We study low-energy properties of spin-$S$ Kitaev models in an anisotropic limit. The effective form of a local conserved quantity is derived in the low-energy subspace. We find this is the same as that of $S=1/2$ case for the half-integer spins but shows a different form for the integer spins. Applying the perturbation theory to the anisotropic Kitaev model, we obtain the effective Hamiltonian. In the integer spin case, the effective model is equivalent to a free spin model under a uniform magnetic field, where quantum fluctuations are quenched. On the other hand, in the half-integer case, the system is described by the toric code Hamiltonian, where quantum fluctuations play a crucial role in the ground state. The boundary effect in the anisotropic Kitaev system is also discussed.
We study low-energy properties of spin-S Kitaev models in an anisotropic limit. The effective form of a local conserved quantity is derived in the low-energy subspace. We find this is the same as that of S=1/2 case for the half-integer spins but shows a different form for the integer spins. Applying the perturbation theory to the anisotropic Kitaev model, we obtain the effective Hamiltonian. In the integer spin case, the effective model is equivalent to a free spin model under a uniform magnetic field, where quantum fluctuations are quenched. On the other hand, in the half-integer case, the system is described by the toric code Hamiltonian, where quantum fluctuations play a crucial role in the ground state. The boundary effect in the anisotropic Kitaev system is also discussed.
Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-S Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for S=1 and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general S, we study the anisotropic spin-S Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer S we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.
We investigate magnetic properties in the S=1 Kitaev model in the anisotropic limit. Performing the fourth-order perturbation expansion with respect to the x-bonds, y-bonds, and magnetic field, we derive the effective Hamiltonian, where the low-energy physics should be described by the free spins with an effective magnetic field. Making use of the exact diagonalization method for small clusters, we discuss ground-state properties in the system complementary.
We study a one-dimensional version of the Kitaev model on a ring of size N, in which there is a spin S>1/2 on each site and the Hamiltonian is $J{\ensuremath{\sum}}_{n}{S}_{n}^{x}{S}_{n+1}^{y}$. The cases where S is integer and half-odd integer are qualitatively different. We show that there is a Z2-valued conserved quantity Wn for each bond (n,n+1) of the system. For integer S, the Hilbert space can be decomposed into 2N sectors, of unequal sizes. The number of states in most of the sectors grows as dN, where d depends on the sector. The largest sector contains the ground state, and for this sector, for S=1, d=(5+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large N indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term $\ensuremath{\lambda}{\ensuremath{\sum}}_{n}{W}_{n}$, and show that this has gapless excitations in the range ${\ensuremath{\lambda}}_{1}^{c}\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}{\ensuremath{\lambda}}_{2}^{c}$. We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points ${\ensuremath{\lambda}}_{1}^{c}$ and ${\ensuremath{\lambda}}_{2}^{c}$.
Abstract We investigate the anisotropic S = 1/2 Kitaev model on the honeycomb lattice with the ordered-flux structure. By diagonalizing the Majorana Hamiltonian for the flux configuration, we find two distinct gapped quantum spin liquids. One of them is the gapped state realized in the large anisotropic case, where low energy properties are described by the toric code. On the other hand, when the system has small anisotropy, the other gapped quantum spin liquid is stabilized by the ordered-flux configuration. Since these two gapped quantum spin liquids are separated by the gapless region, these are not adiabatically connected to each other.
In the first part of this paper, we study the spin-S Kitaev model using spin-wave theory. We discover a remarkable geometry of the minimum-energy surface in the N-spin space. The classical ground states, called Cartesian or CN-ground states, whose number grows exponentially with the number of spins N, form a set of points in the N-spin space. These points are connected by a network of flat valleys in the N-spin space giving rise to a continuous family of classical ground states. Further, the CN-ground states have a correspondence with dimer coverings and with self-avoiding walks on a honeycomb lattice. The zero-point energy of our spin-wave theory picks out a subset from a continuous family of classically degenerate states as the quantum ground states; the number of these states also grows exponentially with N. In the second part, we present some exact results. For arbitrary spin S, we show that localized Z2 flux excitations are present by constructing plaquette operators with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$, which commute with the Hamiltonian. This set of commuting plaquette operators leads to an exact vanishing of the spin-spin correlation functions beyond nearest-neighbor separation found earlier for the spin-1/2 model [G. Baskaran et al., Phys. Rev. Lett. 98, 247201 (2007)]. We introduce a generalized Jordan-Wigner transformation for the case of general spin S and find a complete set of commuting link operators similar to the spin-1/2 model, thereby making the Z2 gauge structure more manifest. The Jordan-Wigner construction also leads, in a natural fashion, to Majorana fermion operators for half-odd-integer spin cases and hard-core boson operators for integer spin cases strongly suggesting the presence of Majorana fermion and boson excitations in the respective low-energy sectors. Finally, we present a modified Kitaev Hamiltonian, which is exactly solvable for all half-odd-integer spins; it is equivalent to an exponentially large number of copies of spin-1/2 Kitaev Hamiltonians.
In this paper, we consider the Kitaev toric code with specific open boundary conditions. Such a physical system has a highly degenerate ground state determined by the degrees of freedom localized at the boundaries. We can write down an explicit expression for the ground state of this model. Based on this, the entanglement properties of the model are studied for two types of bipartition: one, where subsystem A is completely contained in B; and the second, where the boundary of the system is shared between A and B. In the former configuration, the entanglement entropy is the same as for the periodic boundary condition case, which means that the bulk is completely decoupled from the boundary on distances larger than the correlation length. In the latter, deviations from the torus configuration appear due to the edge states and lead to an increase of the entropy. We then determine an effective theory for the boundary of the system. In the case where we apply a small magnetic field as a perturbation, the degrees of freedom on the boundary acquire a dispersion relation. The system can there be described by a Hamiltonian of the Ising type with a generic spin-exchange term.