Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The European Physical Journal B

A new look at Ernst Ising’s thesis

R. Folk

Published Jun 1, 2025• 1 citation
Publisher

Abstract

Abstract The famous paper (Ising in Z Phys 31: 253, A translation to English is by Jane Ising, Tom Cummings, Ulrich Harsch is found Bibliotheca Augustina 2001/02 https://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html , 1928) contains only a greatly reduced version of the material presented in his thesis. Looking at it therefore provides new insights into the Ising model in terms of its later implications. The method of calculating the partition function from the configuration of the system succeeds by finding a polynomial whose largest root defines the result in the thermodynamic limit and which coincides with the characteristic polynomial of the transfer matrix. This matrix works with the states of the elementary elements of the model and was found later in 1941 by Kramers and Wannier. Ising used his method for several one-dimensional models—variants of the Ising chain: a chain with three instead of two states (a forerunner of the Potts model formulated 1952), the Ising ladder given by two interacting chains and the Ising chain with next nearest-neighbor interaction published by Montroll in 1941. The largest root could not be found in all cases and approximations had to be made. Although no phase transition at finite temperature was found for the Ising chain, it turned out in the following decades that there is a critical point at $$T=0$$ T = 0 . Thus, the simple Ising chain can be used to study all properties approaching a transition point in this exactly solvable model. The historical introduction to the problem, given by Ising in his dissertation, goes back to Richard Kirwan (1733–1812), who was the first to associate ferromagnetism with the ordering of interacting magnetic elements similar to crystallization. Graphical abstract

DOI
10.1140/epjb/s10051-025-00954-x

Related papers

Ising’s Roots and the Transfer-Matrix Eigenvalues

R. Folk, Yurij V. Holovatch

Today, the Ising model is an archetype describing collective ordering processes. As such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago (in 1924) contained not only the solution of what we call now the ‘classical 1D Ising model’ but also other problems. Some of these problems, as well as the method of their solution, are the subject of this note. In particular, we discuss the combinatorial method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that ‘Ising’s roots’ that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising’s thesis, which is not included in his famous paper of 1925 (E. Ising, Z. Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now-abundant models with many-component order parameters.

arXiv:2405.05703•Entropy•May 28, 2024•5 citations

Sorting Out the Potts Models

R. J. Baxter

It was a great pleasure to hear that the American Physical Society had awarded me the 2006 Lars Onsager Prize and to read the informative news story “Order Parameter of the Chiral Potts Model Succumbs at Last to Exact Solution” that appeared on page 19 of the November edition of Physics Today. I have a few comments.There are various models named after Ren Potts, who died in Adelaide, Australia, on 9 August 2005. He was a student of Cyril Domb at Oxford University in 1952, and worked on the vector and scalar extensions of the Ising model. 1 1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. The two are often confused. The vector model is described in the Physics Today story, whereas it is the scalar model that has been solved at criticality. The N-state chiral Potts model is a different model again. Its transfer matrices commute, which makes it “solvable” in the sense that one can calculate the free energy. For N = 2 it reduces to the Ising model. It is true that C. N. Yang took six months to calculate the spontaneous magnetization (order parameter) of the Ising model, 2 2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. whereas I have been puzzling over the extension of that problem to the chiral Potts model for 15 years!For both models the results were previously known: Onsager had announced his result for the Ising model at a conference in 1949, 3 3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). and Giuseppe Albertini, Barry McCoy, Jacques Perk, and Shuang Tang gave the formula for chiral Potts in 1989. However, there are some differences.The story refers to both these announcements as conjectures, but it is clear from a later paper by Onsager 4 4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. that he had already derived his answer when he announced it. Being Onsager, he did not rush into print, but worked at refining and polishing his derivation for three years, which gave Yang the opportunity to publish his solution in 1952.By contrast, the chiral Potts result given by Giuseppe Albertini and colleagues really was a conjecture—an elegant formula, inspired by the Ising result, that fitted all known series expansions. It is this conjecture that I have now verified, subject to analyticity assumptions no more drastic than those used for the order parameters of other post-Ising models. From 1998 until recently, I was looking at the Jimbo-Miwa-Nakayashiki method. It depends on solving for a function of two arguments, and I was having trouble. The idea that came to me over two days in December 2004 was to relate those arguments so that there was just one independent variable, in much the same way as in the free energy calculation one starts by looking at the simpler “superintegrable” case. The switch to one variable made the needed analyticity properties fairly obvious, and was sufficient to obtain the order parameters. 5 5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. It was a good illustration of the approach used in these solvable models—one needs to generalize sufficiently to see properties like commuting transfer matrices, but one should beware of generalizing unnecessarily.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. Google ScholarCrossref, ISI2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. Google Scholar3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). Google ScholarCrossref4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. Google Scholar5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. Google ScholarCrossref, ISI© 2005 American Institute of Physics.

Physics Today•Dec 1, 2005•4 citations

On Ising models

Hoede, C., Zandvliet, Henricus J.W.

For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.

•Jun 1, 2006•0 citations

Critical behavior in many body systems

Cyril Domb

Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.

Unknown eBooks•Jul 1, 1974•0 citations

The exact solution of the Axial Next-Nearest-Neighbor Ising model on width 2 spin chains

Pavel Khrapov, Anastasia Loginova

Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.

arXiv:2202.01834•Feb 3, 2022•0 citations

Critical behavior of the ferromagnetic ising chain with interactionJ ij =|i−j|−2

G. V. Matvienko

Theoretical and Mathematical Physics•Jun 1, 1985•11 citations
Abstract The famous paper (Ising in Z Phys 31: 253, A translation to English is by Jane Ising, Tom Cummings, Ulrich Harsch is found Bibliotheca Augustina 2001/02 https://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html , 1928) contains only a greatly reduced version of the material presented in his thesis. Looking at it therefore provides new insights into the Ising model in terms of its later implications. The method of calculating the partition function from the configuration of the system succeeds by finding a polynomial whose largest root defines the result in the thermodynamic limit and which coincides with the characteristic polynomial of the transfer matrix. This matrix works with the states of the elementary elements of the model and was found later in 1941 by Kramers and Wannier. Ising used his method for several one-dimensional models—variants of the Ising chain: a chain with three instead of two states (a forerunner of the Potts model formulated 1952), the Ising ladder given by two interacting chains and the Ising chain with next nearest-neighbor interaction published by Montroll in 1941. The largest root could not be found in all cases and approximations had to be made. Although no phase transition at finite temperature was found for the Ising chain, it turned out in the following decades that there is a critical point at T=0T=0T=0 T = 0 . Thus, the simple Ising chain can be used to study all properties approaching a transition point in this exactly solvable model. The historical introduction to the problem, given by Ising in his dissertation, goes back to Richard Kirwan (1733–1812), who was the first to associate ferromagnetism with the ordering of interacting magnetic elements similar to crystallization. Graphical abstract
Today, the Ising model is an archetype describing collective ordering processes. As such, it is widely known in physics and far beyond. Less known is the fact that the thesis defended by Ernst Ising 100 years ago (in 1924) contained not only the solution of what we call now the ‘classical 1D Ising model’ but also other problems. Some of these problems, as well as the method of their solution, are the subject of this note. In particular, we discuss the combinatorial method Ernst Ising used to calculate the partition function for a chain of elementary magnets. In the thermodynamic limit, this method leads to the result that the partition function is given by the roots of a certain polynomial. We explicitly show that ‘Ising’s roots’ that arise within the combinatorial treatment are also recovered by the eigenvalues of the transfer matrix, a concept that was introduced much later. Moreover, we discuss the generalization of the two-state model to a three-state one presented in Ising’s thesis, which is not included in his famous paper of 1925 (E. Ising, Z. Physik 31 (1925) 253). The latter model can be considered as a forerunner of the now-abundant models with many-component order parameters.
It was a great pleasure to hear that the American Physical Society had awarded me the 2006 Lars Onsager Prize and to read the informative news story “Order Parameter of the Chiral Potts Model Succumbs at Last to Exact Solution” that appeared on page 19 of the November edition of Physics Today. I have a few comments.There are various models named after Ren Potts, who died in Adelaide, Australia, on 9 August 2005. He was a student of Cyril Domb at Oxford University in 1952, and worked on the vector and scalar extensions of the Ising model. 1 1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. The two are often confused. The vector model is described in the Physics Today story, whereas it is the scalar model that has been solved at criticality. The N-state chiral Potts model is a different model again. Its transfer matrices commute, which makes it “solvable” in the sense that one can calculate the free energy. For N = 2 it reduces to the Ising model. It is true that C. N. Yang took six months to calculate the spontaneous magnetization (order parameter) of the Ising model, 2 2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. whereas I have been puzzling over the extension of that problem to the chiral Potts model for 15 years!For both models the results were previously known: Onsager had announced his result for the Ising model at a conference in 1949, 3 3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). and Giuseppe Albertini, Barry McCoy, Jacques Perk, and Shuang Tang gave the formula for chiral Potts in 1989. However, there are some differences.The story refers to both these announcements as conjectures, but it is clear from a later paper by Onsager 4 4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. that he had already derived his answer when he announced it. Being Onsager, he did not rush into print, but worked at refining and polishing his derivation for three years, which gave Yang the opportunity to publish his solution in 1952.By contrast, the chiral Potts result given by Giuseppe Albertini and colleagues really was a conjecture—an elegant formula, inspired by the Ising result, that fitted all known series expansions. It is this conjecture that I have now verified, subject to analyticity assumptions no more drastic than those used for the order parameters of other post-Ising models. From 1998 until recently, I was looking at the Jimbo-Miwa-Nakayashiki method. It depends on solving for a function of two arguments, and I was having trouble. The idea that came to me over two days in December 2004 was to relate those arguments so that there was just one independent variable, in much the same way as in the free energy calculation one starts by looking at the simpler “superintegrable” case. The switch to one variable made the needed analyticity properties fairly obvious, and was sufficient to obtain the order parameters. 5 5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. It was a good illustration of the approach used in these solvable models—one needs to generalize sufficiently to see properties like commuting transfer matrices, but one should beware of generalizing unnecessarily.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. R. Potts, Proc. Camb. Phil. Soc. 48, 106 (1952) https://doi.org/10.1017/S0305004100027419. Google ScholarCrossref, ISI2. C. N. Yang, Collected Papers, Freeman and Co, San Francisco (1983), p. 12. Google Scholar3. L. Onsager, Nuovo Cimento 6(suppl.), 261 (1949). Google ScholarCrossref4. L. Onsager, in Critical Phenomena in Alloys, Magnets and Superconductors, R. E. Mills, E. Ascher, R. I. Jaffee, eds., McGraw-Hill, New York (1971), p. 3. Google Scholar5. R. J. Baxter, J. Stat. Phys. 120, 1 (2005) https://doi.org/10.1007/s10955-005-5534-3. Google ScholarCrossref, ISI© 2005 American Institute of Physics.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.
Using the transfer matrix method for Axial Next-Nearest Neighbor Ising model without an external field on a closed chain of spins of width 2 in the direction of interaction of only nearest neighbors and length L in the direction of interaction of nearest neighbors and next nearest neighbors, we found exact values of the partition function in a finite strip of length L , free energy, internal energy per node, heat capacity in a finite strip of length L and in the thermodynamic limit. The entire spectrum of the 16x16 transfer matrix and the structure of all transfer matrix eigenvectors are found. The problem of finding the spectrum of the transfer matrix is reduced to finding the spectrum of two matrixes of size 8x8. The characteristic polynomial of the first matrix is decomposed into two linear polynomials of multiplicity two and two square trinomials. The characteristic polynomial of the second matrix is decomposed into two polynomials of the fourth degree, the roots of which are found by the Ferrari method. The free energy in the thermodynamic limit has a very simple form and is expressed in terms of the logarithm of the root of the quadratic equation. Two theorems on the physical characteristics of the ANNNI model in a finite closed strip of length L and in the thermodynamic limit are formulated and proved. An example of calculation is given for some values of the model parameters in the thermodynamic limit, heat capacity graphs are plotted. A comparative analysis of the heat capacity for the double-stranded ANNNI model and the ANNNI model on the entire flat lattice is given. The closest connection and similarity of the physical characteristics of the two-chain model and the model on the entire flat lattice is shown.