We analyze a spin- $1/2$ chain with two-spin interactions which are shown to be exactly solvable by Lieb, Schultz, and Mattis [Ann. Phys. 16, 407 (1961)]. We show that the model can be viewed as a generalized Kitaev model that is analytically solvable for all defect sectors. We present an alternate proof that the defect-free sector is the ground state, which is valid for a larger parameter range. We show that the defect sectors have degenerate ground states corresponding to unpaired Majorana fermion modes and that the degeneracy is topologically protected against disorder in the spin-spin couplings. The unpaired Majorana fermions can be manipulated by tuning the model parameters and can hence be used for topological quantum computation.We analyze a spin-$1/2$ chain with two-spin interactions which are shown to be exactly solvable by Lieb, Schultz, and Mattis [Ann. Phys. 16, 407 (1961)]. We show that the model can be viewed as a generalized Kitaev model that is analytically solvable for all defect sectors. We present an alternate proof that the defect-free sector is the ground state, which is valid for a larger parameter range. We show that the defect sectors have degenerate ground states corresponding to unpaired Majorana fermion modes and that the degeneracy is topologically protected against disorder in the spin-spin couplings. The unpaired Majorana fermions can be manipulated by tuning the model parameters and can hence be used for topological quantum computation.

arXiv:1301.5786Physical Review BMay 9, 20135 citations

The quantum Ising chain with a generalized defect

Uwe Grimm

arXiv:hep-th/0310089Nuclear Physics BAug 1, 199029 citations

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

Boris 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.

arXiv:1609.00925Scientific ReportsApr 27, 20178 citations

The Ising quantum chain with an extended defect

Haye Hinrichsen B

Nuclear Physics BJun 1, 199017 citations

We analyze a spin-

1/2

chain with two-spin interactions which are shown to be exactly solvable by Lieb, Schultz, and Mattis [Ann. Phys. 16, 407 (1961)]. We show that the model can be viewed as a generalized Kitaev model that is analytically solvable for all defect sectors. We present an alternate proof that the defect-free sector is the ground state, which is valid for a larger parameter range. We show that the defect sectors have degenerate ground states corresponding to unpaired Majorana fermion modes and that the degeneracy is topologically protected against disorder in the spin-spin couplings. The unpaired Majorana fermions can be manipulated by tuning the model parameters and can hence be used for topological quantum computation.