals tumor cell identity as a defining factor of the surrounding microenvironment [abstract]. In: Proceedings of the AACR Special Conference in Cancer Research: Advances in Pancreatic Cancer Research; 2024 Sep 15-18; Boston, MA. Philadelphia (PA): AACR; Cancer Res 2024;84(17 Suppl_2):Abstract nr B081.

Bundled matrix product states represent low-energy excitations faithfully

Abstract We consider a set of density matrices. All of which are written in the same orbital basis, but the orbital basis size is less than the total Hilbert space size. We ask how each density matrix is related to each of the others by establishing a norm between density matrices based on the truncation error in a partial trace for a small set of orbitals. We find that states with large energy differences must have large differences in their density matrices. Small energy differences are divided into two groups, one where two density matrices have small differences and another where they are very different, as is the case of symmetry. We extend these ideas to a bundle of matrix product states and show that bond dimension of the wavefunction ansatz for two states with large energy differences are larger. Meanwhile, low energy differences can have nearly the same bond dimensions for similar states.

Remote sensing ghost imaging based on Hadamard modulated Gaussian array beam

A ghost imaging (GI) scheme by using Hadamard modulated Gaussian array beam (HMGAB) is proposed. Compared to traditional light sources modulated using ground glass, DMD, or SLM, HMGAB has higher emitting energy, making it advantageous for remote sensing GI. Moreover, the incoherent superposition of array sub-beams makes our scheme avoid the periodic aliasing which usually occurs in GI systems using optical phased array beam. The relationship between imaging maximum distance and target size is established and verified. In addition, GANs are used to address the issue of point-like reconstructed images caused by the spacing between array sub-beams.

Time Resolved Quantum Tomography in Molecular Spectroscopy by the Maximal Entropy Approach.

Attosecond science offers unprecedented precision in probing the initial moments of chemical reactions, revealing the dynamics of molecular electrons that shape reaction pathways. A fundamental question emerges: what role, if any, do quantum coherences between molecular electron states play in photochemical reactions? Answering this question necessitates quantum tomography─the determination of the electronic density matrix from experimental data, where the off-diagonal elements represent these coherences. The Maximal Entropy (MaxEnt) based Quantum State Tomography (QST) approach offers unique advantages in studying molecular dynamics, particularly with partial tomographic data. Here, we explore the application of MaxEnt-based QST on photoexcited ammonia, necessitating the operator form of observables specific to the performed measurements. We present two methodologies for constructing these operators: one leveraging Molecular Angular Distribution Moments (MADMs) which accurately capture the orientation-dependent vibronic dynamics of molecules and another utilizing Angular Momentum Coherence Operators to construct measurement operators for the full rovibronic density matrix in the symmetric top basis. A key revelation of our study is the direct link between Lagrange multipliers in the MaxEnt formalism and the unique set of MADMs. Additionally, we visualize the electron density within the molecular frame, demonstrating charge migration across the molecule. Furthermore, we achieve a groundbreaking milestone by constructing, for the first time, the entanglement entropy of the electronic subsystem─a metric that was previously inaccessible. The entropy vividly reveals and quantifies the effects of coupling between the excited electron and nuclear degrees of freedom. Consequently, our findings open new avenues for research in ultrafast molecular spectroscopy within the broader domain of quantum information science, offering profound implications for the study of molecular systems under excitation using quantum tomographic schemes.

Capacity of entanglement and volume law

We investigate various aspects of capacity of entanglement in certain setups whose entanglement entropy becomes extensive and obeys a volume law. In particular, considering geometric decomposition of the Hilbert space, we study this measure both in the vacuum state of a family of non-local scalar theories and also in the squeezed states of a local scalar theory. We also evaluate field space capacity of entanglement between interacting scalar field theories. We present both analytical and numerical evidences for the volume law scaling of this quantity in different setups and discuss how these results are consistent with the behavior of other entanglement measures including Renyi entropies. Our study reveals some generic properties of the capacity of entanglement and the corresponding reduced density matrix in the specific regimes of the parameter space. Finally, by comparing entanglement entropy and capacity of entanglement, we discuss some implications of our results on the existence of consistent holographic duals for the models in question.

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Generalized prismatic tensegrity derived by dihedral symmetric lines

Classic prismatic tensegrity structures, characterized by dihedral symmetry with one orbit of nodes, are among the simplest and possibly the earliest spatial tensegrity structures invented. This paper introduces a generalized form of the prismatic tensegrity structures by converting a single-loop linkage into truss, in which the lines of joint axes rather than the nodes have dihedral symmetry. Since the vector space formed by the line coordinates of these joints has rank degeneracy one, the generated tensegrity structures are kinematically and statically indeterminate. These tensegrity structures are further proved to be prestress-stable, generally, for the total or partial parameter space of lines based on a second-order analysis of screws, and are called dihedral-line tensegrity structures in this paper. Specifically, this paper focuses on symmetric dihedral-line tensegrity structures, in which the nodes also have dihedral symmetry but in two orbits and members in seven orbits, and are called two-orbit dihedral-line tensegrity structures. It is found that there are at least N struts for the generated tensegrity with DN symmetry. And the classic prismatic tensegrity structures can be recovered from these dihedral-line tensegrity structures by removing certain zero-force members. Symmetric-adapted force density matrices are also provided as well as the relation to that of the classic prismatic tensegrity. Given 4N+6 dimensional parameters inherent to these tensegrity structures, a rich variety of tensegrity structure family is presented.

Quantum state estimation based on deep learning

Abstract We used deep learning techniques to construct various models for reconstructing quantum states from a given set of coincidence measurements. Through simulations, we demonstrated that our approach generates functionally equivalent reconstructed states for a wide range of pure and mixed input states. Compared to traditional methods, our system offers the advantage of faster speed. Additionally, by training our system with measurement results containing simulated noise sources, we could show a significant improvement in average fidelity compared to typical reconstruction methods. We also found that constraining the variational manifold to physical states, i.e., positive semi-definite density matrices, greatly enhances the quality of the reconstructed states in the presence of experimental imperfections and noise. Finally, we validated the correctness and superiority of our model by using data generated on IBMQ, a real quantum computer.

Preserving the Hermiticity of the one-body density matrix for a non-interacting Fermi gas

Abstract The one-body density matrix (ODM) for a zero temperature non-interacting Fermi gas can be approximately obtained in the semiclassical regime through different ℏ -expansion techniques. One would expect that each method of approximating the ODM should yield equivalent density matrices which are both Hermitian and idempotent to any order in ℏ . However, the Kirzhnits and Wigner–Kirkwood methods do not yield these properties, while the Grammaticos–Voros method does. Here we show explicitly, for arbitrary d-dimensions through an appropriate change into symmetric coordinates, that each method is indeed identical, Hermitian, and idempotent. This change of variables resolves the inconsistencies between the various methods, showing that the non-Hermitian and non-idempotent behavior of the Kirzhnits and Wigner–Kirkwood methods is an artifact of performing a non-symmetric truncation to the semiclassical ℏ -expansions. Our work also provides the first explicit derivation of the d-dimensional Grammaticos–Voros ODM, originally proposed by Redjati et al (2019 J. Phys. Chem. Solids 134 313–8) based on their d = 1 , 2 , 3 , 4 expressions.

Revealing discontinuous and continuous quantum phase transitions in shaken optical lattices

We perform the cluster mean-field with density matrix renormalization (CMFT+DMRG) calculation on the two-band extended Bose–Hubbard model to uncover the physics behind the discontinuous transitions observed in the one-dimensional shaken optical lattice with hybridized two lowest Bloch bands. We determine the superfluid, Mott insulator, and staggered superfluid phases associated with this model using the appropriate order parameters. We obtained the phase diagrams for two values of shaking amplitudes and illustrated the phases and nature of phase transitions. The transition from Mott insulator and superfluid to staggered superfluid is discontinuous for small shaking amplitudes. We analyse how the shaking frequency controls the effective chemical potential in the model and, consequently, the ground state energy that drives the discontinuous transition. Finally, we compare our results with those of earlier works.