majorana-268 lecture - Zero-Energy Majorana Modes in CondensedMatter Systems For P268r November 2013 Localized zero-energy Majorana modes Extra zero

majorana-268 lecture - Zero-Energy Majorana Modes in...

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Zero-Energy Majorana Modes in Condensed- Matter Systems For P268r. November 2013
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Localized zero-energy Majorana modes Extra “zero energy” degrees of freedom that have been hypothesized to occur at isolated point defects in some special correlated-electron systems. Majorana modes should have some very peculiar properties , have generated much interest. So far, only modest evidence of their realization in experiments on actual physical systems.
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Localized Majorana Modes Defining Properties Localized Majorana modes are associated with point defects in a system that otherwise would have an energy gap for electronic excitations. In a system with N point defects (Majorana sites) will have a “degenerate” ground state. Set of ground states form a Hilbert space with dimension 2 N/2 . Ground-state degeneracy is “robust” to local perturbations.
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Comparison: Localized spins in an insulator Magnetic impurities in an insulator have localized low-energy degrees of freedom. Simplest case has S=1/2; low-energy states of an isolated impurity form a two dimensional Hilbert space. System with N impurities has low energy Hilbert space with dimension 2 N . Basis states by specifying S z = 1/2 or -1/2 at each site. If no applied magnetic field , and no interactions between spins, states are degenerate in energy. Take into account exchange interactions, degeneracy is split. Exchange interactions fall off exponentially with distance for spins in an insulator, so splitting can be negligible if spins are far apart. But degeneracy is not robust, even if spins are far apart. Spins have magnetic momen t, observable quantity, couples to any local magnetic field. Magnetic moments give rise to dipole interaction between spins, falls off as 1/r 3 , not exponentially.
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Majorana Modes System with N point defects (Majorana sites): Low energy Hilbert space has dimension 2 N/2 . Effectively: one S=1/2 degree of freedom for every two Majorana sites. If sites are far apart , no local observable can distinguish between states in the Hilbert space; energies are precisely degenerate. For finite separation r, energy splittings fall off exponentially: E split e - r / ξ . ( ξ microscopic coherence length) Splitting can be negligible if r is sufficiently large, and no local perturbation can split this degeneracy. In theoretical discussions, often assume that separations are sufficiently large that E split can be set equal to zero.
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Manipulation of Majorana states Suppose there is a way to physically move defects around “adiabatically” as a function of time. (Adiabatically means that motion is slow on the frequency scale defined by the lowest finite-energy excitations in the system, but fast on the scale of the exponentially small energy splittings of states in the “zero- energy” Hilbert space. Electron s ystem will stay in low-energy Hilbert space.
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