Ground State Wave Function Research Articles

Localization of nuclear wave functions of lithium in [Li+@C60]PF6-: molecular insights into two-site disorder-order transition.

Lithium endohedral fullerene, Li+@C60, is a porous system ideal for studying the quantized translational motion of the Li+ nucleus under subnanoscale confinement. The quantized nuclear motion strongly depends on the anharmonic and polarizable adsorbent potential within the C60 cage, which can be perturbed by cage distortion and/or exterior ions. In our recent paper, H. Ando and Y. Nakao, Phys. Chem. Chem. Phys., 2021, 23, 9785-9803, we focused on a [Li+@C60]PF6- salt and theoretically investigated how the Li+ ion in each C60 cage is simultaneously localized at two equivalent disordered sites in 24 K < T ≪ 100 K. At 24 K, the salt exhibits a disorder-order transition, whereby every Li+ ion becomes mostly localized at one of the two disordered sites below that temperature. Herein we discuss the origin of this transition with special attention to the local structural distortion and intermolecular interactions. Using the Fourier grid Hamiltonian method and a model function that fits a post-Hartree-Fock potential energy surface, we obtained hundreds of low-energy nuclear wave functions of Li+ confined within the cage. The weak distortions of the C60 cage and the PF6- coordination sphere below 24 K and concurrent inversion-symmetry breaking affect intermolecular interactions, thus making the wave functions of the nuclear ground state and several low-energy excited states localized around the experimental high-occupancy disordered site. This demonstrates that the distortions correlate closely with the two-site disorder-order transition. Finally, we reveal that two absorption peaks in the terahertz frequency range show substantial blueshifts upon cooling below 24 K, which serve as fingerprints of the transition.

Characterizing Superradiant Phase of the Quantum Rabi Model

Recently, a superradiant phase transition first predicted theoretically in the quantum Rabi model (QRM) has been verified experimentally. This further stimulates the interest in the study of the process of phase transition and the nature of the superradiant phase since the fundamental role of the QRM in describing the interaction of light and matter, and more importantly, the QRM contains rich physics deserving further exploration despite its simplicity. Here we propose a scheme consisting of two successive diagonalizations to accurately obtain the ground-state and excited states wavefunctions of the QRM in full parameter regime ranging from weak to deep-strong couplings. Thus, one is able to see how the phase transition occurs and how the photons populate in Fock space of the superradiant phase. We characterize the photon populations by borrowing the distribution concept in random matrix theory and find that the photon population follows a Poissonian-like distribution once the phase transition takes place and further exhibits the statistics of Gaussian unitary ensemble with increasing coupling strength. More interestingly, the photons in the excited states behave even like the statistics of Gaussian orthogonal ensemble. Our results not only deepen understanding on the superradiant phase transition but also provide an insight on the nature of the superradiant phase of the QRM and related models.

Solving the Schrödinger equation computationally using the Lanczos algorithm

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of particles in terms of wave functions. These wave functions provide a probabilistic description of where the particle is likely to be found, making them a crucial tool for understanding how particles interact with their environment. Methods for solving the Schrödinger equation analytically tend to be very mathematically complex, leading us to look for other ways to solve the Schrödinger equation. Unfortunately, in most cases, the Schrödinger equation cannot be solved for an exact solution. There are methods, however, for approximate solutions. In this paper, we investigated the Lanczos algorithm, a computational method that can solve the 2D Schrödinger equation in cases where finding an analytical solution would not be feasible. We discretized the Schrödinger equation, then ran a Python program using the Lanczos algorithm to solve for the ground state wave function. We hypothesized that the Schrödinger equation could be solved computationally for the ground state wave function using the Lanczos algorithm. We found that this method efficiently solved the Schrödinger equation for complicated 2D potentials. Additionally, we verified the accuracy of the method by comparing the results with wave functions from problems with known, exact solutions.

α-AsW9O33]9- bridged hexagonal clusters of Ln(III) showing field induced SMM behavior: experimental and theoretical insight.

Polyoxometalates (POM), as inorganic polydentate oxygen donors, provide binding opportunities for oxophilic lanthanide metal centers to construct novel Ln-substituted POM materials with exciting structures and attractive properties. Herein, we have reported four arsenotungstate [α-AsW9O33]9- based lanthanide-containing polyoxometalates [CsxK36-x{Ln6(H2O)12(α-AsW9O33)6}]·yH2O (Ln = Er (1), Gd (2), Ho (3), and Tb (4)), which are synthesized in an alkaline medium. Complexes 1-3 are the dimeric structures of [Ln3(H2O)6(α-AsW9O33)3]18- polyanions, whereas complex 4 is a hexamer of the polyanion [Tb (H2O)2(α-AsW9O33)]6- as a building unit. In all the complexes, [α-AsW9O33]9- units are staggered up and down and give rise to the chair conformation, where one [α-AsW9O33]9- unit bridges two Ln(III) centers through four μ2-oxygen and two terminal oxygen atoms, resulting in the hexagonal arrangement of lanthanides. The dynamic magnetic measurement indicates that only complex 1 exhibits slow relaxation of magnetization with an applied dc field (1500 Oe). To gain insight into the slow relaxation of magnetization in complex 1, the ligand-field parameters and the splitting of the ground-state multiplet of the Er(III) ions have been estimated. The ab initio calculation results confirm that the ground state wave function of these molecules (1, 3, and 4) is mainly composed of a mixture of mJ states, and the non-axial crystal field (CF) terms are more predominant than the axial CF term. The solid-state fluorescence spectra of 1-4 reveal that the photoexcitation O → M ligand-to-metal charge-transfer (LMCT) of arsenotungstate fragments is effectively quenched due to the spatial coordination environment around the Ln(III) ion.

Dynamics of spin-tensor-momentum coupled Bose-Einstein condensates

We investigate the dynamics of the plane wave state in one-dimensional spin-tensor-momentum coupled Bose-Einstein condensate. By using the Gaussian variational approximation, we first derive the equations of motion for the variational parameters, including the center-of-mass coordinate, momentum, amplitude, width, chirp, and relative phase. These variational parameters are coupled together nonlinearly by the spin-tensor-momentum coupling, Raman coupling, and the spin-dependent atomic interaction. By minimizing the energy with respect to the variational parameters, we find that the ground state is a biaxial nematic state, the momentum of the ground state decreases monotonically with the increase of the strength of the Raman coupling, and the parity of real part of the ground-state wave function is opposite to that of the imaginary part. The linear stability analysis shows that the ground state is dynamically stable under a perturbation, and exhibits three different oscillation excitation modes, the frequencies of which are related to the strength of the Raman coupling, the aspect ratio of the harmonic trap, and the strength of the atomic interaction. By solving the equations of motion for the variational parameters, we find that the system displays periodical oscillation in the dynamical evolution. These variational results are also confirmed by the direct numerical simulations of the Gross-Pitaevskii equations, and these findings reveal the unique properties given by the spin-tensor-momentum coupling.

Revisited relativistic Dirac-Hartree-Fock X-ray scattering factors. I. Neutral atoms with Z = 2-118.

In this first of a series of publications, the X-ray scattering factors for neutral atoms are revisited. Using the recently developed DBSR_HF program [Zatsarinny & Froese Fischer (2016). Comput. Phys. Comm. 202, 287-303] the fully relativistic Dirac-Hartree-Fock ground-state wavefunctions for all atoms with Z = 2-118 (He-Og) have been calculated using the extended average level scheme and including both the Breit interaction correction to the electronic motion due to magnetic and retardation effects, and the Fermi distribution function for the description of the nuclear charge density. The comparison of our wavefunctions with those obtained in several previous studies in terms of the total and orbital (spinor) electronic energies, and a number of local and integrated total and orbital properties, confirmed the quality of the generated wavefunctions. The employed dense radial grid combined with the DBSR_HF's B-spline representation of the relativistic one-electron orbitals allowed for a precise integration of the X-ray scattering factors using a newly developed Fortran program SF. Following the established procedure [Maslen et al. (2006). International Tables for Crystallography, Vol. C, Section 6.1.1, pp. 554-589], the resulting X-ray scattering factors have been interpolated in the 0 ≤ sin θ/λ ≤ 2 Å-1 and 2 ≤ sin θ/λ ≤ 6 Å-1 ranges using the recommended analytical functions with both the four- (which is a current convention) and five-term expansions. An exhaustive comparison of the newly generated X-ray scattering factors with the International Union of Crystallography recommended values and those from a number of previous studies showed an overall good agreement and allowed identification of a number of typos and inconsistencies in the recommended quantities. A detailed analysis of the results suggests that the newly derived values may represent an excellent compromise among all the previous studies. The determined conventional interpolating functions for the two sin θ/λ intervals show, on average, the same accuracy as the recommended parametrizations. However, an extension of each expansion by only a single term provides a significant improvement in the accuracy of the interpolated values for an overwhelming majority of the atoms. As such, an updated set of the fully relativistic X-ray scattering factors and the interpolating functions for neutral atoms with Z = 2-118 can be easily incorporated into the existing X-ray diffraction software with only minor modifications. The outcomes of the undertaken research should be of interest to members of the crystallographic community who push the boundaries of the accuracy and precision of X-ray diffraction studies.

Phase transitions and topological properties of the 52 quantum Hall states with strong Landau-level mixing

We numerically study a $\frac{5}{2}$ fractional quantum Hall system with even number of electrons using the exact diagonalization where both the strong Landau-level (LL) mixing and a finite width of the quantum well have been considered and adapted into a screened Coulomb interaction. With the principal component analysis, we are able to recognize a compressible-incompressible phase transition in the parameter space made of the magnetic field and the quantum well width by the competition between the first two leading components of the ground-state wave functions, which is consistent with the low-lying spectral feature and previous works in the odd-electron system. In addition, the presence of the subdominant third component suggests an incompressible transition occurring as the LL mixing strength grows into a certain parameter region associated with the ZnO experiments. We further investigate the strongly LL-mixed phase in this emerging region with the Hall viscosity, wave function overlaps, and the entanglement spectra. Results show it can be well described as a particle-hole symmetrized Pfaffian state with the dual topological properties of the Pfaffian and the anti-Pfaffian states.

From Entanglement Generated Dynamics to the Gravitational Anomaly and Chiral Central Charge.

We apply modular flow-entanglement generated dynamics-to characterize quantum orders of ground state wave functions. In particular, we study the linear response of the entanglement entropy of a simply connected region with respect to modular flow. First, we apply it to (1+1)D conformal field theories and demonstrate its relationship to the chiral central charge-or, equivalently, the perturbative gravitational anomaly-which is shown to vanish. Next, we apply it to (2+1)D gapped ground states where it reduces to a recently proposed formula by Kim etal. that is conjectured to compute the edge chiral central charge. Modular flow provides an intuitive picture for this conjecture based on bulk-edge correspondence. We also provide numerics on free-fermion models that lend support to our picture.

Exploring 2 + 2 answers to 3 + 1 questions

We explore potential uses of physics formulated in Kleinian (i.e. 2 + 2) signature spacetimes as a tool for understanding properties of physics in Lorentzian (i.e. 3 + 1) signature. Much as Euclidean (i.e. 4 + 0) signature quantities can be used to formally construct the ground state wavefunction of a Lorentzian signature quantum field theory, a similar analytic continuation to Kleinian signature constructs a state of low particle flux in the direction of analytic continuation. There is also a natural supersymmetry algebra available in 2 + 2 signature, which serves to constrain the structure of correlation functions. Spontaneous breaking of Lorentz symmetry can produce various [Formula: see text] supersymmetry algebras that in 3 + 1 signature correspond to nonsupersymmetric systems. We speculate on the possible role of these structures in addressing the cosmological constant problem.

N-body correlation of Tonks–Girardeau gas

For the well-known exponential complexity it is a giant challenge to calculate the correlation function for general many-body wave function. We investigate the ground state $n$th-order correlation functions of the Tonks-Girardeau (TG) gases. Basing on the wavefunction of free fermions and Bose-Fermi mapping method we obtain the exact ground state wavefunction of TG gases. Utilizing the properties of Vandermonde determinant and Toeplitz matrix, the $n$th-order correlation function is formulated as $(N-n)$-order Toeplitz determinant, whose element is the integral dependent on 2$(N-n)$ sign functions and can be computed analytically. By reducing the integral on domain $[0,2\pi]$ into the summation of the integral on several independent domains, we obtain the explicit form of the Toeplitz matrix element ultimately. As the applications we deduce the concise formula of the reduced two-body density matrix and discuss its properties. The corresponding natural orbitals and their occupation distribution are plotted. Furthermore, we give a concise formula of the reduced three-body density matrix and discuss its properties. It is shown that in the successive second measurements, atoms appear in the regions where atoms populate with the maximum probability in the first measurement.

Variational tensor network operator

We propose a simple and generic construction of the variational tensor network operators to study the quantum spin systems by the synergy of ideas from the imaginary-time evolution and the variational optimization of trial wave functions. By applying these operators to simple initial states, accurate variational ground-state wave functions with extremely few parameters can be obtained. Furthermore, the framework can be applied to spontaneously study symmetry-breaking, symmetry-protected topological, and intrinsic topologically ordered phases, and we show that symmetries of the local tensors associated with these phases can emerge directly after the optimization without any gauge fixing. This provides a universal way to identify quantum phase transitions without prior knowledge of the system.

Classical vertex model dualities in a family of two-dimensional frustrated quantum antiferromagnets

We study a general class of easy-axis spin models on a lattice of corner sharing even-sided polygons with all-to-all interactions within a plaquette. The low energy description corresponds to a quantum dimer model on a dual lattice of even coordination number with a multi dimer constraint. At an appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the ground state wavefunction can be exactly mapped onto a classical vertex model on the dual lattice. When the dual lattice is bipartite, the vertex models are bonded and are self dual under Wegner's duality, with the self dual point corresponding to the RK point of the original multi-dimer model. We argue that the self dual point is a critical point based on known exact solutions to some of the vertex models. When the dual lattice is non-bipartite, the vertex model is arrowed, and we use numerical methods to argue that there is no phase transition as a function of the vertex weights. Motivated by these wavefunction dualities, we construct two other distinct families of frustration-free Hamiltonians whose ground states can be mapped onto these vertex models. Many of these RK Hamiltonians provably host $\mathbb{Z}_2$ topologically ordered phases.

New aspects of the low-energy structure of At211

Lifetimes of low-energy states in the semimagic nucleus $^{211}\mathrm{At}$ were measured employing the recoil-distance Doppler shift and the Doppler-shift attenuation methods. The deduced transition probabilities are compared to two shell-model calculations, one using the modified Kuo-Herling interaction in a multi-$j$ model space and the other using a semiempirical interaction for protons confined to the single-$j 0{h}_{9/2}$ orbital. The Kuo-Herling calculations overestimates some of the ground-state transition probabilities, possibly due to contributions not included in the calculated ground-state wave function of $^{211}\mathrm{At}$. A strong underestimation of the $E2$ strengths involving the $7/{2}_{1}^{\ensuremath{-}}$ state is also observed. Therefore, a modification of a single two-body matrix element of the Kuo-Herling interaction is introduced which improves the agreement with the experimental data significantly. The calculations in the single-$j$ approximation agree very well with the measured transition probabilities, indicating that seniority can be regarded as a good quantum number in $^{211}\mathrm{At}$.

Real-space many-body marker for correlated Z2 topological insulators

Taking the clue from the modern theory of polarization [R. Resta, Rev. Mod. Phys. {\bf 66}, 899 (1994)], we identify an operator to distinguish between ${\mathbb Z}_2$-even (trivial) and ${\mathbb Z}_2$-odd (topological) insulators in two spatial dimensions. Its definition extends the position operator [R. Resta and S. Sorella, Phys. Rev. Lett. {\bf 82}, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of non-interacting models, where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster, where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus allowing us to investigate generic interacting systems, such as strongly-correlated topological insulators.

Quark Condensate in QCD at Nonzero Magnetic Field and Temperature

The basic form of the quark condensate for arbitrary values of external magnetic field and temperature, is derived using the field equations with account of confinement. The resulting expression of the chiral condensate is shown to be proportional to square of the singlet $q\bar q$ ground state wave function at origin, $|\phi_0(0)|^2$. For light quarks without magnetic field the condensates are proportional to $\sigma^{3/2}$ ($\sigma$ is the string tension). Numerical results are presented in 5 Tables and shown to be in good agreement with the lattice data, both for nonzero magnetic field $eB$ and temperature $T$ in the range $0 <T < 120$~MeV,~ $0 <eB <4$~GeV$^2$.

Anisotropic photoelectric properties in GaN/AlN quantum dots under terahertz laser polarization

The linear and nonlinear anisotropic optical absorption coefficients(OACs) and refractive index changes(RICs) of GaN/AlN cubical quantum dots(CQDs) subjected to terahertz laser field(TLF) are investigated. The results show that under the polarization of TLF, with the increase of laser dressing parameters, the peak of OACs and RICs increases nonlinearly and have a redshift. More importantly, our work indicated that with the enhancement of TLF polarization, the redistribution of electron excited state wave function will lead to the “loss” of optical properties in the specific frequency band. The redistribution of electron ground state wave function has a significant modulation effect on the magnitude of OACs(RICs) peak. • The enhancement of the refractive index changes was observed at low frequencies. • The phenomenon of wave function exchange in laser field modulation is found. • The modulation of laser field on optical coefficient cannot cover the whole frequency band.

Spin-orbit excitons and electronic configuration of the 5d4 insulator Sr3Ir2O7F2

Here, we report on the low-energy excitations within the paramagnetic spin-orbit insulator ${\mathrm{Sr}}_{3}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}{\mathrm{F}}_{2}$ studied via resonant inelastic x-ray scattering, ab initio quantum chemical calculations, and model Hamiltonian simulations. This material is a unique ${d}^{4} {\mathrm{Ir}}^{5+}$ analog of ${\mathrm{Sr}}_{3}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$ that forms when F ions are intercalated within the SrO layers spacing the square lattice ${\mathrm{IrO}}_{6}$ bilayers of ${\mathrm{Sr}}_{3}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$. Due to the large distortions about the ${\mathrm{Ir}}^{5+}$ ions, our computations demonstrate that a large single-ion anisotropy yields an $S=1$ ($L\ensuremath{\approx}1, J\ensuremath{\approx}0$) ground-state wave function. Weakly coupled, excitonic modes out of the ${S}_{z}=0$ ground state are observed and are well described by a phenomenological spin-orbit exciton model previously developed for $3d$ and $4d$ transition metal ions. The implications of our results regarding the interpretation of previous studies of hole-doped iridates close to ${d}^{4}$ fillings are discussed.

Error-correction properties of an interacting topological insulator

We analyze the phase diagram of a topological insulator model including antiferromagnetic interactions in the form of an extended Su-Schrieffer Heeger model. To this end, we employ a recently introduced operational definition of topological order based on the ability of a system to perform topological error correction. We show that the necessary error correction statistics can be obtained efficiently using a Monte-Carlo sampling of a matrix product state representation of the ground state wave function. Specifically, we identify two distinct symmetry-protected topological phases corresponding to two different fully dimerized reference states. Finally, we extend the notion of error correction to classify thermodynamic phases to those exhibiting local order parameters, finding a topologically trivial antiferromagnetic phase for sufficiently strong interactions.

Built-in potential and ground state switching in coupled InGaN/GaN quantum wires

Built-in potential and ground state switching in coupled In0.2Ga0.8N/GaN quantum wires (QWRs) is investigated as a function of barrier width between QWRs. In the case of a relatively thin barrier thickness, the ground state wave function of hole exists in the upper QWR region while that of electron exists in the lower QWR region because the lowest and the highest built-in potentials exist in the upper and the lower QWRs, respectively. However, the difference of bottom energies in the lower and upper QWRs is gradually decreased with increasing barrier width between QWRs. As a result, in the case of a relatively thick barrier thickness (Lb≥40Å), we observe that the ground state switching of hole and electron wavefunctions occurs.

Elucidation of LMCT Excited States for Lanthanoid Complexes: A Theoretical and Solid-State Experimental Framework.

Photophysical and magnetic properties arising from both ground and excited states of lanthanoid ions are relevant for numerous applications. These properties can be substantially affected, both adversely and beneficially, by ligand-to-metal charge-transfer (LMCT) states. However, probing LMCT states remains a significant challenge in f-block chemistry, particularly in the solid state. Intriguingly, the europium compounds [EuIII(18-c-6)(X4Cat)(NO3)]·MeCN (18-c-6 = 18-crown-6; X = Cl (tetrachlorocatecholate, 1-Eu) or Br (tetrabromocatecholate, 2-Eu) are distinctly darkly-colored, in marked contrast to the analogues with other lanthanoid ions in the 1-Ln and 2-Ln series (Ln = La, Ce, Nd, Gd, Tb, and Dy). Herein, we report a multi-technique investigation of these compounds that has allowed elucidation of the LMCT character of the relevant absorption bands using magnetometry, absorption and emission spectroscopies, and solid-state electrochemistry. To support experimental observations, we present a semi-quantitative multireference ab initio model that (i) captures the anomalously low-lying LMCT excited state observed in the visible spectrum of 1-Eu (and its absence in the other 1-Ln analogues); (ii) elucidates the contribution of the LMCT excitation to the crystal field split 7FJ ground-state wave functions; and (iii) identifies the crucial role played by radial dynamical correlation of the EuIII 4f electrons in the description of the LMCT excited state, modeled by the inclusion of 4f → 5f excitations in the optimized wave function. By providing a set of experimental and theoretical tools, this work establishes a framework for the elucidation of LMCT excited states in lanthanoid compounds in the solid state.