Kubo Formula Research Articles

Anisotropic magneto-optical absorption and linear dichroism in two-dimensional semi-Dirac electron systems

We present a theoretical study on the Landau levels (LLs) and magneto-optical absorption of a two-dimensional semi-Dirac electron system under a perpendicular magnetic field. Based on an effective k.p Hamiltonian, we find that the LLs are proportional to the two-thirds power law of the magnetic field and level index, which can be understood as a hybridization of the LL of Schrodinger and Dirac electrons with new features. With the help of Kubo formula, we find the selection rule for interband (intraband) magneto-optical transition is anisotropic (isotropic). Specifically, the selection rules for interband magneto-optical transitions are $\Delta n$=0, $\pm2$ ($\pm2$, $\pm4$) for linearly polarized light along the linear (parabolic) dispersion direction, while the selection rules for the intraband transition are $\Delta n$=$\pm1$, $\pm3$ regardless of the polarization direction of the light. Further, the magneto-optical conductivity for interband (intraband) transition excited by linearly polarized light along the linear dispersion direction is two (one) orders of magnitude larger than that along the parabolic dispersion direction. This anisotropic magneto-optical absorption spectrum clearly reflects the structure of the LLs, and results in a strong linear dichroism. Importantly, a perfect linear dichroism with magnetic-field tunable wavelength can be achieved by using the interband transition between the two lowest LLs, i.e, from Ev0 to Ec0. Our results shed light on the magneto-optical property of the two dimensional semi-Dirac electron systems and pave the way to design magnetically controlled optical devices.

Resonant optical topological Hall conductivity from skyrmions

We study the high frequency Hall conductivity in a two-dimensional (2D) model of conduction electrons coupled to a background magnetic skyrmion texture via an effective Hund's coupling term. For an ordered skyrmion crystal, a Kubo formula calculation using the basis of skyrmion crystal Chern bands reveals a resonant Hall response at a frequency set by the Hund's coupling: $\hbar\omega_{\text{res}} \approx J_H$. A complementary real-space Kubo formula calculation for an isolated skyrmion in a box reveals a similar resonant Hall response. A linear relation between the area under the Hall resonant curve and the skyrmion density is discovered numerically and is further elucidated using a gradient expansion which is valid for smooth textures and a local approximation based on a spin-trimer calculation. We point out the issue of distinguishing this skyrmion contribution from a similar feature arising from spin-orbit interactions, as demonstrated in a model for Rashba spin-orbit coupled electrons in a collinear ferromagnet, which is analogous to the difficulty of unambiguously separating the d.c. topological Hall effect from the anomalous Hall effect. The resonant feature in the high frequency topological Hall effect is proposed to provide a potentially useful local optical signature of skyrmions via probes such as scanning magneto-optical Kerr microscopy.

Spin transport on the ferromagnetic checkerboard lattice

I study spin transport on the anisotropic ferromagnetic checkerboard lattice with a Dzyaloshinskii -Moriya interaction and an easy-axis anisotropy term. I show that by fine-tuning the parameters, we can get a flat band. The transverse and longitudinal dynamical spin conductivities are calculated using a Kubo formula. The Drude weight was also calculated as a function of temperature. The Drude weight vanishes at T = 0 but is finite for T > 0. A finite Drude weight indicates an ideal conductor, which we refer to as ballistic transport.

Cubic Hall viscosity in three-dimensional topological semimetals

The nondissipative (Hall) viscosity is known to play an interesting role in two-dimensional (2D) topological states of matter, in the hydrodynamic regime of correlated materials, and in classical active fluids with broken time-reversal symmetry (TRS). However, generalizations of these effects to 3D have remained elusive. In this work, we address this question by studying the Hall viscoelastic response of 3D crystals. We show that for systems with tetrahedral symmetries, there exist new, intrinsically 3D Hall viscosity coefficients that cannot be obtained via a reduction to a quasi-2D system. To study these coefficients, we specialize to a theoretically and experimentally motivated tight-binding model for a chiral magnetic metal in (magnetic) space group [(M)SG] P2$_1$3 (No. 198.9), a nonpolar group of recent experimental interest that hosts both chiral magnets and topological semimetals (TSMs). Using the Kubo formula for viscosity, we compute two forms of the Hall viscosity, phonon and ``momentum'' (conventional) and show that for the tight-binding model we consider, both forms realize the novel cubic Hall viscosity. We conclude by discussing the implication of our results for transport in 2D magnetic metals and discuss some candidate materials in which these effects may be observed.

Theoretical study of extrinsic spin current generation in ferromagnets induced by anisotropic spin-flip scattering

The spin Hall effect (SHE) and the magnetic spin Hall effect (MSHE) are responsible for electrical spin current generation, which is a key concept of modern spintronics. We theoretically investigated the spin conductivity induced by spin-dependent s-d scattering in a ferromagnetic 3d alloy model by employing microscopic transport theory based on the Kubo formula. We derived a novel extrinsic mechanism that contributes to both the SHE and MSHE. This mechanism can be understood as the contribution from anisotropic (spatial-dependent) spin-flip scattering due to the combination of the orbital-dependent anisotropic shape of s-d hybridization and spin flipping, with the orbital shift caused by spin-orbit interaction with the d-orbitals. We also show that this mechanism is valid under crystal-field splitting among the d-orbitals in either the cubic or tetragonal symmetry.

Effects of strong correlations on the nonlinear response in Weyl-Kondo semimetals

Nonlinear responses give rise to various exciting phenomena, which are forbidden in linear responses. Among them, one of the most fascinating phenomena is the recently observed giant spontaneous Hall effect in $\mathrm{Ce_{3}Bi_{4}Pd_{3}}$. This material is a promising candidate for a Weyl-Kondo semimetal, and this experiment implies that strong correlation effects can enhance the nonlinear Hall effect. However, most theoretical studies on nonlinear responses have been limited to free systems, and the connection between nonlinear responses and strong correlation effects is poorly understood. Motivated by these experiments and recent theoretical advances to analyze strong correlation effects on the nonlinear response, we study a periodic Anderson model describing $\mathrm{Ce_{3}Bi_{4}Pd_{3}}$ using the dynamical mean-field theory. We calculate the nonlinear longitudinal conductivity and the nonlinear Hall conductivity using the Kubo formula extended to the nonlinear response regime and clarify their temperature dependences. We numerically show that strong correlations can enhance nonlinear conductivities, and we conclude that the magnitude of the experimentally observed giant nonlinear Hall effect can be explained by strong correlation effects.

Spin Hall effect in the monolayer Janus compound MoSSe enhanced by Rashba spin-orbit coupling

Sizable Rashba spin-orbit coupling (SOC) is of critical importance in potential applications of two-dimensional materials in spintronics devices. However, due to the presence of vertical mirror symmetry, Rashba SOC is absent in 2H transition-metal dichalcogenide monolayers and the spin Hall conductivities are attributed only to Zeeman splitting. We study theoretically the electronic structures and intrinsic spin Hall conductivity of two-dimensional monolayer Janus MoSSe by performing first-principles calculations as well as by using the Kubo formula with Wannier interpolations. We find out that monolayer Janus MoSSe possesses considerable spin Hall conductivities both in conduction and valence bands. In valence bands, the spin Hall conductivity of Janus MoSSe is comparable to that in ${\mathrm{MoS}}_{2}$ and ${\mathrm{MoSe}}_{2}$. Moreover, in the conduction bands, the spin Hall conductivities are enhanced up to two orders of magnitude because of strong Rashba SOC. The spin Hall conductivity can be tuned significantly by adjusting the Fermi level or external strains. Our results show that monolayer Janus MoSSe could be a potential candidate to realize two-dimensional flexible spintronics devices.

Integer quantum Hall effect in Kekulé-patterned graphene

Y-shaped Kekulé bond textures in a honeycomb lattice on a graphene-copper superlattice have recently been experimentally revealed. In this paper, the effects of such a bond modulation on the transport coefficients of Kekulé-patterned graphene are investigated in the presence of a perpendicular magnetic field. Analytical expressions are derived for the Hall and longitudinal conductivities using the Kubo formula. It is found that the Y-shaped Kekulé bond texture lifts the valley degeneracy of all Landau levels except that of the zero mode, leading to additional plateaus in the Hall conductivity accompanied by a split of the corresponding peaks in the longitudinal conductivity. Consequently, the Hall conductivity is quantized as ± ne 2/h for n = 2, 4, 6, 8, 10, …, excluding some plateaus that disappear due to the complete overlap of the Landau levels of different cones. These results also suggest that DC Hall conductivity measurements will allow us to determine the Kekulé bond texture amplitude.

Macroscopic transports in a rotational system with an electromagnetic field

In this work, we explore macroscopic transport phenomena associated with a rotational system in the presence of an external orthogonal electromagnetic field. Simply based on the lowest Landau level approximation, we derive nontrivial expressions for chiral density and various currents consistently by adopting small angular velocity expansion or Kubo formula. While the generation of anomalous electric current is due to the pseudo gauge field effect of the spin-rotation coupling, the chiral density and current can be simply explained with the help of Lorentz boosts. Finally, Lorentz covariant forms can be obtained by unifying our results and the magnetovorticity effect.

Second order equilibrium transport in strongly coupled mathcal{N} = 4 supersymmetric SU(Nc) Yang-Mills plasma via holography

A relativistic fluid in 3+1 dimensions with a global U(1) symmetry admits nine independent static susceptibilities at the second order in the hydrodynamic derivative expansion, which capture the response of the fluid in thermal equilibrium to the presence of external time-independent sources. Of these, seven are time-reversal mathbbm{T} invariant and can be obtained from Kubo formulas involving equilibrium two-point functions of the energy-momentum tensor and the U(1) current. Making use of the gauge/gravity duality along with the aforementioned Kubo formulas, we compute all seven mathbbm{T} invariant second order susceptibilities for the mathcal{N} = 4 supersymmetric SU(Nc) Yang-Mills plasma in the limit of large Nc and at strong ’t-Hooft coupling λ. In particular, we consider the plasma to be charged under a U(1) subgroup of the global SU(4) R-symmetry of the theory. We present analytic expressions for three of the seven mathbbm{T} invariant susceptibilities, while the remaining four are computed numerically. The dual gravitational description for the charged plasma in thermal equilibrium in the absence of background electric and magnetic fields is provided by the asymptotically AdS5 Reissner-Nordström black brane geometry. The susceptibilities are extracted by studying perturbations to the bulk geometry as well as to the bulk gauge field. We also present an estimate of the second order transport coefficient κ, which determines the response of the fluid to the presence of background curvature, for QCD, and compare it with previous determinations made using different techniques.

Invariance principles in the theory and computation of transport coefficients

In this work, we elaborate on two recently discovered invariance principles, according to which transport coefficients are, to a large extent, independent of the microscopic definition of the densities and currents of the conserved quantities being transported (energy, momentum, mass, charge). The first such principle, gauge invariance, allows one to define a quantum adiabatic energy current from density-functional theory, from which the heat conductivity can be uniquely defined and computed using equilibrium ab initio molecular dynamics. When combined with a novel topological definition of atomic oxidation states, gauge invariance also sheds new light onto the mechanisms of charge transport in ionic conductors. The second principle, convective invariance, allows one to extend the analysis to multi-component systems. These invariance principles can be combined with new spectral analysis methods for the current time series to be fed into the Green–Kubo formula to obtain accurate estimates of transport coefficients from relatively short molecular dynamics simulations.Graphic abstract

Microscopic formulas for thermoelectric transport coefficients in lattice systems

A macroscopic description of thermoelectric phenomena involves several tensorial transport coefficients. Textbook microscopic Kubo formulas for them are plagued with ambiguities in the definitions of the current operators and the magnetization. We derive a version of these formulas for lattice systems which is free from ambiguities but contains additional terms compared to the textbook results. For symmetric components of thermoelectric tensors, we identify a large class of lattice systems for which the additional terms vanish with a natural choice of the energy current. To eliminate ambiguities in the skew-symmetric components, one needs to interpret them as relative quantities: only their differences for pairs of materials are well-defined.

Multi-layered parallel plate waveguide with electrically and magnetically biased graphene sheets

The performance of a multi-layered parallel plate waveguide, supported by graphene plates, is evaluated here. The graphene layers were biased with both magnetic and electric fields. The graphene plates weremodelled as an anisotropic surface derived from the Kubo formula. Maxwell's equations were solved for these waveguides; and it is shown when both magnetic and electric biases were applied to the graphene, hybrid modes propagated inside the waveguides. The intensity of each TM and TE mode could be adjusted with an applied external field bias. This study of wave confinement has shown that, by varying the bias, full control can be exerted over the wave's propagation within a waveguide; whereby, increasing the bias causes added phase changes over the propagation length, while increasing the waveguide's attenuation. It must be emphasized that the presented procedure is also applicable to other guiding structures that have boundary with isotropic or anisotropic conductivities.

The time-averaged spin-Hall conductivity in two-dimensional spin–orbit coupled systems: the role of spin dynamics

We investigate the spin-Hall conductivity (SHC) in the two-dimensional spin–orbit coupled systems by taking spin precession into account. The influence of spin precession on the spin-Hall current in the presence of disorder is investigated. Perturbation method leads to the result that the spin dynamics is composed of Larmor (periodic) and non-Larmor (non-periodic) precession. The non-Larmor component of spin is found to be the same result obtained from the Kubo formula in the short time limit. The Larmor component of spin, i.e., spin precession in the unperturbed system, was neglected in the previous studies, but it plays an important role in the linear response theory. The Larmor precession of spin should be zero after time average over several complete periods because there is no specific orientation perpendicular to the plane. By using the requirement of vanishing time-averaged Larmor precession of spin, we calculate the time-averaged non-Larmor SHC for k-cubic Rashba system by using the conventional definition of the spin current. Our result is 2.1(q/8π) which is very close to the experimental value 2.2(q/8π) by Wunderlich et al [2005 Phys. Rev. Lett. 94, 047204], where −q is the electric charge.

Green–Kubo formula for Boltzmann and Fermi–Dirac statistics

Shear viscosity of nuclear matter is extracted via the Green-Kubo formula and the Gaussian thermostated SLLOD algorithm (the shear rate method) in a periodic box by using an improved quantum molecular dynamic (ImQMD) model without mean field, also it is calculated by a Boltzmann-type equation. Here a new form of the Green-Kubo formula is put forward in the present work. For classical limit at nuclear matter densities of $0.4\rho_{0}$ and $1.0\rho_{0}$, shear viscosity by the traditional and new form of the Green-Kubo formula as well as the SLLOD algorithm are coincident with each other. However, for non-classical limit, shear viscosity by the traditional form of the Green-Kubo formula is higher than those obtained by the new form of the Green-Kubo formula as well as the SLLOD algorithm especially in low temperature region. In addition, shear viscosity from the Boltzmann-type equation is found to be less than that by the Green-Kubo method or the SLLOD algorithm for both classical and non-classical limits.

Chiral effects on rotating and accelerated backgrounds

If the gauge quantum chiral anomaly predicts chiral effects related to chemical potentials, then the gravitational chiral anomaly is responsible for the effects of temperature. The prediction of the gravitational anomaly for the temperature term in the chiral vortical effect coincides with the prediction of statistical calculations by the Kubo formula for spin 1/2, but two predictions disagree already for spin 1. A solution to this problem related to the features of infrared regularization is described. Also the thermodynamic instability in systems with acceleration, which occurs at temperatures below the Unruh temperature and the transition through the Unruh temperature are discussed.

Superlattice on the surface of a nanotube

The results of theoretical studies of the thermodynamic, kinetic, and high-frequency properties of the electron gas on the surface of a nanotube in a magnetic field in the presence of a longitudinal superlattice are presented. Nano-dimensions of the motion area lead to energy quantization. Its multiply connected structure in the presence of a magnetic field leads to effects that are derived from the Aharonov–Bohm effect. It is shown that the curvature of a nanotube, even in the absence of a magnetic field, causes new macroscopic oscillation effects such as de Haas–van Alphen oscillations, which are associated with the quantization of the transverse electron motion energy and with the root peculiarities of the density of electron states on the nanotube surface. Thermodynamic potentials and heat capacity of the electron gas on the tube are calculated in the gas approximation. The Kubo formula for the conductivity tensor of the electron gas on the nanotube surface is obtained. The Landau damping regions of electromagnetic waves on a tube are determined and the beats are theoretically predicted on the graph of the dependence of conductivity on tube parameters. In the hydrodynamic approximation, the plasma waves on the surface of a semiconductor nanotube with a superlattice are considered. It is shown that optical and acoustic plasmons can propagate along a tube with one kind of carrier. Electron spin waves on the surface of a semiconductor nanotube with a superlattice in a magnetic field are studied. The spectra and areas of collisionless damping of these waves are found. We have shown that the spin wave damping is absent in these areas if the tubes with a degenerate electron gas have small radius.

Efficient Time-Domain Approach for Linear Response Functions

We derive the general Kubo formula in a form that solely utilizes the time evolution of displacement operators. The derivation is based on the decomposition of the linear response function into its time-symmetric and time-antisymmetric parts. We relate this form to the well-known fluctuation-dissipation formula and discuss theoretical and numerical aspects of it. The approach is illustrated with an analytical example for magnetic resonance as well as a numerical example where we analyze the electrical conductivity tensor and the Chern insulating state of the disordered Haldane model. We introduce a highly efficient time-domain approach that describes the quantum dynamics of the resistivity of this model with an at least 1000-fold better performance in comparison to existing time-evolution schemes.

Calculation of Berry curvature using non-orthogonal atomic orbitals

We present a derivation of the full formula to calculate the Berry curvature on non-orthogonal numerical atomic orbital (NAO) bases. Because usually, the number of NAOs is larger than that of the Wannier bases, we use a orbital contraction method to reduce the basis sizes, which can greatly improve the calculation efficiency without significantly reducing the calculation accuracy. We benchmark the formula by calculating the Berry curvature of ferroelectric BaTiO3 and bcc Fe, as well as the anomalous Hall conductivity for Fe. The results are in excellent agreement with the finite-difference and previous results in the literature. We find that there are corrections terms to the Kubo formula of the Berry curvature. For the full NAO base, the differences between the two methods are negligibly small, but for the reduced bases sets, the correction terms become larger, which may not be neglected in some cases. The formula developed in this work can readily be applied to the non-orthogonal generalized Wannier functions.

Kubo formulae for first-order spin hydrodynamics

We derive Kubo formulae for first-order spin hydrodynamics based on non-equilibrium statistical operators method. In first-order spin hydrodynamics, there are two new transport coefficients besides the ordinary ones appearing in first-order viscous hydrodynamics. They emerge due to the incorporation of the spin degree of freedom into fluids and the spin-orbital coupling. Zubarev's non-equilibrium statistical operator method can be well applied to investigate these quantum effects in fluids. The Kubo formulae, based on the method of non-equilibrium statistical operators, are related to equilibrium (imaginary-time) infrared Green's functions, and all the transport coefficients can be determined when the microscopic theory is specified.