The frustrated magnet $\alpha \text{−}{\mathrm{RuCl}}_3$ is one of the prime candidates for realizing a Kitaev quantum spin liquid (QSL). However, the existence of a field-induced intermediate QSL phase in this material remains under debate. Here, we employ sign-free numerically exact quantum Monte Carlo simulations to investigate the Kitaev-Heisenberg (KH) model on the honeycomb lattice with $K=-2J$ under an applied magnetic field along the $z$ direction. Our findings reveal that the system undergoes a direct quantum phase transition from a zigzag magnetically ordered phase to a spin-polarized phase at zero temperature, which belongs to the 3D XY universality class. At finite temperatures, a Berezinskii-Kosterlitz-Thouless transition line separates the spin-polarized phase from a quasi-long-range ordered state, eventually terminating at the quantum critical point. Our results convincingly show that there is no intermediate QSL phase in the KH model with a $z$-direction magnetic field, which we believe will shed important light on understanding experimental observations in $\alpha \text{−}{\mathrm{RuCl}}_3$.

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• Magnetic order
• Spin liquid
• Honeycomb lattice
• Quantum many-body systems
• Strongly correlated systems
• Monte Carlo methods