Static Mean-field Theory Research Articles

Hund's metal physics: From SrNiO2 to LaNiO2

We study the normal state electronic structure of the recently discovered infinite-layer nickelate superconductor, Nd$_{1-x}$Sr$_x$NiO$_2$. Using SrNiO$_2$ as a reference, we find that Nd$_{1-x}$Sr$_x$NiO$_2$ is a multi-orbital electronic system with characteristic Hund's metal behaviors, such as metallicity, the importance of high-spin configurations, tendency towards orbital differentiation, and the absence of magnetism in regimes which are ordered according to static mean-field theories. In addition, our DFT+DMFT calculations with exact double counting scheme show that despite large charge carrier doping from SrNiO$_2$ to LaNiO$_2$, the Ni-$3d$ total occupancy is barely changed due to the decreased hybridization with the occupied oxygen-2$p$ states, and increased hybridization with the unoccupied La-5$d$ states. Our results are in good agreement with the existing resonant inelastic x-ray scattering measurements and pave the way to understand the pairing mechanism of Nd$_{1-x}$Sr$_x$NiO$_2$.

A self-consistent field formulation of excited state mean field theory.

We show that, as in Hartree-Fock theory, the orbitals for excited state mean field theory can be optimized via a self-consistent one-electron equation in which electron-electron repulsion is accounted for through mean field operators. In addition to showing that this excited state ansatz is sufficiently close to a mean field product state to admit a one-electron formulation, this approach brings the orbital optimization speed to within roughly a factor of two of ground state mean field theory. The approach parallels Hartree Fock theory in multiple ways, including the presence of a commutator condition, a one-electron mean-field working equation, and acceleration via direct inversion in the iterative subspace. When combined with a configuration interaction singles Davidson solver for the excitation coefficients, the self-consistent field formulation dramatically reduces the cost of the theory compared to previous approaches based on quasi-Newton descent.

Core excitations with excited state mean field and perturbation theory

We test the efficacy of excited state mean field theory and its excited-state-specific perturbation theory on the prediction of K-edge positions and x-ray peak separations. We find that the mean field theory is surprisingly accurate, even though it contains no accounting of differential electron correlation effects. In the perturbation theory, we test multiple core-valence separation schemes and find that, with the mean field theory already so accurate, electron-counting biases in one popular separation scheme become a dominant error when predicting K-edges. Happily, these appear to be relatively easy to correct for, leading to a perturbation theory for K-edge positions that is lower scaling and more accurate than coupled cluster theory and competitive in accuracy with recent high-accuracy results from restricted open-shell Kohn-Sham theory. For peak separations, our preliminary data show excited state mean field theory to be exceptionally accurate, but more extensive testing will be needed to see how it and its perturbation theory compare to coupled cluster peak separations more broadly.

N5-Scaling Excited-State-Specific Perturbation Theory

We show that by working in a basis similar to that of the natural transition orbitals and using a modified zeroth-order Hamiltonian, the cost of a recently introduced perturbative correction to excited-state mean field theory can be reduced from seventh to fifth order in the system size. The (occupied)2(virtual)3 asymptotic scaling matches that of ground-state second-order Møller-Plesset theory but with a significantly higher prefactor because the bottleneck is iterative: it appears in the Krylov-subspace-based solution of the linear equation that yields the first-order wave function. Here, we discuss the details of the modified zeroth-order Hamiltonian we use to reduce the cost and the automatic code generation process we used to derive and verify the cost scaling of the different terms. Overall, we find that our modifications have little impact on the method's accuracy, which remains competitive with singles and doubles equation-of-motion coupled cluster.

Nonlocal exchange interactions in strongly correlated electron systems

We study the influence of ferromagnetic nonlocal exchange on correlated electrons in terms of a $SU(2)$-Hubbard-Heisenberg model and address the interplay of on-site interaction induced local moment formation and the competition of ferromagnetic direct and antiferromagnetic kinetic exchange interactions. In order to simulate thermodynamic properties of the system in a way that largely accounts for the on-site interaction driven correlations in the system, we advance the correlated variational scheme introduced in [M. Sch\"uler et al., Phys. Rev. Lett. 111, 036601 (2013)] to account for explicitily symmetry broken electronic phases by introducing an auxiliary magnetic field. After benchmarking the method against exact solutions of a finite system, we study the $SU(2)$-Hubbard-Heisenberg model on a square lattice. We obtain the $U$-$J$ finite temperature phase diagram of a $SU(2)$-Hubbard-Heisenberg model within the correlated variational approach and compare to static mean field theory. While the generalized variational principle and static mean field theory yield transitions from dominant ferromagnetic to antiferromagnetic correlations in similar regions of the phase diagram, we find that the nature of the associated phase tranistions differs between the two approaches. The fluctuations accounted for in the generalized variational approach render the transitions continuous, while static mean field theory predicts discontinuous transitions between ferro- and antiferromagnetically ordered states.

Excited state mean-field theory without automatic differentiation.

We present a formulation of excited state mean-field theory in which the derivatives with respect to the wave function parameters needed for wave function optimization (not to be confused with nuclear derivatives) are expressed analytically in terms of a collection of Fock-like matrices. By avoiding the use of automatic differentiation and grouping Fock builds together, we find that the number of times we must access the memory-intensive two-electron integrals can be greatly reduced. Furthermore, the new formulation allows the theory to exploit the existing strategies for efficient Fock matrix construction. We demonstrate this advantage explicitly via the shell-pair screening strategy with which we achieve a cubic overall cost scaling. Using this more efficient implementation, we also examine the theory's ability to predict charge redistribution during charge transfer excitations. Using the coupled cluster as a benchmark, we find that by capturing orbital relaxation effects and avoiding self-interaction errors, excited state mean field theory out-performs other low-cost methods when predicting the charge density changes of charge transfer excitations.

A Generalized Variational Principle with Applications to Excited State Mean Field Theory.

We present a generalization of the variational principle that is compatible with any Hamiltonian eigenstate that can be specified uniquely by a list of properties. This variational principle appears to be compatible with a wide range of electronic structure methods, including mean field theory, density functional theory, multireference theory, and quantum Monte Carlo. Like the standard variational principle, this generalized variational principle amounts to the optimization of a nonlinear function that, in the limit of an arbitrarily flexible wave function, has the desired Hamiltonian eigenstate as its global minimum. Unlike the standard variational principle, it can target excited states and select individual states in cases of degeneracy or near-degeneracy. As an initial demonstration of how this approach can be useful in practice, we employ it to improve the optimization efficiency of excited state mean field theory by an order of magnitude. With this improved optimization, we are able to demonstrate that the accuracy of the corresponding second-order perturbation theory rivals that of singles-and-doubles equation-of-motion coupled cluster in a substantially broader set of molecules than could be explored by our previous optimization methodology.

Density Functional Extension to Excited-State Mean-Field Theory.

We investigate an extension of excited-state mean-field theory in which the energy expression is augmented with density functional components in an effort to include the effects of weak electron correlations. The approach remains variational and entirely time independent, allowing it to avoid some of the difficulties associated with linear response and the adiabatic approximation. In particular, all of the electrons' orbitals are relaxed state specifically, and there is no reliance on Kohn-Sham orbital energy differences, both of which are important features in the context of charge transfer. Preliminary testing shows clear advantages for single-component charge transfer states, but the method, at least in its current form, is less reliable for states in which multiple particle-hole transitions contribute significantly.

Crossover from conventional to inverse indirect magnetic exchange in the depleted Anderson lattice

We investigate the finite-temperature properties of an Anderson lattice with regularly depleted impurities. The physics of this model is ruled by two different magnetic exchange mechanisms: conventional Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction at weak hybridization strength V and a novel inverse indirect magnetic exchange (IIME) at strong V, both favoring a ferromagnetic ground state. The stability of ferromagnetic order against thermal fluctuations is systematically studied by static mean-field theory for an effective low-energy spin-only model emerging perturbatively in the strong-coupling limit as well as by dynamical mean-field theory for the full model. The Curie temperature is found at a maximum for a half-filled conduction band and at intermediate hybridization strengths in the crossover regime between RKKY and IIME.

Electric polarization in correlated insulators

We derive a formula for the electric polarization of interacting insulators, expressed in terms of the full Green's and vertex functions. We exemplify this method in the half-filled ionic Hubbard model treated within dynamical mean field theory (DMFT). The electric polarization of a correlated band insulator is determined by the interplay of ionicity and covalency, and both quantities are renormalized by the electron-electron interactions. We introduce quasiparticle approximation to the exact equation for the polarization, and compare the results of this approximation with those of the exact DMFT formulation and of static mean field theories such as the LDA+ U. The latter overestimates the electronic contribution to the electric polarization when the quasiparticle weight of the active bands is very small.

Spin-resolved correlations at arbitrary spin polarization and ground state of a quasi-one-dimensional electron gas

We study the spin-resolved correlations at arbitrary spin polarization ζ and the ground state of a quasi-one-dimensional electron gas by using the dynamical self-consistent mean-field theory of Singwi et al. Numerical results are presented for the spin-resolved pair-correlation functions, correlation energies, and ground-state energy at selected ζ and a range of electron density rs. Our results agree nicely with the recent lattice regularized diffusion Monte Carlo simulation data at small rs and ζ. With increasing rs and/or ζ, the spin-resolved correlations become less satisfactory, but the spin-summed quantities show a reasonable agreement, thus implying a cancellation among discrepancies in the components. Interestingly, theory predicts that the like-spin correlation energy becomes little positive for rs>1.5. A comparison between the ground-state energies of the unpolarized and fully polarized spin phases reveals that the exchange–correlations may induce a phase transition to the latter above a critical rs, rsc. Importantly, the transition is in agreement with recent experiments on low density electron quantum wires. However, the stability of partially spin-polarized states could not be ascertained due to difficulty in obtaining the self-consistent density response function beyond a certain rs, preceding rsc. Using the static mean-field theory, we find that the spin-polarization transition is abrupt (i.e., the partially spin-polarized states are energetically unstable) and it occurs at nearly the same rs as in the dynamical approach.

Spin-polarized electron–electron quantum bilayers: Finite width effect

We study the effect of finite width on the ground-state of a spin-polarized electron–electron quantum bilayers (EEBL) system at temperature T=0. Correlations among carriers are treated beyond the static mean-field theories by using the quantum or dynamical version of Singwi, Tosi, Land and Sjölander (qSTLS) theory. Numerical results are presented for the pair-correlation function, the ground-state energy, the static density susceptibility, and the static local-field correction factor as a function of density parameter r sl and interlayer spacing d. Interestingly, we find that the inclusion of finite width lowered the critical density, for the onset of Wigner crystal (WC) ground-state, as compared to the similar recent study of spin-polarized EEBL system without finite width effect. Further, spin-polarization effect is seen to introduce a marked change in the ground-state energy of the EEBL system as compared to the results of unpolarized EEBL system with finite width. Results of ground-state energies are also compared with the recent diffusion Monte Carlo (DMC) and variational Monte Carlo (VMC) simulation studies of spin-polarized EEBL system with zero width.

Modelling relaxation processes for fluids in porous materials using dynamic mean field theory: application to pore networks

We present an application of a recently developed dynamic mean field theory to the study of relaxation dynamics in adsorption and desorption from pore networks. The theory predicts the evolution of density distribution in the system, based on an underlying free energy functional from static mean field theory and the system evolves to equilibrium or metastable equilibrium states consistent with the static theory. The theory makes it possible to follow the evolution of the density distribution with time in response to a change in the bulk pressure or chemical potential. We compare uptake dynamics for a 2D slit pore network with that in a single slit pore. We see more rapid uptake dynamics in the pore network in some cases, due to the greater access of the pore space to the bulk. We also observe that the formation of liquid bridges can slow down the mass transfer in the pore network in certain situations.

Modeling Relaxation Processes for Fluids in Porous Materials Using Dynamic Mean Field Theory: An Application to Partial Wetting

We review a recently developed dynamic mean field theory for fluids confined in porous materials and apply it to a case where the solid-fluid interactions lead to partial wetting on a planar surface. The theory describes the evolution of the density distribution for a fluid in a pore that has contact with the bulk during a quench in the bulk chemical potential. In this way the dynamics of adsorption and desorption can be studied. By focusing on partial wetting situation we can investigate influence of a weaker surface field on the mechanisms of capillary condensation and desorption. We have studied the dynamics of pore filling in a quench of the chemical potential between two states either side of the pore filling step, tracking the density distributions during the process. The pore filling process features an asymmetric density distribution where a liquid droplet appears on one of the walls. The droplet spreads and grows in size and this is followed by the appearance of a liquid bridge between the pore walls (for longer pores two liquid bridges are seen). The density distributions obtained in the dynamics resemble those obtained from static mean field theory in the canonical ensemble for an infinite pore without contact with the bulk.

A spiking network model for learning reward timing in cortex

Event Abstract Back to Event A spiking network model for learning reward timing in cortex The ability to represent time is an essential component of cognition but its neural basis is unknown. Although extensively studied both behaviorally and electrophysiologically, a general theoretical framework describing the elementary neural mechanisms used by the brain to learn temporal representations is lacking. It is commonly believed that the underlying cellular mechanisms reside in high order cortical regions but recent studies show sustained neural activity in primary sensory cortices that can represent the timing of expected reward. In particular, a recent study by Shuler and Bear (2006) has shown learned, reward timing dependent cortical activity in primary visual cortex. We postulate that a network of laterally connected cortical neurons is sufficient for generating the timing dependent neuronal dynamics, that the timing information can be stored within synaptic efficacies, and that this cortical activity can be learned through a novel variant of reinforcement learning. One of the advantages of the theory proposed here is that it does not require specialized processes such as tagged delay lines of phase locked oscillators, as assumed explicitly or implicitly in previous theories of timing. We mathematically formulate the neuronal dynamics and the learning rule, and implement our model computationally both in a rate based implementation and in a model of conductance based spiking neurons. We show that this theory can account for various aspects of the experimental data, but also explore the limitations of this model. We further analyze the spiking neural model using a quasi-steady state mean field theory (e.g: Renart et al, 2003). This analysis reduces the dynamics of the network to a single non-linear dynamical equation. We show that the results of the mean field theory are in close agreement with the simulation results. Our theoretical model can account for existing experimental results, and can also produce novel predictions. We show several such predictions which include change in pair wise correlations as a result of learning, and an increase in spontaneous and evoked activity after learning. Using the original data from the Shuler and Bear (2006) paper we show that the experimental data is consistent with these predictions and that both evoked activity and spontaneous activity are significantly increased by 74% and 40% respectively as a result of training. One prominent experimental feature of temporal interval estimation is the scalar rule (Webber law), which states that the standard deviation of estimation errors increase linearly with the estimated interval. Using a simple population decoding scheme based on the activity of the trained network of spiking neurons we show that this model produces estimation errors that increase approximately linearly with the estimated interval. We explore the origin of these results, and how they compare to experimental findings. Here we present a theory to account for the specific results of Shuler and Bear (2006) which can also form the basis for a general theory of timing estimation in the cortex. We computationally implement and mathematically analyze the theory, and propose experimental tests. We also show here experimental confirmation of some of our theoretical predictions. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Poster Presentation Topic: Poster Presentations Citation: (2009). A spiking network model for learning reward timing in cortex. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.313 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 04 Feb 2009; Published Online: 04 Feb 2009. 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Ground-state properties of a quasi-one-dimensional electron gas within a dynamical self-consistent mean-field approximation

The ground-state properties of the quasi-one-dimensional electron gas are determined theoretically within the quantum/dynamical version of the self-consistent mean-field approximation of Singwi, Tosi, Land, and Sj\"olander (the so-called qSTLS approach). The transverse motion of electrons is assumed to be confined by a harmonic potential. The calculated static structure factor, pair-correlation function, and correlation energy are compared directly with the recent findings of lattice regularized diffusion Monte Carlo simulation study due to Casula et al. It has been found that the qSTLS results are overall in better agreement with the simulation data than the predictions based upon static mean-field theories. Results for the dynamic local-field correction, dynamic structure factor, and plasmon excitation energy are also reported. The qSTLS approach is found to yield an inadequate description of the dynamic properties; for instance, the dynamic structure factor was seen to become negative over a range of frequencies. Our theoretical predictions, seen in conjunction with similar studies on the three- and two-dimensional electron systems, lead us to conclude that the correlation effects are relatively more pronounced in one-dimensional electron gas.

Mean field kinetic theory for a lattice gas model of fluids confined in porous materials

We consider the mean field kinetic equations describing the relaxation dynamics of a lattice model of a fluid confined in a porous material. The dynamical theory embodied in these equations can be viewed as a mean field approximation to a Kawasaki dynamics Monte Carlo simulation of the system, as a theory of diffusion, or as a dynamical density functional theory. The solutions of the kinetic equations for long times coincide with the solutions of the static mean field equations for the inhomogeneous lattice gas. The approach is applied to a lattice gas model of a fluid confined in a finite length slit pore open at both ends and is in contact with the bulk fluid at a temperature where capillary condensation and hysteresis occur. The states emerging dynamically during irreversible changes in the chemical potential are compared with those obtained from the static mean field equations for states associated with a quasistatic progression up and down the adsorption/desorption isotherm. In the capillary transition region, the dynamics involves the appearance of undulates (adsorption) and liquid bridges (adsorption and desorption) which are unstable in the static mean field theory in the grand ensemble for the open pore but which are stable in the static mean field theory in the canonical ensemble for an infinite pore.

Linked-cluster expansion around mean-field theories of interacting electrons.

A general expansion scheme based on the concept of linked-cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field theories at weak (Hartree-Fock) and strong (Hubbard-III) coupling the expansion represents a universal and comprehensive tool for systematic improvements of static mean-field theories. As an example of the general formalism we investigate in detail an analytically tractable series of ring diagrams that correctly capture dynamical fluctuations at weak coupling. We introduce renormalizations of the diagrammatic expansion at various levels and show how the resultant theories are related to other approximations of similar origin. We demonstrate that only fully self-consistent approximations produce global and thermodynamically consistent extensions of static mean-field theories. A fully self-consistent theory for the ring diagrams is reached by summing the so-called noncrossing diagrams.

Static mean-field theory for molecular vibrations: Self-consistent correlation corrections

We present an improvement to the self-consistent field (SCF) scheme for coupled vibrations, assuming that the correction to the SCF wavefunction can also be approximately written as a product of single-mode functions. This defines a self-consistent correlations method, whose computational effort scales linearly with the mode number. We test the method by a series of model computations on nonbending triatomics in local coordinates. The scheme seems to be simple and general for correcting SCF wavefunctions for coupled anharmonic vibrations.

On the Dynamic Effect in Phase Transitions of Finite Systems—Illustrated With a Two-Level SP(4) Model

The phase transition of the two-level SP(4) model was investigated. By comparing the exact results with the approximate results of the static mean field theory, the importance of the dynamic effect in phase transitions of finite systems is exhibited.