Local Ansatz Research Articles

Randomized Quasi-Optimal Local Approximation Spaces in Time

We target time-dependent partial differential equations (PDEs) with heterogeneous coefficients in space and time. To tackle these problems, we construct reduced basis/multiscale ansatz functions defined in space that can be combined with time stepping schemes within model order reduction or multiscale methods. To that end, we propose to perform several simulations of the PDE for a few time steps in parallel starting at different, randomly drawn start points, prescribing random initial conditions; applying a singular value decomposition to a subset of the so obtained snapshots yields the reduced basis/multiscale ansatz functions. This facilitates constructing the reduced basis/multiscale ansatz functions in an embarrassingly parallel manner. In detail, we suggest using a data-dependent probability distribution based on the data functions of the PDE to select the start points. Each local in time simulation of the PDE with random initial conditions approximates a local approximation space in one time point that is optimal in the sense of Kolmogorov. The derivation of these optimal local approximation spaces which are spanned by the left singular vectors of a compact transfer operator that maps arbitrary initial conditions to the solution of the PDE in a later point of time is one other main contribution of this paper. By solving the PDE locally in time with random initial conditions, we construct local ansatz spaces in time that converge provably at a quasi-optimal rate and allow for local error control. Numerical experiments demonstrate that the proposed method can outperform existing methods like the proper orthogonal decomposition even in a sequential setting and is well capable of approximating advection-dominated problems.

Bridging physical intuition and hardware efficiency for correlated electronic states: the local unitary cluster Jastrow ansatz for electronic structure.

A prominent goal in quantum chemistry is to solve the molecular electronic structure problem for ground state energy with high accuracy. While classical quantum chemistry is a relatively mature field, the accurate and scalable prediction of strongly correlated states found, e.g., in bond breaking and polynuclear transition metal compounds remains an open problem. Within the context of a variational quantum eigensolver, we propose a new family of ansatzes which provides a more physically appropriate description of strongly correlated electrons than a unitary coupled cluster with single and double excitations (qUCCSD), with vastly reduced quantum resource requirements. Specifically, we present a set of local approximations to the unitary cluster Jastrow wavefunction motivated by Hubbard physics. As in the case of qUCCSD, exactly computing the energy scales factorially with system size on classical computers but polynomially on quantum devices. The local unitary cluster Jastrow ansatz removes the need for SWAP gates, can be tailored to arbitrary qubit topologies (e.g., square, hex, and heavy-hex), and is well-suited to take advantage of continuous sets of quantum gates recently realized on superconducting devices with tunable couplers. The proposed family of ansatzes demonstrates that hardware efficiency and physical transparency are not mutually exclusive; indeed, chemical and physical intuition regarding electron correlation can illuminate a useful path towards hardware-friendly quantum circuits.

Correlated Electrons and Mass Enhancement on the β-Mn Type Lattice

Electronic structure and electron correlations of the Hubbard model on the Beta-Mn lattice have been investigated on the basis of the momentum dependent local ansatz wavefunction method. It is found that the electrons on the Beta-Mn lattice are significantly correlated as compared with those on the face centered cubic (fcc) and the body centered cubic (bcc) lattices; the correlation energy gain of the Beta-Mn lattice structure is the largest among the Beta-Mn, the fcc, and the bcc structures, and the charge fluctuations show a significant suppression. In particular, it is found that the mass enhancement factor on the Beta-Mn lattice shows a large peak around the electron number per atom n=1.1 when n is varied at the typical Coulomb interaction energy strength of transition metals. It is pointed that the large mass enhancement may explain anomalous electronic specific heat coefficient in Beta-Mn.

The Gibbons–Hawking Ansatz in Generalized Kähler Geometry

We derive a local ansatz for generalized K\"ahler surfaces with nondegenerate Poisson structure and a biholomorphic $S^1$ action which generalizes the classic Gibbons-Hawking ansatz for invariant hyperK\"ahler manifolds, and allows for the choice of one arbitrary function. By imposing the generalized K\"ahler-Ricci soliton equation, or equivalently the equations of type IIB string theory, the construction becomes rigid, and we classify all complete solutions with the smallest possible symmetry group.

Momentum-Dependent Local Ansatz Approach to the Metallic Ferromagnetism

The first principles momentum dependent local ansatz wavefunction method (MLA) has been extended to the ferromagnetic state by introducing spin-dependent variational parameters. The theory is applied to the ferromagnetic Fe, Co, and Ni. It is shown that the MLA yields the magnetizations being comparable to the results obtained by the GGA (generalized gradient approximation) in the density functional theory. The projected momentum distribution functions as well as the mass enhancement factors are also calculated on the same footing, and are compared with those in the paramagnetic state. It is shown that the calculated mass enhancement factor of Fe is strongly suppressed by the spin polarization due to exchange splitting of the e${}_{\rm g}$ flat bands, while those of Co and Ni remain unchanged by the polarization. These results are shown to be consistent with the experimental results obtained from the low-temperature specific heats.

Non-Gaussian CMB and LSS statistics beyond polyspectra

Cosmic inflation may have led to non-Gaussian initial conditions that cannot be fully parametrised by 3- and/or 4-point functions. In this work, we discuss various strategies to search for primordial non-Gaussianity beyond polyspectra with the help of cosmological data. Our starting point is a generalised local ansatz for the primordial curvature perturbation ζ of the form ζ = ζG + ℱNG (ζG), where ζG is a Gaussian random field and ℱNG is an arbitrary function parametrising non-Gaussianity that, in principle, could be reconstructed from data. Noteworthily, in the case of multi-field inflation, the function ℱNG can be shown to be determined by the shape of tomographic sections of the landscape potential responsible for driving inflation. We discuss how this generalised local ansatz leads to a probability distribution functional that may be used to extract information about inflation from current and future observations. In particular, we derive various classes of probability distribution functions suitable for the statistical analysis of the cosmic microwave background and large-scale structure.

AC Stark Effect on Vortex Spectra Generated by Circularly Polarized Pulses

In this paper, we report the results of numerical simulation on vortex-shaped photoelectron momentum spectra of the hydrogen atom irradiated by a pair of time-delayed circularly polarized ultrashort pulses. Spectral alterations including broadening, splitting, and fusion are observed with stronger pulse intensity deploying the quantum wave-packet theory. These alterations of the dynamic interference structure are further investigated to stem from the ac Stark effect, by making the local approximation for an analytical ansatz. In addition, for evaluating the ac Stark effect on the hydrogen atom quantitatively, we propose a nonlinear-curve-fitting algorithm of the ground-state population for extracting the complex Stark coefficient, which we define. We find that the ac Stark effect can be well characterized by the complex coefficient, as supported by the overall agreement between the momentum spectra obtained from the wave-packet theory and the local approximation ansatz. The present research may shed further light on Stark effect in laser atom interactions.

Quasiparticles in Quantum Spin Chains with Long-Range Interactions.

We study quasiparticle excitations for quantum spin chains with long-range interactions using variational matrix product state techniques. It is confirmed that the local quasiparticle ansatz is able to capture those excitations very accurately, even when the correlation length becomes very large and in the case of topological nontrivial excitation such as spinons. It is demonstrated that the breaking of the Lieb-Robinson bound follows from the appearance of cusps in the dispersion relation, and evidence is given for a crossover between different quasiparticles as the long-range interactions are tuned.

Stochastic approach to irreversible thermodynamics.

An extension of classical irreversible thermodynamics pioneered by Ilya Prigogine is developed, in which fluctuations of macroscopic observables accounting for microscopic-scale processes are incorporated. The contribution of the fluctuations to the entropy production is derived from a generalized entropy balance equation and expressed in terms of the fluctuating variables, via an extended local equilibrium Ansatz and in terms of the probability distributions of these variables. The approach is illustrated on reactive systems involving linear and nonlinear steps, and the role of the distance from equilibrium and of the nonlinearities is assessed.

Dealing with the exponential wall in electronic structure calculations.

An alternative to the density functional theory is the use of wavefunction based electronic structure calculations for solids. In order to perform them, the Exponential Wall (EW) problem has to be resolved. It is caused by an exponential increase of the number of configurations with increasing electron number N. There are different routes one may follow. One is to characterize a many-electron wavefunction by a vector in Liouville space with a cumulant metric rather than in Hilbert space. This removes the EW problem. Another is to model the solid by an impurity or fragment embedded in a bath which is treated at a much lower level than the former. This is the case in the Density Matrix Embedding Theory (DMET) or the Density Embedding Theory (DET). The latter two are closely related to a Schmidt decomposition of a system and to the determination of the associated entanglement. We show here the connection between the two approaches. It turns out that the DMET (or DET) has an identical active space as a previously used Local Ansatz, based on a projection and partitioning approach. Yet, the EW problem is resolved differently in the two cases. By studying a H10 ring, these differences are analyzed with the help of the method of increments.

First-Principles Momentum Dependent Local Ansatz Approach to the Momentum Distribution Function in Iron-Group Transition Metals

The momentum distribution function (MDF) bands of iron-group transition metals from Sc to Cu have been investigated on the basis of the first-principles momentum dependent local ansatz wavefunction method. It is found that the MDF for $d$ electrons show a strong momentum dependence and a large deviation from the Fermi-Dirac distribution function along high-symmetry lines of the first Brillouin zone, while the $sp$ electrons behave as independent electrons. In particular, the deviation in bcc Fe (fcc Ni) is shown to be enhanced by the narrow $e_{g}$ ($t_{2g}$) bands with flat dispersion in the vicinity of the Fermi level. Mass enhancement factors (MEF) calculated from the jump on the Fermi surface are also shown to be momentum dependent. Large mass enhancements of Mn and Fe are found to be caused by spin fluctuations due to $d$ electrons, while that for Ni is mainly caused by charge fluctuations. Calculated MEF are consistent with electronic specific heat data as well as the recent angle resolved photoemission spectroscopy data.

Single-field inflation and the local ansatz: Distinguishability and consistency

The single-field consistency conditions and the local ansatz have played separate but important roles in characterizing the non-Gaussian signatures of single- and multifield inflation respectively. We explore the precise relationship between these two approaches and their predictions. We demonstrate that the predictions of the single-field consistency conditions can never be satisfied by a general local ansatz with deviations necessarily arising at order $(n_s-1)^2$. This implies that there is, in principle, a minimum difference between single- and (fully local) multifield inflation in observables sensitive to the squeezed limit such as scale-dependent halo bias. We also explore some potential observational implications of the consistency conditions and its relationship to the local ansatz. In particular, we propose a new scheme to test the consistency relations. In analogy with delensing of the cosmic microwave background, one can deproject the coupling of the long wavelength modes with the short wavelength modes and test for residual anomalous coupling.

First-Principles Momentum Dependent Local Ansatz Approach to the Ground-State Properties of Iron-Group Transition Metals

The ground-state properties of iron-group transition metals from Sc to Cu have been investigated on the basis of the first-principles momentum dependent local ansatz (MLA) theory. Correlation energy gain is found to show large values for Mn and Fe: 0.090 Ry (Mn) and 0.094 Ry (Fe). The Hund-rule coupling energies are found to be 3000 K (Fe), 1400 K (Co), and 300 K (Ni). It is sugested that these values can resolve the inconsistency in magnetic energy between the density functional theory and the first-principles dynamical coherent potential approximation theory at finite temperatures. Charge fluctuations are shown to be suppressed by the intra-orbital correlations and inter-orbital charge-charge correlations, so that they show nearly constant values from V to Fe: 1.57 (V and Cr), 1.52 (Mn), and 1.44 (Fe), which are roughly twice as large as those obtained by the $d$ band model. The amplitudes of local moments are enhanced by the intra-orbital and inter-orbital spin-spin correlations and show large values for Mn and Fe: 2.87 (Mn) and 2.58 (Fe). These values are in good agreement with the experimental values estimated from the effective Bohr magneton number and the inner core photoemission data.

First-Principles Theory of Momentum Dependent Local Ansatz Approach to Correlated Electron System

We have extended the momentum-dependent local-ansatz (MLA) wavefunction method to the first-principles version using the tight-binding LDA+U Hamiltonian for the description of correlated electrons in the real system. The MLA reduces to the Rayleigh-Schr\"odinger perturbation theory in the weak correlation limit, and describes quantitatively the ground state and related low-energy excitations in solids. The theory has been applied to the paramagnetic Fe. The role of electron correlations on the energy, charge fluctuations, amplitude of local moment, momentum distribution functions, as well as the mass enhancement factor in Fe has been examined as a function of Coulomb interaction strength. It is shown that the inter-orbital charge-charge correlations between $d$ electrons make a significant contribution to the correlation energy and charge fluctuations, while the intra-orbital and inter-orbital spin-spin correlations make a dominant contribution to the amplitude of local moment and the mass enhancement in Fe. Calculated partial mass enhancements are found to be 1.01, 1.01, and 3.33 for $s$, $p$, and $d$ electrons, respectively. The averaged mass enhancement 1.65 is shown to be consistent with the experimental data as well as the recent results of theoretical calculations.

First-Principles Momentum-Dependent Local Ansatz Wavefunction and Momentum Distribution Function Bands of Iron

We have developed a first-principles local ansatz wavefunction approach with momentum-dependent variational parameters on the basis of the tight-binding LDA+U Hamiltonian. The theory goes beyond the first-principles Gutzwiller approach and quantitatively describes correlated electron systems. Using the theory, we find that the momentum distribution function (MDF) bands of paramagnetic bcc Fe along high-symmetry lines show a large deviation from the Fermi-Dirac function for the $d$ electrons with $e_{g}$ symmetry and yield the momentum-dependent mass enhancement factors. The calculated average mass enhancement $m^{\ast}/m = 1.65$ is consistent with low-temperature specific heat data as well as recent angle-resolved photoemission spectroscopy (ARPES) data.

Interface Reduction for CMS Methods and Alternative Model Order Reduction

In many cases, complex dynamical systems consist of separable components. In order to be able to stick to a setup with exchangeable modules, the description of the interface or port that is used for the connection is very important. Classic reduction approaches in substructuring, such as the well-know Craig-Bampton procedure, typically use brute-force methods for the interface description. It is shown that instead of maintaining local ansatz functions for each node at the interface, a semi-global approach can be used. For the derivation of important deformation patterns at the interface, multiple possibilities are considered and compared.

First-principles Theory of the Momentum-dependent Local Ansatz for Correlated Electron System

The momentum-dependent local-ansatz (MLA) wavefunction describes well correlated electrons in solids in both the weak and strong interaction regimes. In order to apply the theory to the realistic system, we have extended the MLA to the first-principles version using the tight-binding LDA+U Hamiltonian. We demonstrate for the paramagnetic Fe that the first-principles MLA can describe a reasonable correlation energy gain and suppression of charge fluctuations due to electron correlations. Furthermore, we show that the MLA yields a distinct momentum dependence of the momentum distribution, and thus improves the Gutzwiller wavefunction.

Momentum-dependent local ansatz approach to correlated electrons

The wavefunction method provides us with a useful tool to describe electron correlations in solids at the ground state. In this paper we review the recent development of the momentum-dependent local ansatz (MLA) wavefunction. It is constructed by taking into account two-particle excited states projected onto the local orbitals, and the momentum-dependent amplitudes of these states are chosen as variational parameters. The MLA describes accurately correlated electron states from the weak to the intermediate Coulomb interaction regime in infinite dimensions, and works well even in the strongly correlated region by introducing a new starting wavefunction called the hybrid (HB) wavefunction. The MLA-HB is therefore shown to overcome the limitation of the original local ansatz (LA) wavefunction as well as the Gutzwiller wavefunction. In particular, the calculated quasiparticle weight versus Coulomb interaction curve is shown to be close to that obtained by the numerical renormalization group approach. It is also shown that the MLA is applicable to the first-principles Hamiltonian.

Higher moments of primordial non-Gaussianity and N-bodysimulations

We perform cosmological N-body simulations with non-Gaussian initial conditions generated from two independent fields. The dominant contribution to the perturbations comes from a purely Gaussian field, but we allow the second field to have local non-Gaussianity that need not be weak. This scenario allows us to adjust the relative importance of non-Gaussian contributions beyond the skewness, producing a scaling of the higher moments different from (and stronger than) the scaling in the usual single field local ansatz. We compare semi-analytic prescriptions for the non-Gaussian mass function, large scale halo bias, and stochastic bias against the simulation results. We discuss applications of this work to large scale structure measurements that can test a wider range of models for the primordial fluctuations than is usually explored.

Constraining the initial conditions of the Universe using large scale structure

Primordial non-Gaussianity induces a scale-dependent bias in large scale structure (LSS) data, proportional to fNL/k2 for the exact local ansatz. Recent work has shown that models of inflation that predict a large squeezed limit bispectrum, such as multi-field models and single field inflation with a modified initial state, typically give rise to a generalized local ansatz, with the scale-dependent bias now proportional to \U0001d49cNL/kα. We use photometric measurements of the angular power spectrum of luminous red galaxies and quasars in the Sloan Digital Sky Survey Data Release Eight (SDSS DR8) with the above parameterization to constrain the amplitude \U0001d49cNL and scale-dependence α. We find that the marginalized upper limit on α is 2.0 at the 95% confidence level, consistent with the local ansatz. We also present Fisher forecasts for a survey of the same size as DR8 to assess the role of systematics in current photometric LSS data. Moreover, we present analytic results on the expected mass dependence of \U0001d49cNL for different inflationary models, which can be an important observable for future surveys, if primordial non-Gaussianity is non-zero.