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.
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 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 prominent feature of quantum spin liquids is fractionalization of the spin degree of freedom. Fractionalized excitations have their own dynamics in different energy scales, and hence, affect finite-temperature (
T) properties in a peculiar manner even in the paramagnetic state harboring the quantum spin liquid state. We here present a comprehensive theoretical study of the spin dynamics in a wide
T range for the Kitaev model on a honeycomb lattice, whose ground state is a quantum spin liquid. In this model, the fractionalization occurs to break up quantum spins into itinerant matter fermions and localized
Z2 fluxes, which results in two crossovers at very different
T scales. Extending the previous study for the isotropic coupling case [J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016)], we calculate the dynamical spin structure factor $S(\mathbf{q},\ensuremath{\omega})$, the NMR relaxation rate
1/T1, and the magnetic susceptibility $\ensuremath{\chi}$ while changing the anisotropy in the exchange coupling constants, by using the dynamical mean-field theory based on a Majorana fermion representation. We describe the details of the methodology including the continuous-time quantum Monte Carlo method for computing dynamical spin correlations and the maximum entropy method for analytic continuation. We confirm that the combined method provides accurate results in a wide
T range including the region where the spins are fractionalized. We find that also in the anisotropic cases the system exhibits peculiar behaviors below the high-
T crossover whose temperature is comparable to the average of the exchange constants: $S(\mathbf{q},\ensuremath{\omega})$ shows an inelastic response at the energy scale of the averaged exchange constant,
1/T1 continues to grow even though the equal-time spin correlations are saturated and almost
T independent, and $\ensuremath{\chi}$ deviates from the Curie-Weiss behavior. In particular, when the exchange interaction in one direction is stronger than the other two, the dynamical quantities exhibit qualitatively different
T dependences from the isotropic case at low
T, reflecting the opposite parity between the flux-free ground state and the flux-excited state, and a larger energy cost for flipping a spin in the strong interaction direction. On the other hand, when the exchange anisotropy is in the opposite way, the results are qualitatively similar to those in the isotropic case. All these behaviors manifest the spin fractionalization in the paramagnetic region. Among them, the dichotomy between the static and dynamical spin correlations is unusual behavior hardly seen in conventional magnets. We discuss the relation between the dichotomy and the spatial configuration of the
Z2 gauge fields. Our results could stimulate further experimental and theoretical analyses of candidate materials for the Kitaev quantum spin liquids.
Kitaev's compass model on the honeycomb lattice realizes a spin liquid whose emergent excitations are dispersive Majorana fermions and static
Z2 gauge fluxes. We discuss the proper selection of physical states for finite-size simulations in the Majorana representation, based on a recent paper by F. L. Pedrocchi, S. Chesi, and D. Loss [Phys. Rev. B 84, 165414 (2011)]. Certain physical observables acquire large finite-size effects, in particular if the ground state is not fermion-free, which we prove to generally apply to the system in the gapless phase and with periodic boundary conditions. To illustrate our findings, we compute the static and dynamic spin susceptibilities for finite-size systems. Specifically, we consider random-bond disorder (which preserves the solubility of the model), calculate the distribution of local flux gaps, and extract the NMR line shape. We also predict a transition to a random-flux state with increasing disorder.
Finite-temperature (
T) properties of a Kitaev model defined on a honeycomb lattice are investigated by a quantum Monte Carlo simulation, from the viewpoint of fractionalization of quantum
S=1/2 spins into two types of Majorana fermions, itinerant and localized. In this system, the entropy is released successively at two well-separated
T scales, as a clear indication of the thermal fractionalization. We show that the high-
T crossover, which is driven by itinerant Majorana fermions, is closely related with the development of nearest-neighbor spin correlations. On the other hand, the low-
T crossover originates in thermal fluctuations of fluxes composed of localized Majorana fermions, by which the spectrum of itinerant Majorana fermions is significantly disturbed. As a consequence, in the intermediate-
T range, the system exhibits coherent transport of Majorana fermions and
T-linear behavior in the specific heat, despite the Dirac semimetallic spectrum in the low-
T limit. We also show that the flux fluctuations affect the gapless-gapped phase boundary. Our results indicate that the fractionalization is experimentally observable in spin correlations, specific heat, and transport properties.
The Kitaev model exhibits a canonical quantum spin liquid as a ground state and hosts two fractional quasiparticles, i.e., itinerant Majorana fermion and localized flux excitation. The former can carry heat and spin modulations in the quantum spin liquid, but the role of the latter remains unknown for the transport phenomena. Here, we focus on spin transport in the presence of excited fluxes and report that they yield strong interference in the propagation of the Majorana fermions, which have a gaugelike potential emergent around the fluxes. We examine the transient spin dynamics triggered by a pulsed magnetic field at an edge. In the absence of excited fluxes, the magnetic-field pulse creates the plane wave of the Majorana fermions, which flows in the quantum spin liquid. Although this wave does not accompany the change of local spin moments in bulk, it induces local moments at the side opposite to the edge under the magnetic-field pulse. We observe the spatial modulation of induced spin moments when fluxes are excited in the bulk region. This behavior is more striking than the case of lattice defects. Moreover, we find that although the amplitude of the spatial change is almost independent of the distance between lattice defects, it is strongly enhanced by increasing the distance for the case of excited fluxes. The difference is understood from the influence on the itinerant Majorana fermions; the lattice defects change the system locally, but flux excitations alter all the transfer integrals on the string connecting them. The present results will provide another route to observing intrinsic flux excitations distinguished from extrinsic effects such as lattice defects.