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This report reviews recent progress in computing Kubo formulas for general interacting Hamiltonians. The aim is to calculate electric and thermal magneto-conductivities in strong scattering regimes where Boltzmann equation and Hall conductivity proxies exceed their validity. Three primary approaches are explained. 1. Degeneracy-projected polarization formulas for Hall-type conductivities, which substantially reduce the number of calculated current matrix elements. These expressions generalize the Berry curvature integral formulas to imperfect lattices. 2. Continued fraction representation of dynamical longitudinal conductivities. The calculations produce a set of thermodynamic averages, which can be controllably extrapolated using their mathematical relations to low and high frequency conductivity asymptotics. 3. Hall-type coefficients summation formulas, which are constructed from thermodynamic averages. The thermodynamic formulas are derived in the operator Hilbert space formalism, which avoids the opacity and high computational cost of the Hamiltonian eigenspectrum. The coefficients can be obtained by well established imaginary-time Monte Carlo sampling, high temperature expansion, traces of operator products, and variational wavefunctions at low temperatures. We demonstrate the power of approaches 1–3 by their application to well known models of lattice electrons and bosons. The calculations clarify the far-reaching influence of strong local interactions on the metallic transport near Mott insulators. Future directions for these approaches are discussed. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical framework for exploring the optical properties of layered materials. The derived analytical formulas for reflectance and transmittance spectra incorporate vertical valence-state transitions and appropriate boundary conditions for finite-width bulk materials, integrated with the generalized tight-binding model to account for intrinsic interactions and external fields. The study reveals distinct structures in the spectra of multilayer graphene for different stacking layers at low energies (∼0.2–0.8 eV), attributed to interband transitions at the K point. Furthermore, the high-energy part (≳5 eV) highlights thickness-dependent plasmon effects characterized by unique resonance frequencies in the optical spectra. These features depend on the dimensionality, interlayer coupling, and electronic structure of the material. This work provides a deeper understanding of the optical properties of layered structures, facilitating their exploration and characterization for diverse applications. [ABSTRACT FROM AUTHOR]

In this expository article, we present a systematic formal derivation of the Kubo formula for the linear-response current due to a time-harmonic electric field applied to non-interacting, spinless charged particles in a finite volume in the quantum setting. We model dissipation in a transparent way by assuming a sequence of scattering events occurring at random-time intervals modeled by a Poisson distribution. By taking the large-volume limit, we derive special cases of the formula for free electrons, continuum and tight-binding periodic systems, and the nearest-neighbor tight-binding model of graphene. We present the analogous formalism with dissipation to derive the Drude conductivity of classical free particles. [ABSTRACT FROM AUTHOR]

We describe the low‐temperature optical conductivity as a function of frequency for a quantum‐mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su–Schrieffer–Heeger tight‐binding Hamiltonian for noninteracting spinless electrons on a one‐dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo‐type formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the three‐dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations. [ABSTRACT FROM AUTHOR]

We theoretically study the optical properties of monolayer C 3 N within the framework of linear response theory by using a two band effective k ⋅ p Hamiltonian. We find that both the effective masses and optical conductivities are anisotropic arising from the anisotropic band structure at the M point. Although the inter-band coupling only exists in the armchair (x -)direction, the effective mass along the armchair direction is larger than that along the zigzag (y -)direction. Interestingly, the absorption part of the optical conductivity Re σ x x excited by linearly polarized light along the armchair direction is an order of magnitude larger than Re σ y y excited by linearly polarized light along the zigzag direction, resulting in a strong linear dichroism, because the inter-band optical conductivity is dominated by inter-band coupling which only exists in the armchair direction. Finite doping can adjust the optical conductivities i.e., Re σ x x and Re σ y y and modulate the linear dichroism, but the effects of n -type doping and p -type doping are different because of the electron–hole asymmetry. Our results provide further understanding of the electronic state of monolayer C 3 N and maybe useful to design polarized optical devices based on it. • The optical transitions are dominated by electron states at the M point. • The optical conductivities excited by linearly polarized light are highly anisotropic. • Doping can effectively regulate the optical conductivities the linear dichroism. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the Landau problem in the framework of a two-dimensional dynamical noncommutative (DNC) space. We study the deformed Landau problem using time-independent perturbation theory, wherein the energy shift notably depends on the DNC parameter τ. Using the accuracy of energy measurement, we put an upper bound on the parameter τ. Moreover, we study magnetoconductivity by employing the Kubo formula. This approach has allowed us to test the effects of noncommutative and DNC spaces on the behavior of magnetoconductivity. We show that dynamical noncommutativity of space has no effects on the x-component of the magnetoconductivity, but has a direct effect on its y-component. • Exploring the effects of both NC and DNC spaces on the Landau problem. • Studding the effects of both NC and DNC spaces on the magnetoresistance. • Improving the upper bound on the DNC parameter. • Our results may be helpful tool for investigating more related studies, including thermodynamics properties and other scenarios in DNC spaces and to get more insights into the structure of space–time. [ABSTRACT FROM AUTHOR]

In this review, we survey the current progress in computing transport properties in semimetals which harbour non-Fermi liquid (NFL) phases. We first discuss the widely-used Kubo formalism, which can be applied to the effective theory describing the stable NFL phase obtained via a renormalization group procedure and, hence, is applicable for temperatures close to zero (e.g. optical conductivity). For finite-temperature regimes, which apply to the computations of the generalized DC conductivity tensors, we elucidate the memory matrix approach. This approach is based on an effective hydrodynamic description of the system, and is especially suited for tackling transport calculations in strongly-interacting quantum field theories, because it does not rely on the existence of long-lived quasiparticles. As a concrete example, we apply these two approaches to find the response of the so-called Luttinger-Abrikosov-Benelavskii phase of isotropic three-dimensional Luttinger semimetals, which arises under the effects of long-ranged (unscreened) Coulomb interactions, with the chemical potential fine-tuned to cut exactly the nodal point. In particular, we focus on the electric conductivity tensors, thermal and thermoelectric response, Raman response, free energy, entropy density, and shear viscosity., (© 2024 IOP Publishing Ltd.)

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We first review the problem of a rigorous justification of Kubo's formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo's formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit. [ABSTRACT FROM AUTHOR]

We theoretically study Floquet band structures and transport properties of twisted bilayer graphene at the magic-angle under the irradiation of variously polarized light. The magic-angle bilayer graphene is depicted by the newly proposed ten-band tight-binding model and the iterative continued fraction method is adopted to facilitate the calculations of electronic properties in the low-frequency regime. The transitions between Floquet sidebands induce discontinuous electronic bands and energy gaps which further give rise to the antiresonances in longitudinal conductivity calculated by the Kubo formula. Furthermore, significant Hall conductivity is generated by circularly polarized light and its magnitude and sign are sensitive to light polarization as well as photoinduced bandgap-opening, offering a feasible way to tune Hall conductivity by manipulating light polarization. We finally take into account the interplay between light irradiation and short-range disorder, and reveal that disorder scattering remarkably enhances the photoinduced Hall conductivity and can be viewed as an extrinsic source to Hall conductivity.

We investigate the physical properties of silicene with both staggered sublattice potential and magnetization by using Kubo formalism, the latter arises from the magnetic proximity effect by depositing Fe atoms to silicene or depositing silicene on an appropriate ferromagnetic insulator. Based on the low-energy continuum model of the system where inversion symmetry is broken, we show that the system exhibits spin half metal state when staggered sublattice potential is in the same magnitude with mean and staggered magnetization. Besides, Hall conductivity and magnetic moment are all valley dependent, so we investigate the valley Hall effect of the system further by considering magnetization exclusively. This means carriers in different valleys turning into opposite directions transverse to an in-plane electric field. At last, we prove these results by investigating Berry curvature that characterizing Hall transport, which is also valley dependent. These effects can be used to generate valley-polarized currents solely by magnetization, forming the basis for the valley-based electronics applications. [ABSTRACT FROM AUTHOR]

Structural defects are inherent in solids at a finite temperature, because they diminish free energies by growing entropy. The arrangement of these defects may display long-range orders, as occurring in quasicrystals, whose hidden structural symmetry could greatly modify the transport of excitations. Moreover, the presence of such defects breaks the translational symmetry and collapses the reciprocal lattice, which has been a standard technique in solid-state physics. An alternative to address such a structural disorder is the real space theory. Nonetheless, solving 1023 coupled Schrödinger equations requires unavailable yottabytes (YB) of memory just for recording the atomic positions. In contrast, the real-space renormalization method (RSRM) uses an iterative procedure with a small number of effective sites in each step, and exponentially lessens the degrees of freedom, but keeps their participation in the final results. In this article, we review aperiodic atomic arrangements with hierarchical symmetry investigated by means of RSRM, as well as their consequences in measurable physical properties, such as electrical and thermal conductivities.

The optical conductivity of strained monolayer β 12 -borophene is investigated via the Kubo formula and the five-band tight-binding method. In our model, the hopping energy of electrons is modified by strain following the Harrison rule. We investigate the effect of these modifications on the electronic dispersion and optical conductivity. Our results indicate that inherent massless Dirac fermions become massive with tensile strain and that inherent triplet fermions are doubled with compressive strain. Moreover, the strain-induced longitudinal optical conductivity was found to show a redshift spectrum with both tensile and compressive strains, while a blueshift spectrum was appeared for the transverse/Hall optical conductivity. These results show that strain can tune both electronic properties of β 12 -borophene and relevant practical optoelectronic applications. • Tuning the electro-optical properties of monolayer β 12 - borophene via strain. • Massless to massive transition for the inherent Dirac fermions with tensile strain. • Doubling inherent triplet fermions with compressive strain. • Redshift (blueshift) spectrum with strain for the longitudinal (transverse/Hall) optical conductivity. [ABSTRACT FROM AUTHOR]

In this paper we present two quantum field theoretical analyses on the shear and bulk relaxation times. First, we discuss how to find Kubo formulas for the shear and the bulk relaxation times. Next, we provide results on the shear viscosity relaxation time obtained within the diagrammatic approach for the massless λ ϕ 4 theory. [ABSTRACT FROM AUTHOR]

Based on the Kubo formula approach, the spin Hall conductivity (SHC) of a two-dimensional (2D) Rashba s-wave superconductor in the presence of nonmagnetic impurities is calculated. We will show that by increasing the superconducting gap, the SHC decreases monotonically to zero, while by decreasing the concentration of impurities at zero gap, the SHC closes to the clean limit universal value − e 8 π . As a function of the impurity relaxation rate τ at T c = 0.1 and γ = 0.01 ( γ is the spin–orbit coupling in unit of eV · m), we will show that in the dirty limit ( τ → 0) the SHC vanishes, and by increasing the relaxation time ( τ → ∞) the SHC depends on the value of superconducting gap ( Δ = 1.76 T c 1 − T T c ), is changed from zero for full gap to − e 8 π in zero gap. At low temperatures, the SHC goes to zero exponentially and near the critical temperature depending on the concentration of the scattering centers, the SHC will tend to the value of normal state. We will also show that the SHC is independent of spin–orbit coupling ( γ ) in the clean limit. [ABSTRACT FROM AUTHOR]

Calculations of renormalized perpendicular conductivity within Kubo formula employing single particle temperature dependent Green's function formalism for bilayer graphene has been attempted. On the basis of numerical analysis, perpendicular conductivity as a function of temperature, interlayer coupling, onsite Coulomb interaction and carrier concentration per site has been analyzed for both AA- and AB-stacked bilayer graphene. It is found that perpendicular conductivity increases with interlayer coupling and also with temperature at low temperatures while at higher temperatures, there is saturation in perpendicular conductivity. Influences of onsite Coulomb interaction and carrier concentration per site on perpendicular conductivity is just opposite to each other while onsite Coulomb energy suppresses the rate of increase of σ ⊥ / σ ⊥0 with temperature, on the other hand increase in carrier density per site enhance this rate significantly. Finally, theoretically obtained results on temperature dependent perpendicular conductivity are viewed in terms of electronic transport data as well as recent theoretical works available in bilayer graphene. [ABSTRACT FROM AUTHOR]

We calculate the electrical conductivity of a metallic sample under the effects of distributed impurities and a random distribution of grain boundaries by means of a quantum mechanical procedure based on Kubo formula. Grain boundaries are represented either by a one-dimensional regular array of Dirac delta potentials (Mayadas and Shatzkes model) or by its three-dimensional extension (Szczyrbowski and Schmalzbauer model). We give formulas expressing the conductivity of bulk samples, thin films and thin wires of rectangular cross-sections in the case when the samples are bounded by perfectly flat surfaces. We find that, even in the absence of surface roughness, the conductivity in thin samples is reduced from its bulk value. If there are too many grain boundaries per unit length, or their scattering strength is high enough, there is a critical value R c of the reflectivity R of an individual boundary such that the electrical conductivity vanishes for R > R c . Also, the conductivity of thin wires shows a stepwise dependence on R . The effect of weak random variations in the strength or separation of the grain boundaries is computed by means of the method of correlation length. Finally, the resistivity of nanometric polycrystalline tungsten films reported in Choi et al. J. Appl. Phys. (2014) 115 104308 is tentatively analyzed by means of the present formalism. [ABSTRACT FROM AUTHOR]

Based on Kubo's linear response formalism, temperature dependence of the spin-Hall conductivity of a two-dimensional impure (magnetic and nonmagnetic impurities) Rashba electron gas in the presence of electron-electron and electron-phonon interactions is analyzed theoretically. We will show that the temperature dependence of the spin-Hall conductivity is determined by the relaxation rates due to these interactions. At low temperature, the elastic lifetimes ( $$\tau _\mathrm{n}$$ and $$\tau _\mathrm{m} )$$ are determined by magnetic and nonmagnetic impurity concentrations which are independent of the temperature, while the inelastic lifetimes ( $$\tau _\mathrm{ee}$$ and $$\tau _\mathrm{ep} )$$ related to the electron-electron and electron-phonon interactions, decrease when the temperature increases. We will also show that since the spin-Hall conductivity is sensitive to temperature, we can distinguish the intrinsic and extrinsic contributions. [ABSTRACT FROM AUTHOR]