Self-energy Approach Research Articles

Spectral functions of the half-filled one-dimensional Hubbard chain within the exchange-correlation potential formalism

The spectral functions of the one-band half-filled 1D Hubbard chain are calculated using the exchange-correlation potential formalism developed recently. The exchange-correlation potential is adopted from the exact potential derived from the Hubbard dimer. Within an approximation in which the full Green function is replaced by a non-interacting one, the spectral functions can be calculated analytically. Despite the simplicity of the approximation, the resulting spectra are in favorable agreement with the more accurate results obtained from the dynamic density-matrix renormalization group method. In particular, the calculated band gap as a function of $U$ is in close agreement with the exact gap obtained from the Bethe ansatz. In addition, the formal general solution to the equation of motion of the Green function is presented and the difference between the traditional self-energy approach and the exchange-correlation potential formalism is also discussed and elaborated. A simplified Holstein Hamiltonian is considered to further illustrate the general form of the exchange-correlation potential.

Self-energy method for time-dependent spectral functions of the Anderson impurity model within the time-dependent numerical renormalization group approach

The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation of motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time-evolution of spectral features at the different characteristic time-scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.

Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin's Full Second-Order Self-Energy Contribution.

We present and benchmark a self-energy approach for quasiparticle energy calculations that goes beyond Hedin's GW approximation by adding the full second-order self-energy (FSOS-W) contribution. The FSOS-W diagram involves two screened Coulomb interaction (W) lines, and adding the FSOS-W to the GW self-energy can be interpreted as first-order vertex correction to GW (GWΓ(1)). Our FSOS-W implementation is based on the resolution-of-identity technique and exhibits better than O(N5) scaling with system size for small- to medium-sized molecules. We then present one-shot GWΓ(1) (G0W0Γ0(1)) benchmarks for the GW100 test set and a set of 24 acceptor molecules. For semilocal or hybrid density functional theory starting points, G0W0Γ0(1) systematically outperforms G0W0 for the first vertical ionization potentials and electron affinities of both test sets. Finally, we demonstrate that a static FSOS-W self-energy significantly underestimates the quasiparticle energies.

Self-energy Padé approach for analytic continuation: Application to the zero-gap Kondo lattice model

A modified Pade approach is presented as analytic continuation for numerical methods that evaluate correlation functions in imaginary time. Instead of the direct analytic continuation of the correlation functions, the Pade method is applied for the self-energy that is then used for deriving the Green's functions at real energies. We find that this modified, self-energy Pade approach is more stable and robust against statistical errors compared to the direct way. The characteristics and success of the modified Pade approach are analyzed by actual calculations for the illustrative cases of the non-interacting Anderson lattice and Hubbard model. A zero-gap Kondo lattice model with linearly vanishing conduction electron density of states at the Fermi level is also studied, where we use the modified Pade approach as analytic continuation. We investigate the properties of the Kondo insulating state including the dependence of the insulating gap on the Kondo coupling and coherence effects. Furthermore, we identify two energy scales from dynamic and thermodynamic quantities, that are associated as a direct and an indirect gap in a band hybridization picture.

Classical and quantum dynamics of indirect excitons driven by surface acoustic waves

We perform explicit time-dependent classical and quantum propagation of a spatially indirect exciton (SIX) driven by surface acoustic waves (SAWs) in a semiconductor heterostructure device. We model the SIX dynamics at different levels of description, from the Euler-Lagrange propagation of structureless classical particles to unitary Schr\"odinger propagation of an electron-hole wave packet in a mean field and to the full quantum propagation of the two-particle complex. A recently proposed beyond mean-field self-energy approach, adding internal virtual transitions to the c.m. dynamics, has been generalized to time-dependent potentials and turns out to describe very well full quantum calculations, while being orders of magnitude numerically less demanding. We show that SAW-driven SIXs are a sensitive probe of scattering potentials in the devices originating, for example, from single impurities or metallic gates, due to competing length and energy scales between the SAW elastic potential, the scattering potential, and the internal electron-hole dynamic of the SIX. Comparison between different approximations allow us to show that internal correlation of the electron-hole pair is crucial in scattering from shallow impurities, where tunneling plays a major role. On the other hand, scattering from broad potentials, i.e., with length scales exceeding the SIX Bohr radius, is well described as the classical dynamics of a pointlike SIX. Recent experiments are discussed in light of our calculations.

Variations on the “exact factorization” theme

In a series of publications, Hardy Gross and co-workers have highlighted the interest of an “exact factorization” approach to the interacting electron-nuclei problem, be it time-independent or time-dependent. In this approach, an effective potential governs the dynamics of the nuclei such that the resulting N-body nuclear density is in principle exact. This contrasts with the more usual adiabatic approach, where the effective potential leads to an approximate nuclear density. Inspired by discussions with Hardy, we explore the factorization idea for arbitrary many-body Hamiltonians, generalizing the electron-nuclei case, with a focus on the static case. While the exact equations do not lead to any practical advantage, they are illuminating, and may therefore constitute a suitable starting point for approximations. In particular, we find that unitary transformations that diagonalize the coupling term for one of the sub-systems make exact factorization appealing. The algorithms by which the equations for the separate subsystems can be solved in the time-independent case are also explored. We illustrate our discussions using the two-site Holstein model and the quantum Rabi model. Two factorization schemes are possible: one where the boson field feels a potential determined by the electrons, and the reverse exact factorization, where the electrons feel a potential determined by the bosons; both are explored in this work. A comparison with a self-energy approach is also presented.

Strong Plasmon-Phonon Splitting and Hybridization in 2D Materials Revealed through a Self-Energy Approach

We reveal new aspects of the interaction between plasmons and phonons in 2D materials that go beyond a mere shift and increase in plasmon width due to coupling to either intrinsic vibrational modes of the material or phonons in a supporting substrate. More precisely, we predict strong plasmon splitting due to this coupling, resulting in a characteristic avoided crossing scheme. We base our results on a computationally efficient approach consisting in including many-body interactions through the electron self-energy. We specify this formalism for a description of plasmons based upon a tight-binding electron Hamiltonian combined with the random-phase approximation. This approach is accurate provided vertex corrections can be neglected, as is is the case in conventional plasmon-supporting metals and Dirac-fermion systems. We illustrate our method by evaluating plasmonic spectra of doped graphene nanotriangles with varied size, where we predict remarkable peak splittings and other radical modifications in the spectra due to plasmons interactions with intrinsic optical phonons. Our method is equally applicable to other 2D materials and provides a simple approach for investigating coupling of plasmons to phonons, excitons, and other excitations in hybrid thin nanostructures.

Time-dependent scattering of a composite particle: A local self-energy approach for internal excitations

When composite particles---such as small molecules, nuclei, or photogenerated excitons in semiconductors---are scattered by an external potential, energy may be transferred between the c.m. and the internal degrees of freedom. An accurate dynamical modeling of this effect is pivotal in predicting diverse scattering quantities and reaction cross sections, and allows us to rationalize time-resolved energy and localization spectra. Here, we show that time-dependent scattering of a quantum composite particle with an arbitrary, nonperturbative external potential can be obtained by propagating the c.m. degrees of freedom with a properly designed local self-energy potential. The latter embeds the effect of internal virtual transitions and can be obtained by the knowledge of the stationary internal states. The case is made by simulating Wannier-Mott excitons in one- and two-dimensional semiconductor heterostructures. The self-energy approach shows very good agreement with numerically exact Schr\odinger propagation for scattering potentials where a mean-field model cannot be applied, at a dramatically reduced computational cost.

Three-body losses of repulsively interacting three-component fermionic atoms in optical lattices

We investigate the effects of a repulsive three-body interaction on the Mott transition of the repulsively interacting three-component fermionic atoms in optical lattices by means of the self-energy functional approach. We find that the three-body repulsion hardly affects the qualitative features of the Mott transition, because the three-body repulsion does not compete with the two-body repulsions. When the three-body repulsion is extremely strong, the triple occupancy vanishes in the Fermi liquid state. This situation is equivalent to that caused by strong three-body losses. Our results imply that three-body losses have little influence on the Mott transitions in the repulsively interacting three-component fermionic atoms in optical lattices.

Stochastic self-energy subgrid model for the large eddy simulation of turbulent channel flows

This paper presents the large eddy simulation (LES) of turbulent channel flow using a self-energy (SE) subgrid model with coefficients determined from reference direct numerical simulations (DNSs). In contrast to standard approaches that develop subgrid models based upon physical hypotheses, in the present SE approach the model coefficients are determined from the subgrid statistics of a DNS, with physical interpretations made apostiori. This technique is applied here for the first time to wall-bounded flows, specifically channel flow. The stochastic SE subgrid model consists of a meanfield shift, deterministic drain dissipation acting on the resolved field and a stochastic backscatter force. The deterministic SE subgrid model comprises of a net dissipation representing the net effect of the drain and backscatter. We present LESs that reproduce the time-averaged kinetic energy spectra of the DNS within the resolved scales. The direction and magnitude of the energy transfers in scale space can then be determined from the coefficients of the SE subgrid model. Results are presented for friction velocity based Reynolds numbers up to Reτ = 950.

Extended self-energy functional approach for strongly correlated lattice bosons in the superfluid phase

Among the various numerical techniques to study the physics of strongly correlated quantum many-body systems, the self-energy functional approach (SFA) has become increasingly important. In its previous form, however, SFA is not applicable to Bose-Einstein condensation or superfluidity. In this paper we show how to overcome this shortcoming. To this end we identify an appropriate quantity, which we term $D$, that represents the correlation correction of the condensate order parameter, as it does the self-energy for the Green's function. An appropriate functional is derived, which is stationary at the exact physical realizations of $D$ and of the self-energy. Its derivation is based on a functional-integral representation of the grand potential followed by an appropriate sequence of Legendre transformations. The approach is not perturbative and therefore applicable to a wide range of models with local interactions. We show that the variational cluster approach based on the extended self-energy functional is equivalent to the "pseudoparticle" approach introduced in Phys. Rev. B, 83, 134507 (2011). We present results for the superfluid density in the two-dimensional Bose-Hubbard model, which show a remarkable agreement with those of Quantum-Monte-Carlo calculations.

COLOR SUPERFLUID AND TRIONIC STATE OF ATTRACTIVE THREE-COMPONENT LATTICE FERMIONIC ATOMS AT FINITE TEMPERATURES

We investigate the finite-temperature properties of attractive three-component (colors) fermionic atoms in optical lattices using a self-energy functional approach. As the strength of the attractive interaction increases in the low temperature region, a second-order transition occurs from a Fermi liquid to a color superfluid (CSF). In the strong attractive region, a first-order transition occurs from a CSF to a trionic state. In the high temperature region, a cross-over between a Fermi liquid and a trionic state is observed with increasing the strength of the attractive interaction. The cross-over region for fixed temperature is almost independent of filling.

Three-component Repulsive Fermionic Atoms in Optical Lattices at finite temperatures

We investigate finite-temperature properties of three-component (color) repulsive fermionic atoms in optical lattices at half filling using a self-energy functional approach. For a certain anisotropy of the interactions, we find a color density-wave (DW) state at low temperatures. In the weakly interacting region, the second-order phase transition from the color DW state to the Fermi-liquid state occurs with increasing temperature. In the strongly interacting region, by contrast, the first-order transition from the color DW state to a color selective Mott transition (CSMT) state or a Mott insulating state occurs. As the interaction increases above the critical temperatures, we observe a crossover from the CSMT state to the Mott insulating state.

Thermodynamic properties of two-component fermionic atoms trapped in a two-dimensional optical lattice

We study the finite temperature properties of two-component fermionic atoms trapped in a two-dimensional optical lattice. We apply the self-energy functional approach to the two-dimensional Hubbard model with a harmonic trapping potential, and systematically investigate the thermodynamic properties of this system. We find that entropy and grand potential provide evidence of a crossover between the Mott insulating and metallic phases at certain temperatures. In addition, we find that entropy exhibits a cusp-like anomaly at lower temperatures, suggesting a second or higher order antiferromagnetic transition. We estimate the antiferromagnetic transition temperatures, and clarify how the trapping potential affects this magnetic transition.

A self-energy functional approach for spin systems

We set up a new approach to study the physics of spin systems which uses the resolvent method originally proposed by Keiter and Kuramoto. In analogy to the Baym-Kadanoff formalism the latter introduced the partition sum of a system as a functional of the resolvent of the Hamiltonian and its so called generalized self-energy. This functional is the starting point for a variational cluster method which is based on Potthoffs self-energy functional approach. In a first step we apply our ansatz to a Heisenberg spin chain.

Interacting spin waves in the ferromagnetic Kondo lattice model

We present an approach for the ferromagnetic, three-dimensional, and translational-symmetric Kondo lattice model, which allows us to derive both magnon energies and linewidths (lifetimes) and to study the properties of the ferromagnetic phase at finite temperatures. Both ``anomalous softening'' and ``anomalous damping'' are obtained and discussed. Our method consists of mapping the Kondo lattice model onto an effective Heisenberg model by means of the ``modified RKKY interaction'' and the ``interpolating self-energy approach.'' The Heisenberg model is approximately solved by applying the Dyson-Maleev transformation and using the ``spectral density approach'' with a broadened magnon spectral density.

Finite-temperature properties of attractive three-component fermionic atoms in optical lattices

We investigate the finite-temperature properties of attractive three-component (colors) fermionic atoms in optical lattices using a self-energy functional approach. As the strength of the attractive interaction increases in the low-temperature region, we observe a second-order transition from a Fermi liquid to a color superfluid (CSF), where atoms from two of the three colors form Cooper pairs. In the strong attractive region, we observe a first-order transition from a CSF to a trionic state, where three atoms with different colors form singlet bound states. A crossover between a Fermi liquid and a trionic state is observed in the high-temperature region. We present a phase diagram covering zero to finite temperatures. We demonstrate that the CSF transition temperature is enhanced by the anisotropy of the attractive interaction.

Zero-temperature phase diagram of two dimensional Hubbard model

We investigate the two-dimensional Hubbard model on the triangular lattice with anisotropic hopping integrals at half filling. By means of a self-energy functional approach, we discuss how stable the non-magnetic state is against magnetically ordered states in the system. We present the zero-temperature phase diagram, where the normal metallic state competes with magnetically ordered states with (π, π) and (2π /3, 2π /3) structures. It is shown that a nonmagnetic Mott insulating state is not realized as the ground state, in the present framework, but as a meta-stable state near the magnetically ordered phase with (2π /3, 2π/3) structure.

Magnetic phase diagrams of manganite-like local-moment systems with Jahn-Teller distortions

We use an extended two-band Kondo lattice model (KLM) to investigate the occurrence of different (anti)ferromagnetic phases or phase separation depending on several model parameters. With regard to CMR-materials such as the manganites we have added a Jahn-Teller term, direct antiferromagnetic coupling, and Coulomb interaction to the KLM. The electronic properties are self-consistently calculated in an interpolating self-energy approach with no restriction to classical spins and going beyond mean-field treatments. Further on we do not have to limit the Hund's coupling to low or infinite values. Zero-temperature phase diagrams are presented for large parameter intervals. There are strong influences of the type of Coulomb interaction (intraband, interband) and of the important parameters (Hund's coupling, direct antiferromagnetic exchange, Jahn-Teller distortion), especially at intermediate couplings.

Metal–Insulator Transition in the Two-Orbital Hubbard Model at Fractional Band Fillings: Self-Energy Functional Approach

We investigate the infinite-dimensional two-orbital Hubbard model at arbitrary band fillings. By means of the self-energy functional approach, we discuss the stability of the metallic state in the systems with same and different bandwidths. It is found that the Mott insulating phases are realized at commensurate band fillings. Furthermore, it is clarified that the orbital selective Mott phase with one orbital localized and the other itinerant is stabilized even at fractional band fillings in the system with different bandwidths.