Abstract
In the Kitaev honeycomb model, the quantum spin fractionalizes into itinerant Majorana and gauge flux spontaneously upon cooling, leading to rich experimental ramifications at finite temperature and an upsurge of research interest. In this work, we employ the exponential tensor renormalization group approach to explore the Kitaev model under various perturbations, including the external fields, Heisenberg, and the off-diagonal couplings that are common in the Kitaev materials. Through large-scale manybody calculations, we find a Kitaev fractional liquid at intermediate temperature that is robust against perturbations. The fractional liquid exhibits universal thermodynamic behaviors, including the fractional thermal entropy, metallic specific heat, and an intermediate-temperature Curie law of magnetic susceptibility. The emergent universal susceptibility behavior, with a modified Curie constant, can be ascribed to the strongly fluctuating $\mathbb{Z}_2$ fluxes as well as the extremely short-ranged and bond-directional spin correlations. With this insight, we revisit the susceptibility measurements of Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$, and find evident signatures of finite-temperature fractionalization and ferromagnetic Kitaev couplings. Moreover, the peculiar spin correlation in the fractional liquid corresponds to a stripy structure factor which rotates in the extended Brillouin zone as the spin component changes. Therefore, our findings encourage future experimental exploration of fractional liquid in the Kitaev materials by thermodynamic measurements and spin-resolved structure factor probes.
Highlights
Quantum spin liquids realize a novel class of quantum states of matter [1,2,3] which represent the nonmagnetic states with emergent long-range entanglement and fractionalized excitations in frustrated magnets [4,5]
The well-recognized characteristic features at intermediate temperature, including the fractional thermal-entropy plateau and the linear-T specific-heat scaling, clearly persist in the extended Kitaev model. Through exploring these robust finite-T features, we find a generic scenario in the extended Kitaev models relating to the realistic Kitaev materials: A fractionalization of the quantum spin takes place as temperature decreases, and half of the spin degrees of freedom, i.e., the itinerant Majorana fermions, release a large part of the entropy below TH, accompanied by a saturated short-range spin correlation; the other half, the gauge fluxes, fluctuate strongly until near the much lower temperature TL
In the Kitaev model and related Kitaev materials, an alternative scenario is proposed where a fractionalization of spins takes place upon cooling. This gives rise to an intermediate-T Kitaev fractional liquid and eventually to the low-T quantum spin liquid phase with long-range entanglement
Summary
Quantum spin liquids realize a novel class of quantum states of matter [1,2,3] which represent the nonmagnetic states with emergent long-range entanglement and fractionalized excitations in frustrated magnets [4,5]. Based on our XTRG simulations, we reveal that the fractional liquid regime does exist at intermediate temperature, for an extended range of external fields or non-Kitaev terms including the Heisenberg or couplings, even when the KSL ground state is altered. These results shed light on the exotic finite-T quantum states and intriguing properties of the Kitaev materials. VII we summarize our main conclusions and discuss their implications in future studies of the Kitaev model and materials
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