Two-particle Spectral Functions Research Articles

Subgap two-particle spectral weight in disordered s -wave superconductors: Insights from mode coupling approach

We study the two-particle spectral functions and collective modes of weakly disordered superconductors using a disordered attractive Hubbard model on a square lattice. We show that the disorder-induced scattering between collective modes leads to a finite subgap spectral weight in the long-wavelength limit. In general, the spectral weight is distributed between the phase and the Higgs channels, but as we move towards half filling the Higgs contribution dominates. The inclusion of the density fluctuations lowers the frequency at which this mode occurs and results in the phase channel gaining a larger contribution to this subgap mode. Near half filling, the proximity of the system to the charge density wave instability leads to strong fluctuations of the effective disorder at the commensurate wave vector ($[\ensuremath{\pi},\ensuremath{\pi}]$). We develop an analytical mode coupling approach where the pure Goldstone mode in the long-wavelength limit couples to the collective mode at $[\ensuremath{\pi},\ensuremath{\pi}]$. This provides insight into the location and distribution of the two-particle spectral weights between the Higgs and the phase channels.

Thermal effects on collective modes in disordered s -wave superconductors

We investigate the effect of thermal fluctuations on the two-particle spectral function for a disordered $s$-wave superconductor in two dimensions, focusing on the evolution of the collective amplitude and phase modes. We find three main effects of thermal fluctuations: (1) the phase mode is softened with increasing temperature, reflecting the decrease of superfluid stiffness; (2) remarkably, the nondispersive collective amplitude modes at finite energy near $\mathbf{q}=[0,0]$ and $\mathbf{q}=[\ensuremath{\pi},\ensuremath{\pi}]$ survive even in the presence of thermal fluctuations in the disordered superconductor; and (3) the scattering of the thermally excited fermionic quasiparticles leads to low-energy incoherent spectral weight that forms a strongly momentum-dependent background halo around the phase and amplitude collective modes and broadens them. Due to momentum and energy conservation constraints, this halo has a boundary which disperses linearly at low momenta and shows a strong dip near the $[\ensuremath{\pi},\ensuremath{\pi}]$ point in the Brillouin zone.

Superfluid density and collective modes of fermion superfluid in dice lattice

The superfluid properties of attractive Hubbard model in dice lattice are investigated. It is found that three superfluid order parameters increase as the interaction increases. When the filling factor falls into the flat band, due to the infinite large density of states, the resultant superfluid order parameters are proportional to interaction strength, which is in striking contrast with the exponentially small counterparts in usual superfluid (or superconductor). When the interaction is weak, and the filling factor is near the bottom of the lowest band (or the top of highest band), the superfluid density is determined by the effective mass of the lowest (or highest) single-particle band. When the interaction is strong and filling factor is small, the superfluid density is inversely proportional to interaction strength, which is related to effective mass of tightly bound pairs. In the strong interaction limit and finite filling, the asymptotic behaviors of superfluid density can be captured by a parabolic function of filling factor. Furthermore, when the filling is in flat band, the superfluid density shows a logarithmic singularity as the interaction approaches zero. In addition, there exist three undamped collective modes for strong interactions. The lowest excitation is gapless phonon, which is characterized by the total density oscillations. The two others are gapped Leggett modes, which correspond relative density fluctuations between sublattices. The collective modes are also reflected in the two-particle spectral functions by sharp peaks. Furthermore, it is found that the two-particle spectral functions satisfy an exact sum-rule, which is directly related to the filling factor (or density of particle). The sum-rule of the spectral functions may be useful to distinguish between the hole-doped and particle-doped superfluid (superconductor) in experiments.

Two-particle spectral function for disordered s -wave superconductors: Local maps and collective modes

We make the first testable predictions for the local two-particle spectral function of a disordered s-wave superconductor, probed by scanning Josephson spectroscopy (sjs), providing complementary information to scanning tunneling spectroscopy (sts). We show that sjs provides a direct map of the local superconducting order parameter that is found to be anticorrelated with the gap map obtained by sts. Furthermore, this anticorrelation increases with disorder. For the momentum resolved spectral function, we find the Higgs mode shows a non-dispersive subgap feature at low momenta, spectrally separated from phase modes, for all disorder strengths. The amplitude-phase mixing remains small at low momenta even when disorder is large. Remarkably, even for large disorder and high momenta, the amplitude-phase mixing oscillates rapidly in frequency and hence do not affect significantly the purity of the Higgs and phase dominated response functions.

Parquet approximation for molecules: Spectrum and optical conductivity of the Pariser-Parr-Pople model

We study a simple model system for the conjugated $\pi$-bonds in benzene, the Pariser-Parr-Pople (PPP) model, within the parquet approximation (PA), exemplifying the prospects of the PA for molecules. Advantages of the PA are the polynomial scaling with the number of orbitals, and the natural calculation of one- and two-particle spectral functions as well as of response and correlation functions. We find large differences in the electronic correlations in the PPP model compared to a Hubbard model with only local interactions. The quasiparticle renormalization (or mass enhancement) is much weaker in the PPP than in the Hubbard model, but the static part of the self-energy enhances the band gap of the former. Furthermore, the vertex corrections to the optical conductivity are much more important in the PPP model. Because non-local interactions strongly alter the self-energy, we conclude that the PA is more suitable for calculating conjugated $\pi$-bonds in molecules than single site dynamical mean-field theory.

Quasiparticle interactions in frustrated Heisenberg chains

Interactions between elementary excitations in quasi-one dimensional antiferromagnets are of experimental relevance and their quantitative theoretical treatment has been a theoretical challenge for many years. Using matrix product states, one can explicitly determine the wavefunctions of the one- and two-particle excitations, and, consequently, the contributions to dynamical correlations. We apply this framework to the (non integrable) frustrated dimerized spin-1/2 chain, a model for generic spin-Peierls systems, where low-energy quasi-particle excitations are bound states of topological solitons. The spin structure factor involving two quasi-particle scattering states is obtained in the thermodynamic limit with full momentum and frequency resolution. This allows very subtle features in the two-particle spectral function to be revealed which, we argue, could be seen, e.g., in inelastic neutron scattering of spin-Peierls compounds under a change of the external pressure.

Peierls to superfluid crossover in the one-dimensional, quarter-filled Holstein model

We use continuous-time quantum Monte Carlo simulations to study retardation effects in the metallic, quarter-filled Holstein model in one dimension. Based on results which include the one- and two-particle spectral functions as well as the optical conductivity, we conclude that with increasing phonon frequency the ground state evolves from one with dominant diagonal order—2kF charge correlations—to one with dominant off-diagonal fluctuations, namely s-wave pairing correlations. In the parameter range of this crossover, our numerical results support the existence of a spin gap for all phonon frequencies. The crossover can hence be interpreted in terms of preformed pairs corresponding to bipolarons, which are essentially localized in the Peierls phase, and ‘condense’ with increasing phonon frequency to generate dominant pairing correlations.

Two-particle photoemission from strongly correlated systems: A dynamical mean-field approach

We study theoretically the simultaneous, photo-induced two-particle excitations of strongly correlated systems on the basis of the Hubbard model. Under certain conditions specified in this work, the corre- sponding transition probability is related to the two-particle spectral function which we calculate using three different methods: the dynamical-mean field theory combined with quantum Monte Carlo (DMFT- QMC) technique, the first order perturbation theory and the ladder approximations. The results are analyzed and compared for systems at the verge of the metal-insulator transitions. The dependencies on the electronic correlation strength and on doping are explored. In addition, the account for the orbital degeneracy allows an insight into the influence of interband correlations on the two particle excitations. A suitable experimental realization is discussed.

Electronic dynamic Hubbard model: Exact diagonalization study

A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.

Quasiparticle undressing in a dynamic Hubbard model: Exact diagonalization study

Dynamic Hubbard models have been proposed as extensions of the conventional Hubbard model to describe the orbital relaxation that occurs upon double occupancy of an atomic orbital. These models give rise to pairing of holes and superconductivity in certain parameter ranges. Here we explore the changes in carrier effective mass and quasiparticle weight and in one- and two-particle spectral functions that occur in a dynamic Hubbard model upon pairing, by exact diagonalization of small systems. It is found that pairing is associated with lowering of effective mass and increase of quasiparticle weight, manifested in transfer of spectral weight from high to low frequencies in one- and two-particle spectral functions. This 'undressing' phenomenology resembles observations in transport, photoemission and optical experiments in high T_c cuprates. This behavior is contrasted with that of a conventional electron-hole symmetric Holstein-like model with attractive on-site interaction, where pairing is associated with 'dressing' instead of 'undressing'.

Phase shifts and in-medium cross sections for dressed nucleons in nuclear matter

The dressing of nucleons as embodied in single-particle spectral functions is incorporated in the description of nucleon-nucleon scattering in nuclear matter at a density corresponding to k F51.36 fm 21 . In order to clarify the new features associated with the complete off-shell behavior of the single-particle motion, results involving mean-field particles are also presented with special emphasis on the behavior of the phase shifts when bound pair states occur. Both the 1 S0 and 3 S1- 3 D1 channels exhibit this feature at the considered density for mean-field particles at zero temperature. An important tool to assess the effect of the dressing of the particles is the two-particle density of states. A sizable reduction with respect to the mean-field density of states is obtained. At 2e F this reduction corresponds to z kF 2 , where z kF is the strength of the quasiparticle pole at k F , and it can therefore be as large as 0.5. This reduction has significant consequences for the strength of pairing correlations both in the 3 S1- 3 D1 channel where it leads to a dramatic decrease of the attraction at the Fermi energy and for the 1 S0 channel which no longer shows a pairing signal. Phase shifts and cross sections for dressed particles are determined based on expressions which fold the effective interaction with the dressed but noninteracting two-particle spectral function. This folding procedure yields similar results to an ‘‘on-shell’’ prescription reminiscent of the result for free or mean-field particles, except for cross sections deep in the Fermi sea. Comparison of phase shifts and cross sections to the case of mean-field particles indicates that smaller phase shifts in an absolute sense and considerable reductions of the in-medium cross sections for dressed particles are obtained. It is shown that while in many cases these results imply a weakening of the effective interaction, this is not the case for 1 S0 interactions deep in the Fermi sea. @S0556-2813~99!06612-1#

Spectral functions of quantum dots in the integer and fractional quantum Hall regime

The one- and two-particle spectral functions of quantum Hall droplets are studied for filling factors $2>~\ensuremath{\nu}>~1/3$ using exact diagonalization techniques for up to ten electrons. The spectral functions describe the electron addition and subtraction spectra, coherent resonant tunneling, and Raman-scattering cross section. They reflect the magnetic-field-induced spin and charge transitions in the ground state of quantum Hall droplets.

Critical conductivity of the quantum Hall system at higher Landau levels

The two-particle spectral function S(q,\ensuremath{\omega}) is studied at the mobility edge on the quantum Hall system, in the lowest and higher Landau level. The profile and its scaling properties of S(q,\ensuremath{\omega}) are numerically clarified. Our result shows that, in higher Landau level with the long-range random potential, the critical dissipative conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{xx}}^{\mathrm{c}}$ is approximately equal to the universal value 0.${5\mathrm{e}}^{2}$/h. This suggests that the critical conductivity ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{xx}}^{\mathrm{c}}$ on the quantum Hall system with the long-range potential is independent of the Landau-level index N.

Electron-Phonon Interaction in One Dimension: Extended Tomonaga- Luttinger Model

On the basis of the extended Hubbard model we consider phonon mediated forward scattering in one-dimensional systems. Exact results for the two-particle spectral functions are derived for the case when Umklapp and backward scattering due to electron-electron interaction are neglected. Due to the electron-phonon interaction the phase diagram of the interacting electron system changes remarkably. In dependence on the phonon frequency and on the electron-phonon coupling constant a new divergence may appear in a two-particle spectral function or an existing one may be removed. The inclusion of Umklapp and backward scattering leads to the appearance of an energy gap and a polaron energy shift. The electron-phonon coupling constant is strongly renormalized and vanishes faster than the charge velocity as half-filling is approached. Beyond half-filling the polaron shift may diverge at certain values of the coupling parameters. This divergency is related to the Wentzel-Bardeen singularity.

Spectral functions of the disordered Tomonaga - Luttinger model

Using the functional integral bosonization technique, the spinless Tomonaga - Luttinger model with impurity forward scattering is solved exactly. Explicit analytical results are given for the one- and two-particle spectral functions in terms of a convolution integral representation. Sharp structures in the frequency spectra of an ordered system are smoothed out by the disorder. The existence of a charge - density instability is predicted if the critical exponent is less than -0.5. There is no influence of the disorder on the superconducting spectral function .

Scaling, diffusion, and the integer quantized Hall effect.

The behavior of the two-particle spectral function, $S(q,\ensuremath{\omega})$, is examined in the hydrodynamic regime, at the mobility edge in a model for the integer quantized Hall effect. Results are presented from numerical diagonalization of the Hamiltonian for finite systems. For $\frac{{q}^{2}}{\ensuremath{\omega}}$ small, $S(q,\ensuremath{\omega})$ has a conventional, diffusive form. For $\frac{{q}^{2}}{\ensuremath{\omega}}$ large, the novel dependence $S(q,\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{-}\frac{\ensuremath{\eta}}{2}}{q}^{\ensuremath{-}2+\ensuremath{\eta}}$ is obtained, with $\ensuremath{\eta}=0.38\ifmmode\pm\else\textpm\fi{}0.04$.

Scaling and correlations at a mobility edge in two dimensions

Eigenfunction correlations are discussed at the mobility edge in a two-dimensional system with either spin-orbit scattering or a strong. perpendicular magnetic field. A novel form is proposed for the two-particle spectral function in the hydrodynamic limit, assuming conformal invariance and homogeneity under combined energy and length scale transformations.

Anderson localisation in quantum Hall systems

The behavior of non interacting electrons is studied in two-dimensional disordered systems under a perpendicular magnetic field. The strong-field limit is examined, in which scattering between Landau bands may be neglected. It is shown that the form of the two-particle spectral function is severely restricted by a general inequality for certain eigenfunction correlations, together with the requirements of average translational and rotational invariance. Consequences are: there cannot exist bands of extended states (in the diffusion sense) with finite energy width; and either the exponent nu O (which would be new in two-dimensional localisation).

Two-particle spectral function and ac conductivity of an amorphous system far below the mobility edge: A problem of interacting instantons

We use a variational approach to calculate the two-particle spectral function ${S}_{2}({x}_{1}, {x}_{1}, {x}_{2}, {x}_{2}, {E}_{1}{E}_{2})$ of a Gaussian-disordered electron system in the limit of deeply localized states and small energy difference $\ensuremath{\omega}={E}_{1}\ensuremath{-}{E}_{2}$. The solution of the variational equations yields a two-center potential, each center in lowest order being determined by the square of an instanton function. The two instantons interact via the constraint that the Hamiltonian has to have lowest eigenvalues ${E}_{1}$, ${E}_{2}$. As the two centers approach the minimum distance allowed for given $\ensuremath{\omega}$ by the tunnel effect, we are confronted with a problem of confluent saddle points, which forces us to introduce an additional constraint. Our method is rigorous in the limit of weak disorder $|{E}_{1}+{E}_{2}|\ensuremath{\rightarrow}\ensuremath{\infty}$, $\frac{\ensuremath{\omega}}{|{E}_{1}+{E}_{2}|}=\mathrm{const}\ensuremath{\ll}1$. We also apply it to the hydrodynamic limit $\frac{\ensuremath{\omega}}{|{E}_{1}+{E}_{2}|}\ensuremath{\rightarrow}0, |{E}_{1}+{E}_{2}|$ large. It is found that these limits cannot be interchanged. In both limits we evaluated the ac conductivity. The result $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2}{(\mathrm{ln}\ensuremath{\omega})}^{d+1}$ is found in the hydrodynamic limit.