In the Kitaev honeycomb model, the quantum spin fractionalizes into itinerant Majorana fermion and gauge flux upon cooling, leading to rich experimental ramifications at finite temperature and an upsurge of research interest. However, accurate modelings of the Kitaev materials by including realistic couplings beyond the pure Kitaev model constitute a major challenge for the community. With recently developed exponential tensor-network approach, we clear the pathway and perform finite-temperature simulations of the extended Kitaev model with additional interactions common in real materials. At intermediate temperature, we find an emergent Curie law of magnetic susceptibility and a stripy spin-structure factor characterizing the robust Kitaev fractional liquid. With this insight, we revisit the susceptibility measurements of Na 2 IrO 3 and α-RuCl 3 and find evidence of ferromagnetic Kitaev coupling and finite-temperature fractionalization. Bridging the gap between theories and experiments with the state-of-the-art tensor-network simulations, our findings provide guidance for future experimental exploration of the spin liquids in Kitaev materials by thermodynamic measurements and spin-resolved structure factor probes.

The frustrated magnet α-RuCl3 constitutes a fascinating quantum material platform that harbors the intriguing Kitaev physics. However, a consensus on its intricate spin interactions and field-induced quantum phases has not been reached yet. Here we exploit multiple state-of-the-art many-body methods and determine the microscopic spin model that quantitatively explains major observations in α-RuCl3, including the zigzag order, double-peak specific heat, magnetic anisotropy, and the characteristic M-star dynamical spin structure, etc. According to our model simulations, the in-plane field drives the system into the polarized phase at about 7 T and a thermal fractionalization occurs at finite temperature, reconciling observations in different experiments. Under out-of-plane fields, the zigzag order is suppressed at 35 T, above which, and below a polarization field of 100 T level, there emerges a field-induced quantum spin liquid. The fractional entropy and algebraic low-temperature specific heat unveil the nature of a gapless spin liquid, which can be explored in high-field measurements on α-RuCl3.