Kramers Degeneracy Research Articles

The effects of surface polarity and dangling bonds on the electronic properties of monolayer and bilayer MoS2 on α-quartz

We investigate the electronic properties of monolayer and bilayer MoS2 on α-quartz substrate through first-principles density functional calculations. Due to the coupling of the MoS2 with the substrate, the valence band edge state at the Brillouin zone center tends to shift upward, reducing the indirect band gap of the MoS2, whereas the direct gap at the K valleys is less sensitive to substrate conditions. By taking into account the van der Waals interactions between the MoS2 and the substrate, we find that monolayer MoS2 exhibits a transition from direct-gap to indirect-gap semiconductor in the presence of surface O-dangling bonds. Moreover, a charge transfer occurs from MoS2 to SiO2, inducing p-type doping and lifting the Kramers degeneracy by breaking the time reversal symmetry. In bilayer MoS2, O-dangling bonds break the inversion symmetry by inducing dipole fields across the interface and thereby lower the energy band associated with the one layer relative to the other. Although the time reversal symmetry is also broken, its effect on the spin splitting is extremely small in the K valleys so that a strong coupling between the spin and valley degrees of freedom takes place, similar to that found in free-standing monolayer MoS2.

Single particle calculations for a Woods–Saxon potential with triaxial deformations, and large Cartesian oscillator basis (TRIAXIAL 2014, Third version of the code Triaxial)

Theory and FORTRAN program of the first version of this code (TRIAXIAL) have already been described in detail in Computer Physics Comm. 156 (2004) 241–282. A second version of this code (TRIAXIAL 2007) has been given in CPC 176 (2007) 634–635. The present FORTRAN program is the third version (TRIAXIAL 2014) of the same code. Now, It is written in free format. As the former versions, this FORTRAN program solves the same Schrodinger equation of the independent particle model of the atomic nucleus with the same method. However, the present version is much more convenient. In effect, it is characterized by the fact that the eigenvalues and the eigenfunctions can be given by specific subroutines. The latters did not exist in the old versions (2004 and 2007). In addition, it is to be noted that in the previous versions, the eigenfunctions were only given by their coefficients of their expansion onto the harmonic oscillator basis. This method is needed in some cases. But in other cases, it is preferable to treat the eigenfunctions directly in configuration space. For this reason, we have implemented an additional subroutine for this task. Some other practical subroutines have also been implemented. Moreover, eigenvalues and eigenfunctions are recorded onto several files. All these new features of the code and some important aspects of its structure are explained in the document ‘Triaxial2014 use.pdf’. New version program summaryProgram title: Triaxial2014Catalogue identifier: ADSK_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSK_v3_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 13672No. of bytes in distributed program, including test data, etc.: 217598Distribution format: tar.gzProgramming language: FORTRAN 77/90 (double precision).Computer: PC. Pentium 4, 2600MHz and beyond.Operating system: WINDOWS XP, WINDOWS 7, LINUX.RAM: 256 Mb (depending on nmax). Swap file: 4Gb (depending on nmax)Classification: 17.7.Does the new version supersede the previous version?: YesCatalogue identifier of previous version: ADSK_v2_0Journal reference of previous version: Comput. Phys. Comm. 176 (2007) 634Nature of problem: The Single particle energies and the single particle wave functions are calculated from one-body Hamiltonian including a central field of Woods–Saxon type, a spin–orbit interaction, and the Coulomb potential for the protons. We consider only ellipsoidal (triaxial) shapes. The deformation of the nuclear shape is fixed by the usual Bohr parameters (β,γ).Solution method: The representative matrix of the Hamiltonian is built by means of the Cartesian basis of the anisotropic harmonic oscillator, and then diagonalized by a set of subroutines of the EISPACK library. Two quadrature methods of Gauss are employed to calculate respectively the integrals of the matrix elements of the Hamiltonian, and the integral defining the Coulomb potential. Two quantum numbers are conserved: the parity and the signature. Due to the Kramers degeneracy, only positive signature is considered. Therefore, calculations are made for positive and negative parity separately (with positive signature only).Reasons for new version: Now, there are several ways to obtain the eigenvalues and the eigenfunctions. The eigenvalues can be obtained from the subroutine ‘eigvals’ or from the array ‘energies’ or also from the formatted files ‘valuu.dat’, ‘eigenvalo.dat’, ‘eigenva.dat’ or better from the unformatted file ‘eigenvaunf.dat’. The eigenfunctions can be obtained straightforwardly in configuration space from the subroutine ‘eigfunc’ or by their components on the oscillator basis from the subroutine ‘compnts’. The latter are also recorded on a formatted file ‘componento.dat’ or on an unformatted file ‘componentounf.dat’.Summary of revisions: This version is characterized by the fact that the eigenvalues and the eigenfunctions can be given by specific subroutines which did not exist in the old versions (2004 and 2007) of the program. Moreover, the eigenvalues and the eigenfunctions can also be deduced directly from files. It is to be noted that this version is now written in free format. All these reasons contribute to make the use of this code easier.Restrictions: There are two restrictions for the code: The number of the major shells of the basis should not exceed Nmax=26 (which is very sufficient in usual cases). For the largest values of Nmax (∼23–26), the diagonalization takes the major part of the running time, but the global run-time remains reasonable.Additional comments: Software used: (1) COMPAC VISUAL FORTRAN (with full optimizations in the settings project options on WINDOWS XP); (2) SILVERFROST PLATO VERSION 4.63 (with debug.net option on WINDOWS 7); (3) APPROXIMATRIX SIMPLY FORTRAN VERSION 2.13 BUILD (on WINDOWS XP and WINDOWS 7).Running time: (With full optimization in the project settings of the Compaq Visual Fortran on Windows XP) With NMAX=23, for the neutrons case, the running time is about 50 s on the intel core i5 processor.

Intrinsic spin-orbit effect in substrate-graphene Corbino devices

Electronic transport through the Corbino disk in a doped graphene–substrate system is studied taking into account intrinsic spin–orbit interaction (SOI) and perpendicular magnetic field. By combining the intrinsic SOI and substrate–graphene effects, we are able to break the Kramers degeneracy associated to degrees of freedom and to control the spin-polarized resonant transmission channels. The latter manifests itself in both conductance and Fano factor characteristics. As an astonishing result, we have pointed out a new universal limit for the quantum shot noise, related to this particular topology.

An ab initio study of electronic structure and optical properties of multiferroic perovskites PbVO3 and BiCoO3

Within density functional theory (DFT) with the generalized gradient approximation (GGA), GGA plus on-site Coulomb repulsion method, and improved version of the modified Becke–Johnson exchange potential suggested recently by Tran and Blaha [F. Tran, P. Blaha, Phys. Rev. Lett. 102 (2009) 226401] (TB–mBJ), we investigate the electronic structure and optical properties of noncentrosymmetric multiferroic perovskites PbVO3 and BiCoO3. These two compounds, although similar in lattice distortions and population of crystal-field levels, behave quite differently because of the different interplay between the fundamental Kramers degeneracy and the single-ion anisotropy in them. The main characteristic of the calculated TB–mBJ electronic structures is significant rearrangement of the V and Co 3d states near their valence bands tops when compared to the present and earlier GGA and GGA+U calculations of these compounds. This fact causes the different optical responses of the title compounds as well, which are analyzed and interpreted in terms of the calculated electronic structures. A comparison of the calculated properties with available experimental data indicates that the TB–mBJ approach provides a better description of the electronic and optical properties of PbVO3 and BiCoO3 than the standard GGA and GGA+U approaches.

Kramers degeneracy without eigenvectors

Wigner gave a well-known proof of Kramers degeneracy, for time reversal invariant systems containing an odd number of half-integer spin particles. But Wigner's proof relies on the assumption that the Hamiltonian has an eigenvector, and thus does not apply to many quantum systems of physical interest. This note illustrates an algebraic way to talk about Kramers degeneracy that does not appeal to eigenvectors, and provides a derivation of Kramers degeneracy in this more general context.

Time-Reversal Symmetry in Non-Hermitian Systems

For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys \Theta^2=-1, but no such a degeneracy exists when \Theta^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when \Theta^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.

Magnetic structure of the noncentrosymmetric perovskitesPbVO3andBiCoO3: Theoretical analysis

It is well known that if a crystal structure has no inversion symmetry, it may allow for Dzyaloshinskii-Moriya magnetic interactions, operating between different crystallographic unit cells, which in turn should lead to the formation of long-periodic spin-spiral structures. Such a behavior is anticipated for two simple perovskites PbVO3 and BiCoO3, crystallizing in the noncentrosymmetric tetragonal P4mm structure. Nevertheless, we argue that in reality PbVO3 and BiCoO3 should behave very differently. Due to the fundamental Kramers degeneracy for the odd-electron systems, PbVO3 has no single-ion anisotropy. Therefore, the ground state of PbVO3 will be indeed the spin spiral with the period of about one hundred unit cells. However, the even-electron BiCoO3 has a large single-ion anisotropy, which locks this system in the collinear easy-axis C-type antiferromagnetic ground state. Our theoretical analysis is based on the low-energy model, derived from the first-principles electronic structure calculations.

Phonon-assisted transport through suspended carbon nanotube quantum dots

Motivated by several recent experiments, we investigate phonon-assisted electronic transport through suspended carbon nanotube quantum dots including electron-electron, electron-phonon, and spin-orbit interactions. By effectively decoupling the electron and phonon thermodynamics, we present an explanation for the puzzling thermoinduced vibrational absorption sidebands, as well further predict that these absorption sidebands are obviously diminished in the partially filled Coulomb diamonds due to the destructive superposition of electron and hole states. The spin-orbit coupling is shown to split all phonon-assisted sidebands while the Kramers degeneracy remains. Interestingly, in the strong spin-orbit coupling regime, some split absorption sidebands are suppressed by the Coulomb blockade effect, while all split emission sidebands always survive.

Entanglement and inversion symmetry in topological insulators

Topological band insulators are usually characterized by symmetry-protected surface modes or quantized linear-response functions (like Hall conductance). Here we present a way to characterize them based on certain bulk properties of just the ground-state wave function, specifically, the properties of its entanglement spectrum. We prove that whenever protected surface states exist, a corresponding protected ``mode'' exists in the entanglement spectrum as well. Besides this, the entanglement spectrum sometimes succeeds better at indicating topological phases than surface states. We discuss specifically the example of insulators with inversion symmetry which is found to act as an antiunitary symmetry on the entanglement spectrum. A Kramers degeneracy can then arise even when time-reversal symmetry is absent. This degeneracy persists for interacting systems. The entanglement spectrum is therefore a promising tool to characterize topological band insulators and superconductors beyond the free-particle approximation.

Symmetry-protected -quantization and quaternionic Berry connection with Kramers degeneracy

As for a generic parameter-dependent Hamiltonian with time reversal (TR) invariance, a non-Abelian Berry connection with Kramers (KR) degeneracy is introduced by using a quaternionic Berry connection. This quaternionic structure naturally extends to the many-body system with KR degeneracy. Its topological structure is explicitly discussed in comparison with the one without KR degeneracy. Natural dimensions to have nontrivial topological structures are discussed by presenting explicit gauge fixing. Minimum models to have accidental degeneracies are given with/without KR degeneracy, which describe the monopoles of Dirac and Yang. We have shown that the Yang monopole is literally a quaternionic Dirac monopole.The generic Berry phases with/without KR degeneracy are introduced by the complex/quaternionic Berry connections. As for the symmetry-protected -quantization of these general Berry phases, a sufficient condition of the -quantization is given as the inversion/reflection equivalence.Topological charges of the SO(3) and SO(5) nonlinear σ-models are discussed in relation to the Chern numbers of the CP1 and HP1 models as well.

On the Physical Reasons for the Extension of Symmetry Groups in Molecular Spectroscopy

Several situations of general interest, in which the symmetry groups usually applied to spectroscopy problems need to be extended, are reviewed. It is emphasized that any symmetry group of geometrical operations to be used in Molecular Spectroscopy should be extended for completeness by considering the time reversal operator, as far as the Hamiltonian is invariant with respect to the inversion of the direction of motion. This can explain the degeneracy of pairs of vibrational and rotational states spanning the so-called separably degenerate irreducible representations, in symmetric tops of low symmetry, and Kramers degeneracy in odd electron molecules in the absence of magnetic fields. An extension with account of time reversal is also useful to determine relative phase conventions on vibration-rotation wavefunctions, which render all vibration-rotation matrix elements real. An extension of a molecular symmetry group may be required for molecules which can attain different geometries by large amplitude periodical motions, if such motions are hindered and are not completely free. Special cases involving the internal rotation are discussed in detail. It is observed that the symmetry classification of vibrational modes involving displacements normal to the internal rotation axis is not univocal, but can be done in several ways, which actually correspond to different conventions on the separation of vibration and internal rotation in the adopted basis functions. The symmetry species of the separate vibrational and torsional factors of these functions depend on the adopted convention.

Kramers degeneracy in a magnetic field and Zeeman spin-orbit coupling in antiferromagnetic conductors

In this article, I analyze the symmetries and degeneracies of electron eigenstates in a commensurate collinear antiferromagnet. In a magnetic field transverse to the staggered magnetization, a hidden anti-unitary symmetry protects double degeneracy of the Bloch eigenstates at a special set of momenta. In addition to this `Kramers degeneracy' subset, the manifold of momenta, labeling the doubly degenerate Bloch states in the Brillouin zone, may also contain an `accidental degeneracy' subset, that is not protected by symmetry and that may change its shape under perturbation. These degeneracies give rise to a substantial momentum dependence of the transverse g-factor in the Zeeman coupling, turning the latter into a spin-orbit interaction. I discuss a number of materials, where Zeeman spin-orbit coupling is likely to be present, and outline the simplest properties and experimental consequences of this interaction, that may be relevant to systems from chromium to borocarbides, cuprates, hexaborides, iron pnictides, as well as organic and heavy fermion conductors.

Spin-orbit coupling in singly chargedπ-conjugated polymers

Understanding spin-orbit coupling in singly charged $\ensuremath{\pi}$-conjugated polymer chains is of basic importance to the study of spin transport in organic semiconductors. We show that the spin-orbit coupling is quenched to first order because the diagonal matrix elements vanish and the energy levels in polymers are orbitally nondegenerate. Even the off-diagonal matrix elements are zero or negligibly small unless a twisted nonplanar polymer chain is considered. For such a twisted chain, we calculate these matrix elements using tight-binding wave functions based on ${p}_{z}$ orbitals. We show that the Kramers degeneracy prevents the spin-orbit-induced spin precession. Instead, we propose a phonon-assisted spin-flip process and calculate its rate.

Kramers Degeneracy Theorem in Nonrelativistic QED

Degeneracy of the eigenvalues of the Pauli-Fierz Hamiltonian with spin 1/2 is proven by the Kramers degeneracy theorem. The Pauli-Fierz Hamiltonian at fixed total momentum is also investigated.

Relativistic E×T Jahn–Teller effect in tetrahedral systems

It is shown that (2)E states in tetrahedral systems exhibit a linear ExT Jahn-Teller effect which is of purely relativistic origin (that is, it arises from the spin-orbit-coupling operator). The electrostatic interactions give rise to a Jahn-Teller effect which is quadratic in the T displacements. The 4 x 4 Hamiltonian matrix in a diabatic spin-electron basis is derived by an expansion of the electrostatic electronic Hamiltonian and the Breit-Pauli spin-orbit operator in powers of the Jahn-Teller active normal mode and taking account of symmetry selection rules for the matrix elements. The adiabatic potential-energy functions of the (2)E x T system are doubly degenerate (Kramers degeneracy). For small displacements from the tetrahedral reference geometry, the adiabatic potential-energy surfaces represent a double cone in four-dimensional space, which is a novel topography of Jahn-Teller potential-energy surfaces. The topological phases of the adiabatic electronic wave functions are discussed.

Kramers degeneracy in a magnetic field and zeeman spin-orbit coupling in antiferromagnetic conductors

In this article, I study magnetic response of electron wavefunctions in a commensurate collinear Neel antiferromagnet. I show that, at a special set of momenta, hidden anti-unitary symmetry protects Kramers degeneracy of Bloch eigenstates against a magnetic field, pointing transversely to staggered magnetization. Hence a substantial momentum dependence of the transverse g-factor in the Zeeman coupling, turning the latter into a Zeeman spin-orbit interaction, that may be present in materials from chromium to borocarbides, cuprates, pnictides, as well as organic and heavy fermion conductors.

Kramers Degeneracy in a Magnetic Field and Zeeman Spin-Orbit Coupling in Antiferromagnetic Conductors

In this Letter, I study the magnetic response of electron wave functions in a commensurate collinear antiferromagnet. I show that, at a special set of momenta, hidden antiunitary symmetry protects Kramers degeneracy of Bloch eigenstates against a magnetic field, pointing transversely to staggered magnetization. Hence, a substantial momentum dependence of the transverse g-factor in the Zeeman term, turning the latter into a spin-orbit coupling that may be present in materials from chromium to borocarbides, cuprates, pnictides, as well as organic and heavy fermion conductors.

A proof of the Kramers degeneracy of transmission eigenvalues from antisymmetry of the scattering matrix

In time-reversal-symmetric systems with half integral spins (or more concretely, systems with an antiunitary symmetry that squares to −1 and commutes with the Hamiltonian) the transmission eigenvalues of the scattering matrix come in pairs. We present a proof of this fact that is valid for both even and odd number of modes and relies solely on the antisymmetry of the scattering matrix imposed by time reversal symmetry.

Quasparticle Spin Susceptibility in Heavy-Fermion Superconductors: An NMR Study Compared with Specific Heat Results

Quasi-particle spin susceptibility ($\chi^{qp}$) for various heavy-fermion (HF) superconductors are discussed on the basis of the experimental results of electronic specific heat ($\gamma_{el}$), NMR Knight shift ($K$) and NMR relaxation rate ($1/T_1$) within the framework of the Fermi liquid model for a Kramers doublet crystal electric field (CEF) ground state. $\chi^{qp}_{\gamma}$ is calculated from the enhanced Sommerfeld coefficient $\gamma_{el}$ and $\chi^{qp}_{T_1}$ from the quasi-particle Korringa relation $T_1T(K^{qp}_{T_1})^2=const.$ via the relation of $\chi^{qp}_{T_1}=(N_A\mu_B/A_{hf})K^{qp}_{T_1}$ where $A_{hf}$ is the hyperfine coupling constant, $N_A$ the Abogadoro's number and $\mu_B$ the Bohr magneton. For the even-parity (spin-singlet) superconductors CeCu$_2$Si$_2$, CeCoIn$_5$ and UPd$_2$Al$_3$, the fractional decrease in the Knight shift, $\delta K^{obs}$, below the superconducting transition temperature ($T_c$) is due to the decrease of the spin susceptibility of heavy quasi-particle estimated consistently from $\chi^{qp}_{\gamma}$ and $\chi^{qp}_{T_1}$. This result allows us to conclude that the heavy quasi-particles form the spin-singlet Cooper pairs in CeCu$_2$Si$_2$, CeCoIn$_5$ and UPd$_2$Al$_3$. On the other hand, no reduction in the Knight shift is observed in UPt$_3$ and UNi$_2$Al$_3$, nevertheless the estimated values of $\chi^{qp}_{\gamma}$ and $\chi^{qp}_{T_1}$ are large enough to be probed experimentally. The odd-parity superconductivity is therefore concluded in these compounds. The NMR result provides a convincing way to classify the HF superconductors into either even- or odd- parity paring together with the identification for the gap structure, as long as the system has Kramers degeneracy.

Formation of magnetic moments in the cuprate superconductor Hg 0.8Cu 0.2Ba 2Ca 2Cu 3O 8+ δ below Tc seen by NQR

We report pure zero field nuclear magnetic resonance (NQR) measurements on the optimally doped three layer high- T c-compounds HgBaCaCuO and HgBaCaCuO(F) with T c=134 K. Above T c two Cu NQR line pairs are observed in the spectra corresponding to the two inequivalent Cu lattice sites. Below T c the Cu NQR spectra show additional lines leading to the extreme broadened Cu NQR spectra at 4.2 K well known for the HgBaCaCuO compounds. The spin–lattice relaxation curves follow a triple exponential function with coefficients depend onto the saturation time (number of saturation pulses), whereas the spin–spin relaxation curve is described by a single exponential function. From the spin–lattice relaxation we deduced a complete removal of the Kramers degeneracy of the Cu quadrupole levels indicating that the additional lines are due to a Zeemann splitting of the 63/65Cu lines due to the spontaneous formation of magnetic moments within the CuO layers. Below 140 K, the spectra are well fitted by a number of 6 63/65Cu line pairs. From the number of the Cu lines, the position of the lines relative to each other and the complete removal of the Kramers degeneracy we deduced an orientation of the magnetic moments parallel to the c-axis with magnitudes of the order of 1000 G. We also discuss the possible microscopic origin of the observed internal magnetic fields.