Kitaev quantum spin liquids host Majorana fermions via the fractionalization of spins. In a magnetic field, the Majorana fermions were predicted to comprise a topological state, which has attracted great attention by the discovery of the half-quantized thermal Hall conductivity. Nevertheless, a reliable theory remains elusive for the field effect, especially at finite temperature. Here we present unbiased large-scale numerical results for the Kitaev model in a wide range of magnetic field and temperature. We find that the unconventional paramagnetic region showing fractional spin dynamics extends at finite temperature, far beyond the field range where the topological state is expected at zero temperature. Our results show the confinement-deconfinement behavior between the fractional Majorana excitations and the conventional magnons.
Recent discovery of the half quantized thermal Hall conductivity in [Formula: see text]-RuCl[Formula: see text], a candidate material for the Kitaev spin liquid, suggests the presence of a highly entangled quantum state in external magnetic fields. This field induced phase appears between the low field zig-zag magnetic order and the high field polarized state. Motivated by this experiment, we study possible field induced quantum phases in theoretical models of the Kitaev magnets, using the two dimensional tensor network approach or infinite tensor product states. We find various quantum ground states in addition to the chiral Kitaev spin liquid occupying a small area in the phase diagram. They form a band of emergent quantum phases in an intermediate window of external magnetic fields, somewhat reminiscent of the experiment. We discuss the implications of these results in view of the experiment and previous theoretical studies.
Recent experiments on Kitaev spin liquid candidate materials reported nonmonotonic behavior of thermal conductivity as a function of magnetic field, which lead to conflicting interpretations of its origin. Motivated by this development, we study the magnetic field dependence of thermal conductivity of a generalized Kitaev model, which allows the phase transitions between different flux sectors as a function of the magnetic field. The thermal conductivity due to Majorana fermions shows dip-bump structures as the magnetic field increases, which is caused by either the transitions between different flux sectors of Kitaev spin liquids or the topological transitions that change the Majorana Chern number within the same flux sector. It is shown that the change of Chern number is closely related to the four-Majorana-fermion interaction induced by the magnetic field. The nonmonotonic behavior in thermal conductivity emerges at finite temperature, and it becomes weaker when temperature decreases toward zero. Our model provides a generic mechanism for the Kitaev spin liquids to develop nonmonotonic magnetic-field dependence of thermal conductivity while the comparison to realistic materials remains an open question for future investigation.
We investigate the thermal Hall conductivity in the Kitaev model with additional interactions under a magnetic field, employing a finite-temperature tensor network method benchmarked by a thermal pure quantum state technique. We find that the thermal Hall conductivity divided by temperature,
κxy/T, significantly overshoots the value of the half-integer quantization and exhibits a pronounced hump while decreasing temperature. Moreover, we show that the field-direction dependence of
κxy/T is consistent with the sign of the Chern number associated with the Majorana fermions across a wide range of magnetic fields. We also demonstrate that the additional off-diagonal interactions, known as the
Γ and
Γ′ terms, considerably affect
κxy/T. In particular, we show that positive
Γ and negative
Γ′ lead to a remarkable enhancement in the intermediate temperature region. From the comparison with the classical counterpart, we reveal that the effects of the
Γ term go beyond the classical picture, indicating significant quantum fluctuation effects, while those of the
Γ′ term are well captured at the classical level. These comprehensive analyses indicate that the enhanced thermal Hall response is consistently explained by dominant contributions from topological Majorana fermions, even within the polarized regime beyond the critical field. Our approach not only establishes a robust theoretical framework for understanding the thermal Hall transport in Kitaev materials such as
α-RuCl
3, but also offers a promising pathway to bridge the gap between theories and experiments across a wide range of strongly correlated materials.
In this work we study the phase diagram of the Kekul\'e-Kitaev model. The model is defined on a honeycomb lattice with bond-dependent anisotropic exchange interactions, making it exactly solvable in terms of Majorana representation of spins, in close analogy to the Kitaev model. However, the energy spectrum of Majorana fermions has a multiband structure characterized by Chern numbers 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We obtained the phase diagram of the model in the plane of exchange couplings and in the presence of a magnetic field and found chiral topological and trivial spin-liquid ground states. In the absence of magnetic field most of the phase diagram is a trivial gapped phase continuously connected to an Abelian phase, while in the presence of the magnetic field a topological phase arises. Furthermore, motivated by recent thermal measurements on the spin-liquid candidate $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$, we calculated the thermal Hall conductivity at different regimes of parameters and temperatures and found the latter is quantized over a wide range of temperatures.
Motivated by the recent experimental observation of the half-quantized thermal Hall conductivity for the candidate material of the Kitaev spin liquid, $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ [Y. Kasahara et al., Nature 559, 227 (2018)], we investigate effects of non-Kitaev exchange interactions, which exist in the real material, on the gapped chiral spin-liquid state realized in the case with an applied magnetic field. It is found that off-diagonal exchange interactions enhance significantly the mass gap of Majorana fermions. This result provides a possible explanation for robust quantization of the thermal Hall conductivity observed in the above-mentioned experiment. Furthermore, we demonstrate that if the Kitaev spin-liquid state and the zigzag antiferromagnetic order coexist around the border of these two phases, off-diagonal exchange interactions can induce Fermi surfaces of Majorana fermions, leading to a state similar to the U(1) spin liquid.