We study the Kitaev model with direction-dependent interactions to investigate how the flux configuration and/or the anisotropy in the exchanges affect the Majorana excitations. It is found that the triangular flux configurations with unit length
q induce the quantum spin liquid states with gapped Majorana excitations. Systematic numerical calculations demonstrate that $(q\ensuremath{-}1)(q\ensuremath{-}2)/2$ gapped states bounded by gapless states are realized in the vicinity of the isotropic limit. The induced gapped quantum spin liquid states are not adiabatically connected to the gapped one realized in the large anisotropic limit. Therefore, these two states are distinct from each other. The nature of the gapped states can be explained by the fact that the hybridization between the gapless points, which originates from the superlattice potential due to the flux configuration, opens the gap in the Majorana excitations.
The Kitaev model on a honeycomb lattice with bond-dependent Ising interactions offers an exactly solvable model of a quantum spin liquid (QSL) with fractionalized excitations: gapped
Z2 fluxes and gapless linearly dispersing Majorana fermions in the isotropic limit (
Kx=Ky=Kz). We explore the phase diagram along two axes, an external magnetic field
h, applied out of plane of the honeycomb, and anisotropic interactions
Kz larger than the other two (${K}_{x}={K}_{y}\ensuremath{\equiv}K$). For ${K}_{z}/K\ensuremath{\gg}2$ and
h=0, the matter Majorana fermions have the largest gap, and the system is described by a gapped
Z2 toric code. One of the central questions we address is whether the fractionalized excitations in the different phases have sharp signatures that can be detected in experiments. We show that while the response to single-spin excitations is broad, the spectral function corresponding to two-spin excitations across a bond has sharp signatures that can be attributed to specific anyons. In the toric code regime, the $\ensuremath{\epsilon}=e\ifmmode\times\else\texttimes\fi{}m$ fermion, formed from the bosonic Ising electric (
e) and magnetic (
m) charges, disperses along a specific one-dimensional direction that provides a fingerprint of fractionalization. At lower
Kz in the center of the Abelian phase, in a regime we dub the primordial fractionalized regime, the field generates a hybridization between the $\ensuremath{\epsilon}$ fermion and the Majorana matter fermion, resulting in a $\ensuremath{\psi}$ fermion which too has a distinct quasi-one-dimensional dispersion. All the other phases in the field-anisotropy plane are naturally obtained from this primordial soup. These highly constrained fractonlike dispersions can be observable by inelastic light and neutron scattering, thereby providing ``smoking gun'' signatures of fractionalization in the QSL phase. Our analysis is based on calculations of susceptibilities, topological entanglement entropy, and excitation dynamics, obtained using exact diagonalization and density matrix renormalization group, and supported by perturbation theory.
We study the effects of the flux configurations on the emergent Majorana fermions in the
S=21 Kitaev model on a honeycomb lattice, where quantum spins are fractionalized into itinerant Majorana fermions and localized fluxes. A quantum spin liquid appears as the ground state of the Kitaev model in the flux-free sector, which has intensively been investigated so far. In this flux sector, the Majorana fermion system has linear dispersions and shows power-law behavior in the Majorana correlations. On the other hand, periodically arranged flux configurations yield low-energy excitations in the Majorana fermion system, which are distinctly different from those in the flux-free state. We find that one of the periodically arranged flux states results in the gapped Majorana dispersion and the exponential decay in the Majorana correlations. The Kitaev system with another flux configuration exhibits a semi-Dirac-type dispersion, leading to the power-law decay with a smaller power than that in the flux-free sector along symmetry axes. We also examine the effect of the randomness in the flux configurations and clarify that the Majorana density of states is filled by increasing the flux density, and power-law decay in the Majorana correlations remains. The present results could be important to control the motion of Majorana fermions, which carries the spin excitations, in the Kitaev candidate materials.
A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. This model can be rewritten as a free Majorana fermion system coupled with Z 2 variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the temperature dependence of the Majorana spectra. We find that the semimetallic dip is filled as temperature increases in the gapless region, but surprisingly, the spectrum develops an excitation gap in the region near the gapless-gapped boundary. Such changes of the low-energy spectrum appear sharply at the transition temperature from the spin liquid to the paramagnetic state. The results indicate that thermal fluctuations of the Z 2 fields significantly influence the low-energy state of Majorana fermions, especially in the spin liquid formation.
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ gauge symmetry. We mapped out the phase diagram of the model and found gapped and gapless spin-liquid phases. While the low-energy sector of the gapped phase corresponds to the well-known topological color code model on a honeycomb lattice, the low-energy sector of the gapless phases is described by an effective spin model with three-body interactions on a triangular lattice. A gap is opened in the spectrum in small magnetic fields, where we showed that the ground state has a finite topological entanglement entropy. We argue that the gapped phases could be possibly described by exotic excitations, and their corresponding spectrum is richer than the Ising phase of the Kitaev model.
We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories based on different Majorana fermion representations of the S=1/2 spin operators. First, we use a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite snake string operator. To ensure that the fermionized Hamiltonian remains local, we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Second, we use the conventional Kitaev representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both representations, we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin liquids, the JWT fails to correctly describe the magnetic instability and finite-temperature behavior. Our results show that the magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale, we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures, in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures. Published by the American Physical Society 2024