Ice-type model

In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice.^[1] Variants have been proposed as models of certain ferroelectric^[2] and antiferroelectric^[3] crystals.

In 1967, Elliott H. Lieb found the exact solution to a two-dimensional ice model known as "square ice".^[4] The exact solution in three dimensions is only known for a special "frozen" state.^[5]

Description

An ice-type model is a lattice model defined on a lattice of coordination number 4. That is, each vertex of the lattice is connected by an edge to four "nearest neighbours". A state of the model consists of an arrow on each edge of the lattice, such that the number of arrows pointing inwards at each vertex is 2. This restriction on the arrow configurations is known as the ice rule. In graph theoretic terms, the states are Eulerian orientations of an underlying 4-regular undirected graph. The partition function also counts the number of nowhere-zero 3-flows.^[6]

For two-dimensional models, the lattice is taken to be the square lattice. For more realistic models, one can use a three-dimensional lattice appropriate to the material being considered; for example, the hexagonal ice lattice is used to analyse ice.

At any vertex, there are six configurations of the arrows which satisfy the ice rule (justifying the name "six-vertex model"). The valid configurations for the (two-dimensional) square lattice are the following:

The energy of a state is understood to be a function of the configurations at each vertex. For square lattices, one assumes that the total energy $E$ is given by

E=n_{1}\epsilon _{1} n_{2}\epsilon _{2} \ldots  n_{6}\epsilon _{6},

for some constants $\epsilon _{1},\ldots ,\epsilon _{6}$ , where $n_{i}$ here denotes the number of vertices with the $i$ th configuration from the above figure. The value $\epsilon _{i}$ is the energy associated with vertex configuration number $i$ .

One aims to calculate the partition function $Z$ of an ice-type model, which is given by the formula

Z=\sum \exp(-E/k_{\rm {B}}T),

where the sum is taken over all states of the model, $E$ is the energy of the state, $k_{\rm {B}}$ is the Boltzmann constant, and $T$ is the system's temperature.

Typically, one is interested in the thermodynamic limit in which the number $N$ of vertices approaches infinity. In that case, one instead evaluates the free energy per vertex $f$ in the limit as $N\to \infty$ , where $f$ is given by

f=-k_{\rm {B}}TN^{-1}\log Z.

Equivalently, one evaluates the partition function per vertex $W$ in the thermodynamic limit, where

W=Z^{1/N}.

The values $f$ and $W$ are related by

f=-k_{\rm {B}}T\log W.

Physical justification

Several real crystals with hydrogen bonds satisfy the ice model, including ice^[1] and potassium dihydrogen phosphate KH
₂PO
₄^[2] (KDP). Indeed, such crystals motivated the study of ice-type models.

In ice, each oxygen atom is connected by a bond to four hydrogens, and each bond contains one hydrogen atom between the terminal oxygens. The hydrogen occupies one of two symmetrically located positions, neither of which is in the middle of the bond. Pauling argued^[1] that the allowed configuration of hydrogen atoms is such that there are always exactly two hydrogens close to each oxygen, thus making the local environment imitate that of a water molecule, H
₂O. Thus, if we take the oxygen atoms as the lattice vertices and the hydrogen bonds as the lattice edges, and if we draw an arrow on a bond which points to the side of the bond on which the hydrogen atom sits, then ice satisfies the ice model. Similar reasoning applies to show that KDP also satisfies the ice model.

In recent years, ice-type models have been explored as descriptions of pyrochlore spin ice^[7] and artificial spin ice systems,^[8]^[9] in which geometrical frustration in the interactions between bistable magnetic moments ("spins") leads to "ice-rule" spin configurations being favoured. Recently such analogies have been extended to explore the circumstances under which spin-ice systems may be accurately described by the Rys F-model.^[10]^[11]^[12]^[13]

Specific choices of vertex energies

On the square lattice, the energies $\epsilon _{1},\ldots ,\epsilon _{6}$ associated with vertex configurations 1-6 determine the relative probabilities of states, and thus can influence the macroscopic behaviour of the system. The following are common choices for these vertex energies.

The ice model

When modeling ice, one takes $\epsilon _{1}=\epsilon _{2}=\ldots =\epsilon _{6}=0$ , as all permissible vertex configurations are understood to be equally likely. In this case, the partition function $Z$ equals the total number of valid states. This model is known as the ice model (as opposed to an ice-type model).

The KDP model of a ferroelectric

Slater^[2] argued that KDP could be represented by an ice-type model with energies

\epsilon _{1}=\epsilon _{2}=0,\epsilon _{3}=\epsilon _{4}=\epsilon _{5}=\epsilon _{6}>0

For this model (called the KDP model), the most likely state (the least-energy state) has all horizontal arrows pointing in the same direction, and likewise for all vertical arrows. Such a state is a ferroelectric state, in which all hydrogen atoms have a preference for one fixed side of their bonds.

Rys F model of an antiferroelectric

The Rys $F$ model^[3] is obtained by setting

\epsilon _{1}=\epsilon _{2}=\epsilon _{3}=\epsilon _{4}>0,\epsilon _{5}=\epsilon _{6}=0.

The least-energy state for this model is dominated by vertex configurations 5 and 6. For such a state, adjacent horizontal bonds necessarily have arrows in opposite directions and similarly for vertical bonds, so this state is an antiferroelectric state.

The zero field assumption

If there is no ambient electric field, then the total energy of a state should remain unchanged under a charge reversal, i.e. under flipping all arrows. Thus one may assume without loss of generality that

\epsilon _{1}=\epsilon _{2},\quad \epsilon _{3}=\epsilon _{4},\quad \epsilon _{5}=\epsilon _{6}

This assumption is known as the zero field assumption, and holds for the ice model, the KDP model, and the Rys F model.

History

The ice rule was introduced by Linus Pauling in 1935 to account for the residual entropy of ice that had been measured by William F. Giauque and J. W. Stout.^[14] The residual entropy, $S$ , of ice is given by the formula

S=k_{\rm {B}}\log Z=k_{\rm {B}}\,N\,\log W,

where $k_{\rm {B}}$ is the Boltzmann constant, $N$ is the number of oxygen atoms in the piece of ice, which is always taken to be large (the thermodynamic limit) and $Z=W^{N}$ is the number of configurations of the hydrogen atoms according to Pauling's ice rule. Without the ice rule we would have $W=4$ since the number of hydrogen atoms is $2N$ and each hydrogen has two possible locations. Pauling estimated that the ice rule reduces this to $W=1.5$ , a number that would agree extremely well with the Giauque-Stout measurement of $S$ . It can be said that Pauling's calculation of $S$ for ice is one of the simplest, yet most accurate applications of statistical mechanics to real substances ever made. The question that remained was whether, given the model, Pauling's calculation of $W$ , which was very approximate, would be sustained by a rigorous calculation. This became a significant problem in combinatorics.

Both the three-dimensional and two-dimensional models were computed numerically by John F. Nagle in 1966^[15] who found that $W=1.50685\pm 0.00015$ in three-dimensions and $W=1.540\pm 0.001$ in two-dimensions. Both are amazingly close to Pauling's rough calculation, 1.5.

In 1967, Lieb found the exact solution of three two-dimensional ice-type models: the ice model,^[4] the Rys $F$ model,^[16] and the KDP model.^[17] The solution for the ice model gave the exact value of $W$ in two-dimensions as

W_{2D}=\left({\frac {4}{3}}\right)^{3/2}=1.5396007....

which is known as Lieb's square ice constant.

Later in 1967, Bill Sutherland generalised Lieb's solution of the three specific ice-type models to a general exact solution for square-lattice ice-type models satisfying the zero field assumption.^[18]

Still later in 1967, C. P. Yang^[19] generalised Sutherland's solution to an exact solution for square-lattice ice-type models in a horizontal electric field.

In 1969, John Nagle derived the exact solution for a three-dimensional version of the KDP model, for a specific range of temperatures.^[5] For such temperatures, the model is "frozen" in the sense that (in the thermodynamic limit) the energy per vertex and entropy per vertex are both zero. This is the only known exact solution for a three-dimensional ice-type model.

Relation to eight-vertex model

The eight-vertex model, which has also been exactly solved, is a generalisation of the (square-lattice) six-vertex model: to recover the six-vertex model from the eight-vertex model, set the energies for vertex configurations 7 and 8 to infinity. Six-vertex models have been solved in some cases for which the eight-vertex model has not; for example, Nagle's solution for the three-dimensional KDP model^[5] and Yang's solution of the six-vertex model in a horizontal field.^[19]

Boundary conditions

This ice model provide an important 'counterexample' in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions.^[20] The model was analytically solved for periodic boundary conditions, anti-periodic, ferromagnetic and domain wall boundary conditions. The six vertex model with domain wall boundary conditions on a square lattice has specific significance in combinatorics, it helps to enumerate alternating sign matrices. In this case the partition function can be represented as a determinant of a matrix (whose dimension is equal to the size of the lattice), but in other cases the enumeration of $W$ does not come out in such a simple closed form.

Clearly, the largest $W$ is given by free boundary conditions (no constraint at all on the configurations on the boundary), but the same $W$ occurs, in the thermodynamic limit, for periodic boundary conditions,^[21] as used originally to derive $W_{2D}$ .

3-colorings of a lattice

The number of states of an ice type model on the internal edges of a finite simply connected union of squares of a lattice is equal to one third of the number of ways to 3-color the squares, with no two adjacent squares having the same color. This correspondence between states is due to Andrew Lenard and is given as follows. If a square has color i = 0, 1, or 2, then the arrow on the edge to an adjacent square goes left or right (according to an observer in the square) depending on whether the color in the adjacent square is i 1 or i−1 mod 3. There are 3 possible ways to color a fixed initial square, and once this initial color is chosen this gives a 1:1 correspondence between colorings and arrangements of arrows satisfying the ice-type condition.

Notes

^ ^a ^b ^c Pauling, L. (1935). "The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement". Journal of the American Chemical Society. 57 (12): 2680–2684. doi:10.1021/ja01315a102.
^ ^a ^b ^c Slater, J. C. (1941). "Theory of the Transition in KH₂PO₄". Journal of Chemical Physics. 9 (1): 16–33. Bibcode:1941JChPh...9...16S. doi:10.1063/1.1750821.
^ ^a ^b Rys, F. (1963). "Über ein zweidimensionales klassisches Konfigurationsmodell". Helvetica Physica Acta. 36: 537.
^ ^a ^b Lieb, E. H. (1967). "Residual Entropy of Square Ice". Physical Review. 162 (1): 162–172. Bibcode:1967PhRv..162..162L. doi:10.1103/PhysRev.162.162.
^ ^a ^b ^c Nagle, J. F. (1969). "Proof of the first order phase transition in the Slater KDP model". Communications in Mathematical Physics. 13 (1): 62–67. Bibcode:1969CMaPh..13...62N. doi:10.1007/BF01645270. S2CID 122432926.
^ Mihail, M.; Winkler, P. (1992). "On the Number of Eularian Orientations of a Graph". SODA '92 Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics. pp. 138–145. ISBN 978-0-89791-466-6.
^ Bramwell, Steven T; Harris, Mark J (2020-09-02). "The history of spin ice". Journal of Physics: Condensed Matter. 32 (37): 374010. Bibcode:2020JPCM...32K4010B. doi:10.1088/1361-648X/ab8423. ISSN 0953-8984. PMID 32554893.
^ Wang, R. F.; Nisoli, C.; Freitas, R. S.; Li, J.; McConville, W.; Cooley, B. J.; Lund, M. S.; Samarth, N.; Leighton, C.; Crespi, V. H.; Schiffer, P. (January 2006). "Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands". Nature. 439 (7074): 303–306. arXiv:cond-mat/0601429. Bibcode:2006Natur.439..303W. doi:10.1038/nature04447. ISSN 1476-4687. PMID 16421565. S2CID 1462022.
^ Perrin, Yann; Canals, Benjamin; Rougemaille, Nicolas (December 2016). "Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice". Nature. 540 (7633): 410–413. arXiv:1610.01316. Bibcode:2016Natur.540..410P. doi:10.1038/nature20155. ISSN 1476-4687. PMID 27894124. S2CID 4409371.
^ Jaubert, L. D. C.; Lin, T.; Opel, T. S.; Holdsworth, P. C. W.; Gingras, M. J. P. (2017-05-19). "Spin ice Thin Film: Surface Ordering, Emergent Square ice, and Strain Effects". Physical Review Letters. 118 (20): 207206. arXiv:1608.08635. Bibcode:2017PhRvL.118t7206J. doi:10.1103/PhysRevLett.118.207206. ISSN 0031-9007. PMID 28581768. S2CID 118688211.
^ Arroo, Daan M.; Bramwell, Steven T. (2020-12-22). "Experimental measures of topological sector fluctuations in the F-model". Physical Review B. 102 (21): 214427. arXiv:2010.05839. Bibcode:2020PhRvB.102u4427A. doi:10.1103/PhysRevB.102.214427. ISSN 2469-9950. S2CID 222290448.
^ Nisoli, Cristiano (2020-11-01). "Topological order of the Rys F-model and its breakdown in realistic square spin ice: Topological sectors of Faraday loops". Europhysics Letters. 132 (4): 47005. arXiv:2004.02107. Bibcode:2020EL....13247005N. doi:10.1209/0295-5075/132/47005. ISSN 0295-5075. S2CID 221891692.
^ Schánilec, V.; Brunn, O.; Horáček, M.; Krátký, S.; Meluzín, P.; Šikola, T.; Canals, B.; Rougemaille, N. (2022-07-07). "Approaching the Topological Low-Energy Physics of the F Model in a Two-Dimensional Magnetic Lattice". Physical Review Letters. 129 (2): 027202. Bibcode:2022PhRvL.129b7202S. doi:10.1103/PhysRevLett.129.027202. ISSN 0031-9007. PMID 35867462. S2CID 250378329.
^ Giauque, W. F.; Stout, Stout (1936). "The entropy of water and third law of thermodynamics. The heat capacity of ice from 15 to 273K". Journal of the American Chemical Society. 58 (7): 1144–1150. Bibcode:1936JAChS..58.1144G. doi:10.1021/ja01298a023.
^ Nagle, J. F. (1966). "Lattice Statistics of Hydrogen Bonded Crystals. I. The Residual Entropy of Ice". Journal of Mathematical Physics. 7 (8): 1484–1491. Bibcode:1966JMP.....7.1484N. doi:10.1063/1.1705058.
^ Lieb, E. H. (1967). "Exact Solution of the Problem of the Entropy of Two-Dimensional Ice". Physical Review Letters. 18 (17): 692–694. Bibcode:1967PhRvL..18..692L. doi:10.1103/PhysRevLett.18.692.
^ Lieb, E. H. (1967). "Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric". Physical Review Letters. 19 (3): 108–110. Bibcode:1967PhRvL..19..108L. doi:10.1103/PhysRevLett.19.108.
^ Sutherland, B. (1967). "Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals". Physical Review Letters. 19 (3): 103–104. Bibcode:1967PhRvL..19..103S. doi:10.1103/PhysRevLett.19.103.
^ ^a ^b Yang, C. P. (1967). "Exact Solution of a Two-Dimensional Model for Hydrogen-Bonded Crystals". Physical Review Letters. 19 (3): 586–588. Bibcode:1967PhRvL..19..586Y. doi:10.1103/PhysRevLett.19.586.
^ Korepin, V.; Zinn-Justin, P. (2000). "Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions". Journal of Physics A. 33 (40): 7053–7066. arXiv:cond-mat/0004250. Bibcode:2000JPhA...33.7053K. doi:10.1088/0305-4470/33/40/304. S2CID 2143060.
^ Brascamp, H. J.; Kunz, H.; Wu, F. Y. (1973). "Some rigorous results for the vertex model in statistical mechanics". Journal of Mathematical Physics. 14 (12): 1927–1932. Bibcode:1973JMP....14.1927B. doi:10.1063/1.1666271.