On the sum of the index of a parabolic subalgebra and of its nilpotent radical

In this short note, we investigate the following question of Panyushev : ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra $\mathfrak{g}$ and the index of its nilpotent radical always greater than or equal to the rank of $\mathfrak{g}$?''. Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author, and proved by Millet-Fauquant and Joseph, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of $\mathfrak{g}$. This provides new examples of direct sum decomposition of a semisimple Lie algebra verifying the ``index additivity condition'' as stated by Ra{\"\i}s.Comment: Author's affiliation added in the second versio

Theory optical excitation spectra and depolarization dynamics in bilayer WS2_2 from viewpoint of excimers

We investigate the optical excitation spectra and the photoluminescence depolarization dynamics in bilayer WS$_2$. A different understanding of the optical excitation spectra in the recent photoluminescence experimentby Zhu {\em et al.} [arXiv:1403.6224] in bilayer WS$_2$ is proposed. In the experiment, four excitations (1.68, 1.93, 1.99 and 2.37 eV) are observed and identified to be indirect exciton for the $\Gamma$ valley, trion, A exciton and B exciton excitations, respectively, with the redshift for the A exciton energy measured to be 30$\sim$50 meV when the sample synthesized from monolayer to bilayer. According to our study, by considering there exist both the intra-layer and charge-transfer excitons in the bilayer WS$_2$, with inter-layer hopping of the hole, there exists excimer state composed by the superposition of the intra-layer and charge-transfer exciton states. Accordingly, we show that the four optical excitations in the bilayer WS$_2$ are the A charge-transfer exciton, ${\rm A}'$ excimer, ${\rm B}'$ excimer and B intra-layer exciton states, respectively, with the calculated resonance energies showing good agreement with the experiment. In our picture, the speculated indirect exciton, which involves a high-order phonon absorption/emission process, is not necessary. Furthermore, the binding energy for the excimer state is calculated to be 40 meV, providing reasonable explanation for the experimentally observed energy redshift of the A exciton. Based on the excimer states, we further derive the exchange interaction Hamiltonian. Then the photoluminescence depolarization dynamics due to the electron-hole exchange interaction is studied in the pump-probe setup by the kinetic spin Bloch equations. We find that ......Comment: 14 pages, 2 figure

Gauge-invariant theory of optical response to THz pulses in s-wave and (ss+pp)-wave superconducting semiconductor quantum wells

We investigate the optical response to the THz pulses in the s-wave and ($s$+$p$)-wave superconducting semiconductor quantum wells by using the gauge-invariant optical Bloch equations, in which the gauge structure in the superconductivity is explicitly retained. By using the gauge transformation, not only can the microscopic description for the quasiparticle dynamics be realized, but also the dynamics of the condensate is included, with the superfluid velocity and the effective chemical potential naturally incorporated. We reveal that the superfluid velocity itself can contribute to the pump of quasiparticles (pump effect), with its rate of change acting as the drive field to drive the quasiparticles (drive effect). Specifically, the drive effect can contribute to the formation of the blocking region for the quasiparticle, which directly suppresses the anomalous correlation of the Cooper pairs. We find that both the pump and drive effects contribute to the oscillations of the Higgs mode with twice the frequency of the optical field. However, it is shown that the contribution from the drive effect to the excitation of Higgs mode is dominant as long as the driven superconducting momentum is less than the Fermi momentum. This is in contrast to the conclusion from the Liouville or Bloch equations in the literature, in which the drive effect on the anomalous correlation is overlooked with only the pump effect considered.Furthermore, in the gauge-invariant optical Bloch equations, the charge neutrality condition is {\em consistently} considered based on the two-component model for the charge, in which the charge imbalance of quasiparticles can cause the fluctuation of the effective chemical potential. ......Comment: 33 pages, 16 figure

Hot-electron effect in spin relaxation of electrically injected electrons in intrinsic Germanium

The hot-electron effect in the spin relaxation of electrically injected electrons in intrinsic Germanium is investigated by the kinetic spin Bloch equations both analytically and numerically. It is shown that in the weak-electric-field regime with $E\lesssim 0.5$~kV/cm, our calculations has reasonable agreement with the recent transport experiment in the spin-injection configuration [Phys. Rev. Lett. {\bf 111}, 257204 (2013)]. We reveal that the spin relaxation is significantly enhanced at low temperature in the presence of weak electric field $E\lesssim 50$~V/cm, which originates from the obvious center-of-mass drift effect due to the weak electron-phonon interaction, whereas the hot-electron effect is demonstrated to be less important. This can explain the discrepancy between the experimental observation and the previous theoretical calculation [Phys. Rev. B {\bf 86}, 085202 (2012)], which deviates from the experimental results by about two orders of magnitude at low temperature. It is further shown that in the strong-electric-field regime with $0.5\lesssim E \lesssim 2$~kV/cm, the spin relaxation is enhanced due to the hot-electron effect, whereas the drift effect is demonstrated to be marginal. Finally, we find that when $1.4 \lesssim E\lesssim 2$~kV/cm which lies in the strong-electric-field regime, a small fraction of electrons ($\lesssim 5\%$) can be driven from the L to $\Gamma$ valley, and the spin relaxation rates are the same for the $\Gamma$ and L valleys in the intrinsic sample without impurity. With the negligible influence of the spin dynamics in the $\Gamma$ valley to the whole system, the spin dynamics in the L valley can be measured from the $\Gamma$ valley by the standard direct optical transition method.Comment: 10 pages, 3 figures, to be published in JPC

Gapped triplet pp-wave superconductivity in strong spin-orbit-coupled semiconductor quantum wells in proximity to ss-wave superconductor

We show that the {\it gapped} triplet superconductivity, i.e., a triplet superconductor with triplet order parameter, can be realized in strong spin-orbit-coupled quantum wells in proximity to $s$-wave superconductor. It is revealed that with the singlet order parameter induced from the superconducting proximity effect, in quantum wells, not only can the triplet pairings arise due to the spin-orbit coupling, but also the triplet order parameter can be induced due to the repulsive effective electron-electron interaction, including the electron-electron Coulomb and electron-phonon interactions. This is a natural extension of the work of de Gennes, in which the repulsive-interaction-induced singlet order parameter arises in the normal metal in proximity to $s$-wave superconductor [Rev. Mod. Phys. {\bf 36}, 225 (1964)]. Specifically, we derive the effective Bogoliubov-de Gennes equation, in which the self-energies due to the effective electron-electron interactions contribute to the singlet and triplet order parameters. It is further shown that for the singlet order parameter, it is efficiently suppressed due to this self-energy renormalization; whereas for the triplet order parameter, it is the $p$-wave ($p_x\pm ip_y$) one with the ${\bf d}$-vector parallel to the effective magnetic field due to the spin-orbit coupling. Finally, we perform the numerical calculation in InSb (100) quantum wells. Specifically, we reveal that the Coulomb interaction is much more important than the electron-phonon interaction at low temperature. Moreover, it shows that with proper electron density, the minimum of the renormalized singlet and the maximum of the induced triplet order parameters are comparable, and hence can be experimentally distinguished.Comment: 15 pages, 8 figures, PRB in pres

Anomalous D'yakonov-Perel' spin relaxation in InAs (110) quantum wells under strong magnetic field: role of Hartree-Fock self-energy

We investigate the influence of the Hartree-Fock self-energy, acting as an effective magnetic field, on the anomalous D'yakonov-Perel' spin relaxation in InAs (110) quantum wells when the magnetic field in the Voigt configuration is much stronger than the spin-orbit-coupled field. The transverse and longitudinal spin relaxations are discussed both analytically and numerically. For the transverse configuration, it is found that the spin relaxation is very sensitive to the Hartree-Fock effective magnetic field, which is very different from the conventional D'yakonov-Perel' spin relaxation. Even an extremely small spin polarization ($P=0.1\%$) can significantly influence the behavior of the spin relaxation. It is further revealed that this comes from the {\em unique} form of the effective inhomogeneous broadening, originated from the mutually perpendicular spin-orbit-coupled field and strong magnetic field. It is shown that this effective inhomogeneous broadening is very small and hence very sensitive to the Hartree-Fock field. Moreover, we further find that in the spin polarization dependence, the transverse spin relaxation time decreases with the increase of the spin polarization in the intermediate spin polarization regime, which is also very different from the conventional situation, where the spin relaxation is always suppressed by the Hartree-Fock field. It is revealed that this {\em opposite} trends come from the additional spin relaxation channel induced by the HF field. For the longitudinal configuration, we find that the spin relaxation can be either suppressed or enhanced by the Hartree-Fock field if the spin polarization is parallel or antiparallel to the magnetic field.Comment: 10 pages, 2 figure

Spin diffusion in ultracold spin-orbit coupled 40^{40}K gas

We investigate the steady-state spin diffusion for ultracold spin-orbit coupled $^{40}$K gas by the kinetic spin Bloch equation approach both analytically and numerically. Four configurations, i.e., the spin diffusions along two specific directions with the spin polarization perpendicular (transverse configuration) and parallel (longitudinal configuration) to the effective Zeeman field are studied. It is found that the behaviors of the steady-state spin diffusion for the four configurations are very different, which are determined by three characteristic lengths: the mean free path $l_{\tau}$, the Zeeman oscillation length $l_{\Omega}$ and the spin-orbit coupling oscillation length $l_{\alpha}$. It is analytically revealed and numerically confirmed that by tuning the scattering strength, the system can be divided into {\it five} regimes: I, weak scattering regime ($l_{\tau}\gtrsim l_{\Omega}, l_{\alpha}$); II, Zeeman field-dominated moderate scattering regime ($l_{\Omega}\ll l_{\tau}\ll l_{\alpha}$); III, spin-orbit coupling-dominated moderate scattering regime ($l_{\alpha}\ll l_{\tau}\ll l_{\Omega}$); IV, relatively strong scattering regime ($l_{\tau}^c\ll l_{\tau}\ll l_{\Omega}, l_{\alpha}$); V, strong scattering regime ($l_{\tau}\ll l_{\Omega}, l_{\alpha},l_{\tau}^c$), with $l_{\tau}^c$ representing the crossover length between the relatively strong and strong scattering regimes. In different regimes, the behaviors of the spacial evolution of the steady-state spin polarization are very rich, showing different dependencies on the scattering strength, Zeeman field and spin-orbit coupling strength. The rich behaviors of the spin diffusions in different regimes are hard to be understood in the framework of the simple drift-diffusion model or the direct inhomogeneous broadening picture in the literature. ...Comment: 19 pages, 5 figure

Spin relaxation in ultracold spin-orbit coupled 40^{40}K gas

We report the anomalous Dyakonov-Perel' spin relaxation in ultracold spin-orbit coupled $^{40}$K gas when the coupling between $|9/2,9/2\ >$ and $|9/2,7/2\ >$ states (atcing as the effective Zeeman magnetic field) is much stronger than the spin-orbit coupled field. Both the transverse and longitudinal spin relaxations are investigated with small and large spin polarizations. It is found that with small spin polarization, the transverse (longitudinal) spin relaxation is divided into four (two) regimes: the normal weak scattering regime, the anomalous Dyakonov-Perel'-like regime, the anomalous Elliott-Yafet-like regime and the normal strong scattering regime (the anomalous Elliott-Yafet-like regime and the normal strong scattering regime), with only the normal weak scattering regime being in the weak scattering limit. This is very different from the conventional situation under the weak magnetic field, which is divided into the weak and strong scattering regimes according to the weak/strong scattering limit. With large spin polarization, we find that the Hartree-Fock self-energy, which acts as an effective magnetic field, can markedly suppress the transverse spin relaxation in both weak and strong scattering limits. Moreover, by noting that as both the momentum relaxation time and the Hartree-Fock effective magnetic field vary with the scattering length in cold atoms, the anomalous Dyakonov-Perel'-like regime is suppressed and the transverse spin relaxation is hence divided into three regimes in the scattering length dependence: the normal weak scattering regime, the anomalous Elliott-Yafet-like regime and the strong scattering regime. On the other hand, the longitudinal spin relaxation is again divided into the anomalous EY-like and normal strong scattering regimes. ...Comment: 12 pages, 3 figure

Valley depolarization due to inter- and intra-valley electron-hole exchange interactions in monolayer MoS2_{2}

We investigate the valley depolarization due to the electron-hole exchange interaction in monolayer MoS$_{2}$. Both the long- and short-range parts of the intra- and inter-valley electron-hole exchange interactions are calculated. We find that both the long- and short-range exchange interactions can cause the inter- and intra-valley bright exciton transitions. With the intra-valley bright exciton transition channel nearly forbidden due to the large splitting of the valence bands, the inter-valley channel due to the exchange interaction can cause the valley depolarization efficiently by the Maialle-Silva-Sham mechanism [Phys. Rev. B {\bf 47}, 15776 (1993)]. With only the long-range exchange interaction, the calculations show good agreement with the recent valley polarization experiments, including the time-resolved valley polarization measurement, the pump-probe experiment and the steady-state PL polarization measurement. We further show that for the A-exciton with large (small) center-of-mass momentum, the long-range exchange interaction can cause the {\em fast} ({\em slow}) inter-valley exciton transition.Comment: 8 pages, 2 figur

Novel valley depolarization dynamics and valley Hall effect of exciton in mono- and bilayer MoS2_2

We investigate the valley depolarization dynamics and valley Hall effect of exciton due to the electron-hole exchange interaction in mono- and bilayer MoS$_2$ by solving the kinetic spin Bloch equations. The effect of the exciton energy spectra by the electron-hole exchange interaction is explicitly considered. For the valley depolarization dynamics, in the monolayer MoS$_2$, it is found that in the strong scattering regime, the conventional motional narrowing picture is no longer valid, and a novel valley depolarization channel is opened. For the valley Hall effect of exciton, in both the mono- and bilayer MoS$_2$, with the exciton equally pumped in the K and K' valleys, the system can evolve into the equilibrium state where the valley polarization is parallel to the effective magnetic field due to the exchange interaction. With the drift of this equilibrium state by applied uniaxial strain, the exchange interaction can induce the {\it momentum-dependent} valley/photoluminesence polarization, which leads to the valley/photoluminesence Hall current. Specifically, the disorder strength dependence of the valley Hall conductivity is revealed. In the strong scattering regime, the valley Hall conductivity decreases with the increase of the disorder strength; whereas in the weak scattering regime, it saturates to a constant, which can be much larger than the one in Fermi system due to the absence of the Pauli blocking.Comment: 14 pages, 7 figure