The classification of loop symmetries in Kitaev's honeycomb lattice model provides a natural framework to study the abelian topological degeneracy. We derive a perturbative low-energy effective Hamiltonian, that is valid to all orders of the expansion and for all possible toroidal configurations.Using this form we demonstrate at what order the system's topological degeneracy is lifted by finite size effects and note that in the thermodynamic limit it is robust to all orders. Further, we demonstrate that the loop symmetries themselves correspond to the creation, propagation and annihilation of fermions. Importantly, we note that these fermions, made from pairs of vortices, can be moved with no additional energy cost. Recently, Kitaev introduced a spin-1/2 quantum lattice model with abelian and non-abelian topological phases [1]. This model is relevant to on-going research into topologically fault-tolerant quantum information processing [2,3,4]. The system comprises of two-body interactions and is exactly solvable, which makes it attractive both theoretically [5,6,7,8,9,10,11,12,13,14,15,16] and experimentally [17,18,19,20].Here, by classifying the loop symmetries of the system according to their homology, we derive a convenient form of the effective Hamiltonian on the torus. The result is valid for all orders of the Brillouin-Wigner perturbative expansion around the fully dimerized point as well as for all toroidal configurations. This allows the system's topological degeneracy to be addressed and shows at what order in the expansion the degeneracy is lifted. In the thermodynamic limit the system's topological degeneracy remains to all orders. In a separate analysis, valid for the full parameter space, we examine the pairedvortex excitations created by applying certain open string operations to the ground state. These vortex-pairs are fermions and can be freely transported in a way that keeps additional unwanted excitations to a minimum.The Hamiltonian for the system can be written as H = − α∈{x,y,z} i,j
We develop a rigorous and highly accurate technique for the calculation of the Berry phase in systems with a quadratic Hamiltonian within the context of the Kitaev honeycomb lattice model. The method is based on the recently found solution of the model that uses the Jordan-Wigner-type fermionization in an exact effective spin-hardcore boson representation. We specifically simulate the braiding of two non-Abelian vortices (anyons) in a fourvortex system characterized by a twofold degenerate ground state. The result of the braiding is the non-Abelian Berry matrix, which is in excellent agreement with the predictions of the effective field theory. The most precise results of our simulation are characterized by an error of the order of 10 −5 or lower. We observe exponential decay of the error with the distance between vortices, studied in the range of one to nine plaquettes. We also study its correlation with the involved energy gaps and provide a preliminary analysis of the relevant adiabaticity conditions. The work allows one to investigate the Berry phase in other lattice models including the Yao-Kivelson model and particularly the square-octagon model. It also opens up the possibility of studying the Berry phase under nonadiabatic and other effects that may constitute important sources of errors in topological quantum computation.
We consider a square optical lattice in two dimensions and study the effects of both the strength and symmetry of spin-orbit coupling and Zeeman field on the ground-state, i.e., Mott-insulator (MI) and superfluid (SF), phases and phase diagram, i.e., MI-SF phase-transition boundary, of the two-component Bose-Hubbard model. In particular, based on a variational Gutzwiller ansatz, our numerical calculations show that the spin-orbit-coupled SF phase is a nonuniform (twisted) one, with its phase (but not the magnitude) of the order parameter modulating from site to site. Fully analytical insights into the numerical results are also given.
Parametrization of qutrits on the complex projective plane CP 2 = SU (3)/U (2) is given explicitly. A set of constraints that characterize mixed state density matrices is found.
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