The long-standing question of whether the residual entropy of hexagonal ice ($S_h$) equals that of cubic ice ($S_c$) remains unresolved despite decades of research on ice-type models. While analytical studies have established the inequality $S_h\ge S_c$, numerical investigations suggest that the two values are very close. In this work, we revisit this problem using high-precision tensor network methods. In the Monte Carlo approaches used most commonly, the residual entropy cannot be directly obtained by sampling the ground-state degeneracy space. However, the tensor network framework enables an explicit encoding of the “ice rule” into local tensors, and then the residual entropy is transformed into finding the largest eigenvalue of a transfer operator in the form of a projected entangled-pair operator, which allows high-accuracy numerical evaluation. Meanwhile, we propose a perspective based on analyzing the normality of the transfer operator and examine it numerically with variational tensor network methods. This analysis allows for a direct computation for both residual entropies with our recently developed split corner transfer matrix renormalization group algorithm, providing a numerical evidence supporting the equality between $S_h$ and $S_c$.

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• Classical statistical mechanics
• Ice
• Lattice models in statistical physics
• Projected entangled pair states
• Tensor network methods