Inspired by the experimental realization of direct kagome spin ice [Yue et al., Nat. Nanotechnol. 19, 1101 (2024)], the theoretical six-vertex model on the kagome lattice is systematically simulated using the directed loop Monte Carlo method. Four distinct vortex lattice phases are identified: (i) antiferromagnetic leg states and vortex lattice order on both triangular and honeycomb faces, with a winding number $k=1$; (ii) ferromagnetic leg states and vortex lattice order on both types of faces, with $k=-2$ on the honeycomb faces and $k=1$ on the triangular faces; (iii) paramagnetic leg states and vortex lattice order on the triangular faces with $k=1$; and (iv) paramagnetic leg states and vortex lattice order on the honeycomb faces with $k=1$. As for ferromagnetic to different types of paramagnetic phase, besides the Ising universality class with $y_t=1$, varying critical exponents have also been found with different values of vertex weights. The transition between the third type and fourth type of vortex lattice phases occurs with the new exponent $y_t=1.340\left(3\right)$. The third and fourth types of the vortex lattice phase to the vortex disorder phase are found to be of the Berezinskii-Kosterlitz-Thouless type. These findings contribute to the search for and understanding of ice on complex lattices.

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• Magnetic phase transitions
• Artificial spin ice
• Kagome lattice
• Monte Carlo methods