Ground State Wave Function Research Articles

Hartree-Fock critical nuclear charge in two-electron atoms.

Electron correlation effects play a key role in stabilizing two-electron atoms near the critical nuclear charge, representing the smallest charge required to bind two electrons. However, deciphering the importance of these effects relies on fully understanding the uncorrelated Hartree-Fock description. We investigate the properties of the ground state wave function in the small nuclear charge limit using various symmetry-restricted Hartree-Fock formalisms. We identify the nuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts an inner and outer electron. We also identify closed-shell and unrestricted critical nuclear charges where the highest occupied orbital energy becomes zero and the electron density detaches from the nucleus. Finally, we identify the importance of fractional spin errors and static correlation for small nuclear charges.

Lanczos recursion on a quantum computer for the Green's function and ground state

A state-preserving quantum counting algorithm is used to obtain coefficients of a Lanczos recursion from a single ground state wavefunction on the quantum computer. This is used to compute the continued fraction representation of an interacting Green's function for use in condensed matter, particle physics, and other areas. The wavefunction does not need to be re-prepared at each iteration. The quantum algorithm represents an exponential reduction in memory over known classical methods. An extension of the method to determining the ground state is also discussed.

Bayesian Optimization of Bose-Einstein Condensates

Machine Learning methods are emerging as faster and efficient alternatives to numerical simulation techniques. The field of Scientific Computing has started adopting these data-driven approaches to faithfully model physical phenomena using scattered, noisy observations from coarse-grained grid-based simulations. In this paper, we investigate data-driven modelling of Bose-Einstein Condensates (BECs). In particular, we use Gaussian Processes (GPs) to model the ground state wave function of BECs as a function of scattering parameters from the dimensionless Gross Pitaveskii Equation (GPE). Experimental results illustrate the ability of GPs to accurately reproduce ground state wave functions using a limited number of data points from simulations. Consistent performance across different configurations of BECs, namely Scalar and Vectorial BECs generated under different potentials, including harmonic, double well and optical lattice potentials pronounces the versatility of our method. Comparison with existing data-driven models indicates that our model achieves similar accuracy with only a small fraction (frac{1}{50}th) of data points used by existing methods, in addition to modelling uncertainty from data. When used as a simulator post-training, our model generates ground state wave functions 36 times faster than Trotter Suzuki, a numerical approximation technique that uses Imaginary time evolution. Our method is quite general; with minor changes it can be applied to similar quantum many-body problems.

Unconventional filling factor of 4/11: A closed-form ground-state wave function

The ground state at 4/11 filling factor is very well understood, Mukherjee et al. [Phys. Rev. Lett. 112, 016801 (2014)] in terms of the 1/3-filled second effective Landau level of the composite fermions whose correlations resemble that of electrons in the ground state of a two-body Haldane pseudo-potential of relative angular momentum 3, ${V}_{3}$. We here propose a closed-form ground-state wave function for ${V}_{3}$ at 1/3 filling factor. We successfully compare it with the exact wave function for systems with a few electrons by calculating their mutual overlap, pair-correlation function, and entanglement spectra. By numerical exact diagonalization for a few electron systems, we find a window of nonzero ${V}_{3}$ is essential together with ${V}_{1}$ for being 4/11 state incompressible. The constructed wave function for the 4/11 state using this proposed wave function has satisfactorily high overlap with previously studied composite-fermion-diagonalized ground-state wave function.

Solution of two-qubit Rabi model with extended generalized rotating-wave approximation

The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.

Analytic ground state energy and wave function of a quantum system in an exponential-form anharmonic interaction potential

An anharmonic oscillator with a perturbed quadratic potential which is coupled with an exponential term is being investigated in this paper by isolating an anharmonic oscillator interaction potential from the actual interaction experienced by the quantum system studied, and using standard perturbative techniques. The anharmonic potential considered is of interest because of its usefulness in the study of non-centrosymmetric materials which have applications in piezoelectricity. The ground state energy eigenvalue and its associated eigenstates were calculated for the quantum system using an analytical approach. Results obtained are compared to those of quantised harmonic oscillator to show the effect of the perturbation.

Extraction of the many-body Chern number from a single wave function

The quantized Hall conductivity of integer and fractional quantum Hall (IQH and FQH) states is directly related to a topological invariant, the many-body Chern number. The conventional calculation of this invariant in interacting systems requires a family of many-body wave functions parameterized by twist angles in order to calculate the Berry curvature. In this paper, we demonstrate how to extract the Chern number given a single many-body wave function, without knowledge of the Hamiltonian. For FQH states, our method requires one additional integer invariant as input: the number of $2\pi$ flux quanta, $s$, that must be inserted to obtain a topologically trivial excitation. As we discuss, $s$ can be obtained in principle from the degenerate set of ground state wave functions on the torus, without knowledge of the Hamiltonian. We perform extensive numerical simulations involving IQH and FQH states to validate these methods.

Applying a four-body continuum-distorted-wave formalism in calculating the fully differential cross section for single ionization of helium in collision with high-energy protons

The present study aimed to calculate the fully differential cross section for single ionization of helium atom by fast protons impact. These calculations are performed as four-body formalism by applying the continuum-distorted-wave method, considering the wave functions symmetrically in the initial and final channels. The dependence of this theory on the ground-state wave functions of the helium in the initial channel is investigated through performing calculations for the three wave functions Hylleraas, Silverman and Silverman-Platas-Matse, respectively. In addition, the effect of inter-nuclear interaction in determining the dynamics of collision for all three wave functions is investigated. Comparing the results of the present theory with the experimental results in the 1 MeV impact energies of protons, the 6.5 eV electron ejection energies and 0.75 a.u transferred momentum in the scattering plane and azimuthal plane indicates that this theory can accurately predict binary and recoil peaks and determine the dynamics of collision at the energies discussed in the paper.

Studies of local structures for Cu2+ centers in M2Zn(SO4)2·6H2O (M = NH4 and Rb) crystals

Abstract The electron paramagnetic resonance (EPR) parameters (g factor g i , and hyperfine structure constants A i , with i = x, y, z) and local structures for Cu2+ centers in M2Zn(SO4)2·6H2O (M = NH4 and Rb) are theoretically investigated using the high order perturbation formulas of these EPR parameters for a 3d 9 ion under orthorhombically elongated octahedra. In the calculations, contribution to these EPR parameters due to the admixture of d-orbitals in the ground state wave function of the Cu2+ ion are taken into account based on the cluster approach, and the required crystal-field parameters are estimated from the superposition model which enables correlation of the crystal-field parameters and hence the studied EPR parameters with the local structures of the Cu2+ centers. Based on the calculations, the Cu–H2O bonds are found to suffer the axial elongation ratio δ of about 3 and 2.9% along the z-axis, meanwhile, the planar bond lengths may experience variation ratio τ (≈3.8 and 1%) along x- and y-axis for Cu2+ center in (NH4)2Zn(SO4)2·6H2O and Rb2Zn(SO4)2·6H2O, respectively. The theoretical results show good agreement with the observed values.

Non-Equilibrium Skewness, Market Crises, and Option Pricing: Non-Linear Langevin Model of Markets with Supersymmetry

This paper presents a tractable model of non-linear dynamics of market returns using a Langevin approach. Due to non-linearity of an interaction potential, the model admits regimes of both small and large return fluctuations. Langevin dynamics are mapped onto an equivalent quantum mechanical (QM) system. Borrowing ideas from supersymmetric quantum mechanics (SUSY QM), a parameterized ground state wave function (WF) of this QM system is used as a direct input to the model, which also fixes a non-linear Langevin potential. Using a two-component Gaussian mixture as a ground state WF with an asymmetric double well potential produces a tractable low-parametric model with interpretable parameters, referred to as the NES (Non-Equilibrium Skew) model. Supersymmetry (SUSY) is then used to find time-dependent solutions of the model in an analytically tractable way. Additional approximations give rise to a final practical version of the NES model, where real-measure and risk-neutral return distributions are given by three component Gaussian mixtures. This produces a closed-form approximation for option pricing in the NES model by a mixture of three Black-Scholes prices, providing accurate calibration to option prices for either benign or distressed market environments, while using only a single volatility parameter. These results stand in stark contrast to the most of other option pricing models such as local, stochastic, or rough volatility models that need more complex specifications of noise to fit the market data.

Extended Hořava Gravity with Physical Ground-State Wavefunction

We propose a new extended theory of Hořava gravity based on the following three conditions: (i) power-counting renormalizable, (ii) healthy IR behavior and (iii) a stable vacuum state in a quantized version of the theory. Compared with other extended theories, we stress that any realistic theory of gravity must have physical ground states when quantization is performed. To fulfill the three conditions, we softly break the detailed balance but keep its basic structure unchanged. It turns out that the new model constructed in this way can avoid the strong coupling problem and remains power-counting renormalizable, moreover, it has a stable vacuum state by an appropriate choice of parameters.

Quantum phase diagrams of matter-field Hamiltonians I: Fidelity, Bures distance, and entanglement

A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for N a = 1 and N a = 4 particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with 2 modes of electromagnetic field. For N a = 1, the reduced density matrix of the matter allows us to determine the phase regions in a 2-simplex (associated to a general three dimensional density matrix), on the different 3-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.

Characterizing Topological Order with Matrix Product Operators

One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of {mathsf {G}}- and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

Many-Body Invariants for Chern and Chiral Hinge Insulators.

We construct new many-body invariants for 2D Chern and 3D chiral hinge insulators characterizing quantized pumping of bulk dipole and quadrupole moments. The many-body invariants are written entirely in terms of many-body ground state wave functions on a torus geometry with twisted boundary conditions and a set of unitary operators. We present a number of supporting arguments for the invariants via topological field theory interpretation, adiabatic pumping argument, and direct mapping to free-fermion band indices. Therefore, the invariants explicitly encircle several different pillars of theoretical descriptions of topological phases. Furthermore, our many-body invariants are written in forms which can be directly employed in various numerics including the exact diagonalization and the density-matrix renormalization group simulations. We finally confirm our invariants by numerical computations including an infinite density-matrix renormalization group on quasi-one-dimensional systems.

Possible multiorbital ground state in CeCu2Si2

The crystal-field ground state wave function of CeCu$_2$Si$_2$ has been investigated with linear polarized $M$-edge x-ray absorption spectroscopy from 250mK to 250K, thus covering the superconducting ($T_{\text{c}}$=0.6K), the Kondo ($T_{\text{K}}$$\approx$20K) as well as the Curie-Weiss regime. The comparison with full-multiplet calculations shows that the temperature dependence of the experimental linear dichroism is well explained with a $\Gamma_7^{(1)}$ crystal-field ground-state and the thermal population of excited states at around 30meV. The crystal-field scheme does not change throughout the entire temperature range thus making the scenario of orbital switching unlikely. Spectroscopic evidence for the presence of the Ce 4$f^0$ configuration in the ground state is consistent with the possibility for a multi-orbital character of the ground state. We estimate from the Kondo temperature and crystal-field splitting energies that several percents of the higher lying $\Gamma_6$ state and $\Gamma_7^{(2)}$ crystal-field states are mixed into the primarily $\Gamma_7^{(1)}$ ground state. This estimate is also supported by re-normalized band-structure calculations that uses the experimentally determined crystal-field scheme.

Approaching the full configuration interaction ground state from an arbitrary wavefunction with gradient descent and quasi-Newton algorithms.

We consider gradient descent and quasi-Newton algorithms to optimize the full configuration interaction (FCI) ground state wavefunction starting from an arbitrary reference state |0⟩. We show that the energies obtained along the optimization path can be evaluated in terms of expectation values of |0⟩, thus avoiding explicit storage of intermediate wavefunctions. This allows us to find the energies after the first few steps of the FCI algorithm for systems much larger than what standard deterministic FCI codes can handle at present. We show an application of the algorithm with reference wavefunctions constructed as linear combinations of non-orthogonal determinants.

A simple variational quantum Monte Carlo-effective mass approach for excitons and trions in quantum dots

A computational model is presented to calculate the ground state energy of neutral and charged excitons confined in semiconductor quantum dots. The model is based on the variational Quantum Monte Carlo method and effective mass Hamiltonians. Through an iterative Newton–Raphson process, minimizing the local energy, and (optional) parallelization of random walkers, fast and accurate estimates of both confinement and Coulomb binding energies can be obtained in standard desktop computers. To illustrate the reach of the model, we provide Fortran programs and illustrative calculations for colloidal CdSe nanoplatelets with large lateral dimensions and dielectric confinement, where electronic correlations are strong. The results compare well with exact variational calculations and largely outperform configuration interaction calculations in computational efficiency. Program summaryProgram title: vqmc-emaCPC Library link to program files:https://doi.org/10.17632/dtpyc2pffg.1Licensing provisions: GPLv3Programming language: Fortran 90Nature of problem: Calculation of either exciton or trion ground state energy and wave function in a cuboidal semiconductor nanoplatelet. Hard wall quantum confinement coexists with dielectric confinement (self-energy and Coulomb polarization terms).Solution method: Variational quantum Monte Carlo with effective mass Hamiltonians, integrated into a Newton–Raphson solver. OMP parallelization library can be (optionally) linked.

Energy scales in 4f1 delafossite magnets: Crystal-field splittings larger than the strength of spin-orbit coupling in KCeO2

Ytterbium-based delafossites with effective S=1/2 moments are investigated intensively as candidates for quantum spin-liquid ground states. While the synthesis of related cerium compounds has also been reported,many important details concerning their crystal, electronic, and magnetic structures are unclear. Here we analyze the S=1/2 system KCeO2, combining complementary theoretical methods. The lattice geometry was optimized and the band structure investigated using density functional theory extended to the level of a GGA+U calculation in order to reproduce the correct insulating behavior. The Ce 4f1 states were then analyzed in more detail with the help of ab initio wave-function-based computations. Unusually large effective crystal-field splittings of up to 320 meV are predicted, which puts KCeO2 in the strong field coupling regime. Our results reveal a subtle interplay between ligand-cage electrostatics and the trigonal field generated by the extended crystalline surroundings, relevant in the context of recent studies on tuning the nature of the ground-state wave function in 4f triangular-lattice and pyrochlore compounds. It also makes KCeO2 an interesting model system in relation to the effect of large crystal-field splittings on the anisotropy of intersite exchange in spin-orbit coupled quantum magnets.

Sensitivity of one-neutron knockout of halo nuclei to their nuclear structure

Halo nuclei are located far from stability and exhibit a very peculiar structure. Due to their very short lifetime, they are often studied through reactions. Breakup reactions are of particular interest since their cross sections are large for these loosely-bound nuclei. Inclusive measurements of breakup–also called knockout reactions-have even higher statistics. In this proceeding, we study which nuclear-structure information can be inferred from the parallel-momentum distribution of the core of one-neutron halo nuclei after the knockout of its halo neutron. In particular, we analyse the influence of the ground-state wavefunction, the presence of excited states within the halo-nucleus spectrum and resonances in the core-neutron continuum. Our analysis shows that such observables are sensitive to the tail of the ground-state wavefunction. The presence of excited state decreases the breakup strength, and this flux is transferred to the inelastic-scattering channel. This indicates a conservation of the flux within each partial wave. We also show that the parallel-momentum distributions are insensitive to the existence of resonances within the continuum, they can thus be ignored in practice. This independence of the continuum argues that the parallel-momentum distributions are ideal observables to extract very precisely the ANCs of halo nuclei.

Exactly Solvable System of One-Dimensional Trapped Bosons with Short- and Long-Range Interactions.

We introduce a model of trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction in one spatial dimension. We find the exact ground-state energy and many-body wave function. The density profile and the pair-correlation function are sampled using MonteCarlo method and show a rich variety of regimes with crossovers between them. Strong attraction leads to a trapped McGuire quantum soliton. Weak repulsion results in an incompressible Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interactions. Stronger repulsion induces Friedel oscillations and the eventual formation of a Wigner crystal.