Anomalous Self-energy Research Articles

Maximum entropy analytic continuation of anomalous self-energies

Maximum entropy analytic continuation of anomalous self-energies

Maximum entropy analytic continuation of anomalous self-energies

Maximum entropy analytic continuation of anomalous self-energies

Eliashberg theory of the Jahn-Teller-Hubbard model

We study a multiorbital Hubbard model coupled to local Jahn-Teller phonons to investigate the superconducting state realized in fullerides. A weak-coupling approach is employed in combination with a local self-energy approximation. In addition to the normal and anomalous self-energies of the electrons, we consider the phonon self-energy, which allows a self-consistent treatment of the energetics. The frequency dependence of the self-energies and their characteristic coefficients, such as renormalization factors and dampings, are investigated in detail using numerical calculations. It is clarified that the anisotropic phonons play an important role in the stabilization of the superconducting state. By comparing the full results to those without phonon self-energies, we show that the superconductivity is stabilized by the softening of the phonon frequency. The effects of electronic fluctuations are also considered, which leads to the coupling to orbitons, an analog of plasmons in the electron gas. This additional contribution further stabilizes the superconducting state.

Hidden self-energies as origin of cuprate superconductivity revealed by machine learning

Experimental data are the source of understanding matter. However, measurable quantities are limited and theoretically important quantities are sometimes hidden. Nonetheless, recent progress of machine-learning techniques opens possibilities of exposing them only from available experimental data. In this paper, after establishing the reliability of the method in various careful benchmark tests, the Boltzmann-machine method is applied to the angle-resolved photoemission spectroscopy spectra of cuprate high temperature superconductors, Bi$_2$Sr$_2$CuO$_{6+\delta}$ (Bi2201) and Bi$_2$Sr$_2$CaCuO$_{8+\delta}$ (Bi2212). We find prominent peak structures both in normal and anomalous self-energies, but they cancel in the total self-energy making the structure apparently invisible, while the peaks make universally dominant contributions to superconducting gap, hence evidencing the signal that generates the high-$T_{\rm c}$ superconductivity. The relation between superfluid density and critical temperature supports involvement of universal carrier relaxation associated with dissipative strange metals, where enhanced superconductivity is promoted by entangled quantum-soup nature of the cuprates. The present achievement opens avenues for innovative machine-learning spectroscopy method to reveal fundamental properties hidden in direct experimental accesses.

Thermal fluctuations in superconducting phases with chiral d + id and s symmetry on a triangular lattice

The behavior of thermal fluctuations of a superconducting order parameter with extended s and chiral d + id symmetry is investigated. The study is carried out on a triangular lattice within the framework of the quasi-two-dimensional single-band model with attraction between electrons at neighboring sites. The method of consistent consideration of the order parameter fluctuations and the charge carrier scattering by fluctuations of coupled electron pairs, based on the theory of functional integration is used. The distribution functions of the phase fluctuation probabilities depending on temperature and charge carrier concentration are obtained. The temperature dependences of the amplitudes of the averaged superconducting order parameter are calculated. A phase diagram of superconducting states is constructed for the entire range of variation in the charge carrier concentration 0 < n < 2. Near the boundaries of this range, topologically trivial superconducting states with extended s symmetry are realized, while a superconducting state with topologically nontrivial chiral d + id symmetry is realized between them. The calculated anomalous self-energies are compared with the experimental ones obtained using machine learning techniques

Peak structure in the self-energy of cuprate superconductors

The recently deduced normal and anomalous self-energies from photoemission spectra of cuprate superconductors via the machine learning technique are calling for an explanation. Here the normal and anomalous self-energies in cuprate superconductors are analyzed within the framework of the kinetic-energy-driven superconductivity. It is shown that the exchanged spin excitations give rise to the well-pronounced low-energy peak-structures in both the normal and anomalous self-energies, however, they do not cancel in the total self-energy. In particular, the peak-structure in the normal self-energy is mainly responsible for the peak-dip-hump structure in the single-particle excitation spectrum, and can persist into the normal-state, while the sharp peak in the anomalous self-energy gives rise to a crucial contribution to the superconducting gap, and vanishes in the normal state. Moreover, the evolution of the peak structure with doping and momentum are also analyzed.

Anisotropic dressing of electrons in electron-doped cuprate superconductors

The recent experiments revealed a remarkable possibility for the absence of the disparity between the phase diagrams of the electron- and hole-doped cuprate superconductors, while such an aspect should be also reflected in the dressing of the electrons. Here the phase diagram of the electron-doped cuprate superconductors and the related exotic features of the anisotropic dressing of the electrons are studied based on the kinetic-energy driven superconductivity. It is shown that although the optimized Tc in the electron-doped side is much smaller than that in the hole-doped case, the electron- and hole-doped cuprate superconductors rather resemble each other in the doping range of the superconducting dome, indicating an absence of the disparity between the phase diagrams of the electron- and hole-doped cuprate superconductors. In particular, the anisotropic dressing of the electrons due to the strong electron's coupling to a strongly dispersive spin excitation leads to that the electron Fermi surface is truncated to form the disconnected Fermi arcs centered around the nodal region. Concomitantly, the dip in the peak-dip-hump structure of the quasiparticle excitation spectrum is directly associated with the corresponding peak in the quasiparticle scattering rate, while the dispersion kink is always accompanied by the corresponding inflection point in the total self-energy, as the dip in the peak-dip-hump structure and dispersion kink in the hole-doped counterparts. The theory also predicts that both the normal and anomalous self-energies exhibit the well-pronounced low-energy peak-structures.

Lattice and spin excitations of YFeO3 : A Raman and density functional theory study

We report a Raman scattering study of $\mathrm{Y}\mathrm{Fe}{\mathrm{O}}_{3}$ orthoferrite. Polarized Raman measurements in wide temperature range and first-principles calculations provided the assignment of the observed phonon modes to vibrational symmetries and atomic displacements. We did not observe previously reported anomalies in the behavior of phonon frequencies near the temperature of the antiferromagnetic transition. Temperature behavior of frequencies and linewidths of most phonons may be described by known expressions for cubic and quartic anharmonicity, which implies a significant contribution of phonon-phonon interaction. At the same time, several phonons exhibit asymmetric profiles and anomalies in the behavior of their widths, which indicates an interaction with a continuum of some excitations. The most striking evidence of this is the observation of the coupling of the ${B}_{2g}$ phonon at 640 ${\mathrm{cm}}^{\ensuremath{-}1}$ with the two-magnon continuum, expressed in anomalous phonon self-energies in the vicinity of ${T}_{N}$. The appearance of a broad continuum of low-frequency excitations when approaching ${T}_{N}$, which is possibly induced by the damping of spin excitations due to interaction with the lattice, has been revealed. This interaction can lead to additional contributions to the renormalization of phonon self-energies, formally described by formulas for anharmonic effects. Phonon scattering induced by defects has been identified. The temperature behavior of resonant second-order phonon and two-magnon scattering was studied in detail.

Vortex bound state of a Kondo lattice coupled to a compensated metal

We theoretically study physical properties of the low-energy quasiparticle excitations at the vortex core in the full-gap superconducting state of the Kondo lattice coupled to compensated metals. Based on the mean-field description of the superconducting state, we numerically solve the Bogoliubov-de Gennes (BdG) equations for the tight-binding Hamiltonian. The isolated vortex is characterized by a length scale independent of the magnitude of the interaction and the energy level of the core bound state is the same order as the bulk gap. These properties are in strong contrast to the conventional s-wave superconductor. To gain further insights, we also consider the effective Hamiltonian in the continuous limit and construct the theoretical framework of the quasiclassical Green's function of conduction electrons. With the use of the Kramer-Pesch approximation, we analytically derive the spectral function describing the quasiparticle excitations which is consistent with the numerics. It has been revealed that the properties of the vortex bound state are closely connected to the characteristic odd frequency dependence of both the normal and anomalous self-energies which is proportional to the inverse of frequency.

Superconductivity in systems exhibiting the Altshuler-Aronov anomaly

Making use of generalized Eliashberg equations, we describe the Altshuler-Aronov (AA) effect and superconductivity on equal footing. We derive explicit expressions for the Coulomb pseudopotential in 3D, taking into account also the anomalous diffusion. We present a full numerical solution for two normal-state and two anomalous self-energies. In the normal state, we amend the known results for the purely electronic AA effect; with electron-phonon coupling turned on, we find additional anomalies in the density of states close to the phonon energy. We study how the critical temperature and density of states of strongly disordered 3D superconductors change with normal-state resistivity. We find that the type of transition from the superconducting to the insulating state depends on the strength of electron-phonon coupling: at weak coupling there exists an intermediate normal state, whereas at strong coupling the transition is direct.

Frequency-dependent structure of superconducting gap function in two-dimensional Hubbard model

The frequency dependence of the superconducting gap function represents the dynamics of the electrons forming the Cooper pairs. Bearing the high-temperature superconducting cuprates in mind, we calculate the superconducting gap function in the two-dimensional Hubbard model by means of a cluster extension of the dynamical mean-field theory. We scrutinize the frequency dependence and its change with temperature, doping concentration and interaction strength. The calculated superconducting gap function shows various different behaviors from the anomalous self-energy, owing to a strong frequency dependence of the normal self-energy. The calculated results also suggest that the superconductivity is stronger when the interaction strength is as large as or slightly smaller than the bare band width and is suppressed for larger interactions.

Color Superconductivity and Charge Neutrality in Yukawa Theory.

It is generally believed that when Cooper pairing occurs between two different species of fermions, their Fermi surfaces become locked together so that the resultant state remains "neutral," with equal number densities of the two species, even when subjected to a chemical potential that couples to the difference in number densities. This belief is based on mean-field calculations in models with a zero-range interaction, where the anomalous self-energy is independent of energy and momentum. Following up on an early report of a deviation from neutrality in a Dyson-Schwinger calculation of color-flavor-locked quark matter, we investigate the neutrality of a two-species condensate using a Yukawa model which has a finite-range interaction. In a mean field calculation we obtain the full energy-momentum dependence of the self-energy and find that the energy dependence leads to a population imbalance in the Cooper-paired phase when it is stressed by a species-dependent chemical potential. This gives some support to the suggestion that the color-flavor-locked phase of quark matter might not be an insulator.

Low energy impurity kink in the normal and anomalous self-energies in Bi-cuprate superconductors

The sharp low energy kink (LEK) in quasiparticle (qp) spectra well below the superconducting energy gap observed in the angle-resolved photo-emission spectroscopy (ARPES) of the Bi-cuprates may be understood in terms of the forward scattering impurities located off the Cu–O planes. The relevance of the idea has been established by comparing the calculated normal self-energy from the off-plane impurity effects and the extracted one from the self-energy analysis of [Formula: see text] (Bi2212) ARPES data in Hong et al. [Phys. Rev. Lett. 113, 057001 (2014)]. In addition to the explanation of the LEK, this is a necessary step to analyze ARPES data, to reveal the spectrum of fluctuations promoting superconductivity. We also present the extracted anomalous self-energy from the self-energy analysis, which is its first experimental determination as far as we are aware of. The extracted anomalous self-energy and its implications are discussed in comparison with the calculated impurity self-energy term.

What is the pairing glue in the cuprates? Insights from normal and anomalous propagators

Both the pairing and the pair-breaking modes lead to similar kinks of the electron dispersion curves in superconductors, and therefore the photoemission spectroscopy cannot be straightforwardly applied in search for their pairing glue. If the momentum dependence of the normal and anomalous self-energy can be neglected, manipulation with the data does allow us to extract the gap function and therefore also the pairing glue. However, in the superconducting cuprates, such procedure may not be justified. In this paper, we point out that the pairing glue is more directly visible in the spectral function of the Nambu–Gor’kov anomalous propagator and we demonstrate that this function is in principle experimentally observable.

Renormalization group flow for fermions into antiferromagnetically ordered phases: Method and mean-field models

We present a functional renormalization group (fRG) flow for many-fermion lattice models into phases with broken spin-rotational symmetry. The flow is expressed purely in terms of fermionic vertex functions. The symmetry breaking is seeded by a small initial anomalous self-energy which grows at the transition scale and which prevents a runaway flow at nonzero scales. Focusing on the case of commensurate antiferromagnetism, we discuss how the interaction vertex can be parametrized efficiently. For reduced models with long-range bare interactions, we show the results of standard mean-field theory are reproduced by the fRG and how anisotropies in the spin sector change the flows. We then describe a more efficient decomposition of the interaction vertex that should allow for the treatment of more general models.

Bose-Einstein-condensed systems of hard spheres at high density

The properties of Bose-Einstein-condensed systems of hard spheres at high density are studied by taking into account higher-order corrections beyond the Hartree-Fock-Bogoliubov approximation. Based on a diagrammatic technique of the perturbation expansion starting from the Bogoliubov approximation, we show that the effective interactions separate into two distinct parts: the screened and condensate-mediated interactions. The latter is described by the exchange of virtual quasiparticles between noncondensate bosons via the Bose-Einstein condensate and is attractive for small energy transfer. We calculate the condensation temperature, excitation spectrum, and condensate density in the whole region of the gas parameter. It is shown that the condensate temperature decreases compared to that of an ideal Bose gas due to hard-sphere interactions. We find that the condensate-mediated attraction between noncondensate bosons leads to an enhancement of the anomalous self-energy and, as a consequence, to the emergence of a roton minimum in the excitation spectrum and to the strong suppression of the condensate fraction in the region of high density.

Self-consistentt-matrix theory of the Hartree-Fock-Bogoliubov approximation for Bose-Einstein-condensed systems

We present a self-consistent $t$-matrix theory for Bose-Einstein-condensed systems within the Hartree-Fock-Bogoliubov (HFB) approximation. Using the Lippmann-Schwinger equation for a $t$ matrix describing the collision between two particles via an interparticle potential, we derive a set of equations for the normal and anomalous self-energies in the HFB approximation expressed in terms of the $t$ matrix. These equations are solved for a hard-sphere potential. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low densities and to the Brueckner-Sawada theory at high densities. The spectrum is gapless and linear in small momentum, but does not have any roton minimum in the large-momentum region even for high densities such as those of ${}^{4}$He.

Nambu-Eliashberg theory for multiscale quantum criticality: Application to ferromagnetic quantum criticality in the surface of three-dimensional topological insulators

We develop an Eliashberg theory for multi-scale quantum criticality, considering ferromagnetic quantum criticality in the surface of three dimensional topological insulators. Although an analysis based on the random phase approximation has been performed for multi-scale quantum criticality, an extension to an Eliashberg framework was claimed to be far from triviality in respect that the self-energy correction beyond the random phase approximation, which originates from scattering with $z = 3$ longitudinal fluctuations, changes the dynamical exponent $z = 2$ in the transverse mode, explicitly demonstrated in nematic quantum criticality. A novel ingredient of the present study is to introduce an anomalous self-energy associated with the spin-flip channel. Such an anomalous self-energy turns out to be essential for self-consistency of the Eliashberg framework in the multi-scale quantum critical point because this off diagonal self-energy cancels the normal self-energy exactly in the low energy limit, preserving the dynamics of both $z = 3$ longitudinal and $z = 2$ transverse modes. This multi-scale quantum criticality is consistent with that in a perturbative analysis for the nematic quantum critical point, where a vertex correction in the fermion bubble diagram cancels a singular contribution due to the self-energy correction, maintaining the $z = 2$ transverse mode. We also claim that this off diagonal self-energy gives rise to an artificial electric field in the energy-momentum space in addition to the Berry curvature. We discuss the role of such an anomalous self-energy in the anomalous Hall conductivity.

Critical particle-hole composites at twice the Fermi wave vector in a U(1) spin liquid with a Fermi surface

We find ``chiral symmetry breaking'' at finite energies in a U(1) spin liquid, corresponding to critical particle-hole composite states with twice the Fermi momentum (2${k}_{F}$). We investigate this Fermi-surface problem based on the Nambu-Eliashberg theory, where the off-diagonal pairing self-energy is introduced to catch the Aslamasov-Larkin vertex correction. This approach is quite similar to the case of superconductivity, where such Aslamasov-Larkin quantum corrections in the particle-particle channel are well known to be responsible for superconducting instability, formulated as the Nambu-Eliashberg theory in an elegant way. We obtain the pairing self-energy, which vanishes at zero energy but displays the same power-law dependence for frequency as the normal Eliashberg self-energy. As a result, even the pairing self-energy correction does not modify the Eliashberg dynamics without the Nambu spinor representation, where thermodynamics is described by the typical $z=3$ scaling free energy. We discuss the physical implications of the anomalous self-energy, which is identical to the conventional Eliashberg normal self-energy, focusing on thermodynamics.

Functional renormalization-group approach to interacting bosons at zero temperature

We investigate the single-particle spectral density of interacting bosons within the non-perturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U(1) symmetry constraints, and are in accordance with the Nepomnyashchy relation which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasi-free bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozieres and discuss the nature of the non-analyticities of the self-energies in the very small momentum regime.