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

Using high-precision tensor network methods to explicitly encode ice rules and verify the normality of the transfer operator, this study provides rigorous numerical evidence that the residual entropies of hexagonal and cubic ices are equal.

The Gist

Imagine you have a giant, three-dimensional puzzle made of water molecules. In this puzzle, the oxygen atoms form a rigid skeleton, but the hydrogen atoms are like tiny, restless guests who can sit in different spots between the oxygen atoms.

There are strict rules for how these guests can sit (called "ice rules"): every oxygen atom must have exactly two guests sitting close to it and two sitting a bit farther away. Even when the temperature drops so low that the water turns to solid ice, the guests don't freeze into a single, perfect arrangement. Instead, they can still shuffle around in trillions of different ways while obeying the rules.

This "shuffling" creates a leftover amount of disorder, known as residual entropy. Scientists have been arguing for decades about a specific question: Does the amount of disorder differ depending on the shape of the ice skeleton?

There are two main shapes of ice:

1. Hexagonal Ice (Ih): The most common form found in nature (like snowflakes).
2. Cubic Ice (Ic): A rarer form with a slightly different 3D structure.

For years, mathematicians proved that Hexagonal ice must have at least as much disorder as Cubic ice ($S_h \ge S_c$). However, computer simulations suggested the numbers were so close they might actually be identical. The problem was that the computers used to check this (called "Monte Carlo" methods) were like trying to count every possible shuffle by randomly guessing; they couldn't see the whole picture clearly enough to tell if the numbers were truly equal or just very close.

The New Approach: The "Tensor Network" Lens

The authors of this paper used a powerful new mathematical tool called Tensor Networks. You can think of this as a high-definition lens that doesn't just guess the answer but maps out the entire landscape of possibilities at once.

Instead of randomly shuffling the guests, they built a mathematical "transfer machine" (called a transfer operator). This machine takes a layer of the ice, applies the rules, and passes it to the next layer. By finding the "strongest signal" (the largest eigenvalue) coming out of this machine, they could calculate the exact amount of disorder without needing to guess.

The Big Discovery: The "Mirror" Test

Here is the clever part of their discovery. They realized that for the two types of ice to have the exact same disorder, the mathematical machine used for Cubic ice had to behave in a very specific way: it had to be normal.

In simple terms, a "normal" machine is one where the order in which you run its steps doesn't change the final outcome. It's like a mirror that reflects light perfectly; if you look at it from the front or the side, the reflection is consistent.

The authors ran a high-precision test to see if the Cubic ice machine was "normal." They found that it is 99.99% normal. It's not a perfect mirror (there's a tiny, tiny flaw), but it's so close to perfect that, for all practical purposes, it acts like one.

The Final Result

Because the machine is so close to being "normal," the authors were able to run their calculation directly without having to force the numbers to fit a specific shape (a trick previous researchers had to use).

When they did the math:

• The disorder of Hexagonal ice ($S_h$) came out to be 0.4104251.
• The disorder of Cubic ice ($S_c$) came out to be 0.4104248.

The difference between these two numbers is so small (about 5 parts in a million) that it is likely just a tiny error in the calculation method, not a real difference in the physics.

Conclusion

In everyday language: Hexagonal ice and Cubic ice have the exact same amount of leftover disorder.

The authors didn't just guess this; they used a sophisticated mathematical "lens" to prove that the rules governing the two ice types are so similar that they result in the same level of chaos, finally settling a long-standing debate in physics. They also noted that this method could be used to study other, stranger forms of ice that scientists have recently discovered, though that is a job for future research.

Technical Summary

Technical Summary: Equivalence of Residual Entropy in Hexagonal and Cubic Ices via Tensor Network Methods

Problem Statement
The paper addresses a long-standing unresolved question in statistical physics regarding the residual entropy of water ice. Specifically, it investigates whether the residual entropy of hexagonal ice ($S_h$) equals that of cubic ice ($S_c$). While analytical studies have established the inequality $S_h \geq S_c$, numerical investigations using Monte Carlo methods and series expansions have consistently suggested the two values are extremely close, leaving the question of strict equality open. A primary challenge in resolving this is that standard Monte Carlo approaches cannot directly sample the ground-state degeneracy space to obtain residual entropy; instead, they rely on integrating heat capacity over finite temperatures. Furthermore, previous tensor network attempts to solve this were limited because the transfer operator for cubic ice is non-Hermitian, requiring artificial symmetry constraints that obscured the distinction between the two ice phases.

Methodology
The authors employ high-precision tensor network methods to reformulate the residual entropy problem.

1. Tensor Network Representation: The "ice rule" (two hydrogen atoms close and two far from each oxygen) is encoded into local tensors. The 3D lattice structures of hexagonal ($I_h$) and cubic ($I_c$) ice are mapped onto tensor networks where the total number of configurations is determined by the dominant eigenvalue of a layer-to-layer transfer operator ($\hat{T}$).
   • For cubic ice, $\hat{T}_c = \hat{M}$.
   • For hexagonal ice, $\hat{T}_h = \hat{M}\hat{M}^T$.
2. Normality Analysis: The authors investigate the "normality" of the cubic ice transfer operator $\hat{M}$ (where a matrix is normal if $\hat{M}\hat{M}^T = \hat{M}^T\hat{M}$). They numerically calculate the fidelity between $\hat{M}\hat{M}^T$ and $\hat{M}^T\hat{M}$ using the Variational Uniform Matrix Product State (VUMPS) algorithm. High normality implies that the eigenvalues of $\hat{M}\hat{M}^T$ correspond to the squared norms of $\hat{M}$'s eigenvalues, theoretically supporting $S_h = S_c$.
3. Variational Optimization: To compute the residual entropies directly, the authors utilize the recently developed split Corner Transfer Matrix Renormalization Group (split-CTMRG) algorithm. Crucially, they argue that because $\hat{M}$ is non-negative and highly normal, the variational principle (Rayleigh quotient) can be applied to find the dominant eigenvector without imposing the spatial symmetry constraints used in prior works. This allows for a direct, unconstrained comparison of $S_h$ and $S_c$.

Key Contributions

• Direct Computation: The paper provides a framework to directly compute residual entropy in the ground-state degeneracy space using tensor networks, bypassing the limitations of finite-temperature integration required by Monte Carlo methods.
• Normality Perspective: The authors introduce the analysis of transfer operator normality as a theoretical and numerical tool. They demonstrate that the transfer operator for cubic ice is "extremely close" to being normal, which justifies the use of variational methods without artificial symmetry constraints.
• Algorithmic Application: The work applies the split-CTMRG method to optimize infinite Projected Entangled Pair States (iPEPS) for non-Hermitian transfer operators, achieving high bond dimensions ($D=7$) and CTM dimensions ($\chi=150$).

Results

• Normality Measure: The numerical fidelity between $\hat{M}\hat{M}^T$ and $\hat{M}^T\hat{M}$ was extrapolated to the infinite bond dimension limit, yielding a value of $F \approx 0.999903$. This indicates a very high degree of normality, though not strictly perfect.
• Residual Entropy Values: Using the split-CTMRG method and extrapolating to the thermodynamic limit ($D, \chi \to \infty$), the authors obtained:
  • $S_h = 0.4104251(6)$
  • $S_c = 0.4104248(14)$
• Comparison: The relative difference between the two values is on the order of $5 \times 10^{-6}$. Within the numerical accuracy of their method, the results indicate that $S_h$ and $S_c$ are equal. The unconstrained optimization yielded slightly higher values for hexagonal ice compared to previous constrained tensor network studies, confirming the importance of the larger variational space.

Significance and Claims
The paper claims to provide strong numerical evidence supporting the equality $S_h = S_c$. The significance of this work lies in:

1. Resolving the Inequality: It moves beyond the analytical lower bound ($S_h \geq S_c$) to provide numerical evidence that the gap is effectively zero within current computational precision.
2. Methodological Advancement: It demonstrates that the normality of a transfer operator is a sufficient condition to apply variational tensor network methods to non-Hermitian systems without imposing artificial symmetries. This offers a new perspective for studying 3D statistical physics models where layer-to-layer transfer matrices are not Hermitian.
3. Future Applicability: The authors suggest that this approach is promising for investigating the residual entropy of other, more complex ice phases (of which over 20 exist) where oxygen atoms form different regular lattices but maintain a coordination number of 4.

The authors remain modest, noting that while the high degree of normality supports the equality, the tiny violation of strict normality does not mathematically disprove the equality, but rather necessitates the high-precision numerical approach they employed to confirm it.