Two-Dimensional Hydrogen Bonded Crystals without the Ice Rule
01 Nov 1970-Journal of Mathematical Physics (American Institute of Physics)-Vol. 11, Iss: 11, pp 3183-3186
TL;DR: In this article, the singularities of the ground state energy of a related ring of interacting spins were found for 2D hydrogen-bonded crystals obeying the ice rule.
Abstract: Models of 2‐dimensional hydrogen bonded crystals obeying the ice rule, which previously have been solved exactly, are generalized by removing the ice rule. Many of the peculiar and unique properties of the solutions for the constrained models are now explained by showing that these models, above critical temperature, are equivalent to new unconstrained models at critical temperature. In addition to locating the critical temperature for the general but unsolved models, we locate the singularities of the ground state energy of a related ring of interacting spins.
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TL;DR: In this article, the eigenwert problem involved in the corresponding computation for a long strip crystal of finite width, joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum.
Abstract: The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$
5,081 citations
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TL;DR: In this article, two genuinely quantum models for an antiferromagnetic linear chain with nearest neighbor interactions are constructed and solved exactly, in the sense that the ground state, all the elementary excitations and the free energy are found.
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TL;DR: In this article, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics, and a method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite.
Abstract: In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at $\frac{J}{k{T}_{c}}=0.8814$ if we denote by $J$ the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at ${T}_{c}$, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.
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TL;DR: Bethe's hypothesis for the ground state of a one-dimensional cyclic chain of anisotropic nearest-neighbor spin-spin interactions was proved for any fixed number of down spins as mentioned in this paper.
Abstract: Bethe's hypothesis is proved for the ground state of a one-dimensional cyclic chain of anisotropic nearest-neighbor spin-spin interactions. The proof holds for any fixed number of down spins.
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TL;DR: In this article, the authors consider the problem of determining the position of a hydrogen atom relative to an oxygen atom in a periodic crystal lattice, where the hydrogen atoms are not at the centers of the bonds, however, so that there are two possible states for each bond corresponding to the two positions of the hydrogen atom.
Abstract: At low temperatures ice has a residual entropy caused, presumably, by an indeterminacy of the crystal structure. The oxygen atoms constitute a periodic crystal lattice that is hydrogen bonded. The hydrogen atoms are not at the centers of the bonds, however, so that there are two possible states for each bond corresponding to the two positions of the hydrogen atom relative to the bond midpoint. Nevertheless, not all bond configurations are allowed, for there is a constraint called the “ice condition” such that for the four bonds emanating from each oxygen atom, exactly two of the bonds must have the hydrogen atoms close to the oxygen atom.
366 citations