Exact analytical solution of a time-reversal-invariant topological superconducting wire

Abstract

We consider a model proposed before for a time-reversal-invariant topological superconductor (TRITOPS) which contains a hopping term t, a chemical potential µ, an extended s-wave pairing ∆ and spin-orbit coupling λ. We show that for |∆| = |λ|, µ = t = 0, the model can be solved exactly defining new fermion operators involving nearest-neighbor sites. The many-body ground state is four-fold degenerate due to the existence of two zero-energy modes localized exactly at the first and the last site of the chain. These four states show entanglement in the sense that creating or annihilating a zero-energy mode at the first site is proportional to a similar operation at the last site. By continuity, this property should persist for general parameters. Using these results we correct some statements related with the so called "time-reversal anomaly". Addition of a small hopping term for a chain with an even number of sites breaks the degeneracy and the ground state becomes unique with an even number of particles. We also consider a small magnetic field applied to one end of the chain. We compare the many-body excitation energies and spin projection along the spin-orbit direction for both ends of the chains with numerical results obtaining good agreement.

Magnetism and anomalous apparently inverse Jahn-Teller effect in Sr2CrO4

“…The experimental vector in the plane is found at [0.5; 0.5], while the susceptibility yields an incommensurate [∼0.4; ∼0.4] value, which may suggest a "lock-in" of the AFM spin density wave towards the commensurate value. In conclusion, a purely localised theoretical approach, both numerical Lanczos calculations [10] and through an effective Kugel-Khomski Hamiltonian [26], has been shown to be successful in fully understanding the magnetism and the orbital physics of Sr 3 Cr 2 O 7 . Up to now the compound Sr 2 CrO 4 has been approached from the same localised point of view [7,8,15,16].…”

One-dimensional spin–orbit coupled Dirac system with extended s-wave superconductivity: Majorana modes and Josephson effects

“…We would like to emphasize here that our model has only one zero energy mode at each end of a long system, in contrast to conventional TRITOPS which have a Kramers pair of zero energy modes at each end [56][57][58][59][60][61][62][63][64]. While we have shown this numerically above, we can also show this analytically for the special case corresponding to μ = Δ 0 = 0.…”

“…We emphasize here that we are proposing to study a purely Dirac Hamiltonian with a spin-orbit coupled form, in contrast to the earlier models which generally begin with a Schrödinger Hamiltonian and add a spin-orbit term to that; it is known that the latter kind of models with combinations of p-wave and swave pairings [55] or extended s-wave pairing [56][57][58][59][60][61][62][63][64] can host Majorana end modes. These models studied earlier are known as time-reversal-invariant topological superconductors (TRITOPS), and they have four branches of electrons near the Fermi energy: spin-up and spin-down (or two channels) each of which has both right-moving and left-moving branches (see reference [63] for a review).…”

Influence of B‐site cation in lattice dynamics of bilayered Ruddlesden–Popper Sr3B2O7 (B = Zr, Mo, Sn, Hf) compounds

“…Oxygen atoms O3 under D 4h site symmetry are located at the equatorial position of the Sr2O layer as well as at the apical positions of double-layered HfO 6 octahedra. 32 The square lattice of the layers HfO 2 has been created by the Sr2 atoms under the D 4h site symmetry and O3 atoms under the D 4h site symmetry.…”