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Register a new userExperimental evidence for Zeeman spin-orbit coupling in layered antiferromagnetic conductors
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Most of solid-state spin physics arising from spin-orbit coupling, from fundamental phenomena to industrial applications, relies on symmetry-protected degeneracies. So does the Zeeman spin-orbit coupling, expected to manifest itself in a wide range of antiferromagnetic conductors. Yet, experimental proof of this phenomenon has been lacking. Here, we demonstrate that the Neel state of the layered organic superconductor $kappa$-(BETS)$_2$FeBr$_4$ shows no spin modulation of the Shubnikov-de Haas oscillations, contrary to its paramagnetic state. This is unambiguous evidence for the spin degeneracy of Landau levels, a direct manifestation of the Zeeman spin-orbit coupling. Likewise, we show that spin modulation is absent in electron-doped Nd$_{1.85}$Ce$_{0.15}$CuO$_4$, which evidences the presence of Neel order in this cuprate superconductor even at optimal doping. Obtained on two very different materials, our results demonstrate the generic character of the Zeeman spin-orbit coupling.
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Motivated by the recently synthesized insulating nickelate Ni$_2$Mo$_3$O$_8$, which has been reported to have an unusual non-collinear magnetic order of Ni$^{2+}$ $S=1$ moments with a nontrivial angle between adjacent spins, we construct an effective spin-1 model on the honeycomb lattice, with the exchange parameters determined with the help of first principles electronic structure calculations. The resulting bilinear-biquadratic model, supplemented with the realistic crystal-field induced anisotropy, favors the collinear Neel state. We find that the crucial key to explaining the observed noncollinear spin structure is the inclusion of the Dzyaloshinskii--Moriya (DM) interaction between the neighboring spins. By performing the variational mean-field and linear spin-wave theory (LSWT) calculations, we determine that a realistic value of the DM interaction $Dapprox 2.78$ meV is sufficient to quantitatively explain the observed angle between the neighboring spins. We furthermore compute the spectrum of magnetic excitations within the LSWT and random-phase approximation (RPA) which should be compared to future inelastic neutron measurements.
A strongly spin-orbital coupled systems could be in a magnetic ordered phase at zero field. However, a Zeeman field could drive it into different quantum or topological phases. In this work, starting from general symmetry principle, we construct various effective actions to study all these quantum phases and phase transitions which take different forms depending on the condensation momenta are commensurate or in-commensurate. We not only recover all these quantum phases and their excitations achieved by the microscopic calculations, but also discover several novel classes of quantum phase transitions with dynamic exponents $ z=1, z=2 $ and anisotropic ones $ (z_x=3/2, z_y=3) $ respectively. We determine the relations between the quantum spin and the order parameters of the effective actions which display rich spin-orbital structures. We find a new type of dangerously irrelevant operator we name type-II, in distinction from the known one we name type-I. We explore a new phenomena called order parameter fractionization where one complex order parameter split into two which is different than quantum spin fractionization into a spinon and a $ Z_2 $ flux. Finite temperature transitions are presented. The dynamic spin-spin correlation functions are evaluated. Thermal Hall conductivities are discussed. The cases with the $ U(1)_{soc} $ symmetry explicitly broken are briefly outlined. In view of recent experimental advances in generating 2d SOC for cold atoms in optical lattices, these new many-body phenomena can be explored in the near future cold atom experiments. Implications to various SOC materials such as MnSi, Fe$_{0.5}$Co$_{0.5}$Si, especially 4d Kitaev materials $alpha$-RuCl$_3$ in a Zeeman field are outlined.
We study the ground state properties of spin-half bosons subjected to the Rashba spin-orbit coupling in two dimensions. Due to the enhancement of the low energy density of states, it is expected that the effect of interaction becomes more important. After reviewing several possible ideal condensed states, we carry out an exact diagonalization calculation for a cluster of the bosons in the presence of strong spin-orbit coupling on a two-dimensional disk and reveal strong correlations in its ground state. We derive a low-energy effective Hamiltonian to understand how states with strong correlations become energetically more favorable than the ideal condensed states.
The origin of the g-factor of the two-dimensional (2D) electrons and holes moving in the periodic crystal lattice potential with the perpendicular magnetic and electric fields is discussed. The Pauli equation describing the Landau quantization accompanied by the Rashba spin-orbit coupling (RSOC) and Zeeman splitting (ZS) for 2D heavy holes with nonparabolic dispersion law is solved exactly. The solutions have the form of the pairs of the Landau quantization levels due to the spinor-type wave functions. The energy levels depend on amplitudes of the magnetic and electric fields, on the g-factor {g-h}, and on the parameter of nonparabolicity C. The dependences of two energy levels in any pair on the Zeeman parameter {Z_h}={g_h}{m_h}/4{m_0}, where {m_h} is the hole effective mass, are nonmonotonous and without intersections. The smallest distance between them at C=0 takes place at the value {Z_h}=n/2, where n is the order of the chirality terms determined by the RSOC and is the same for any quantum number of the Landau quantization.
When an electron or hole is in a conduction band of a crystal, it can be very different from 2, depending upon the crystalline anisotropy and the direction of the applied magnetic induction ${bf B}$. In fact, it can even be 0! To demonstrate this quantitatively, the Dirac equation is extended for a relativistic electron or hole in an orthorhombically-anisotropic conduction band with effective masses $m_j$ for $j=1,2,3$ with geometric mean $m_g=(m_1m_2m_3)^{1/3}$. The appropriate Foldy-Wouthuysen transformations are extended to evaluate the non-relativistic Hamiltonian to $O({rm m}c^2)^{-4}$, where ${rm m}c^2$ is the particles Einstein rest energy. For ${bf B}||hat{bf e}_{mu}$, the Zeeman $g_{mu}$ factor is $2{rm m}sqrt{m_{mu}}/m_g^{3/2} + O({rm m}c^2)^{-2}$. While propagating in a two-dimensional (2D) conduction band with $m_3gg m_1,m_2$, $g_{||}<<2$, consistent with recent measurements of the temperature $T$ dependence of the parallel upper critical induction $B_{c2,||}(T)$ in superconducting monolayer NbSe$_2$ and in twisted bilayer graphene. While a particle is in its conduction band of an atomically thin one-dimensional metallic chain along $hat{bf e}_{mu}$, $g<<2$ for all ${bf B}={bf abla}times{bf A}$ directions and vanishingly small for ${bf B}||hat{bf e}_{mu}$. The quantum spin Hall Hamiltonian for 2D metals with $m_1=m_2=m_{||}$ is $K[{bf E}times({bf p}-q{bf A})]_{perp}sigma_{perp}+O({rm m}c^2)^{-4}$, where ${bf E}$ and ${bf p}-q{bf A}$ are the planar electric field and gauge-invariant momentum, $q=mp|e|$ is the particles charge, $sigma_{perp}$ is the Pauli matrix normal to the layer, $K=pmmu_B/(2m_{||}c^2)$, and $mu_B$ is the Bohr magneton.
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