Proximate Kitaev Physics
Candidate Kitaev Materials: α-RuCl3
Kitaev's exactly soluble bond-selective Ising model on the honeycomb lattice delivered a new paradigm for obtaining both gapped and gapless quantum spin-liquid states, as well as opening a new direction for gaining insight into the topological properties of spin liquids. It became more than a mathematical curiosity when it was shown that Kitaev-type interactions between selected spin components can exist in compounds of 5d transition-metal ions, most notably Ir. Less expected was the possibility of Kitaev physics in 4d systems, specifically of Ru, and in the meantime some Co (3d) Kitaev materials have also been proposed. The key problem is that the Kitaev interactions appear together with Heisenberg and other interactions, such that Kitaev physics has to compete with magnetism and other instabilities, whence the term "proximate Kitaev."
We began by trying to interpret experimental results for the candidate Kitaev mater ial α-R uCl3, in which the competing magnetic order is suppressed by an applied magnetic field of 7 T. Thermal conductivity measurements [1] demonstrated the presence of a gapless excitation contributing to thermal transport in the field-induced quantum disordered phase. NMR measurements then demonstrated further that it is the spin excitations which are gapless [2], and suggested a field-induced algebraic quantum spin liquid whose power-law temperature-dependence of 1/T1 indicated point nodes in the 2D Brillouin zone of the honeycomb lattice.
Proximate Kitaev Models
To understand this behaviour we developed a projected spinon formalism in which to analyse the "K-J-Γ plus field" models required to describe the full physics of the insulating magnetic 4d and 5d materials that are candidates for displaying some form of proximate Kitaev physics. The projection was enforced by a variational Monte Carlo treatment, but the analytical key was the use of the projective symmetry group to limit the otherwise rather large number of allowed mean-field parameters. We focused first on the field-induced QSL phase, which is obtained with large Γ, and showed [3] that the field direction is the key to obtaining a Dirac spectrum and topological spin liquids with different chiralities.
To understand this behaviour we developed a projected spinon formalism in which to analyse the "K-J-Γ plus field" models required to describe the full physics of the insulating magnetic 4d and 5d materials that are candidates for displaying some form of proximate Kitaev physics. The projection was enforced by a variational Monte Carlo treatment, but the analytical key was the use of the projective symmetry group to limit the otherwise rather large number of allowed mean-field parameters. We focused first on the field-induced QSL phase, which is obtained with large Γ, and showed [3] that the field direction is the key to obtaining a Dirac spectrum and topological spin liquids with different chiralities.
To place this result in its proper perspective, we studied the extended K-J-Γ phase diagram at zero field [4]. Except at small J, we found magnetically ordered phases with first-order transitions. Adjacent to the "generic" Kitaev-type spin-liquid regime (gKSL, the phase surrounding the Kitaev point) we found one completely new and different gapless quantum spin-liquid state, which we named the PKSL. In contrast to the gKSL, the spinon excitation spectrum of the PKSL has 14 nodes, and these spinons are complex mixtures of Kitaev's b and c Majorana modes, leading to a gapless spin response at many k-points.
We were not able to obtain clear results for the effects of fields imposed in arbitrary directions, but for a field normal to the honeycomb plane we found a rich cascade of previously unrealised topological states from Kitaev's "16-fold way" classification [4]. Specifically, these new topological states have Chern numbers of 5 and 4, and hence the states trapped around vortex cores are respectively non-Abelian and Abelian anyons.
[1] Anomalous Thermal Conductivity and Magnetic Torque Response in the Honeycomb Magnet RuCl3,
I. A. Leahy, C. A. Pocs, P. E. Siegfried, D. Graf, S.-H. Do, K.-Y. Choi, B. Normand and M. Lee,
Phys. Rev. Lett. 118, 187203 (2017).
[2] Gapless Spin Excitations in the Field-Induced Quantum Spin-Liquid Phase of RuCl3,
J. Zheng, K. Ran, T. Li, J. Wang, P. Wang, B. Liu, Z.-X. Liu, B. Normand, J. Wen and W. Yu,
Phys. Rev. Lett. 119, 227208 (2017).
[3] Dirac and Chiral Quantum Spin Liquids on the Honeycomb Lattice in a Magnetic Field,
Z.-X. Liu and B. Normand,
Phys. Rev. Lett. 120, 187201 (2018).
[4] One Proximate Kitaev Spin Liquid in the K-J-Γ Model on the Honeycomb Lattice,
J. Wang, B. Normand and Z.-X. Liu,
Phys. Rev. Lett. 123, 197201 (2019).
I. A. Leahy, C. A. Pocs, P. E. Siegfried, D. Graf, S.-H. Do, K.-Y. Choi, B. Normand and M. Lee,
Phys. Rev. Lett. 118, 187203 (2017).
[2] Gapless Spin Excitations in the Field-Induced Quantum Spin-Liquid Phase of RuCl3,
J. Zheng, K. Ran, T. Li, J. Wang, P. Wang, B. Liu, Z.-X. Liu, B. Normand, J. Wen and W. Yu,
Phys. Rev. Lett. 119, 227208 (2017).
[3] Dirac and Chiral Quantum Spin Liquids on the Honeycomb Lattice in a Magnetic Field,
Z.-X. Liu and B. Normand,
Phys. Rev. Lett. 120, 187201 (2018).
[4] One Proximate Kitaev Spin Liquid in the K-J-Γ Model on the Honeycomb Lattice,
J. Wang, B. Normand and Z.-X. Liu,
Phys. Rev. Lett. 123, 197201 (2019).