Realizing Majorana fermion modes in the Kitaev model

We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work. To achieve this we first apply a uniform [111] magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p+ip topological superconductor of spinons. We then study possible vortex binding in such system to a topologically trivial spot in the ground state. We consider two cases in the system: one is a vacancy and the other is a fully polarized spin. We show that in both cases, the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform [111] magnetic field. The distribution and asymptotic behavior of these Majorana zero modes are studied. The Majorana zero modes in both cases decay exponentially in space, and are robust against local perturbations and other Majorana zero modes far away, which makes them promising candidates for braiding in topological quantum computing.     

Topological phase diagrams and Majorana zero modes of the Kitaev ladder and tube

In this paper,we study two quasi-one-dimensional(1 D) Kitaev models with ladder-like and tube-like spatial structures,respectively.Our results provide the phase diagrams and explicit expressions of the Majorana zero modes.The topological phase diagrams are obtained by decomposing the topological invariants and the topological conditions for topologically nontrivial phases are given precisely.For systems which belongs to topological class BDI,we obtain the regions in the phase diagrams where the topological numbers show even-odd effect.For the Kitaev tube model a phase factor induced by the magnetic flux in the axial direction of the tube is introduced to alter the classification of the tube Hamiltonian from class BDI to D.The Kitaev tube of class D is characterized by the Z_2 index when the number of chains is odd while 0,1,2 when the number of chains is even.The phase diagrams show periodic behaviors with respect to the magnetic flux.The bulk-boundary correspondence is demonstrated by the observations that the topological conditions for the bulk topological invariant to take nontrivial values are precisely those for the existence of the Majorana zero modes.     

Instability of Interfaces in the Antiferromagnetic XXZ Chain at Zero Temperature

 For the antiferromagnetic, highly anisotropic XZ and XXZ quantum spin chains, we impose periodic boundary conditions on chains with an odd number of sites to force an interface (or kink) into the chain. We prove that the energy of the interface depends on the momentum of the state. This shows that at zero temperature the interface in such chains is not stable. This is in contrast to the ferromagnetic XXZ chain for which the existence of localized interface ground states has been proven for any amount of anisotropy in the Ising-like regime. Received: 15 August 2002 / Accepted: 8 January 2003 Published online: 14 April 2003 RID="⋆" ID="⋆" ? Copyright rests with the authors. Reproduction of the entire article for non-commercial purposes is permitted. Communicated by M. Aizenman     

Topological Invariants in Terms of Green's Function for the Interacting Kitaev Chain

A one-dimensional closed interacting Kitaev chain and the dimerized version are studied. The topological invariants in terms of Green's function are calculated by the density matrix renormalization group method and the exact diagonalization method. For the interacting Kitaev chain, we point out that the calculation of the topological invariant in the charge density wave phase must consider the dimerized configuration of the ground states. The variation of the topological invariant is attributed to the poles of eigenvalues of the zero-frequency Green functions. For the interacting dimerized Kitaev chain, we show that the topological invariant defined by Green's functions can distinguish more topological nonequivalent phases than the fermion parity.     

Effect of Magnetic-Impurity Scattering in Disordered Two-Dimensional Square Lattices Around Half Filling

Weak-localization effect in the presence of magnetic impurities is studied in disordered two-dimensional tight-binding square lattices around half filling. Both the magnetic and nonmagnetic impurities are assumed to be randomly distributed on small fractions of the sites, while the nonmagnetic impurities have a strong potential yielding a unitary-limit scattering. We derive in details the expressions of diffusive π modes in the retarded-retarded (or advancedadvanced) channel, which result from the existence of particle-hole symmetry. The quantum interference correction to the density of states is calculated. While the magnetic-impurity scattering suppresses the quantum correction from π-mode cooperon, it does not affect the contribution of π-mode diffuson.     

Effect of Magnetic-Impurity Scattering in Disordered Two-Dimensional Square Lattices Around Half Filling

Weak-localization effect in the presence of magnetic impurities is studied in disordered two-dimensional tight-binding square lattices around half filling. Both the magnetic and nonmagnetic impurities are assumed to be randomly distributed on small fractions of the sites, while the nonmagnetic impurities have a strong potential yielding a unitary-limit scattering. We derive in details the expressions of diffusive π modes in the retarded-retarded (or advanced-advanced) channel, which result from the existence of particle-hole symmetry. The quantum interference correction to the density of states is calculated. While the magnetic-impurity scattering suppresses the quantum correction from π-mode cooperon, it does not affect the contribution of π-mode diffuson.     

Chain of Majorana states from superconducting Dirac fermions at a magnetic domain wall

We study theoretically a strongly type-II s-wave superconducting state of two-dimensional Dirac fermions in proximity to a ferromagnet having in-plane magnetization. It is shown that a magnetic domain wall can host a chain of equally spaced vortices in the superconducting order parameter, each of which binds a Majorana-fermion state. The overlap integral of neighboring Majorana states is sensitive to the position of the chemical potential of the Dirac fermions. Thermal transport and scanning tunneling microscopy experiments to probe the Majorana fermions are discussed.     

Drude Weight, Optical Conductivity of Two-Dimensional Hubbard Model at Half Filling

We study the Drude weight D and optical conductivity of the two-dimensional （2D） Hubbard model at half filling with staggered magnetic flux （SMF）. When SMF being introduced, the hopping integrals are modulated by the magnetic flux. The optical sum rule, which is related to the mean kinetic energy of band electrons, is evaluated for this 2D Hubbard Hamiltonian. Our present result gives the dependence of the kinetic energy, D and the optical conductivity on SMF and U. At half filling D vanishes exponentially with system size. We also find in the frequency dependence of the optical conductivity, there is δ-function peak at ω ≈ 2｜m｜U and the incoherent excitations begin to present themselves extended to a higher energy region.     

Drude Weight,Optical Conductivity of Two-Dimensional Hubbard Model at Half Filling

We study the Drude weight D and optical conductivity of the two-dimensional (2D) Hubbard model at half filling with staggered magnetic flux (SMF). When SMF being introduced, the hopping integrals are modulated by the magnetic flux. The optical sum rule, which is related to the mean kinetic energy of band electrons, is evaluated for this 2D Hubbard Hamiltonian. Our present result gives the dependence of the kinetic energy, D and the optical conductivity on SMF and U. At half filling D vanishes exponentially with system size. We also find in the frequency dependence of the optical conductivity, there is δ-function peak at ω≈2|m|U and the incoherent excitations begin to present themselves extended to a higher energy region.     

Destruction of Fermion Zero Modes on Cosmic Strings

I examine the existence of zero-energy fermionsolutions (zero modes) on cosmic strings in an SO (10)grand unified theory. The current-carrying capability ofa cosmic string formed at one phase transition can be modified at subsequent phasetransitions. I show that the zero modes may be destroyedand the conductivity of the string altered. I discussthe cosmological implications of this, and show that it allows vorton bounds to be relaxed.     

Charge Transport Properties of the Majorana Zero Mode Induced Noncollinear Spin Selective Andreev Reflection

We study the charge transport properties of the spin-selective Andreev reflection(SSAR)effect between a spin polarized scanning tunneling microscope(STM)tip and a Majorana zero mode(MZM).Considering both the MZM and the excited states,we calculate the conductance and the shot noise power of the noncollinear SSAR using scattering theory.We find that the excited states give rise to inside peaks.Moreover,we numerically calculate the shot noise power and the Fano factor of the SSAR effect.Our calculation shows that the shot noise power and the Fano factor are related to the angle between the spin polarization direction of the STM tip and that of the MZM,which provide additional characteristics to detect the MZM via SSAR.     

Fermionic Zero Modes in Self-dual Vortex Background on a Torus

We study fermionic zero modes in the self-dual vortex background on an extra two-dimensional Riemann surface in （5＋1） dimensions. Using the generalized Abelian-Higgs model, we obtain the inner topological structure of the self-dual vortex and establish the exact self-duality equation with topological term. Then we analyze the Dirac operator on an extra torus and the effective Lagrangian of four-dimensional fermions with the self-dual vortex background. Solving the Dirac equation, the fermionic zero modes on a torus with the self-dual vortex background in two simple cases are obtained.     

Effect of Majorana Phases in Neutrino Oscillation

Quantum gravity (Planck scale effects) lead to an effective SU(2) _L ×U(1) invariant dimension-5 Lagrangian involving neutrino and Higgs fields. On symmetry breaking, this operator gives rise to correction to the above masses and mixing. The gravitational interaction M _X =M _pl , we find that for degenerate neutrino mass spectrum, it is shown that the Majorana phase of the neutrino mixing matrix can effects in neutrino oscillation probability.     

Fermionic Zero Modes in Gauge and Gravity Backgrounds on a Sphere

In this letter we study fermionic zero modes in gauge and gravity backgrounds taking a two-dimensional compact manifold S2 as extra dimensions. The result is that there exist massless Dirac fermions which have normalizable zero modes under quite general assumptions about these backgrounds on the bulk. Several special cases of gauge background on the sphere axe discussed and some simple fermionic zero modes are obtained.     

Magnetic Properties of the Infinitely Repulsive Hubbard Model Near Half Filling

Many theoretical works discuss the magnetic properties of the strongly repulsive Hubbard model near half filling. Several authors suggest that the system which contains a few holes does not behave as a ferromagnet, unlike in the one-hole case described by Nagaoka's famous result. We discuss the finite temperature properties of finite lattices. Using an efficient Monte Carlo method magnetic and specific heat results are presented for a 10 by 10 lattice with two holes. Our results show that the magnetization increases with decreasing temperature for such a finite system.     

Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

We study the ground state of the Pauli Hamiltonian with a magnetic field in ${\mathbb{R}^{2d}}$ , d > 1. We consider the case where a scalar potential W is present and the magnetic field B is given by ${B=2i\partial\bar{\partial} W}$ . The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.     

Zero Energy Modes of Massless Fermions in S3R Spacetime

The Dirac-type equation on topology is worked out and the complete set of solutions in the particular physical case of the zero-energy modes of the massless field quanta is derived. Unlike the Minkowskian case, the 1/2fermionic vacua on the manifold is made of nontrivial static modes of defined chirality.     

Dynamics of the Entanglement Zero Modes in the Haldane Model under a Quantum Quench

We investigate evolution of entanglement spectra of the Haldane model for Chern insulators upon a sudden quench within the same topological phase. In particular, we focus on the location of the entanglement spectrum crossing, which signifies the bulk topology. It is shown that the coplanarity condition for the pseudomagnetic field of the model, which can be used to determine the crossing in the equilibrium case, needs to be relaxed.We analytically derive the non-equilibrium condition with the he...     

Creation and manipulation of anyons in the Kitaev model

We analyze the effect of local spin operators in the Kitaev model on the honeycomb lattice. We show, in perturbation around the isolated-dimer limit, that they create Abelian anyons together with fermionic excitations which are likely to play a role in experiments. We derive the explicit form of the operators creating and moving Abelian anyons without creating fermions and show that it involves multispin operations. Finally, the important experimental constraints stemming from our results are discussed.