Magnetic field direction dependence of the spin liquid state in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mrow> <mml:mtext>−</mml:mtext> <mml:mi>RuC</mml:mi> </mml:mrow> <mml:msub> <mml:mi mathvarian

Numerous experiments have shown that <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>α</a:mi> <a:mrow> <a:mtext>−</a:mtext> <a:mi>RuC</a:mi> </a:mrow> <a:msub> <a:mi mathvariant="normal">l</a:mi> <a:mn>3</a:mn> </a:msub> </a:mrow> </a:math> exhibits a spin liquid (SL) state near 7.1 T when an in-plane magnetic field is applied perpendicular to the Ru-Ru bond directions ([110]), while SL states are rarely observed under fields applied along the Ru-Ru bond ([100]). The underlying mechanism for this pronounced anisotropy remains elusive. In this work, we employ semiclassical spin dynamics simulations to investigate the magnetization evolution of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>α</c:mi> <c:mrow> <c:mtext>−</c:mtext> <c:mi>RuC</c:mi> </c:mrow> <c:msub> <c:mi mathvariant="normal">l</c:mi> <c:mn>3</c:mn> </c:msub> </c:mrow> </c:math> with lattice distortion under varying external magnetic fields. It is found that each [100] direction has its own dominant orientation, and each [110] orientation marks the boundary, where the two alignments compete with each other. The magnetizations are influenced by multiple interactions and perturbations from lattice distortions, leading to the occurrence of the SL state in the [110] direction. These results provide new insights into the field-induced phase transitions in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>α</e:mi> <e:mrow> <e:mtext>−</e:mtext> <e:mi>RuC</e:mi> </e:mrow> <e:msub> <e:mi mathvariant="normal">l</e:mi> <e:mn>3</e:mn> </e:msub> </e:mrow> </e:math> and the microscopic mechanisms underlying the emergence of spin liquid behavior.