Particle-hole Propagator Research Articles

Variational and parquet-diagram calculations for neutron matter. III. S -wave pairing

We apply parquet-diagram summation methods for the calculation of the superfluid gap in $S$-wave pairing in neutron matter for realistic nucleon-nucleon interactions such as the Argonne $v_6$ and the Reid $v_6$ potentials. It is shown that diagrammatic contributions that are outside the parquet class play an important role. These are, in variational theories, identified as so-called "commutator contributions". Moreover, using a particle-hole propagator appropriate for a superfluid system results in the suppression of the spin-channel contribution to the induced interaction. Applying these corrections to the pairing interaction, our results agree quite well with Quantum Monte Carlo data.

Bosonization of the Q=0 continuum of Dirac fermions

We develop a bosonization formalism that captures non-perturbatively the interaction effects on the $\mathbf{Q}=0$ continuum of excitations of nodal fermions above one dimension. Our approach is a natural extension of the classic bosonization scheme for higher dimensional Fermi surfaces to include the $\mathbf{Q}=0$ neutral excitations that would be absent in a single-band system. The problem is reduced to solving a boson bilinear Hamiltonian. We establish a rigorous microscopic footing for this approach by showing that the solution of such boson bilinear Hamiltonian is exactly equivalent to performing the infinite sum of Feynman diagrams associated with the Kadanoff-Baym particle-hole propagator that arises from the self-consistent Hartree-Fock approximation to the single particle Green's function. We apply this machinery to compute the interaction corrections to the optical conductivity of 2D Dirac Fermions with Coulomb interactions reproducing the results of perturbative renormalization group at weak coupling and extending them to the strong coupling regime.

Nuclear electromagnetic dipole response with the self-consistent Green's function formalism

Microscopic calculations of the electromagnetic response of medium-mass nuclei are now feasible thanks to the availability of realistic nuclear interactions with accurate saturation and spectroscopic properties, and the development of large-scale computing methods for many-body physics. The purpose is to compute isovector dipole electromagnetic (E1) response and related quantities, i.e. integrated dipole cross section and polarizability, and compare with data from photoabsorption and Coulomb excitation experiments. The single-particle propagator is obtained by solving the Dyson equation, where the self-energy includes correlations non-perturbatively through the Algebraic Diagrammatic Construction (ADC) method. The particle-hole ($ph$) polarization propagator is treated in the Dressed Random Phase Approximation (DRPA), based on an effective correlated propagator that includes some $2p2h$ effects but keeps the same computation scaling as the standard Hartree-Fock propagator. The E1 responses for $^{14,16,22,24}$O, $^{36,40,48,52,54,70}$Ca and $^{68}$Ni have been computed: the presence of a soft dipole mode of excitation for neutron-rich nuclei is found, and there is a fair reproduction of the low-energy part of the experimental excitation spectrum. This is reflected in a good agreement with the empirical dipole polarizability values. For a realistic interaction with an accurate reproduction of masses and radii up to medium-mass nuclei, the Self-Consistent Green's Function method provides a good description of the E1 response, especially in the part of the excitation spectrum below the Giant Dipole Resonance. The dipole polarizability is largely independent from the strategy of mapping the dressed propagator to a simplified one that is computationally manageable

In medium dispersion relation effects in nuclear inclusive reactions at intermediate and low energies

In a well-established many-body framework, successful in modeling a great variety of nuclear processes, we analyze the role of the spectral functions (SFs) accounting for the modifications of the dispersion relation of nucleons embedded in a nuclear medium. We concentrate in processes mostly governed by one-body mechanisms, and study possible approximations to evaluate the particle–hole propagator using SFs. We also investigate how to include together SFs and long-range RPA-correlation corrections in the evaluation of nuclear response functions, discussing the existing interplay between both type of nuclear effects. At low energy transfers (≤50MeV), we compare our predictions for inclusive muon and radiative pion captures in nuclei, and charge–current (CC) neutrino–nucleus cross sections with experimental results. We also present an analysis of intermediate energy quasi-elastic neutrino scattering for various targets and both neutrino and antineutrino CC driven processes. In all cases, we pay special attention to estimate the uncertainties affecting the theoretical predictions. In particular, we show that errors on the σμ∕σe ratio are much smaller than 5%, and also much smaller than the size of the SF+RPA nuclear corrections, which produce significant effects, not only in the individual cross sections, but also in their ratio for neutrino energies below 400 MeV. These latter nuclear corrections, beyond Pauli blocking, turn out to be thus essential to achieve a correct theoretical understanding of this ratio of cross sections of interest for appearance neutrino oscillation experiments. We also briefly compare our SF and RPA results to predictions obtained within other representative approaches.

Resummation of fermionic in-medium ladder diagrams to all orders

A system of fermions with a short-range interaction proportional to the scattering length $a$ is studied at finite density. At any order $a^n$, we evaluate the complete contributions to the energy per particle $\bar E(k_f)$ arising from combined (multiple) particle-particle and hole-hole rescatterings in the medium. This novel result is achieved by simply decomposing the particle-hole propagator into the vacuum propagator plus a medium-insertion and correcting for certain symmetry factors in the $(n-1)$-th power of the in-medium loop. Known results for the low-density expansion up to and including order $a^4$ are accurately reproduced. The emerging series in $a k_f$ can be summed to all orders in the form of a double-integral over an arctangent function. In that representation the unitary limit $a\to \infty$ can be taken and one obtains the value $\xi= 0.5067$ for the universal Bertsch parameter. We discuss also applications to the equation of state of neutron matter at low densities and mention further extensions of the resummation method.

Tracing molecular electronic excitation dynamics in real time and space

We present a method for studying the movement of electrons and energy within and between electronically excited molecules. The dynamically changing state is a many-electron wavepacket, for which we numerically integrate the Schrodinger equation using the ADC(2) effective Hamiltonian for the particle-hole propagator. We develop the tools necessary for following the separate motions of the particles and holes. Total particle and hole densities can be used to give an overview of the dynamics, which can be atomically decomposed in a Mulliken fashion, or individual particle and hole states give a more detailed look at the structure of an excitation. We apply our model to a neon chain, as an illustrative example, projecting an excited eigenstate of an isolated atom onto the coupled system as the initial state. In addition to demonstrating our propagation and analysis machinery, the results show a dramatic difference in excitation-energy transfer rates as a consequence of initial polarization. Furthermore, already in a system with three constituents, an important aspect of multiple coupled systems appears, in that one absorbing system essentially shields another, changing the effective sitewise coupling parameters.

On the relations between the polarization propagator, the principal propagator, and the particle-hole propagator

It is demonstrated that the polarization propagator, the principal propagator, and in some cases also the particle-hole propagator calculated through the same order in perturbation theory all yield the same excitation energies, while only the residues of the polarization can be identified with electronic transition moments in methods which go beyond the random-phase approximation. The extra terms appearing in the second-order polarization propagator approach account for up to 10% of the individual electronic transition moments.

On the calculation of generalized antisymmetric geminal power (GAGP) wavefunctions

It has recently been shown that a wavefunction formed from an antisymmetrized product of identical geminals times a single determinant is, when optimized properly, a consistent ground state of the particle-hole propagator. This optimized generalized antisymmetrized geminal power (GAGP) wavefunction is reviewed and its computational implementation described. The procedure is applied to the X2S state of Li, 3S and X1S states of Be, and X1S state of Mg.

Extension of the random phase approximation including the self-consistent coupling to two-phonon contributions

A microscopic formalism is developed that includes the coupling to two particle-hole phonons in the particle-hole propagator by extending the dressed random phase approximation (DRPA) equation for a finite system. The resulting formalism is applied to study the low-lying excitation spectrum of 16O. It is observed that the coupling to two-phonon states at low energy generates excited states with quantum numbers that cannot be obtained in the DRPA approach. Nevertheless, the two-phonon states mix weakly with particle-hole configurations and participate only partially in the formation of the lowest-lying positive-parity excited states. The stability of the present calculation is tested vs. the truncation of model space. It is demonstrated that when single-particle strength fragmentation is properly considered, the present formalism exhibits convergence with respect to the chosen model space within the confines of the chosen approximation scheme.

Finite expansion of the inverse matrix in the polarization propagator method

An alternative theoretical approach to the polarization propagator based on a new finite expansion of a finite-dimensional matrix is presented. The general equations for such an expansion are derived and the validity conditions stated. This method is used to accomplish an approximate scheme for the self-energy of the particle–hole propagator within the superoperator formalism. Within this scheme each contribution includes corrections to infinite order in electronic interaction and so describes collective effects in a natural way. Individual contributions can be interpreted as describing the propagation of the interaction through a particular subset of electronic excitations. Comparison with other known approximation levels, such as the random-phase approximation, is also analyzed.

Study of normal liquid 3He using the method of collective description of interparticle interactions and a realistic form of two-body potential

Visualizing 3He liquid at zero temperature as an interacting system of quasi-particles and the quanta of its zero-sound mode, calculation of the binding energy of the system in its normal phase is made using many-body techniques with a realistic form of interparticle potential constituting both repulsive and attractive components. The new potential has a form similar to the Walecka potential used by us some time back in the study of nuclear matter, and unlike the Lennard–Jones potential, which is most commonly used for liquid 3He, it has a Fourier transform. The calculated binding energy per particle in liquid 3He is found to be −2.478 K, which exactly agrees with the experimental result −2.48 K. This shows a substantial improvement over our earlier result obtained for a contact form of interparticle potential. For the new potential, we arrive at a velocity of 203.8 m/s for the zero-sound mode by following a self-consistent method of calculation, which is close to the experimental value of 194.4 m/s obtained by Abel et al. (Phys. Rev. 147 (1966) 111). By accounting for the local field corrections, beyond the random-phase approximation, to the bare particle-hole propagator in a parametric way, we exactly reproduce the experimental result for the zero-sound mode. The present calculation also indicates the propagation of a new kind of sound wave in liquid 3He at T=0 K, with a velocity of 153.7 m/s, similar to the one found in an earlier calculation.

On the dynamic polarizability of atoms

The positive frequency dependent polarizability of atoms α(ω) is discussed in terms of the particle-hole polarization propagator. It is considered in the simplest approximation defined by the Bethe-Salpeter equation which includes a subset of particle-hole interactions to all orders in the Coulomb potential. Its solution is used to show that when the polarizability is formulated in terms of an effective dipole matrix element of the atomic sub-shells, the interaction kernel is symmetric and hence makes the central quantity in the formulation, the polarization function, hermitian.

Can there be a T3 ln T kind of behaviour of the low temperature specific heat of liquid 3He without the paramagnons?

It is shown that even without invoking the concepts like paramagnons, the temperature dependence of the interparticle correlations in a system of liquid 3He can give rise a T 3 ln T kind of behaviour of its low temperature fermionic specific heat. It is found to be coming from the self-energy corrections to the bare single particle energy involving the particle-hole propagator. Looking at the similar kind of behaviour observed by us recently for an electron liquid, one may conclude that the T 3 ln T behaviour is perhaps universal for all fermi systems, although for liquid 3He the very dependence also follows from the paramagnon effects. It is interesting to see that unlike earlier theories, an extremely good fit is obtained with the experimental data over the entire range of low temperatures.

Study of excited states of polyethylene in the Hartree–Fock, Tamm–Dancoff, and random-phase approximations

We report calculations of the excitonic spectra of all-trans polyethylene in the Hartree–Fock, Tamm–Dancoff, and random-phase approximations. For the formulation of approximations beyond Hartree–Fock level, we have used the particle–hole propagator technique. In the first case the interaction between the excited electron and hole is neglected and in the latter two cases it is taken into account at first order in electron–electron interaction. Using the 3–21G atomic basis set, the elementary gap is reduced from about 17 eV at the Hartree–Fock level to 13 eV in the Tamm–Dancoff approximation. The spectra in both approximations are also different. The difference between the results obtained in the Tamm–Dancoff and the random-phase approximations is very small. We discuss the densities of exciton states for different exciton quasi momenta (K=0 or K=π/a), different band interactions and different spins.

Self-energies for the particle-hole propagator from Feynman-Dyson equations

Correlation-relaxation corrections to electronic transition energies are obtained through the self-energy fields for the particle-hole propagator. These objects are generated from Feynman-Dyson like equations within the scenario of the superoperator algebra. A Hartree-Fock reference state and a partition of the operator space along with a particle conserving manifold are used to recast the lower order perturbative expressions for the self-energies. Correlated states are used to make explicit the appearance of higher order correction terms to the self-energy. It is explicitly shown that the present scheme leads to a proper classification of the order of the contribution from both the intrinsic interaction operator and the improved reference state.

Response function of asymmetric nuclear matter

The charge longitudinal response function is examined in the framework of the random-phase approximation in an isospin-asymmetric nuclear matter where proton and neutron densities are different. This asymmetry changes the response through both the particle-hole interaction and the free particle-hole polarization propagator. We discuss these two effects on the response function on the basis of our numerical results in detail.

A self-consistent scheme for evaluating the velocity of the zero-sound in normal liquid 3He and the study of its variation with temperature

Using the model Hamiltonian which has been derived by us recently for the study of liquid 3He, we have succeeded in reproducing a value of 204.2 m/s for the velocity of the zero-sound in liquid 3He at temperature T = 0, by following a self-consistent method. This is close to the experimental result of 194.4 m/s as obtained by Abel et al. An exact agreement with the experimental value is obtained by introducing a term into our theory to account for the local field corrections to the bare particle-hole propagator beyond the random phase approximation (RPA) in a parametric way. By considering the effects of damping of the zero-sound mode, we have been able to explain fairly well the variation of the zero-sound velocity with temperature. The present theory seems to also indicate the propagation of a new kind of sound wave in liquid 3He having a velocity of 153.7 m/s.

Important modifications near a magnetic instability in the particle-hole propagator of weakly localized two-dimensional fermion systems

In the Hubbard-type paramagnon model important changes arise in the particle-hole propagator structure for weakly disordered, two-dimensional (2d), strongly interacting itinerant-fermion systems near a magnetic instability, when all momenta up to twice the Fermi momentum (0\ensuremath{\le}q\ensuremath{\le}2${k}_{F}$) are equally relevant. In 3d systems, the only vanishing momentum values (q\ensuremath{\sim}0) are the key ones, for which only the diffusive regime arises in the case of weak localization; in contrast, in 2d nearly magnetic systems, other regimes, in addition to the diffusive one, must be taken into account and be considered as playing equally important roles. Different temperature dependences follow for the physical properties of the system which furthermore do not vary the same way with the impurity concentration. Their relative weights are affected accordingly, which will be of importance for comparison with experiments.

The response function of a slab of noninteracting nuclear matter

Analytic expressions of the density and of the particle-hole polarization propagator are derived for a slab of noninteracting nuclear matter. The nondiagonal behaviour of the polarization propagator in the momentum components perpendicular to the slab is explored.

Time‐dependent density operator in the random impulse approximation: I. Connections with the “generalized sudden approximation” for photoionization and electron impact ionization; and II. Relevance to reduced density operators, particle–hole matrices, and propagators for the initial state

AbstractA density operator analysis is made of the particle‐ejection (e.g., photoionization) problem in the “generalized sudden approximation” for the case in which the ejected particle is detected but its dynamical properties are not observed. It is shown that the resultant statistical density operator may be derived from that obtained in a particle‐conserving Hamiltonian‐evolution linear‐response model with a random impulse perturbation, and that the physical applicability of the latter model embraces the range of applicability of the former. The initial value of the statistical density operator for the above particle‐ejection problem is shown to be the N – (1)‐particle reduced density operator of the initial state; and the initial value of the statistical density operator for the excitations in the random impulse model is shown to be the N‐particle Hermitian operator whose matrix representation is the G matrix of Garrod and Percus. The significance of the eigenvalue spectrum of these operators to the excitation properties of the system is discussed, especially for the random impulse model, where a large eigenvalue of the G matrix can signal strong preferential excitation to its corresponding particle‐hole collective state, even for a random perturbation. Extensions of these ideas to excitations from states with a large eigenvalue of the two‐particle reduced density operator (e.g., superconducting states) are mentioned. The The applicability of these density matrices to the description of the excitation spectrum due to a well‐defined perturbation is discussed. The relationship of these time‐dependent density operators to the one‐particle propagator and the particle–hole propagator (polarization propagator) is established.