Electronic Hilbert Space Research Articles

Embedding vertex corrections in GW self-energy: Theory, implementation, and outlook.

The vertex function (Γ) within the Green's function formalism encapsulates information about all higher-order electron-electron interaction beyond those mediated by density fluctuations. Herein, we present an efficient approach that embeds vertex corrections in the one-shot GW correlation self-energy for isolated and periodic systems. The vertex-corrected self-energy is constructed through the proposed separation-propagation-recombination procedure: the electronic Hilbert space is separated into an active space and its orthogonal complement denoted as the "rest;" the active component is propagated by a space-specific effective Hamiltonian different from the rest. The vertex corrections are introduced by a rescaled time-dependent nonlocal exchange interaction. The direct Γ correction to the self-energy is further updated by adjusting the rescaling factor in a self-consistent post-processing cycle. Our embedding method is tested mainly on donor-acceptor charge-transfer systems. The embedded vertex effects consistently and significantly correct the quasiparticle energies of the gap-edge states. The fundamental gap is generally improved by 1-3eV upon the one-shot GW approximation. Furthermore, we provide an outlook for applications of (embedded) vertex corrections in calculations of extended solids.

Zero-temperature phases of the two-dimensional Hubbard-Holstein model: A non-Gaussian exact diagonalization study

We propose a numerical method which embeds the variational non-Gaussian wavefunction approach within exact diagonalization, allowing for efficient treatment of correlated systems with both electron-electron and electron-phonon interactions. Using a generalized polaron transformation, we construct a variational wavefunction that absorbs entanglement between electrons and phonons into a variational non-Gaussian transformation; exact diagonalization is then used to treat the electronic part of the wavefunction exactly, thus taking into account high-order correlation effects beyond the Gaussian level. Keeping the full electronic Hilbert space, the complexity is increased only by a polynomial scaling factor relative to the exact diagonalization calculation for pure electrons. As an example, we use this method to study ground-state properties of the two-dimensional Hubbard-Holstein model, providing evidence for the existence of intervening phases between the spin and charge-ordered states. In particular, we find one of the intervening phases has strong charge susceptibility and binding energy, but is distinct from a charge-density-wave ordered state, while the other intervening phase displays superconductivity at weak couplings. This method, as a general framework, can be extended to treat excited states and dynamics, as well as a wide range of systems with both electron-electron and electron-boson interactions.

Nonadiabatic dynamics with intersystem crossings: A time-dependent density functional theory implementation

In this work, we derive a method to perform trajectory-based nonadiabatic dynamics that is able to describe both nonadiabatic transitions and intersystem crossing events (transitions between states of different spin-multiplicity) at the same level of theory, namely, time-dependent density functional theory (TDDFT). To this end, we combined our previously developed TDDFT-based trajectory surface hopping scheme with an accurate and efficient algorithm for the calculation of the spin-orbit coupling (SOC) matrix elements. More specifically, we designed two algorithms for the calculation of intersystem crossing transitions, one based on an extended Tully's surface hopping scheme including SOC and the second based on a Landau-Zener approximation applied to the spin sector of the electronic Hilbert space. This development allows for the design of an efficient on-the-fly nonadiabatic approach that can handle, on an equal footing, nonadiabatic and intersystem crossing transitions. The method is applied to the study of the photophysics of sulfur dioxide (SO2) in gas and liquid phases.

Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system

We investigate the entanglement between electronic and vibrational degrees of freedom produced by a vibronic coupling in a molecular system described in the Born-Oppenheimer approximation. Entanglement in a pure state of the Hilbert space $\mathcal{H}={\mathcal{H}}_{\mathrm{el}}⨂{\mathcal{H}}_{\mathrm{vib}}$ is quantified using the von Neumann entropy of the reduced density matrix and the reduced linear entropy. Expressions for these entanglement measures are derived for the $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ and $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ cases of the bipartite entanglement, where 2 and 3 are the dimensions of the electronic Hilbert space $\mathcal{H}$${}_{\mathrm{el}}$, and ${N}_{v}$ is the dimension of $\mathcal{H}$${}_{\mathrm{vib}}$. We study the entanglement dynamics for two electronic states coupled by a laser pulse (a $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case), taking as an example a coupling between the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$ and ${1}_{g}(6s,6{p}_{3/2})$ states of the Cs${}_{2}$ molecule. The reduced linear entropy expression obtained for the $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case is used to follow the entanglement evolution in a scheme proposed for the control of the vibronic dynamics in a Cs${}_{2}$ cold molecule, implying the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$, ${0}_{g}{}^{\ensuremath{-}}(6s,6{p}_{3/2})$, and ${0}_{g}{}^{\ensuremath{-}}(6s,5d)$ electronic states, which are coupled by a nonadiabatic radial coupling and a sequence of chirped laser pulses.

Generalized electronic diabatic scheme: Diagonalizing the electronic Hamiltonian for artificial molecular systems. How do molecular meccanos move?

AbstractA quantum–classic model is presented and used to describe systems ranging from normal molecules up to electronic systems sensed in real space. The quantum system is a set of n‐electrons; a positive background in real space completes the model. A generalized electronic diabatic (GED) theory is introduced. The diabatic functions diagonalize the electronic Hamiltonian for any arrangement of the positive background. Physical quantum states are represented as linear superpositions in the diabatic basis; this latter is always fixed. For systems sensed in real space, the coefficients of the linear superposition are functions of the real space configuration coordinates. Physical changes are produced by interactions with external sources/sinks of energy. An interaction couples different diabatic states; diagonalizing the electronic Hamiltonian plus the couplings leads to new coefficients describing physical states. Among other things, these couplings can be used to simulate the effects produced by scanning tunneling microscopy, atomic force, and transmission electron microscopy on substrates located in real space. The important thing is that time–evolution in electronic Hilbert space can be related to actual motion in real space. The experiment of lateral hopping of a substrate on a metallic surface induced by vibration excitation and followed with scanning tunneling microscope is discussed. A result of the present work is that motion of molecular meccanos reflects then time–evolution in electronic Hilbert space. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

The manipulation of massive ro-vibronic superpositions using time–frequency-resolved coherent anti-Stokes Raman scattering (TFRCARS): from quantum control to quantum computing

Molecular ro-vibronic coherences, joint energy-time distributions of quantum amplitudes, are selectively prepared, manipulated, and imaged in time–frequency-resolved coherent anti-Stokes Raman scattering (TFRCARS) measurements using femtosecond laser pulses. The studies are implemented in iodine vapor, with its thermally occupied statistical ro-vibrational density serving as initial state. The evolution of the massive ro-vibronic superpositions, consisting of 10 3 eigenstates, is followed through two-dimensional images. The first- and second-order coherences are captured using time-integrated frequency-resolved CARS, while the third-order coherence is captured using time-gated frequency-resolved CARS. The Fourier filtering provided by time-integrated detection projects out single ro-vibronic transitions, while time-gated detection allows the projection of arbitrary ro-vibronic superpositions from the coherent third-order polarization. A detailed analysis of the data is provided to highlight the salient features of this four-wave mixing process. The richly patterned images of the ro-vibrational coherences can be understood in terms of phase evolution in rotation–vibration–electronic Hilbert space, using time-circuit diagrams. Beside the control and imaging of chemistry, the controlled manipulation of massive quantum coherences suggests the possibility of quantum computing. We argue that the universal logic gates necessary for arbitrary quantum computing – all single qubit operations and the two-qubit controlled-NOT (CNOT) gate – are available in time-resolved four-wave mixing in a molecule. The molecular rotational manifold is naturally “wired” for carrying out all single qubit operations efficiently, and in parallel. We identify vibronic coherences as one example of a naturally available two-qubit CNOT gate, wherein the vibrational qubit controls the switching of the targeted electronic qubit.

Algebraic Bethe ansatz for the anisotropic supersymmetric U model

We present an algebraic Bethe ansatz for the anisotropic supersymmetric U model for correlated electrons on the unrestricted 4(L)-dimensional electronic Hilbert space x(n=l)(L)C(4)(where L is the lattice length). The supersymmetry algebra of the local Hamiltonian is the quantum superalgebra U-q[gl(2\1)] and the model contains two symmetry-preserving free real parameters; the quantization parameter q and the Hubbard interaction parameter U. The parameter U arises from the one-parameter family of inequivalent typical four-dimensional irreps of U-q[gl(2\1)]. Eigenstates of the model are determined by the algebraic Bethe ansatz on a one-dimensional periodic lattice.

Algebraic Bethe ansatz for the supersymmetric U model.

We present an algebraic Bethe ansatz for the anisotropic supersymmetric U model for correlated electrons on the unrestricted 4(L)-dimensional electronic Hilbert space x(n=l)(L)C(4)(where L is the lattice length). The supersymmetry algebra of the local Hamiltonian is the quantum superalgebra U-q[gl(2\1)] and the model contains two symmetry-preserving free real parameters; the quantization parameter q and the Hubbard interaction parameter U. The parameter U arises from the one-parameter family of inequivalent typical four-dimensional irreps of U-q[gl(2\1)]. Eigenstates of the model are determined by the algebraic Bethe ansatz on a one-dimensional periodic lattice. ©1996 The American Physical Society

New Supersymmetric and Exactly Solvable Model of Correlated Electrons.

A new lattice model is presented for correlated electrons on the unrestricted $4^L$-dimensional electronic Hilbert space $\otimes_{n=1}^L{\bf C}^4$ (where $L$ is the lattice length). It is a supersymmetric generalization of the Hubbard model, but differs from the extended Hubbard model proposed by Essler, Korepin and Schoutens. The supersymmetry algebra of the new model is superalgebra $gl(2|1)$. The model contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter $U$, and has its origin here in the one-parameter family of inequivalent typical 4-dimensional irreps of $gl(2|1)$. On a one-dimensional lattice, the model is exactly solvable by the Bethe ansatz.

Spin flux and magnetic solitons in an interacting two-dimensional electron gas: Topology of two-valued wave functions.

It is suggested that an interacting many-electron system in a two-dimensional lattice may condense into a topological magnetic state distinct from any discussed previously. This condensate exhibits local spin-1/2 magnetic moments on the lattice sites but is composed of a Slater determinant of single-electron wave functions which exist in an orthogonal sector of the electronic Hilbert space from the sector describing traditional spin-density-wave or spiral magnetic states. These one-electron spinor wave functions have the distinguishing property that they are antiperiodic along a closed path encircling any elementary plaquette of the lattice. This corresponds to a 2\ensuremath{\pi} rotation of the internal coordinate frame of the electron as it encircles the plaquette. The possibility of spinor wave functions with spatial antiperiodicity is a direct consequence of the two-valuedness of the internal electronic wave function defined on the space of Euler angles describing its spin. This internal space is the topologically, doubly-connected, group manifold of SO(3). Formally, these antiperiodic wave functions may be described by passing a flux which couples to spin (rather than charge) through each of the elementary plaquettes of the lattice. When applied to the two-dimensional Hubbard model with one electron per site, this new topological magnetic state exhibits a relativistic spectrum for charged, quasiparticle excitations with a suppressed one-electron density of states at the Fermi level.For a topological antiferromagnet on a square lattice, with the standard Hartree-Fock, spin-density-wave decoupling of the on-site Hubbard interaction, there is an exact mapping of the low-energy one-electron excitation spectrum to a relativistic Dirac continuum field theory. In this field theory, the Dirac mass gap is precisely the Mott-Hubbard charge gap and the continuum field variable is an eight-component Dirac spinor describing the components of physical electron-spin amplitude on each of the four sites of the elementary plaquette in the original Hubbard model. Within this continuum model we derive explicitly the existence of hedgehog Skyrmion textures as local minima of the classical magnetic energy. These magnetic solitons carry a topological winding number \ensuremath{\mu} associated with the vortex rotation of the background magnetic moment field by a phase angle 2\ensuremath{\pi}\ensuremath{\mu} along a path encircling the soliton. Such solitons also carry a spin flux of \ensuremath{\mu}\ensuremath{\pi} through the plaquette on which they are centered. The \ensuremath{\mu}=1 hedgehog Skyrmion describes a local transition from the topological (antiperiodic) sector of the one-electron Hilbert space to the nontopological sector. We derive from first principles the existence of deep level localized electronic states within the Mott-Hubbard charge gap for the \ensuremath{\mu}=1 and 2 solitons. The spectrum of localized states is symmetric about E=0 and each subgap electronic level can be occupied by a pair of electrons in which one electron resides primarily on one sublattice and the second electron on the other sublattice. It is suggested that flux-carrying solitons and the subgap electronic structure which they induce are important in understanding the physical behavior of doped Mott insulators.