Perturbative approach to an exactly solved problem: Kitaev honeycomb model

Abstract

We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method which allows one to compute the spectrum as well as matrix elements of operators between eigenstates, at high order. The starting point of our study consists in an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hardcore boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the groundstate energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hardcore boson with an attached spin-and phase-operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multiplaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.

Kitaev model and dimer coverings on the honeycomb lattice

“…We computed the spectrum of H perturbatively up to high orders using the perturbative continuous unitary transformation (PCUT) method [26][27][28][29][30] around the isolated-dimer limit, where one of the couplings is much larger than the others. Such a strategy has been shown to be very successful in the original Kitaev model 23,31,32 although restricted to Abelian phases. Here, we proceed along the same line as in Ref.…”

“…Here, we proceed along the same line as in Ref. 23 and we skip all technical details which can be found in this reference.…”

Quantum memory coupled to cavity modes

“…18 The toric code can be realized as a low-energy effective Hamiltonian of the Kitaev honeycomb model, 21,22 and long-range interactions between anyons appear in the presence of a non-local coupling with cavity modes extending over the whole memory. 18 In analogy to the spin-electric coupling in molecular magnets, [23][24][25] we consider a modification of the Ising couplings of the type J x,y → J x,y + δ x,y (a + a † ).…”

Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

“…For a more rigorous treatment, see the original work [10] and the subsequent developments [22], [33][34][35][36][37][38]. We show how to relate the manipulation of the vortices to the manipulation of the model's physical parameters.…”