Kubo Linear Response Theory Research Articles

Fast Fourier-Chebyshev Approach to Real-Space Simulations of the Kubo Formula.

The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over 10^{7} sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.

Electron-phonon correlations inducing excitonic excitations in semimetal and semiconducting materials

The influence of phonons on the low-energy excitonic excitations at zero temperature in the extended Falicov-Kimball model has been investigated. In the framework of the unrestricted Hartree-Fock approximation, a set of self-consistent equations for the excitonic condensate order parameter and a lattice distortion is derived when both electron-phonon coupling and electron-hole Coulomb interaction are treated on an equal footing. The low-energy excitation properties of the excitonic condensate are addressed in signatures of the optical conductivity and the dynamical excitonic susceptibility function. The real part of the optical conductivity is evaluated by the Kubo linear response theory and the imaginary part of the dynamical excitonic susceptibility is found by adapting the random phase approximation. In the semimetal state, one always finds a sharp peak in the optical conductivity spectrum indicating the stability of the excitonic condensation in the BCS type if the correlation between electrons and phonons becomes significant. In contrast, the peak is smeared out on the semiconducting side indicating the stability of the BEC-type excitonic condensate. In this semiconducting side, the sharp peak signature appears and the system turns to the BCS-type excitonic condensation state by increasing the electron-phonon correlations. In either the semimetal or the semiconducting normal state, increasing the electron-phonon correlations always reinforces a low-energy sharp peak in the dynamical excitonic susceptibility spectrum, indicating the existence of the tightly bound excitonic excitations before the condensation state. Specifically, on the semiconducting side, the ``halo'' phase with the preformed excitons exiting outside of the BEC-excitonic condensation state has been specified. The halo phase becomes more recognizable by raising the electron-phonon correlations.

Magnetization Dynamics in FexCo1-x in Presence of Chemical Disorder

In this paper, we present a theoretical formulation of magnetization dynamics in disordered binary alloys, based on the Kubo linear response theory, interfaced with a seamless combination of three approaches: density functional-based tight-binding linear muffin-tin orbitals, generalized recursion and augmented space formalism. We applied this method to study the magnetization dynamics in chemically disordered FexCo1−x (x = 0.2, 0.5, 0.8) alloys. We found that the magnon energies decreased with an increase in Co concentration. Significant magnon softening was observed in Fe20Co80 at the Brillouin zone boundary. Magnon–electron scattering increased with increasing Co content, which in turn modified the hybridization between the Fe and Co atoms. This reduced the exchange energy between the atoms and softened down the magnon energy. The lowest magnon lifetime was found in Fe50Co50, where disorder was at a maximum. This clearly indicated that the damping of magnon energies in FexCo1−x was governed by hybridization between Fe and Co, whereas the magnon lifetime was controlled by disorder configuration. Our atomistic spin dynamics simulations show reasonable agreement with our theoretical approach in magnon dispersion for different alloy compositions.

Excitonic properties in a double-layer graphene

This paper investigates theoretically the excitonic condensation state at zero temperature in a double-layer graphene structure. In the framework of the unrestricted Hartree–Fock approximation, the electron-hole system in the structure described in the two-band electronic model is analyzed and one finds a set of self-consistent equations determining the excitonic order parameter. The optical properties of the excitonic condensation state then are examined in the Kubo linear optical response theory. Our results indicate that in the case of sufficiently large Coulomb interaction, the BEC excitonic condensation state might occur at low electronic excitation density. By turning the external electric field, the superfluid state stabilizes in the BCS-type excitonic condensate. The optical conductivity spectrum also provides us more insight into the excitonic condensation states.

Far-field radiation of biased monolayer PbBiI

We use the Kubo formula and linear response theory to study the radiation of energy and angular momentum from biased monolayer PbBiI. Our results show that a critical bias field Vbc≃0.326 eV emerges for the radiations. For bias fields below Vbc, there is a negligible shift of the threshold frequency at which the spin-dependent energy radiation is maximum, while above which, it significantly moves to the higher frequencies. At all the threshold frequencies, the sign of possible angular momentum radiations is switched. Further, we find that contrary to Planck's law which is valid for the unbiased energy radiation (proportionality to T4), the biased one is proportional to T5. Also, it is possible to confirm the behaviors of the system below and above Vbc via the temperature-dependent angular momentum radiation. This work illuminates/engineers the application of angular momentum degree of freedom in processing data.

Thermal fluctuations in a realistic ionic-crystal model

We investigate the thermal fluctuations of the ionic motions in a Born model of ionic crystals, namely, a model in which the electrons are eliminated, being replaced by suitable effective potentials among the ions. The model is studied in its classical version, computing the Newtonian trajectories of the ions. The general motivation is that, although being an essential ingredient within Green–Kubo linear response theory, thermal fluctuations apparently were not studied systematically by molecular dynamics methods, as was done instead for the approach to equilibrium in the Fermi–Pasta–Ulam problem. The time evolution of the fluctuations is studied in terms of the time-changes of the mode-energies of the system. The stages of the “regression” of the fluctuations are described, from a first stage of strong time-correlations up to a final decorrelation, and a comparison with the process of approach to equilibrium is performed. Finally, the dependence on specific energy is investigated.

Open Quantum System Response from the Hierarchy of Pure States.

The spectral properties of a quantum system are essential when probing theoretical predictions against experimental data. For an open quantum system strongly interacting with its environment, spectral features are challenging to calculate. Here we demonstrate that the stochastic Hierarchy of Pure States (HOPS) approach is well suited to calculate the response of an open quantum system to a, possibly strong, coherent probe driving. For weak driving, where Kubo's linear response theory is applicable, it turns out that the HOPS method is highly efficient since fluctuations inherent to the stochastic dynamics cancel for the response function and, thus, allow us to obtain the susceptibility easily. Our results are in agreement with experimental data for a strongly damped spin system showing that the transition from oscillatory to overdamped motion is also reflected by the transmission spectrum. As a further application we demonstrate that the susceptibility, quantifying the amplitude of the response, as a function of temperature exhibits a maximum which is the hallmark of stochastic resonance. Beyond the linear regime, the exact open system dynamics shows the asymptotic Floquet state. We use the topic of probe driving and response to present the HOPS approach in a novel and self-contained way. This includes the importance sampling scheme which yields the nonlinear HOPS as well as the stochastic treatment of a thermal initial environmental state within the zero temperature formalism. Special attention is given to the exponential representation of the algebraic Ohmic bath correlation function and the truncation condition for the hierarchy.

A comparative study on the influence of intra- and inter-valley scattering to Hall conductivity of Dirac semimetals

Within the theoretical framework of Kubo linear response theory as well as the self-consistent Born approximation(SCBA), we perform a comparative study about the Hall conductivities of a Dirac semimetal(DSM) subject to, respectively, the intra- and inter-valley scattering. By calculating the Hall conductivity as a function of Fermi level, we find that when the Fermi level lies in the region around the Dirac node, the Hall conductivities between the two cases, i.e. subject to intra- and inter-valley scattering respectively, do not show appreciable difference, regardless of the scattering strength. As the Fermi level goes away from the Dirac node, the Hall conductivity in the case of inter-valley scattering is smaller than that in the case of intra-valley scattering if the scattering has relatively weak or moderate strength. In contrast, as scattering strength increases further, the Hall conductivity spectra between the two cases look almost identical no matter whether the Fermi level is close to or far away from the Dirac node. The underlying physics of these numerical results is that the Fermi surface has nontrivial contribution to the Hall conductivity in the case of weak or moderate scattering. However, in the case of strong scattering, it becomes much smaller than the contribution arising from all the occupied states. In addition, our study indicates that the cone tilt brings about the effective enhancement of Hall conductivity of DSMs.

Electronic Structure and Conductivity of a Disordered A1 – xBx Binary Alloy in the Cluster Approach for the Hubbard Model

We propose a method for calculating the electronic band structure of disordered systems with strong electron correlations. Various approaches to the description of electrical conductivity of disordered systems are considered. Calculations are based on determining the one-particle Green function of the system, which is averaged over different configurations of a cluster, on the Boltzmann formalism, and the Kubo linear response theory. As the basic model, we use the Hubbard model for an A1 – xBx binary alloy.

Semirealistic tight-binding model for spin-orbit torques

We compute the spin-orbit torque in a transition metal heterostructure using Slater-Koster parameterization in the two-center tight-binding approximation and accounting for d-orbitals only. In this method, the spin-orbit coupling is modeled within Russel-Saunders scheme, which enables us to treat interfacial and bulk spin-orbit transport on equal footing. The two components of the spin-orbit torque, dissipative (damping-like) and reactive (field-like), are computed within Kubo linear response theory. By systematically studying their thickness and angular dependence, we were able to accurately characterize these components beyond the traditional "inverse spin galvanic" and "spin Hall" effects. Whereas the conventional field-like torque is purely interfacial, we unambiguously demonstrate that the conventional the damping-like torque possesses both an interfacial and a bulk contribution. In addition, both field-like and damping-like torques display substantial angular dependence with strikingly different thickness behavior. While the planar contribution of the field-like torque decreases smoothly with the nonmagnetic metal thickness, the planar contribution of the damping-like torque increases dramatically with the nonmagnetic metal thickness. Finally, we investigate the self-torque exerted on the ferromagnet when the spin-orbit coupling of the nonmagnetic metal is turned off. Our results suggest that the spin accumulation that builds up inside the ferromagnet can be large enough to induce magnetic excitations.

Drude conductivity of a granular metal

We present a complete derivation of the analog to Drude conductivity in granular metals using diagrammatic methods. The derivation of Drude conductivity in a homogeneous metal using Kubo linear response theory is somewhat involved, with the evaluation of a singular double integral being required to cancel the diamagnetic response. We first provide a history and careful derivation of this standard result, before analyzing the equivalent situation in the model of a granular metal as a lattice of metallic grains connected by tunneling. A singular triple integral occurs here, and its evaluation gives the result expected from the Einstein relation and Fermi’s golden rule.

Anisotropy of resistivity in hexagonal late transition metals and their alloys

Transport properties of hexagonal transition metals Co, Ru, and Os at finite temperatures are studied by means of ab initio electronic structure techniques and the Kubo linear response theory. An alloy analogy model for a quantitative treatment of the electrical conductivities due to temperature-induced lattice vibrations (phonons) and spin fluctuations is applied with focus on anisotropy induced by the hexagonal structure. The resistivity anisotropy in Co is found opposite to that in Ru and Os, in agreement with existing experimental data. This result is ascribed to the strong itinerant ferromagnetism of Co which leads to profound differences in the electronic structure and conductivities in the majority and minority spin channels. A similar sensitivity to spin polarization is predicted for the anisotropy of residual resistivity in random hexagonal Co-rich Co–Ni and Co–Ni–Fe alloys.

Anomalous Hall effect in dense QCD matter

In this letter, we investigate the anomalous Hall effect in dense QCD matter. We use here the Nambu–Jona Lasinio model as an effective model of QCD. When the dual chiral density wave which is the spatially modulated chiral condensate appears in the medium, it gives rise to two Weyl points to the single-particle energy-spectrum and then the anomalous Hall conductivity becomes nonzero. Then, dense QCD matter is analogous to the Weyl semimetal. The direct calculation of the Hall conductivity by way of Kubo's linear response theory gives the term proportional to the distance between the Weyl points. Unlike the Weyl semimetal, there appears the additional contribution induced by axial anomaly.

Anomalous thermoelectric phenomena in lattice models of multi-Weyl semimetals

The thermoelectric transport coefficients are calculated in a generic lattice model of multi-Weyl semimetals with a broken time-reversal symmetry by using the Kubo's linear response theory. The contributions connected with the Berry curvature-induced electromagnetic orbital and heat magnetizations are systematically taken into account. It is shown that the thermoelectric transport is profoundly affected by the nontrivial topology of multi-Weyl semimetals. In particular, the calculation reveals a number of thermal coefficients of the topological origin which describe the anomalous Nernst and thermal Hall effects in the absence of background magnetic fields. Similarly to the anomalous Hall effect, all anomalous thermoelectric coefficients are proportional to the integer topological charge of the Weyl nodes. The dependence of the thermoelectric coefficients on the chemical potential and temperature is also studied.

Negative differential transmission in graphene

By using the Kubo linear response theory with the Keldysh Green function approach, we investigate the mechanism leading to the negative differential transmission in system with the equilibrium electron density much smaller than the photon-excited one. It is shown that the negative differential transmission can appear at low probe-photon energy (in the order of the scattering rate) or at high energy (much larger than the scattering rate). For the low probe-photon energy case, the negative differential transmission is found to come from the increase of the intra-band conductivity due to the large variation of electron distribution after the pumping. As for the high probe-photon energy case, the negative differential transmission is shown to tend to appear with the hot-electron temperature being closer to the equilibrium one and the chemical potential higher than the equilibrium one but considerably smaller than half of the probe-photon energy. We also show that this negative differential transmission can come from both the inter- and the intra-band components of the conductivity. Especially, for the inter-band component, its contribution to the negative differential transmission is shown to come from both the Hartree-Fock self-energy and the scattering. Furthermore, the influence of the Coulomb-hole self-energy is also addressed.

Orbital-selective conductance of Co adatom on the Pt(111) surface

We propose an orbital-selective model for the transport and magnetic properties of the individual Co impurity deposited on the Pt(111). Using the combination of the Anderson-type Hamiltonian and the Kubo's linear response theory we show that the magnetization and dI/dV spectrum of Co adatom are originated from the 3d states of the different symmetry. A textbook expression for the spin-dependent differential conductance provides a natural connection between magnetic and transport properties of Co/Pt(111). We found that it is possible to detect and to manipulate the different 3d states of the Co adatom by tuning the spin polarization of the tip and tip-impurity distance in STM experiments.

Electronic transport coefficients fromab initiosimulations and application to dense liquid hydrogen

Using Kubo's linear response theory, we derive expressions for the frequency-dependent electrical conductivity (Kubo-Greenwood formula), thermopower, and thermal conductivity in a strongly correlated electron system. These are evaluated within ab initio molecular dynamics simulations in order to study the thermoelectric transport coefficients in dense liquid hydrogen, especially near the nonmetal-to-metal transition region. We also observe significant deviations from the widely used Wiedemann-Franz law which is strictly valid only for degenerate systems and give an estimate for its valid scope of application towards lower densities.

Spin Hall and longitudinal conductivity of a conserved spin current in two dimensional heavy-hole gases

The spin Hall and longitudinal conductivity of a 2D heavy-hole gas with {\it k}-cubic Rashba and Dresselhaus spin-orbit interaction is studied in the ac frequency domain. Using Kubo linear-response theory and a recently proposed definition for the (conserved) spin current operator suitable for spin-3/2 holes, it is shown that the spin conductivity tensor exhibit very distinguishable features from those obtained with the standard definition of the spin current. This is due to a significant contribution of the spin-torque term arisen from the alternative definition of spin current which strongly affects the magnitude and the sign of the dynamic spin current. In the dc (free of disorder) limit, the spin Hall conductivity for only (or dominant) {\it k}-cubic Rashba coupling is $\sigma^{s,z}_{xy}(0)=-9e/8\pi$, whereas $\sigma^{s,z}_{xy}(0)=-3e/8\pi$ for only (or dominant) {\it k}-cubic Dresselhaus coupling. Such anisotropic response is understood in terms of the absence of mapping the {\it k}-cubic Rashba $\leftrightarrow$ Dresselhaus Hamiltonians. This asymmetry is also responsible for the non-vanishing dc spin Hall conductivity ($\sigma^{s,z}_{xy}(0)=-6e/8\pi$) when the Rashba and Dresselhaus parameters have the same strength, in contrast with its corresponding case for electrons. These results are of relevance to validate the alternative definition of spin-current through measurements in the frequency domain of the spin accumulation and/or spin currents in 2D hole gases.

The spin dynamics of molecular magnets beyond Kubo's linear response theory

The description of quantum dynamics of nanomagnets is a central issue in most applications proposed for those systems. In this paper, we put forward a modified perturbation approach to study the spin dynamics of a molecular magnet in the presence of time-dependent magnetic fields.The non-perturbed Hamiltonian H0, which defines the interaction picture, may be time-dependent proviso it can be diagonalized at all times by the same basis of states. We probe the method using a simple model Hamiltonian, that contains the important anisotropy terms relevant for Fe8 molecular clusters, and solve as an example the case with the smallest non trivial spin value (S=1). Our modified perturbation approach converges rapidly to the exact solution, goes beyond the Kubo linear response theory, and is well defined even at resonance. Temperature effects in the spin dynamics are taken into account in the context of the density matrix.

Predicting the thermal conductivity of inorganic and polymeric glasses: The role of anharmonicity

The thermal conductivity of several amorphous solids is numerically evaluated within the harmonic approximation from Kubo linear-response theory following the formalism developed by Allen and Feldman [Phys. Rev. B 48, 12581 (1993)]. The predictions are compared to the results of molecular dynamics (MD) simulations with realistic anharmonic potentials and to experimental measurements. The harmonic theory accurately predicts the thermal conductivity of amorphous silicon, a model Lennard-Jones glass, and a bead-spring Lennard-Jones glass. For polystyrene and amorphous silica at room temperature, however, the harmonic theory underestimates the thermal conductivity by a factor of about 2. This result can be explained by the existence of additional thermal transport via anharmonic energy transfer. More surprisingly, the thermal conductivity of polystyrene and amorphous silica at low temperature (MD and experimental) are significantly below the predictions of the harmonic theory. Potential reasons for the failure of the harmonic theory of disordered solids to predict the thermal conductivity of glassy polymers are discussed.