Direct Diagonalization Research Articles

Combining the in-medium similarity renormalization group with the density matrix renormalization group: Shell structure and information entropy

We propose a novel many-body framework combining the density matrix renormalization group (DMRG) with the valence-space (VS) formulation of the in-medium similarity renormalization group. This hybrid scheme admits for favorable computational scaling in large-space calculations compared to direct diagonalization. The capacity of the VS-DMRG approach is highlighted in ab initio calculations of neutron-rich nickel isotopes based on chiral two- and three-nucleon interactions, and allows us to perform converged ab initio computations of ground and excited state energies. We also study orbital entanglement in the VS-DMRG, and investigate nuclear correlation effects in oxygen, neon, and magnesium isotopes. The explored entanglement measures reveal nuclear shell closures as well as pairing correlations.

Non-coplanar magnetism, topological density wave order and emergent symmetry at half-integer filling of moiré Chern bands

Twisted double- and mono-bilayer graphene are graphene-based moiré materials hosting strongly correlated fermions in a gate-tunable conduction band with a topologically non-trivial character. Using unbiased exact diagonalization complemented by unrestricted Hartree-Fock calculations, we find that the strong electron-electron interactions lead to a non-coplanar magnetic state, which has the same symmetries as the tetrahedral antiferromagnet on the triangular lattice and can be thought of as a skyrmion lattice commensurate with the moiré scale, competing with a set of ferromagnetic, topological charge density waves featuring an approximate emergent O(3) symmetry, “rotating” the different charge density wave states into each other. Direct comparison with exact diagonalization reveals that the ordered phases are accurately described within the unrestricted Hartree-Fock approximation. Exhibiting a finite charge gap and Chern number |C|=1, the formation of charge density wave order which is intimately connected to a skyrmion lattice phase is consistent with recent experiments on these systems.

Quantum phase transitions in non-Hermitian [formula omitted]-symmetric transverse-field Ising spin chains

We present a theoretical study of quantum phases and quantum phase transitions occurring in non-Hermitian PT-symmetric superconducting qubits chains described by a transverse-field Ising spin model. A non-Hermitian part of the Hamiltonian is implemented via imaginary staggered longitudinal magnetic field, which corresponds to a local staggered gain and loss terms. By making use of a direct numerical diagonalization of the Hamiltonian for spin chains of a finite size N, we explore the dependencies of the energy spectrum, including the energy difference between the first excited and the ground states, the spatial correlation function of local polarization (z-component of local magnetization) on the adjacent spins interaction strength J and the local gain (loss) parameter γ. A scaling procedure for the correlation length ξ allows us to establish a complete quantum phase diagram of the system. We obtain two quantum phases for J<0, namely, PT-symmetry broken antiferromagnetic state and PT-symmetry preserved paramagnetic state, and the quantum phase transition line between them is the line of exception points. For J>0 the PT-symmetry of the ground state is retained in a whole region of parameter space of J and γ, and a system shows two intriguing quantum phase transitions between ferromagnetic and paramagnetic states for a fixed parameter γ>1. We also provide the qualitative quantum phase diagram γ−J derived in the framework of the Bethe–Peierls approximation that is in a good accord with numerically obtained results.

Revisiting semiconductor bulk hamiltonians using quantum computers

With the advent of near-term quantum computers, it is now possible to simulate solid-state properties using quantum algorithms. By an adequate description of the system's Hamiltonian, variational methods enable to fetch of the band structure and other fundamental properties as transition probabilities. Here, we describe semiconductor structures of the III-V family using k·p Hamiltonians and obtain their band structures using a state vector solver, a probabilistic simulator, and a real noisy-device simulator. The resulting band structures are in good agreement with those obtained by direct diagonalization of the Hamiltonian. The simulation times depend on the optimizer, circuit depth, and simulator used. Finally, with the optimized eigenstates, we convey the inter-band absorption probability, demonstrating the possibility of analyzing the fundamental properties of crystalline systems using quantum computers.

Interaction of two Rydberg atoms in the vicinity of an optical nanofibre

We consider two rubidium atoms, prepared in the same S or P Rydberg states, near an optical nanofibre, and we determine their van der Waals interaction potential as a function of their separation along the nanofibre axis, their distance to the nanofibre axis, and their relative azimuthal angle. We compare results obtained through direct diagonalisation of the Hamiltonian (including quadrupolar interaction terms) with second-order perturbation calculations, and we identify which couplings mainly contribute to the potential in the presence of the nanofibre and in free-space. We relate the appearance of new allowed couplings to the broken rotation symmetry around the interatomic axis due to the presence of the fibre. These couplings induce novel features and cause a reshaping of the interaction anisotropy and formation of an interaction potential well for P states near the nanofibre. Our work constitutes an important step in the assessment of Rydberg atom-nanofibre quantum interfaces and devices.

Kondo resonance effects in emergent flat band materials

Macroscopic degrees of freedom that are involved in the transport of carriers through mesoscopic electronic devices are susceptible to the effects of strong many-body correlations. The presence of magnetic impurities in dilute magnetic alloys typically allow for insights into Kondo effect from the scattering of free carriers by localized electron states of the magnetic impurities but this effect is not well understood when there are no d-band electron states. Herein, the signatures of Kondo resonance effect are elucidated in quantum dots derived from a carbon-nanoline embedded monolayer hexagonal boron nitride whose electron states host flat band ferromagnetism as distinct broken symmetry states. Quantum transport state of mesoscopic devices modelled as quantum dots tunnel coupled to metallic leads is computed by direct diagonalization of the Hamiltonian. The possibility of realizing quantum dots with highly tunable electron states in energy interconversion devices is discussed to show the importance of screening effects on single-electron energy levels. The quantum master equation is solved within different formalisms to determine the stationary-state particle and energy currents. Stability diagrams are calculated to show the dependence of the conductance on experimental control variables of the quantum dot device. The computed responses of the stationary-state transport signatures are used to characterize Kondo resonance effects from flat band states of embedded carbon nanoline-based quantum dots. It is found that the local network structure of the hexagonal ring carbon cluster-based quantum dot has a broken particle-hole symmetry in the transport state. This signals the formation of the quasiparticle states expected in second order scattering when the macroscopic “charge” pseudospin symmetry of the tunnelling electron state is broken dynamically due to charging. The results are discussed to show the implications of a vanishing particle-hole symmetry in the carrier transport state of quantum dots for energy conversion applications.

Electrostatic and Environmental Control of the Trion Fine Structure in Transition Metal Dichalcogenide Monolayers.

Charged excitons or trions are essential for optical spectra in low-dimensional doped monolayers (ML) of transitional metal dichalcogenides (TMDC). Using a direct diagonalization of the three-body Hamiltonian, we calculate the low-lying trion states in four types of TMDC MLs as a function of doping and dielectric environment. We show that the fine structure of the trion is the result of the interplay between the spin-valley fine structure of the single-particle bands and the exchange interaction. We demonstrate that by variations of the doping and dielectric environment, the fine structure of the trion energy can be tuned, leading to anticrossing of the bright and dark states, with substantial implications for the optical spectra of the TMDC ML.

Charge and spin currents in mesoscopic rings: Role of next nearest-neighbors Rashba spin–orbit coupling using the Green’s functions equation of motion technique

The electronic states of a mesoscopic ring are assessed using the Green functions approach and the equation of motion method. We put forward the Green’s function formalism to evaluate precisely persistent charge and spin currents in a ring exhibiting nearest-neighbors (NN) and next-nearest (NNN) neighbors Rashba spin–orbit interactions. Our present scheme circumvents direct diagonalization of the Hamiltonian, allowing to determine numerically persistent currents with both spin–orbit interactions simultaneously. Our results for the spin currents show that the NN Rashba interaction breaks the spin degeneracy producing a square-like shape; while the NNN Rashba interaction creates an energy-dependent splitting of the levels that generates a pattern of small peaks. The merging of both interactions rises a periodic squared-peaked pattern at half integer values of the quantum flux, which suggests the possibility of controlling the spin current magnitude and polarization by modulating the NNN Rashba parameter.

Direct diagonalization method for a few particles trapped in harmonic potentials

We describe a procedure to systematically improve direct diagonalization results for few-particle systems trapped in one-dimensional harmonic potentials interacting by contact interactions. We start from the two-body problem to define a renormalization method for the interparticle interactions. The procedure is benchmarked with state-of-the-art numerical results for three and four symmetric fermions.

Research on current decoupling control of six‐phase permanent magnet linear synchronous motor for electromagnetic launch

AbstractSix‐phase permanent magnet linear synchronous motor (PMLSM) for electromagnetic launch (EML) system presents the characteristics of a high order, nonlinearity, multivariable, strong coupling, and nonperiodic transient operation in the synchronous rotating coordinate system, posing a great challenge to the dynamic response ability of the current loop. Existing research on current decoupling control (CDC) mainly focuses on cross decoupling within a three‐phase system, even though there are neither decoupling methods for multiphase systems nor effective evaluation criteria for the decoupling and dynamic response performances. From this perspective, this paper first presents an equivalent reduced‐order complex‐matrix dynamic mathematical model of six‐phase PMLSM and analyze its transient coupling characteristics during the process of EML. Then, the CDC methods of six‐phase PMLSM based on direct compensation and matrix diagonalization principles are realized, respectively, to accomplish the cross decoupling and back electromotive force decoupling within and between different three‐phase windings. Finally, an all‐round method is proposed, for the first time, to evaluate the decoupling performances and dynamic response performances of different CDC strategies for six‐phase PMLSM. Significant superiority of deviation decoupling regulator in decoupling performance and robustness are verified based on high‐speed EML experimental platform of six‐phase PMLSM.

Space and time averaged quantum stress tensor fluctuations

We extend previous work on the numerical diagonalization of quantum stress tensor operators in the Minkowski vacuum state, which considered operators averaged in a finite time interval, to operators averaged in a finite spacetime region. Since real experiments occur over finite volumes and durations, physically meaningful fluctuations may be obtained from stress tensor operators averaged by compactly supported sampling functions in space and time. The direct diagonalization, via a Bogoliubov transformation, gives the eigenvalues and the probabilities of measuring those eigenvalues in the vacuum state, from which the underlying probability distribution can be constructed. For the normal-ordered square of the time derivative of a massless scalar field in a spherical cavity with finite degrees of freedom, analysis of the tails of these distributions confirms previous results based on the analytical treatment of the high moments. We find that the probability of large vacuum fluctuations is reduced when spatial averaging is included, but the tail still decreases more slowly than exponentially as the magnitude of the measured eigenvalues increases, suggesting vacuum fluctuations may not always be subdominant to thermal fluctuations and opening up the possibility of experimental observation under the right conditions.

Analytic approach for the number statistics of non-Hermitian random matrices.

We introduce a powerful analytic method to study the statistics of the number N_{A}(γ) of eigenvalues inside any smooth Jordan curve γ∈C for infinitely large non-Hermitian random matrices A. Our generic approach can be applied to different random matrix ensembles of a mean-field type, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable, and obtain explicit results for the diluted real Ginibre ensemble. The main outcome is an effective theory that determines the cumulant generating function of N_{A} via a path integral along γ, with the path probability distribution following from the numerical solution of a nonlinear self-consistent equation. We derive expressions for the mean and the variance of N_{A} as well as for the rate function governing rare fluctuations of N_{A}(γ). All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.

Time-reversal symmetry breaking in a square lattice

The bulk conductivity of a two-dimensional system is studied assuming that time-reversal symmetry is broken by internal mechanisms. The study is carried out by direct diagonalization in order to explore the nonlinear-response provoked by the inclusion of an electric field in the system’s Hamiltonian. The system displays a quantized conductivity that depends on the intensity of the field and under specific conditions the conductivity limit at zero electric field displays a nonvanishing value.

Frequency filtering and vibrational spectrum of multiple-valley one-dimensional phononic crystals

In the present study, we have fixed the number of atoms in a one-dimensional phononic chain and changed the number of the valleys (or numbers of wells NOW) in it (by adding new terms to the system Hamiltonian) and studied the transmission properties of his system. Then, we have obtained the phononic transmission coefficient by using a one-dimensional transfer matrix method. We have also investigated the vibrational modes of the assumed system by using a direct diagonalization technique. We show that in our assumed system, there is an Omni-k bandgap within which bandgap exists for all values of the studied spring constant k. In our assumed system, we have also obtained some bi-convex lens-shaped phononic bandgaps as a function of the parameter $$V_{{{\text{conf}}}}$$ which their surface areas can be tuned by changing the number of valleys NOW. An interesting fact is that the transmission coefficient is found to be quantized versus variation of the number of wells NOW which may help experimentalists to use lower values of the material to fabricate their desired system (see the text). We observe that the transmission coefficients as a function of the NOW are some Cantor-like ones that obey self-similar bandgap structure patterns. Besides, we show that NOW and $$V_{conf}$$ are good tuning parameters that can localize the eigenmodes of the system in any part of the chain. We have provided some animation to more clearly illustrate the effect of different parameters on the transmission and vibrational properties of the system. Finally, by using a few parameters, which we have presented in this paper, we can do bandgap engineering and determine the bandgap widths, as well as their places on the frequency axis.

Exploring the State Space Structure of Multiple Spins via Modular Tensor Diagram Approach: Going beyond the Exciton Pair State.

Fully understanding of multistate quantum systems could become formidable if not impossible as the system dimensionality increases. One ideal strategy to comprehend complex systems is to transform the system representation into a more structural one so that major characteristics, connections, and even underlying mechanisms can stand out from the huge unstructured information, e.g., the construction of spin eigenfunctions for a system of multiple spins through the diagonalization of the system Hamiltonian matrix. Here, instead of direct matrix diagonalization, the recently developed modular tensor diagram approach is applied to reorganize the state space structure of multispin systems, extending previous investigations on exciton pair states to exciton trimer states. This implementation demonstrates that the proposed approach not only provides a systematical way to transform the high dimensional multistate system into a well organized structure based on basic (exciton) modules but also paves the way to further analysis on potential applications. For example, the analysis on the state space of the exciton trimer system suggests a possible scheme to improve the laser performance via single fission involving multiexcitations and/or multiple fission steps.

Electronic structure with direct diagonalization on a D-wave quantum annealer

Quantum chemistry is regarded to be one of the first disciplines that will be revolutionized by quantum computing. Although universal quantum computers of practical scale may be years away, various approaches are currently being pursued to solve quantum chemistry problems on near-term gate-based quantum computers and quantum annealers by developing the appropriate algorithm and software base. This work implements the general Quantum Annealer Eigensolver (QAE) algorithm to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on a D-Wave 2000Q quantum annealer. The approach is based on the matrix formulation, efficiently uses qubit resources based on a power-of-two encoding scheme and is hardware-dominant relying on only one classically optimized parameter. We demonstrate the use of D-Wave hardware for obtaining ground and excited electronic states across a variety of small molecular systems. The approach can be adapted for use by a vast majority of electronic structure methods currently implemented in conventional quantum-chemical packages. The results of this work will encourage further development of software such as qbsolv which has promising applications in emerging quantum information processing hardware and has expectation to address large and complex optimization problems intractable for classical computers.

Localization, transport, and edge states in a two-strand ladder network in an aperiodically staggered magnetic field

We investigate the spectral and transport properties of a two-arm tight-binding ladder perturbed by an external magnetic field following an Aubry-Andr\'e-Harper profile. The varying magnetic flux trapped in consecutive ladder-cells simulates an axial twist that enables us, in principle, to probe a wide variety of systems ranging from a ribbon Hofstadter geometry to helical DNA chains. We perform an in-depth numerical analysis, using a direct diagonalization of the lattice Hamiltonian to study the electronic spectra and transport properties of the model. We show that such a geometry creates a self-similar multifractal pattern in the energy landscape. The spectral properties are analyzed using the local density of states and a Green's function formalism is employed to obtain the two-terminal transmission probability. With the standard multifractal analysis and the evaluation of inverse participation ratio we show that, the system hosts both critical and extended phase for a slowly varying aperiodic sequence of flux indicating a possible mobility edge. Finally, we report signatures of topological edge modes that are found to be robust against a correlated perturbation given to the nearest neighbor hopping integrals. Our results can be of importance in experiments involving ladder-like quantum networks, realized with cold atoms in an optical trap setup.

H2O-CO and D2O-CO complexes: Intra- and intermolecular rovibrational states from full-dimensional and fully coupled quantum calculations.

We present efficient yet rigorous, full-dimensional quantum bound-state calculations of the fully coupled J = 0 and one intra- and intermolecular rovibrational levels of H2O-CO and D2O-CO complexes. The new ab initio nine-dimensional (9D) potential energy surface (PES) [Y. Liu and J. Li, Phys. Chem. Chem. Phys. 21, 24101 (2019)] is employed. In the spirit of the recently introduced general procedure [P. M. Felker and Z. Bačić, J. Chem. Phys. 151, 024305 (2019)], the 9D rovibrational Hamiltonian is partitioned into a 5D (rigid-monomer) intermolecular Hamiltonian, two intramolecular vibrational Hamiltonians-one for the water monomer (3D) and another for the CO monomer (1D), and a 9D remainder term. The low-energy eigenstates of the three reduced-dimension Hamiltonians are used to build up the 9D product contracted basis, in which the matrix of the full rovibrational Hamiltonian is diagonalized. In line with the findings of our earlier study referenced above, the 5D intermolecular eigenstates included in the 9D bases extend up to at most 230 cm-1 above the lowest-energy state of the given parity, much less than the intramolecular fundamentals of the two complexes that span the range of energies from about 1200 cm-1 to 3800 cm-1. The resulting Hamiltonian matrices are small for the 9D quantum problem considered, ≈ 10 000 for J = 0 and 13 500 for J = 1 calculations, allowing for direct diagonalization. The 9D calculations permit exploring a number of features of the rovibrational level structure of H2O-CO and D2O-CO that are beyond the quantum 5D rigid-monomer treatments reported to date. These include the differences in the magnitudes of the hydrogen-exchange tunneling splittings computed in 9D and 5D, the sensitivity of the tunneling splittings to the intramolecular vibrational excitation, the frequency shifts of the intramolecular vibrational modes, which, depending on the mode, can be either blue- or redshifts, and the effects of the excitation of the intramolecular fundamentals on the low-lying intermolecular eigenstates. Also examined is the extent of the eigenstate delocalization over the two minima on the PES. Whenever possible, a comparison is made with the experimental data in the literature.

Scaling up the lattice dynamics of amorphous materials by orders of magnitude

We generalise the non-affine theory of viscoelasticity for use with large, well-sampled systems of arbitrary chemical complexity. Having in mind predictions of mechanical and vibrational properties of amorphous systems with atomistic resolution, we propose an extension of the Kernel Polynomial Method (KPM) for the computation of the vibrational density of states (VDOS) and the eigenmodes, including the $\Gamma$-correlator of the affine force-field, which is a key ingredient of lattice-dynamic calculations of viscoelasticity. We show that the results converge well to the solution obtained by direct diagonalization (DD) of the Hessian (dynamical) matrix. As is well known, the DD approach has prohibitively high computational requirements for systems with $N=10^4$ atoms or larger. Instead, the KPM approach developed here allows one to scale up lattice dynamic calculations of real materials up to $10^6$ atoms, with a hugely more favorable (linear) scaling of computation time and memory consumption with $N$.

Static and Dynamic Properties of a Few Spin 1/2 Interacting Fermions Trapped in a Harmonic Potential

We provide a detailed study of the properties of a few interacting spin 1 / 2 fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The N = 2 case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for N = 2 , 3 , 4 and 5 particles, e.g., low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.