Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods

TL;DR

The paper investigates whether the residual entropies of hexagonal and cubic ice are equal by encoding ice-rule constraints into tensor networks and recasting the problem as a transfer-operator eigenproblem. It introduces the normality of the transfer operator as a sufficient condition for $S_h=S_c$ and provides strong numerical evidence for normality via a fidelity measure exceeding $0.9999$, then directly computes residual entropies using a variational iPEPS approach with split-CTMRG, finding $S_h$ and $S_c$ to agree within $10^{-5}$ precision. The work demonstrates that normality enables a non-Hermitian transfer operator to be treated with variational TN methods without symmetry constraints, and reports near-identical residual entropies for $I_h$ and $I_c$ within numerical accuracy. This supports the longstanding conjecture of equality and offers a robust framework for residual entropy calculations in other ice-like lattices and in 3D non-Hermitian transfer problems.

Abstract

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 $S_h \geq S_c$, numerical investigations suggest that the two values are very close. In this work, we revisit this problem using high-precision tensor-network methods. In Monte Carlo approaches 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 new perspective based on analyzing the normality of the transfer operator, and demonstrate that if the operator is normal, the equality $S_h = S_c$ follows directly. Then the variational tensor network methods are employed to numerically verify this normality. Finally both residual entropies are directly computed by using our recently developed split corner transfer matrix renormalization group algorithm, providing a rigorous evidence supporting the equality between $S_h$ and $S_c$.

Figures (6)

• Figure 1: Structure of hexgonal ice $I_h$ (a) and cubic ice $I_c$ (b). The cyan (green) layer denotes the crinkled honeycomb layer (and its mirror counterpart). The dashed circle marks a tetrahedral unit, and one of the hydrogen configurations satisfying the ice rule is shown to the right.
• Figure 2: Tensor network representation of the residual entropy problem for ice $I_h$ and $I_c$. (a) local tetrahedral structure of water molecules and the corresponding local tensors encoding the ice rule. (b) Tensor network representation for the layer-to-layer transfer operator. Dashed black lines denote intralayer bonds. (c) Simplified representation obtained by absorbing the $X$ matrix into $T$ tensors on one of the two sublattices of the honeycomb lattice. (d) Transfer operator for $I_c$, consisting of a uniform local tensor $O$, where $O$ is built by blocking neighboring $T$ tensors. (e) Transfer operator for $I_h$.
• Figure 3: Tensor network method for calculating the measure of normality. (a) Tensor network representation for the numerator in Eq. \ref{['eqn:fidelity_definition']}. (b) The network is transformed into an equivalent 2D tensor network consisting of uniform tensor $\tilde{O}$ by tracing the vertical bonds. (c) Contracting the 2D tensor network with boundary MPS $\ket{\psi(A)}$.
• Figure 4: Possible gauge transformation between local tensors of $\hat{M}\hat{M}^T$ and $\hat{M}^T\hat{M}$. (a) Local tensor $O_1$ ($O_2$) forming the transfer operators $\hat{M}\hat{M}^T$ ($\hat{M}^T\hat{M}$). $O^T$ denotes the tensor obtained by swapping the vertical indices of tensor $O$. (b) Invertible matrix transformation induced by invertible matrices $X$ and $Y$ acting on the links. (c) Invertible MPO transformation induced by invertible MPO $\hat{\mathcal{P}}(X)$. (d) Pulling-through transformation induced by local tensor $X$ and $Y$.
• Figure 5: The fixed point equation for transfer operator $\hat{T}$ with iPEPS $\ket{\psi(A)}$ as an ansatz.