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The third-order nonlinear Hall effect induced by Berry-connection polarizability tensor has been observed in Weyl semimetals T$_d$-MoTe$_2$ as well as T$_d$-TaIrTe$_4$. The experiments were performed on bulk samples, and the results were interpreted with the semiclassical Boltzmann approach. Beyond the bulk limit, we develop a quantum nonlinear transport theory to investigate the third-order Hall response of a four-terminal setup with time-reversal symmetry in quantum regime. The quantum nonlinear theory is verified on a model system of monolayer MoTe$_2$, and numerical results on the angle-resolved Hall currents are qualitatively consistent with the experiment. More importantly, quantum signatures of the third-order Hall effect are revealed, which are independent of the system symmetry. The first quantum signature is quantum enhancement of the third-order Hall current, which is characterized by sharp current peaks whose magnitudes are three orders larger than the first-order Hall current. Such quantum enhancement originates from quantum interference in coherent transport, and it can be easily destroyed by dephasing effect. The second quantum signature is disorder-induced enhancement of the third-order Hall current for weak disorders. Our findings reveal quantum characteristics of the third-order Hall effect, and we propose feasible ways to enhance it in nanoscale systems. The quantum third-order theory developed in this work provides a general formalism for describing nonlinear coherent transport properties in multi-terminal devices, regardless of the system symmetry.

Topological invariants play an important role in characterizing topological phases. However, the topological invariants of Hermitian systems usually fail to characterize non-Hermitian topological systems due to non-Hermiticity. In this work, we generalize the Majorana polarization, which is initially defined to describe Hermitian topological superconductors, as a topological edge invariant to characterize the non-Hermitian topological superconductors. The definition of the generalized Majorana polarization depends upon two inequivalent particle-hole symmetries in non-Hermitian systems. The spinless Kitaev chain model and topological superconductor model on the honeycomb lattice are considered to examine the reliability and validity of the generalized Majorana polarization. We find that the phase transitions obtained by using Majorana polarization are consistent with the commonly used complex-energy point gap descriptions, which indicates the Majorana polarization is a reliable topological invariant to characterize the topological phase transition in non-Hermitian topological superconductors. In addition, the non-Hermitian skin effect on Majorana bound states in non-Hermitian topological superconductors is also discussed.

Twisted bilayer graphene is one of the simplest van der Waals structures, and its inhomogeneous interlayer coupling can induce rich electronic properties. In twisted bilayer graphene nanoribbons (tBLGNRs), the interlayer coupling strengths are different for two ribbon edges due to the inhomogeneous bonding, which splits the edge states into two individuals in energy. The lower-energy state, localizing at the ribbon edge with the stronger interlayer coupling, is a good candidate to generate one-dimensional (1D) topological superconductivity in the presence of Rashba spin-orbit coupling, Zeeman field, and $s$-wave superconductivity. Majorana zero modes (MZMs) are found to be localized at both ends of this edge. The topological invariants of the system are explored by evaluating the Berry phase for infinite-length ribbons and Majorana polarization for quasi-1D ribbons, giving the same topological phase diagram. More importantly, by adjusting interlayer dislocation and uniaxial strain of tBLGNRs across the critical values, the lower-energy edge changes and 1D topological superconductivity can ``jump'' from one ribbon edge to the other one. Finally, by applying a gate voltage bias between bilayers or changing the interlayer distance, a MZM can transfer along the ribbon edge. The tBLGNRs provide an alternative platform to study 1D topological superconductivity and MZMs.

Spin-orbit coupling in graphene is able to induce various topological phases and is also crucial for potential application in graphene-based spintronics. However, graphene itself exhibits extremely weak spin-orbit coupling, and it is rather challenging to enhance the spin-orbit coupling without drastically affecting its fundamental physical property in graphene via external means. In this paper, we show that the charge-compensated $n\text{\ensuremath{-}}p$ codoping approach not only can overcome the main shortcomings arising from single-element adsorption in graphene but can also result in a large Rashba spin-orbit splitting. As an example, we codope heavy adatoms with outer-shell $p$ electrons (e.g., Tl atoms acting as $n$-type dopants) on $p$-type doped graphene (e.g., by substituting carbon atoms with B atoms). We find the following: (1) Electrostatic attraction between $n$- and $p$-type dopants effectively enhances the adsorption and diffusion barrier of metallic adatoms and suppresses the undesirable formation of clustering. (2) Large Rashba spin-orbit splitting ($\ensuremath{\sim}130\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ for 6.25% B-Tl-codoped graphene) is produced due to the electrostatic interaction. (3) The charge-compensated nature and mutual screening of $n\text{\ensuremath{-}}p$ codopants preserve the Dirac dispersion of charge carriers to some extent.

We report a first-principles investigation of spin-dependent transport properties in two different graphene-based molecular junctions. By applying different temperatures between two leads without bias voltage, spin-dependent currents are driven which depend on reference temperature $T$, temperature gradient $\mathrm{\ensuremath{\Delta}}T$, and gate voltage ${V}_{g}$. Moreover, pure spin currents without charge currents can be obtained by adjusting $T,\mathrm{\ensuremath{\Delta}}T$, and ${V}_{g}$ for both molecular junctions. The directions of pure spin currents in these two molecular junctions are opposite, which can be understood by analyzing the transmission coefficients under equilibrium states. Spin thermopower, thermal conductance, and the figure of merit as functions of $T,{V}_{g}$, and chemical potential $\ensuremath{\mu}$ were also investigated in the linear response regime. Large spin thermopower and spin figure of merit can be obtained by adjusting ${V}_{g}$ and $\ensuremath{\mu}$ for each junction, which indicates proper application of spin caloritronic devices of our graphene-based molecular junctions.

We numerically investigate the statistical properties of electrochemical capacitance in disordered two-dimensional mesoscopic capacitors. Based on the tight-binding Hamiltonian, the Green's function formalism is adopted to study the average electrochemical capacitance, its fluctuation as well as the distribution of capacitance of the disordered mesoscopic capacitors for three different ensembles: orthogonal (symmetry index \beta=1), unitary (\beta=2), and symplectic (\beta=4). It is found that the electrochemical capacitance in the disordered systems exhibits universal behavior. In the case of single conducting channel, the electrochemical capacitance follows a symmetric Gaussian distribution at weak disorders as expected from the random matrix theory. In the strongly disordered regime, the distribution is found to be a sharply one-sided form with a nearly-constant tail in the large capacitance region. This behavior is due to the existence of the necklace states in disordered systems, which is characterized by the multi-resonance that gives rise to a large density of states. In addition, it is found that the necklace state also enhances the fluctuation of electrochemical capacitance in the case of single conducting channel. When the number of conducting channels increases, the influence of necklace states becomes less important. For large number of conducting channels, the electrochemical capacitance fluctuation develops a plateau region in the strongly disordered regime. The plateau value is identified as universal electrochemical capacitance fluctuation, which is independent of system parameters such as disorder strength, Fermi energy, geometric capacitance, and system size. Importantly, the universal electrochemical capacitance fluctuation is the same for all three ensembles, suggesting a super-universal behavior., Comment: 9 pages, 7 figures

Waiting time is an important transport quantity that is complementary to average current and its fluctuation. So far all the studies of waiting time distribution (WTD) are limited to steady state transport (either dc or ac). In this work, we present a theory to calculate WTD for coherent electronic systems in transient regime. We express the generating function of full counting statistics using Keldysh non-equilibrium Green's functions formalism. Our analysis goes beyond the wideband approximation and is suitable for first principles calculation on realistic systems. Analytic solution has been obtained for short and long time behaviors of waiting times. At short times, the WTD shows a linear dependence on the waiting time while in the long time limit, WTD follows Poisson distribution. We have applied this theory to a quantum dot connected by two leads and calculated cumulants of transferred charge as well as WTD in the transient regime. We have demonstrated how to relate WTD to experimental measured data.