Several scenarios for the realization of edge Majorana modes in quantum chain systems, spin chains, chains of Josephson junctions, and chains of coupled cavities in quantum optics, are considered. For all these systems excitations can be presented as superpositions of a spinless fermion and a hole, characteristic of a Majorana fermion. We discuss the features of our exact solution with respect to possible experiments, in which edge Majorana fermions can be directly observed when studying magnetic, superconducting, and optical characteristics of such systems.Several scenarios for the realization of edge Majorana modes in quantum chain systems, spin chains, chains of Josephson junctions, and chains of coupled cavities in quantum optics, are considered. For all these systems excitations can be presented as superpositions of a spinless fermion and a hole, characteristic of a Majorana fermion. We discuss the features of our exact solution with respect to possible experiments, in which edge Majorana fermions can be directly observed when studying magnetic, superconducting, and optical characteristics of such systems.

arXiv:1307.6329Physical Review LettersMay 1, 201325 citations

RELAXATION IN QUANTUM SPIN CHAINS: FREE FERMIONIC MODELS

Dragi Karevski

WORLD SCIENTIFIC eBooksMar 1, 20042 citations

Manipulating unpaired Majorana fermions in a quantum spin chain

Abhinav Saket, Syed R. Hassan, Ravi Shankar

We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a ${2}^{N/4}$ -fold degeneracy, where $N$ is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink) ${Z}_{2}$ flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the ${Z}_{2}$ flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a ${2}^{N/4}$-fold degeneracy, where $N$ is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink) ${Z}_{2}$ flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the ${Z}_{2}$ flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.

arXiv:1007.4091Physical Review BNov 8, 201022 citations

Exact solution of the SUq (n)-invariant quantum spin chains

Hector J. de Vega, A. González-Ruiz

arXiv:hep-th/9309022Nuclear Physics BApr 1, 199490 citations

Majorana fermions in the nonuniform Ising-Kitaev chain : exact solution

B. N. Narozhny

A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are “delocalized” between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the “nearest-neighbor” form of the effective low-energy Hamiltonian.A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are “delocalized” between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the “nearest-neighbor” form of the effective low-energy Hamiltonian.

Jan 1, 20178 citations

We analyze an exactly solvable spin-1/2 chain which is a generalized version of Kitaev's honeycomb model. We show that every state of the system has a

{2}^{N/4}

-fold degeneracy, where

N

is the number of sites. We present analytic solutions for the zero energy modes of the Majorana fermions. Localized, unpaired Majorana modes occur even in the bulk of the chain and they are bound to kink (antikink)

{Z}_{2}

flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the

{Z}_{2}

flux configurations can be controlled. We delineate the regions in parameter space for homogenous chains where the zero modes occur. We further show that there is a large parameter space for inhomogenous chains where the unpaired modes occur and that their wave functions can be tuned if the couplings of the model can be tuned.