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arXiv:1007.4091 · Physical Review B

Manipulating unpaired Majorana fermions in a quantum spin chain

Abhinav Saket, Syed R. Hassan, Ravi Shankar

Published Nov 8, 2010• 22 citations
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Abstract

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.

DOI
10.1103/physrevb.82.174409

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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 2N/4{2}^{N/4}2N/4-fold degeneracy, where NNN 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) Z2{Z}_{2}Z2​ flux configurations. The unpaired Majorana modes can therefore be created and manipulated if the Z2{Z}_{2}Z2​ 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 a spin-1/21/21/2 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 study the gapped phase of Kitaev's honeycomb model (a Z2{Z}_{2}Z2​ spin liquid) on a lattice with topological defects. We find that some dislocations and string defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z2{Z}_{2}Z2​ gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z2{Z}_{2}Z2​ vortex around one of the dislocations. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes. To facilitate these theoretical developments, we recast the degenerate perturbation theory for spins in the language of Majorana fermions.
We study the gapped phase of Kitaev's honeycomb model (a Z2{Z}_{2}Z2​ spin liquid) in the presence of lattice defects. We find that some dislocations and bond defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z2{Z}_{2}Z2​ gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z2{Z}_{2}Z2​ vortex around a dislocation. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes.
We unveil an interesting example of topological flat bands of Majorana fermions in quantum spin liquids. We study the Kitaev model on a periodically depleted honeycomb lattice, under a magnetic field within the perturbation theory. The model can be straightforwardly extended while maintaining the exact solvability, and its ground state is a quantum spin liquid as on the honeycomb lattice. As fractionalized excitations, there are unpaired localized Majorana fermions in addition to the itinerant Majorana fermions and Z2{\mathbb{Z}}_{2}Z2​ fluxes. We show that, in the absence of the magnetic field, the Majorana fermions exhibit completely flat bands at zero energy, and by applying the magnetic field, these bands turn into topological flat bands with nonzero Chern number. By varying the anisotropy of the interactions and the magnitude of the magnetic field, we clarify that the system hosts a variety of topological phases that do not appear in the original honeycomb model. We emphasize that the topological flat bands underlying this rich topology come from the hybridization of the Majorana flat bands and unpaired Majorana fermions, which is unique to the flat bands of fractionalized excitations in quantum spin liquids. Our findings would stimulate the exploration of a new class of Kitaev materials exhibiting rich topology arising from topological Majorana flat bands.
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.
The proposal is to probe Majorana modes in the system when its parameters are tuned to bring it into a disentangled ground state, which occur on the parametric curves known as disorder lines (DL). In such state certain correlation functions do not depend on separation. The exact results are presented for the XY spin chain in transverse field, also called the Kitaev chain in the Majorana representation. The single Majorana modes are shown to be localized near the ends of the chain, as in the states off the DL, while the disentangled nnn-particle Majorana modes (n≥2n \geq 2n≥2) penetrate into the bulk without attenuation. The predicted bulk-edge effects can be detected in the specially engineered optical chains and lattices.