###### Abstract

The life of Ernst Ising and the steps to solving the model named after him are
reported in parallel. Wilhelm Lenz suggested his student Ernst Ising to
explain the existence of ferromagnetism on the basis of his publication in
1920. The result, published in 1925 was disappointing, because only the one
dimensional case could be solved with a negative result about the absence of
ferromagnetism. Wolfgang Pauli who was an assistant of Lenz in Hamburg
published in the same year his ‘nonclassical ambiguity’, later identified as
the spin of the electron, and the exclusion principle. He was the first - at
the Solvay Conference in 1930 - to present the Hamiltonian of the Ising model
as we know it today.

Meanwhile Ising had left university research and due to the political
situation in 1938 had to leave Germany and fled to Luxemburg. This went in
hand with damaging the network of researchers dealing with the problem of
ferromagnetism and more generally with phase transitions and statistical
physics. In 1944, the year when Luxemburg was liberated by the American troops
and Ising and his family was rescued, Lars Onsager presented a solution of the
two-dimensional case. In 1952 Chen-Ning Yang solved the problem of Ising’s
thesis in two dimensions; one year later Ising became the US citizenship. The
following development showed, that the model turned out to be a highway to
modern physics concepts applicable also in other fields, although the final
exact solution in three dimensions has not yet been reached.

##  Chapter 0 The Survival of Ernst Ising and
the Struggle to Solve His Model

###### Contents

  1. 0 The Survival of Ernst Ising and the Struggle to Solve His Model
    1. 1 Introduction
    2. 2 Another Introduction
    3. 3 Starting Ising’s academic education
      1. 1 Hamburg University and the Institute for Theoretical Physics
      2. 2 At Lenz’ institute
    4. 4 The way to Ising’s thesis
      1. 1 Kirwan’s hypothesis - a lost historical root
      2. 2 Lenz’s idea of directed magnetons
      3. 3 Stern-Gerlach experiment and spatial quantization
      4. 4 Schottky’s synchronous electrons
    5. 5 A hard task for Ising
    6. 6 The evaluation of Ising’s result
      1. 1 Lenz approval of Ising’s thesis
      2. 2 The reduction made in the publication
      3. 3 Naming the model
    7. 7 Ising leaving Hamburg and his new phase of life until 1930
    8. 8 The model after the thesis until its modern formulation in 1930
      1. 1 Herzfeld 1925
      2. 2 Pauli and the New QM
      3. 3 The New QM and Heisenbergs exchange interaction
      4. 4 Pauli and Heisenberg on the Ising model and the thermodynamic limit
    9. 9 Ising as a teacher from 1933 until the November pogrom in 1938
    10. 10 The impact and development of the Ising model until 1939
      1. 1 Nordheim, Bethe and Peierls
      2. 2 The Ising model in textbooks
      3. 3 The Van der Waals Centenary Congress in Amsterdam November 23 - 26, 1937
    11. 11 Escaping to Luxembourg
    12. 12 The international Conference on Magnetism in Strasbourg 1939
    13. 13 Waiting time over- Consulate closed - Trembling in Mersch
    14. 14 The new approach to the Ising model
      1. 1 Transfer Matrix for the Ising chain
      2. 2 Duality (“the small miracle”)
    15. 15 D-Day and rescue
    16. 16 On the steps to the solution
      1. 1 Onsager’s solution 1944 (“the big miracle”)
    17. 17 Onsager announced in 1948 and Yang published in 1952 the solution for the magnetization
      1. 1 Reactions on Onsager’s solution
      2. 2 Working on the solution
    18. 18 After the rescue
      1. 1 Experimental proof and understanding of the rational value
    19. 19 Conclusion
    20. 0.A Reviews concerning Ising and his model

###  1 Introduction

History is the most fundamental science, for there is no human knowledge which
cannot lose its scientific character when men forget the conditions under
which it originated, the questions which it answered, and the functions it was
created to serve.[Benjamin Farrington [1] cited by Erwin Schrödinger [2]]

The quest to understand the emergence of magnetism in solids after the Great
war is entangled to the worldwide changes in the social and political
environment of the involved scientists. It started from a simple idea for
three-dimensional magnets of a German university professor in 1920 after the
First World War and ended in 1952 in the United States with the solution for
the two-dimensional case by a university professor born 1922 in China. No
solution for the three-dimensional case is known so far. But the model
formulated by Ernst Ising has become a physical standard model reaching out in
other physical fields and other disciplines different from physics like
chemistry, biology, medicine, ecology, computer sciences, economy, social
sciences, linguistic and much more. In physics it lead to merging ideas of
solid state physics with ideas of elementary particle physics and is even
fruitful for modelling our understanding of space and time. Nevertheless one
find’s opinions like this:

This model was suggested to Ising by his thesis adviser, Lenz. Ising solved
the one-dimensional model, …, and on the basis of the fact that the one-
dimensional model had no phase transition, he asserted that there was no phase
transition in any dimension. As we shall see, this is false. It is ironic that
on the basis of an elementary calculation and erroneous conclusion, Ising’s
name has become among the most commonly mentioned in the theoretical physics
literature. But history has had its revenge. Ising’s name, which is correctly
pronounced “E-zing,” is almost universally mispronounced “I-zing.” [Barry
Simon [3]]

One goal of this publication is to make understandable why Ernst Ising could
only solve the one dimensional model and why this was at his time not just a
student exercise as it is nowadays. On the other hand the development of
physics is followed and the steps made by now well known scientists to reach a
solution are shown. To solve the model in higher dimensions Alexander M.
Polyakov[4] named two miracles, which were necessary to happen, a small one -
named duality - and a large one - the Onsager solution (impossible to predict
the mathematical apparatus necessary). From this it became clear that Ising’s
task was a mission impossible and it remains so far.

In 2011 McCoy said about The Romance of the Ising Model [5]: ”Fortunately for
romance, there are many mysteries of the Ising model which are far from being
understood. The romantic in me says that, even when these mysteries have been
understood, the understanding of the mysteries will generate new mysteries and
the romance of the Ising model will be everlasting.”

And in the introduction to his talk The Once and Future Ising Model given at
the Simons Center for Geometry and Physics on December 15, 2015 he cites the
mathematician Charles Dodgson (better known from literature as Lewis Carroll)
when he explained that his talk is not about results but problems with Ising’s
model:

Ising 1976

©Peoria Journal Star

> “You are old, Father William,” the young man said,
> “And your hair has become very white;
> And yet you incessantly stand on your head –
> Do you think, at your age, it is right?”
> “In my youth,” Father William replied to his son,
> “I feared it might injure the brain;
> But, now that I’m perfectly sure I have none,
> Why, I do it again and again.”
> (Lewis Carroll 1865)

###  2 Another Introduction

History uncovers the strange stew of concepts that were necessary for the
development of physics.[M. Stanley [6]]

In the year 1922 when Chen Ning Yang - who solved and published the solution
of Ising’s problem in two dimensions - was born, Wilhelm Lenz asked his
doctoral candidate Ernst Ising at Hamburg University to start with his thesis.
Ising had to formulate a model which should replace [supersede] the Curie-
Weiss theory [7] explaining the existence of a ferromagnetic phase in solids.
It was already known [8, 9] that ferromagnetism is a quantum effect but when
Ising started his thesis the formalism of quantum mechanics was just in the
revolutionary change from the old pseudo-classical version of Bohr-Sommerfeld
to its new abstract version of Pauli, Heisenberg and Schrödinger.

Two years earlier another young man named Lars Onsager was admitted to start
his studies at the Technical University of Trondheim. He came in 1928 to the
United States where he later in 1944 published the exact solution of the Ising
model in two dimensions and zero magnetic field.

In 1938 Ising had to flee from the Nazis to Luxemburg and emigrated in 1947 to
the States. Yang went to the States in 1946 not returning to the communist
China. Similar fates can be told from some of those who filled the gap between
the formulation of the problem and its solution, like for Bethe[10] “You will
probably not know that my mother is Jewish: According to the civil service
law, I am ’not of Aryan descent’ and consequently not worthy of being a civil
servant in the German Reich. […] So I have to face the consequences and try to
find somewhere abroad.” He left Germany in 1935 as Peierls in 1936 and worked
with him in Manchester. Others got in touch with the net which was thrown from
Europe to the United States in that time like Montroll, Wannier or Onsager.
Again others who stayed in Europe tried to survive the Second world War and
keep the contact they have established before, like Kramers.

In 1930 physics was very different from how it looks today. Only two forces
(gravity and electromagnetism), and three particles (the proton, the electron
and the photon) were known at that time. In order to uncover these concepts we
have to look at the past but avoid to look at it through the lens of the
present. But we also have to uncover which concepts known in the past have
been used by the scientists, to understand how they tried to solve the open
problems and to learn from their errors.

Marc Kac who was granted a scholarship to the United States after finishing
his Ph.D. in Lviv gave a combinatorial solution to the two-dimensional Ising
model remarked in his autobiography[11]:

“To explain phase transitions from ‘first principles’ on the basis of
statistical mechanics is still a largely unsolved problem. It continues to
fascinate physicists and since it can be stated in purely mathematical terms
it also attracted the attention of mathematicians.

Because of the inherent complexity of matter one is compelled to work with
simplified models which, one hopes, capture the qualitative and even some
quantitative features of their counterparts in nature.

The most celebrated class of models proposed for the purpose of understanding
magnetic phase transitions is the Ising models. Introduced in 1925 by Ernst
Ising, they are still vigorously studied today and the literature on them is
staggering.”

And quite recently Ising’s one-dimensional model is reconsidered and a
solution by mathematical induction has been suggested [12] because: “The one-
dimensional (1D) Ising model is of fundamental importance in an introductory
course on statistical mechanics, because it is connected to many interesting
physical concepts… Despite the absence of a genuine phase transition, the 1D
Ising model still plays a central role in the comprehension of many
principles, phenomena, and numerical methods in statistical physics.”

In the following alternately the lifetime of Ising and the development in
physics related to his thesis is described.

###  3 Starting Ising’s academic education

Ernst Ising in Bochum

©Tom Ising

  * 1900

born in Cologne as the son of Gustav and Thekla Ising, a Jewish merchant
family

  * 1904

living in Bochum, Goethestraße 18

  * 1918

Matura (general qualification for university entrance), military training

  * 1919

begin of the studies of physics and mathematics at the University of
Göttingen, one semester interruption, then studies of astronomy at the
University of Bonn

  * 1921

April 21 starting studies at the University of Hamburg

  * 1924

December finishing the PhD thesis at the University of Hamburg

Ernst Ising was born on May 10, 1900 to the Jewish family of Gustav Ising and
his wife Thekla Löwe. His father was a merchant in Cologne [Heumarkt 64/66].
Ernst completed his schooltime in Bochum and after the military education
started his study in mathematics and physics in the summer semester 1919 at
the University of Göttingen[13]. “Under Klein, Hilbert, Minkowski, Landau and
Runge, together with the astronomer Karl Schwarzschild and the physicists
Ludwig Prandtl, Peter Debye and Emil Wiechert, there formed the creative
atmosphere, described vividly by Max Born, Harald Bohr and Richard Courant,
which attracted scientists and students from all over the world and made
Göttingen the Mecca of mathematics.”

After a break of one semester may be a so called ‘Zwischensemester’ [14] he
proceeded with mathematics at the university of Bonn [15] (22.04.1920) where
he also studied astronomy[16]. Two semesters later he moved [17] on the 21st
April 1921 to the new University of Hamburg founded in 2019. There he turned
toward theoretical physics on recommendation of professor Wilhelm Lenz (1888 -
1957) and started end of 1922 with his thesis under Lenz’ instruction.

Meyenn [18] describes the situation of physics in Germany: “Through a skilful
science policy based on the principle of self-government, and the strong
mathematical orientation of the exact natural sciences, which originated
especially from Göttingen, the theoretical physicists of the Weimar Republic
were able, despite the general misery after the First World War, extremely
successfully to contribute to the clarification of the atomic riddle. This
glittering epoch of theoretical physics, which culminated with the
establishment of quantum mechanics, has therefore been described as the golden
twenties.

A development that could neither be stopped by the war nor by the economic
plight of the post-war years, then came to a standstill within a few years due
to the anti-science policy of the National Socialists. Arnold Sommerfeld, a
physicist who had been at the center of this development through his research
and teaching, wrote resignedly to Einstein in September 1937: The future looks
bleak for German physics. I must console myself with having experienced her
golden age 1905 - 1930.”

One reason for the success might have been the fact that Theoretical Physics
was ‘cheap’ compared to the upcoming large research facilities in the
experimental sector.

####  1 Hamburg University and the Institute for Theoretical Physics

Despite the great misery in Germany after the First World War, a new
university was democratically founded in Hamburg[19]: “1919, the year the
University was founded, saw a longstanding debate come to an end in Imperial
Germany about whether a university was of any use to the port and trading city
of Hamburg. Before the German Revolution of 1918/19, the majority of the
city’s decision-makers had persistently rejected the idea. Yet on March 28,
1919, the time was ripe. A social democrat majority in the Hamburg parliament
succeeded in founding the University with a sudden shift in their argument: It
was no longer about the usefulness of a university for merchants and shipping
companies but whether university education could dismantle class differences.
This was precisely what the protagonists of the University’s foundation had in
mind: by opening a university for all, it would be possible to guarantee
education and, as a consequence, socioeconomic progress for all, which would
ultimately end class differences.”

One year after the foundation an Institute for Theoretical Physics was created
and Wilhelm Lenz was appointed 1921 as Ordinarius Professor. He had graduated
in Munich under Arnold Sommerfeld, became his assistant and completed there
his Habilitation. In the following he got the position of an extra-ordinarius
professor in Rostock from where he moved to Hamburg as director of the
Institute of Theoretical Physics. There Lenz established the reputation of his
research with a small group of doctorands together with Wolfgang Pauli, who
also came from Sommerfeld’s group in Munich as research assistant for
habilitation in Hamburg.

In general the university of Hamburg and especially the physics department
became internationally recognized. Rabi who was one year at Hamburg University
wrote[20]: “When I was at Hamburg University, it was one of the leading
centers of physics in the world. There was a close collaboration between Stern
and Pauli, between experiment and theory. For example, Stern’s questions were
important in Pauli’s theory of magnetism of free electrons in metals.
Conversely, Pauli’s theoretical researches were important influences in
Stern’s thinking. Further, Stern’s and Pauli’s presence attracted many
illustrious visitors to Hamburg. Bohr and Ehrenfest were frequent visitors.”

####  2 At Lenz’ institute

The conditions for doctoral candidates have been described[21] as very good:
“Joh[annes Hans Daniel] Jensen [(1907 - 1973)], at the time a student at the
seminary, told me that Lenz never stifled his time when it came to helping his
disciples; you could sit with him until late into the night and consult with
him. He did not settle for anything until the basic features of a problem were
clearly and simply worked out.”

When Ernst Ising joined the group of Lenz Werner Schröder had already finished
his thesis[22] and also Lucy Mensing joined the group. Access for women to
universities was made easier in the 1920s, but was tightened again in the
1930s under the National Socialist dictatorship. She worked on a quantum
mechanical problem[23] and was mainly supervised by Wolfgang Pauli, with whom
she later worked on the new quantum mechanics[24]. This time together at the
institute with Ernst led to a lifelong friendship. Ising himself describes his
situation[25] at the institute as follows: “At the time I wrote my doctor
thesis Stern and Gerlach were working in the same institute on their famous
experiment on space quantization[26]. The ideas we had at that time were that
atoms or molecules of magnets had magnetic dipoles and that these dipoles had
a limited number of orientations. We assumed that the field of these dipoles
would die down fast enough so that only interactions should be taken into
account, at least in the first order …I discussed the result of my paper
widely with Professor Lenz and with Dr. Wolfgang Pauli, who at that time was
teaching in Hamburg. There was some disappointment that the linear model did
not show the expected ferromagnetic properties.” Beside those discussions
Ising could also communicate with Mensing on the quantum mechanical background
of his model.

Regarding Pauli Meyenn [27] commented Pauli’s aversion to solid-state physics
but explained: “This is all the more remarkable given that he himself gave the
first impetus to the development of the quantum theory of the solid with his
theory of paramagnetism and also suggested a solid-state theme for Peierl’s
dissertation [Zur kinetischen Theorie der Wärmeleitung in Kristallen (On the
kinetic theory of heat conduction in crystals)]. Later, too, Pauli included
solid state physics in his lecture (in the summer semester of 1938) and also
took part in such events with interest. What Pauli fundamentally rejected was
not solid-state physics as such, but the description of physical effects by
approximation methods or as a consequence of seemingly uncontrollable
circumstances.”

###  4 The way to Ising’s thesis

Science progresses discovery by discovery.[St. Thurner, W. Liu, P. Klimek, S.
A. Cheong[28]]

At the beginning of the 20th century thermodynamic theories were developed to
give a practical quantitative description of the change of phases in condensed
matter with temperature. The first was the van der Waals theory with its
equation of state for gaseous and liquid phases in a fluid. The key of finding
an equation of state was the introduction of an internal pressure resulting
from the interaction between atoms or molecules. As a result two phases of
different density appear, which at the so-called critical point at a certain
pressure and temperature could continuously change into each other by going
from the high temperature gas phase to the low temperature liquid phase.

A quite similar phenomenon was observed in solids which could be magnetized by
an external field and stay magnetized when the field is turned off. Increasing
the temperature, the magnetization became weaker and vanished at a certain
temperature, the Curie temperature. This was called ferromagnetism in contrast
to para- and diamagnetism where the solid system remained unmagnetized when
the external field was turned off.

It was Pierre Weiss who introduced in 1908 the concept of the internal
molecular field[7, 29] in which the elementary magnets find their ordering and
later he related this field to the internal pressure in fluids. This brought
already at the beginning of studying phase transitions a kind of universality
to the phenomena[30].

####  1 Kirwan’s hypothesis - a lost historical root

The Curie Weiss theory was a big advance because of its eminent practical use
to explain the experimental findings in ferromagnets like the existence of a
phase transition, the change in the specific heat when changing the phase, the
qualitative increase of the magnetization in the ferromagnetic phase, and the
behavior of the susceptibility. However its central assumption of an internal
magnetic field resulting from the arrangement of rotatable elementary magnets
and their interactions was unclear and criticizable. Moreover quantitative
deviation especially near the Curie point remained.

A report[31] of a committee of the U.S. National Research Council issued by
the National Academy of Sciences, Washington on Theories of Magnetism was
published at the time when Ising started with his thesis. A translation[32]
was published in 1925 where the authors reworked the copy of the translation
and made a number of improvements and additions. One of the authors, E. M.
Terry, explains the nature of the molecular field: “Just as in the case of a
gas, to account for the transition to the liquid state, there must be added to
the external pressure an internal one due to the mutual attractions between
the molecules, so in the case of a ferromagnetic substance, as it is cooled in
a magnetic field from a temperature which has rendered it paramagnetic, the
transition to the ferromagnetic state is explained by assuming that, due to
the overlapping of the fields of the individual molecules, there comes into
existence an internal or molecular field, which added to the external field,
accounts for the very large intensity characteristic of this state.”

However the nature of the internal (molecular) field remained unclear and its
strength turned out to be too small in order to lead to the observed Curie
temperatures. Terry[31] wrote: “It thus appears that the molecular field can
have neither a magnetic nor an electromagnetic origin and must therefore be of
a nature different from the ordinary magnetic fields with which we are
familiar.” Moreover no model in the context of quantum mechanics was
available. J. Kunz[31] discusses different values of the magneton within the
Bohr atom and his comment ”No theory explains so far why the electron, moving
on a circular or elliptic orbit, about a nucleus, does not lose the energy by
radiation, or why the orbit remains stationary or free from radiation” shows
the difficulties of applying classical models in quantum mechanics.

Thus also Ising started his thesis directly with the statement that
ferromagnetism is a sofar unexplained phenomenon and opens his historic time
line of the different ideas on ferromagnetism with Richard Kirwan (1733 -
1812). The first paragraph of Kirwan’s paper[33] starts with a statement which
may be considered as a motto to Ising’s model:

There are two ways of explaining a natural phenomenon; the first, is by
discovering the conditions and circumstances of its production and the laws by
which its action is governed; the second, is by shewing its analogy,
similarity or coincidence with some general facts with whose laws and
existence we are already acquainted; this last mode is by far the most perfect
and satisfactory. In the first sense of the word electricity and magnetism
have been in some measure explained, but in the last sense neither; the
primary cause of magnetism in particular has hitherto been supposed to iron
alone, or its ores, and to stand unconnected with all other natural phenomena.

Kirwan an Irish scientists working in the fields of meteorology, geology,
chemistry, physics and philology was Fellow of the Royal Society, supporter of
the Copley medal and from 1799 until his death president of the Royal Irish
Academy. His publication Thoughts on Magnetism, which Ising cites, appeared
1797 in the Transactions of the Royal Irish Academy[33] and was later
translated into German. However the editor remarks: “I give these ideas, just
a little abbreviated as I find them in English, since I believe right away
that they are more witty than correct.”. This judgement is relativized by the
editor at the end of the article and he points to Kirwan’s idea to relate the
magnetic force to the force of crystallization ”yet their direction in all its
varieties being exactly the same”.

One might speculate how Ising came to recognize Kirwan’s paper. He may have
found it already in Felix Auerbach’s Geschichtstafeln der Physik (Historic
Tables of Physics) from 1910. Auerbach collected important progresses in
physics ordered in time from -650 to 1900. In this list he cites in the year
1797 for the topic: “Richtungshypothese für den Magnetismus” [Directional
hypothesis for magnetism] and mentioned Kirwan. Auerbach was already in 1895
the author of the section on magnetism in the Handbuch der Physik[34] [second
ed. 1908]. There on page 31 [48] he explained the history of the directional
hypothesis starting with Kirwan and further developed by Ohm[35] and
Weber[36]. It assumes that there are already elementary magnets but randomly
oriented so that the whole magnet is unmagnetized. By orientation in one
common direction the solid becomes magnetized.

In this way Ising connected the old ideas of Kirwan directly to his model
based on the modern development made by Lenz and Schottky.

####  2 Lenz’s idea of directed magnetons

In order to explain the ferromagnetic phase in solids Wilhelm Lenz
suggested[37] in 1920 that the elementary magnets in solids are not free
rotatable but are directed in certain directions defined by the geometry of
the solid. In order to stay simple he assumed that two directions are allowed,
namely the original direction and its reverse. Thus elementary magnets in the
solid have only the possibility to be parallel or antiparallel to each other.
For such a two state situation he calculated the susceptibility and reproduced
Curie’s law.

For ferromagnetic solids he then assumes that there is an energy difference
between parallel oriented and antiparallel oriented neighboring magnets. In
consequence an overall ferromagnetic phase may be explained. The energy
difference should result from a nonmagnetic interaction between the elementary
magnets.

Lenz was extraordinarius professor at the University Rostock when he published
the paper. In December 1920 the procedure for filling the new position of a
full professor of Theoretical Physics started for which Lenz applied. Einstein
was asked about the candidates and wrote in a letter[17] to Edgar Meyer: “I
estimate Lenz similar to Debye and especially consider his [Lenz’s] last, for
the time being only very incomplete publicized works on band spectra and
magnetism extremely important.” Finally Wilhelm Lenz got the position and
started his job in Hamburg 1922 by organizing a so called Vortragsseminar
(lecturing seminar) where also notable external guests participated.

####  3 Stern-Gerlach experiment and spatial quantization

An open question in the Bohr-Sommerfeld quantum mechanics was the quantization
of the orientation of the angular momentum and the orientation of the
corresponding magnetic momentum in an external field. In order to solve this
problem Otto Stern 1919 suggested an experiment which was performed in 1922
together with Walther Gerlach at the University of Frankfurt[38]: “At the time
when the SGE [Stern Gerlach experiment] was performed, the physics community
did not understand why and how the internal magnetic moment (i.e. angular
momentum) of each atom collapses in the SG-apparatus into well-defined angular
orientations with respect to the direction of the outer magnetic field. This
clearly contradicted classical physics where a Larmor precession of the
magnetic moments was expected. For most physicists this was a miraculous
interaction between moving atoms and the SG apparatus.” The outcome of the
experiment was interpreted to show spatial or directional quantization but in
fact it an additional property of the electron later recognized as the spin of
the electron was observed. Pauli sent a post card on February 17, 1922 to
Gerlach with the remark[39]: “This should convert even the nonbeliever [in
spatial quantization] Stern.”

####  4 Schottky’s synchronous electrons

On September 19, 1922 Walter Schottky came from the University of Würzburg for
the winter semester 1922/23 to the University of Hamburg as a salaried
assistant at the Physical State Laboratory. At the Verhandlungen der
Gesellschaft Deutscher Naturforscher und Arzte (Meeting of the Society of
German Natural Scientists and Physiciens) in Leipzig he presented his
results[40, 41] published in 1922.

He corroborated Lenz’s idea by suggesting a quantum mechanical model for the
energy difference of parallel and antiparallel elementary magnets. This paper
is nowadays known for a phenomenon in the specific heat he calculated for the
two state system presented in his paper named Schottky-anomaly. The specific
heat as a function of temperature shows a peak at low temperature before it
goes to zero instead of going to zero continuously.

Schottky considered neighboring electrons circulating like planets around
their atomic nucleus and asked the question: how have two neighboring
electrons to behave in the Bohr Sommerfeld picture in order to minimize their
Coulomb energy? He came to the conclusion: “In the field of magnetism, the
introduced roto-active interactions of the atoms seem to provide the
previously unknown type of directional force, which produces the self-ordering
of the magnetic atom axes and thereby becomes the cause of the spontaneous
magnetization and the ferromagnetic behavior of certain substances.” In order
to show this he calculated the Coulomb energy of the two electron circling on
their atomic path (‘planetary’ circles) around the nucleus of the atoms and
showed that a minimum is reached, when the electrons rotate in the same
direction [synchronism]. In such a case the elementary magnets would be
oriented in the same direction, i.e. they would prefer to be parallel. He gave
a result for the energy gain $\Delta E$ when being parallel

$\Delta
E=\frac{e^{2}}{a}\Big{(}1-\frac{1}{\sqrt{1+k(\frac{r}{a})^{2}}}\Big{)}$ (1)

where $e$ is the electron charge, $a$ the distance of the atoms nuclei in the
solid and $r$ the radius of the electron around the nucleus. The value of $k$
depends on the geometry of the electron orbits. If they are above each other
$k=4$ if they are in the same plane $k$ is about 2. No explicit calculation is
given. A more detailed discussion has been published recently[42].

When one now considers two neighboring atoms in the solid, they may be
arranged besides each other in the same plane or above each other. The
clockwise or counter clockwise rotation of the electron corresponds to a
parallel or antiparallel orientation of the magnetic moment created by the
moving charge. During their rotation the two electrons approach each other and
the minimal distance defines the energy necessary for the continuos rotation.
This minimal distance depends on the phase difference of the circulation. In
each case the electrons are synchronized [that means the phase adjusts itself]
in such a way that the minimal distance is as large as possible. It turns out
that in each case either above or beside each other the energy necessary is
smallest if the rotation is in the same direction and the magnetic moments are
parallel.

In consequence their induced elementary magnetization is parallel. The energy
difference $\Delta E$ to the antiparallel configuration turned out to be of
the order of the Curie temperature if one inserts reasonable values for the
respective parameters of the solid ferromagnets ($T_{c}\approx\Delta
E/k_{B}$). He called this interaction roto-active directivity [rotoaktive
Richtwirkung]. Finally the reason for this behavior of the electrons remained
unknown.

However Schottky [41] suggested that this interaction is of electrostatic
nature and based his considerations on an already general principle the
synchronismus used in the planetary motion of electrons in atoms or between
atoms [43]. The idea was that in crystals the structure results from the
electrostatic attraction and repulsion between the atomic nucleus and the
(planetary like) moving electron. Landé wrote: “The regular structure of the
monatomic crystals also leads to the conclusion that all particles are ordered
in their electron movement phases.”. This phenomenon was called lattice
synchronism (Gittersynchronismus) and Schottky used it now for the ordering of
the magnetons.

Figure 1: The specific heat of a system with two different energy levels. The
peak is called Schottky anomaly. Fig. from Ref. [41].

As already mentioned, Schottky’s paper is nowadays know for the calculation of
the specific heat in a two-state system. The presence of this two states
differing in energy lead to a peak in the specific heat at low temperature,
named Schottky anomaly (see Fig. 1) which is given by

$C/k_{B}=\frac{(\Theta/T)^{2}e^{-\Theta/T}}{(1+e^{-\Theta/T})^{2}}=\Big{[}\frac{(\Theta/T)}{\cosh(\Theta/2T)}\Big{]}^{2}$

As late as 1937 the specific heat of the Ising model was calculated by
Bitter[44] without recognizing that it shows an anomaly as calculated by
Schottky. In fact the Ising chain in zero external field is dual to the two
state model (see Sect. 2 below).

###  5 A hard task for Ising

The more complicated the system considered, the more simplified must its
theoretical description be. …A good theory of complicated systems should
represent only a good ”caricature” of these systems, exaggerating the
properties that are most difficult, and purposely ignoring all the remaining
inessential properties.[Frenkel 1946 cited in 1962 by Tamm[45]]

After the results of Schottky’s ideas on a possible interaction were published
Lenz proposed Ernst Ising to find out if his idea from 1920 can be formulated
in a model showing a ferromagnetic phase for his dissertation. One may cite
here Heisenberg who said[46]: “What we observe is not nature itself, but
nature exposed to our method of questioning.” How do we find out if a
magnetized phase exists? At the end of the year 1922 Ernst Ising started with
his thesis to answer this question.

In principle it was clear how to attack the problem. In fact it looked easy
since the formalism was already developed, namely one had to calculate the
partition function of the system in an external magnetic field. Then one could
look if a magnetized phase could be found by deriving the free energy by the
magnetic field and setting the external magnetic field to zero.

Indeed Ising followed[47] the standard path known in statistical physics
developed by Boltzmann and/or Gibbs. The magnetization $\mathfrak{J}$ (his
notation is used) is obtained by deriving the free energy (a function of
temperature $T$ and external field $\mathfrak{H}$) after the external field.
The free energy in turn is calculated from the partition function $Z$ thus

$\mathfrak{J}=\mathfrak{m}\frac{\partial}{\partial\alpha}\log Z(T,\alpha)$ (2)

with $\alpha={(\mathfrak{m}\mathfrak{H})}/{k_{B}T}$, $\mathfrak{m}$ the
elementary magnets, which can only be parallel or antiparallel and $k_{B}$ the
Boltzmann constant.

The partition function contains the number $\cal N$ of microstates multiplied
by their Boltzmann’s weight given by the energy of the microstate. It has to
be noted that no Hamiltonian could be written down for the microstates due to
the vague microscopic model of Schottky. In order to define the microstates of
the sytstem Ising introduced the notation shown in Fig. 2.

Figure 2:  Ising’s representation of the microstates: Antiparallel
configurations cost energy (red arrows), the direction of the arrows are
irrelevant and therefore Ising replaced them by + and -. . The picture is
taken from the original Ising’s thesis, red arrows are added by the author.

He counted the elementary magnets in (against) the external field direction by
$\nu_{1}$ ($\nu_{2}$) and assigned pluses and minuses to them. The places
where two elementary magnets are antiparallel he called ‘energy places’ (red
arrows) since this configuration with respect to the parallel one needs the
additional energy $\epsilon$. There are $\sigma$ energy places counted. These
three numbers $\nu_{1}$, $\nu_{2}$ and $\sigma$ define now the microstates and
one has to find their number

${\cal N}(\nu_{1},\nu_{2},\sigma).$

Ising starts from $\nu_{r}(m)$ the number of ways in which the integer $m$ may
be represented as a sum of $r$ integer numbers. The result is the binomial
coefficient

$\nu_{r}(m)=\binom{m-1}{r-1}.$

Now he applied this to positive and negative elements in the chain and
observed the configuration of the last boundary element [$\delta=0,1$] if it
is parallel or antiparallel to the first boundary element. This gives the
result for the number of microstates

${\cal
N}(\nu_{1},\nu_{2},\sigma=2s+\delta)=\binom{\nu_{1}-1}{s}\binom{\nu_{2}-1}{s+\delta-1}+\binom{\nu_{2}-1}{s}\binom{\nu_{1}-1}{s+\delta-1}.$

The energy of the system is given by the difference in number of elementary
magnets in and against the external field $(\nu_{1}-\nu_{2})$. The number of
energy states is $\sigma=2s+\delta$, where $s$ is the number of the ‘internal’
energy states. Then the partition function $Z(n)$ for $n$ elementary magnets
in the chain [$\alpha\sim\mathfrak{H}$; $\beta\sim\epsilon$] reads

$Z({n})=\sum_{\nu_{1}+\nu_{2}=0}^{n}\sum_{s=0}^{\infty}\sum_{\delta=0}^{1}{\cal
N}(\nu_{1},\nu_{2},\sigma=2s+\delta)e^{(\nu_{1}-\nu_{2})\alpha-(2s+\delta)\beta}.$

Ising now proceeded with a clever trick. First he introduced an auxiliary
variable $x$ and defined a function $F({x})$

$F({x})=\sum_{{n}=0}^{\infty}Z({n}){x}^{n}$

by adding up the partition functions for all different length $n$ of chains
and calculated the sum over $n$:

$\displaystyle{F}(x)=$
$\displaystyle\sum_{s=0}^{\infty}\sum_{\delta=0}^{1}B^{2s+\delta}\left[\left(\sum_{\nu_{1}=0}^{\infty}\binom{\nu_{1}-1}{s}(xA_{1})^{\nu_{1}}\right)\left(\sum_{\nu_{2}=0}^{\infty}\binom{\nu_{2}-1}{s+\delta-1}(xA_{2})^{\nu_{2}}\right)\right.$
$\displaystyle\qquad\quad\left.+\left(\sum_{\nu_{1}=0}^{\infty}\binom{\nu_{1}-1}{s+\delta-1}(xA_{1})^{\nu_{1}}\right)\left(\sum_{\nu_{2}=0}^{\infty}\binom{\nu_{2}-1}{s}(xA_{2})^{\nu_{2}}\right)\right].$

The number of elementary magnets $n$ has disappeared and was replaced by the
auxiliary variable $x$. In order to get back the partition function for length
$n$ he now expanded the function in powers of $x$ and looked for the
coefficient of $x^{n}$ for large ${n}$. This gives back the partition function
for the length $n$:

${F}({x})=\frac{2{x}[\cosh\alpha-(1-e^{-\epsilon/k_{B}T}){x}]}{1-2\cosh\alpha\cdot{x}+(1-e^{-2\epsilon/k_{B}T}){x}^{2}}=\sum_{\mathbf{n}=0}^{\infty}\left(\frac{a_{1}}{w_{1}}{w_{1}}^{\mathbf{n}}+\frac{a_{2}}{w_{2}}w_{2}^{\mathbf{n}}\right){x}^{\mathbf{n}}$

with $w_{1}=\cosh\alpha+\sqrt{\sinh^{2}\alpha+e^{-2\beta}}$ and $w_{2}<1$. In
this way he recovered the partition function for the chain in the
thermodynamic limit by neglecting the terms $w_{2}$ which decays for large
$n$.

The magnetization $\mathfrak{J}$ is then given by the derivative of the
partition function $Z$ with respect to the field
$\alpha={(\mathfrak{m}\mathfrak{H})}/{k_{B}T}$

$\mathfrak{J}=\mathfrak{m}\frac{\partial}{\partial\alpha}\log
Z=\mathfrak{m}n\frac{\partial}{\partial\alpha}\log{w_{1}}$

with the result

${\mathfrak{J}=\mathfrak{m}n\frac{\sinh\alpha}{\sqrt{\sinh^{2}\alpha+e^{-2\beta}}}}.$

From this one concludes that in the case of zero external field, $\alpha=0$,
there is no finite magnetization at temperatures larger than zero. In
consequence there is no ferromagnetic phase at any finite temperature in the
linear chain.

Ising tried of course to apply his method to more complicated cases in order
to see if he could calculate the partition function in higher dimensions in
presence of an external magnetic field. This is even today not possible, thus
he could only extend his calculations by making further simplifications and/or
approximations in order to proceed to results. So he looked for the cases of

  * •

the ladder or double chain,

  * •

the chain with next nearest neighbor interaction, and

  * •

a chain with more than two possible directions of the elementary magnets at
one place.

None of these extensions made a ferromagnetic phase possible and so Ising
ended his thesis with the supposition that also in the three dimensional case
the model does not lead to a ferromagnetic phase at low temperatures.

###  6 The evaluation of Ising’s result

####  1 Lenz approval of Ising’s thesis

Of course Lenz was also disappointed by this result and as can be seen by his
approval of the thesis[48]: “The more than hundred-year-old theory of
ferromagnetism has come to a certain formal conclusion in the papers of P.
Weiss. These makes it possible to express the thermal behavior of a
ferromagnet in a formal way. However, the Weissian developments can not be
regarded as a satisfactory theory, since their foundation, the so-called
molecular field, themselves has the character of an auxiliary approach. A
satisfactory theory must be based on the behavior of the atoms of a solid
body, for which Bohr’s theory provides the necessary clues. Following a theory
of thermal behavior of the paramagnetic salts that satisfies these
requirements, I have suggested to the candidate to extended these ideas to
ferromagnetism. The necessary rather complicated probabilistic considerations
have been carried out by the author with remarkable skill. They led to the
conclusion that ferromagnetism is not realized on the path taken, and the
reasons for this are being discussed. Since a change of the basic ideas about
the atomic properties and their interaction is not an option, the question
arises as to whether the ferromagnetic state can at all be regarded as a state
of thermal equilibrium. The study of this possibility, however, would have far
exceeded the scope of this thesis.

The overall negative result must be viewed as an important statement for the
continuation of the theory. The work (deserves the) also provides some new
methodological aspects and deserves the predicate: ,,very,,good” Lenz”

Cancellation by Lenz, he also first[17] gave the mark ,,good” but changed it
into ,,very good” as seen by the quotation marks. He supported his decision by
the remark on the new methodical aspect of Ising’s work. Indeed one finds
later citations of Ising’s paper pointing to his combinatorial methods. One
was published in 1939 by A. H. Mood in the paper[49] based on his thesis and
the other one in 1942 by T. S. Chang and C. C. Ho [50]. T.S. Chang was a
student of Ralph H. Fowler working in statistical physics at the University of
Cambridge, United Kingdom. Fowler and Guggenheim published in 1939 the well
known textbook Statistical Thermodynamics where the combinatorial methods are
used explaining order-disorder phase transitions in alloys.

Surprisingly Lenz did not explicit notice that Ising has shown that the Curie-
Weiss theory is wrong in the one-dimensional case. This failure of the mean
field approximation was corrected ten years later by Bethe (see Sect. 10).

####  2 The reduction made in the publication

After finishing his thesis Ising submitted on December 9, 1924 a short summary
of his thesis to Zeitschrift für Physik[51]. In order to be short he
drastically reduced the references. Thus one may count about 18 papers in the
thesis and only 3 papers in the publication. Most important he did not cite
the oldest and the newest reference namely those which concerns Kirwan’s[33]
and Schottky’s paper[41]. He instead just mentioned: “We assume that the
internal energy is smallest when all elements are in the same direction.” Both
references in fact dropped out of the history of ferromagnetism. Thus only two
references of Weiss[29, 52] to the molecular field theory (in French and
German) and Lenz’s paper[37] remained.

Although the result of Ising’s publication is generally seen as negative - no
magnetic phase in the three dimensional system - on the other hand it is
positive in the formulation of a model and by calculating an example of the
failure of the mean field theory in the case of one dimensional systems.
Fortunately the negative result of the model was published leading to an
unforeseen success in the following decades. The tendency that negative
results nowadays are disappearing in most disciplines has been studied by
bibliometric methods[53].

####  3 Naming the model

There is on and off a discussion how to name the model which was suggested by
Wilhelm Lenz and mathematically defined and solved in one dimensions by Ernst
Ising. In the first review on this model Brush [25] noted: “Lenz himself
apparently never made any attempt later on to claim credit for suggesting the
model, and even his colleagues at Hamburg University were not aware of his
contribution [Professor Dr. H. Raether in a private communication to Brush].”
However Brush decided to name the model “Lenz-Ising Model” possibly due to the
correspondence with Ising. But in 1971 in Vol 2 of The Kind of Motion We Call
Heat (p.12) he wrote[54]:

…it is usually futile for historians of science to try to change established
nomenclature,…

He added in footnote 21:

My attempt to add the name of Wilhelm Lenz to the “Ising Model” in statistical
mechanics [Rev. Mod. Phys. 39, 883 (1967)] has been a failure.

Indeed looking at citations in the publications dealing with this model almost
all authors name it “Ising model”. A rare exceptional case is the
publication[55] by H. A. Kramers at the magnetism conference in Strasbourg
1939 but later in 1941 he cited the model as Ising model too. However others
kept the naming in later publications [referring to Brush’s review, but
without knowing his withdrawal]. In the relevant scientific papers looking for
its solution the model is always cited as Ising Model.

###  7 Ising leaving Hamburg and his new phase of life until 1930

Johanna and Ernst ©Tom Ising

  1. 1925

Job at the patent office of the Allgemeine Elektrizitäsgesellschaft (AEG)

  2. 1926

Ernst Ising meets Johanna Ehmer

  3. 1927

Teacher at the boarding school in Salem

  4. 1928

Studying philosophy and pedagogy at Berlin University

  5. 1930

State examen; teaching position Strausberg near Berlin as Studienassessor
[higher civil service]; marrying Johanna Ehmer who held a doctorate in
economics

  6. 1932

Ernst transfered from Strausberg to Crossen/Oder

After Ernst Ising had finished his thesis and his publication he left Hamburg
in 1925 and went to Berlin. He found a job in the patent office of the General
Electric Company [Allgemeine Elektrizitäsgesellschaft (AEG)]. There in Berlin
on June 6, 1926 he met Johanna Ehmer at a Sunday outing of the Pacifistic
Student organization[56].

Johanna was born in 1902 in Berlin, her father being a drugstore owner her
mother a bookkeeper with various businesses. After graduation from the
gymnasium Johanna enrolled in mathematics and physics at the University of
Berlin since in these subjects she was excellent in school. However after
three semesters she switched to economics and proceeded her studies at the
College of Commerce (Handelshochschule) where she earned the commercial
teacher’s diploma in spring 1924. In 1926 she finished her thesis supervised
by Charlotte Leubuscher, who was the first woman in Germany habilitated in
humanities. Looking for a position, Johanna became Research Assistant at the
Department of Economics assigned to Professor Franz Eulenburg.

Ernst Ising was not satisfied by his job and decided to switch to teaching.
Therefore he resigned and found a job at the boarding school in Salem (South
Baden) in 1927 and took some classes at the University of Berlin in order to
finish the the Prussian State examination in 1928. After two years more of
teacher training he became “Studienassesor” (the civil servant position at a
high school) and on December 23, 1930 (six years after finishing his thesis)
Ernst and Johanna married. Ernst had to teach in Strausberg near Berlin and
later in fall 1932 in Crossen/Oder far east of Berlin.

###  8 The model after the thesis until its modern formulation in 1930

####  1 Herzfeld 1925

Karl Herzfeld was the first at the Annual Meeting of German Physicists in
Danzig to refer [57] to Ising’s publication in the year of its publication.
Herzfeld had studied in Vienna and his thesis was supervised by Hasenöhrl.
After five years in Munich he became in 1925 an extraordinary professor there.
A year later he went as a visiting professor to the United States and remained
there until his death in 1978 [58]. Herzfeld gave a review with the title
Molecular- and Atomic Theory of Magnetism in which he also refered to
Schottky’s idea of synchronism and roto-active interaction citing several
publications of Landé [43], Born, Heisenberg[59] und Nordheim [60]. Herzfeld
was not convinced in the argumentation concerning the interaction assumed. He
criticized the restriction to only nearest neighbor interaction and argued
that in such a case the energy is the same irrespective of only one dipole or
a whole group of dipoles lies antiparallel.

####  2 Pauli and the New QM

The “old” Bohr-Sommerfeld quantum mechanics used three quantum integer numbers
for the classical orbital motion of an electron corresponding to characterize
the planetary-like path of the electron around the nucleus. This model could
explain the simplest situation in an atom with one electron but failed in more
complicated cases e.g when more than one electron is present or when external
fields are present.

Pauli was struggling to understand atomic magnetism. He tried to explain the
anomalous Zeeman effect but did not succeed [61]. He used the neighborhood to
Copenhagen and worked on sabbatical at Bohr’s institute from September 1922 to
October 1923. On 17 January 1924 he submitted his application for habilitation
and on February 23, 1924 he already held his inaugural lecture [17]. In the
rest of the year 1924 he developed further his idea of a fourth quantum number
for the electron[62] \- his “classically not describable two-valuedness in the
quantum mechanical properties of the valence electron” (for the non-classical
character see W. Pauli’s Scientific Work by Ch. Enz[63] p. 29). It was in the
same issue of Zeitschrift für Physik received seven days earlier as, and
printed 120 pages behind of Ising’s paper. Shortly later on January 16, 1925
the exclusion principle [64] was received at the journal. This principle says:
“There can never be two or more equivalent electrons in the atom for which the
values of all quantum numbers … coincide in strong fields. If there is an
electron in the atom for which these quantum numbers (in the outer field) have
certain values, this state is occupied…. We cannot give a more detailed
explanation for this rule, but it seems to be very natural on its own.” No
relation between Ising’s model for explaining macroscopic ferromagnetism and
Pauli’s new quantum degree of freedom of the electron to explain the
microscopic magnetism in the atom was seen at that time.

Pauli describes his motivation in the lecture[65] given in Stockholm after the
award of the Nobel Prize in Physics in 1945: (1) “One was an effort to bring
abstract order to the new ideas by looking for a key to translate classical
mechanics and electrodynamics into quantum language which would form a logical
generalization of these.” (2) The second was “a direct interpretation, as
independent of models as possible, of the laws of spectra in terms of integral
numbers, following, as Kepler once did in his investigation of the planetary
system, an inner feeling for harmony.”

Already at that time Pauli was in close contact with Werner Heisenberg. In the
years 1925 to 1928 forty letters from Heisenberg to Pauli are archived, but
unfortunately Pauli’s answers are not conserved. C. F. von Weizsäcker
remembers[66] a remark of Heisenberg: “I never have published a work without
Pauli reading it first. If he said it was wrong, it could still be right; but
then caution was needed.” Both Pauli and Heisenberg shared the doubts on the
classical picture of the atom as planetary system. Thus Heisenberg wrote to
Pauli on June 9, 1925: “It is really my conviction that an interpretation of
the Rydberg formulas in lines of circles and elliptical orbits in classical
geometry does not have the slightest physical sense and all my poor
calculations go to the point to kill completely the term orbits that cannot be
observed anyway, and to replace it properly.” Twenty days later Heisenberg
submitted his famous paper[67] on the new interpretation of quantum mechanics
for publication.

The same reservation had Pauli against a classical interpretations of the new
degree of freedom of the electron. A first interpretation given by Ralph
Kronig was not submitted due to the criticism of Pauli and a contribution[68]
to the spin appeared later. It was Goudsmit’s and Uhlenbeck’s paper[69] of the
year 1925 on the spinning electron which explained the fourth degree of
freedom. With the rotating charged electron a magnetic moment was connected.
The problem with the size of this moment (the electron had to rotate with a
velocity at the circumference higher than the velocity of light) was solved
later by taking into account relativistic effects[70] by Thomas.

Meanwhile also the new quantum mechanics was further developed by Born, Jordan
and Heisenberg and Pauli contributed[71] to that development by solving the
hydrogen atom within the new formalism. Moreover Schrödinger formulated
another interpretation[72] of quantum mechanics - the wave mechanics in
contrast to Heisenberg’s matrix mechanics and showed its equivalence.“A new
mathematically essential more convenient access to the field of quantum
mechanics” as Heisenberg[73] agreed.

####  3 The New QM and Heisenbergs exchange interaction

As early as on November 4, 1926 Heisenberg communicated some ideas about
ferromagnetism to Pauli. He wrote (underlining due to the original): “The idea
is this: In oder to use Langevin’s theory of ferromagn., one has to assume a
strong coupling force between the spinning electrons (only these turn around).
As with helium, this force should be supplied indirectly by the resonance. I
think one can generally prove: parallel positioning of the spin vectors always
gives smallest energy. The energy differences that come into consideration are
of the electrical order of magnitude, but decrease very rapidly with
increasing distances. I have the feeling (without even knowing the material
remotely) that this could in principle be sufficient for an interpretation of
ferromagnetism.”

But it needed one and a half year more until he was able to bring these ideas
into the mathematical formalism of the new quantum mechanics. An important
step was to include the symmetry property of an system of electrons. This was
done almost simultaneously by Heisenberg[73] (June 11) and Dirac[74] (August
26) who explained: ”If the positions of two of the electrons are interchanged,
the new state of the atom is physically indistinguishable from the original
one.” A second important step further was the introduction of the spin by
2$\times$2 matrices into the formalism of quantum mechanics by Pauli[75].

On May 20, 1928 the paper of Heisenberg was received by the journal
Zeitschrift für Physik. In his one sentence abstract he proclaimed[76, 77]:
“Weiss’s molecular forces will be attributed to a quantum-mechanical exchange
phenomenon, and indeed, it will be treated as the exchange process that was
successfully enlisted in recent times by Heitler and London in order to
interpret homopolar valence forces.” He based his theory on the Pauli-Fermi-
Dirac statistic property of the electron and the Pauli principle. The
eigenfunctions of the quantum mechanical system are constructed by allowing
electrons to be exchanged between neighboring atoms. Then the energy depends
on the “very important constant that enter into the perturbation calculations
is the purely static [Coulomb] interaction”:

$J_{E}=\int d\tau_{1}d\tau_{2}\ldots
d\tau_{2n}(\psi^{1}_{1})^{2}(\psi^{2}_{2})^{2}\ldots(\psi^{2n}_{2n})^{2}\big{(}\sum\frac{e^{2}}{r_{kl}}+\sum\frac{e^{2}}{r_{\kappa\lambda}}-\sum\frac{e^{2}}{r_{k\lambda}}\big{)}$
(3)

where $k$ and $l$ denote the electrons and $\kappa$ and $\lambda$ the atoms
and $\psi^{\kappa}_{k}$ the corresponding spatial part of the wave function.
This energy is the Coloumb energy and Heisenberg argues: “due to their
smallness, we can leave the magnetic interaction outside of consideration.” In
fact this establishes within the formalism of the new quantum mechanics, what
has been suggested by Schottky within the old quantum mechanics[42] (see
section 4). Due to the Pauli principle the wave functions lead to a charge
distribution of the electrons which minimizes the Coulomb energy for two
neighbouring electrons with parallel spin. Thus the vague synchronism was
replaced by new nonclassical physical principles.

With this Heisenberg calculated the partition function in Gaussian
approximation in order to find the Curie temperature $\Theta$. A finite value
results in this approximation

$\Theta=\frac{2J_{0}}{k_{B}\big{(}1-\sqrt{1-8/z}\big{)}}$ (4)

where $k_{B}$ is the Boltzmann constant and $z$ the number of nearest
neighbours. From this expression for $\Theta$ Heisenberg concluded that in
three dimensions a phase transition is possible only for $z>8$. This was
considered to confirm the result of Ising since no real finite transition
temperature exists for the chain where $z=2$. The value for the magnetization
turned out to be essentially in agreement with Weiss’s theory.

A final overall picture on the microscopic level for the description of the
electron was given in 1928 by Dirac[78]. He wrote: I “would like to find some
incompleteness in the previous methods of applying quantum mechanics to the
point-charge electron such that, when removed, the whole of the duplexity
phenomena [the spin of the electron] follow without arbitrary assumptions.” In
an interview around 1982 by Hund[79] he further explained: “that forced me
into a different kind of equation [now the famous Dirac equation] and this
different equation brings in the spin of the electron. It was very unexpected
to me to see the spin appearing in that way. [spin means both moment of
momentum and magnetic moment Hund adds] I thought one would have a
satisfactory theory without spin and then one would proceed to a more
complicated theory.” Thus the spin and magnetism of the electron turned out to
be an inevitable property emerging out of quantum theory and special
relativity[80].

Now all the necessary microscopic details have been developed including an
appropriate mathematical method to describe the non-classical two-valuedness
and more than five years after Ising had formulated his model it became
possible to set up a mathematical description by defining a Hamiltonian for
it.

####  4 Pauli and Heisenberg on the Ising model and the thermodynamic limit

##### Heisenberg’s comments on Ising’s model 1928

Ising tried to set up a model in more than one dimension, but this was based
in fact on simplifications and/or approximations in the calculation of the
partition function, which did not lead to ferromagnetism and in fact not to a
true three-dimensional model. After the new developments of the microscopic
basis of possible macroscopic magnetism Heisenberg [81] criticized Ising’s
conclusion:

“The whole question [about ferromagnetism] seems important to me because of
the similarity between my model and that of Ising. According to my current
view, Ising would have gotten ferromagnetism, if he had accepted enough
neighbours (about $z\leq 8$) [Authors remark: Even if one accumulates sheets
of planar Ising planes to a cubic model it would have only $z=6$]. After the
argument that Ising has published for the spatial model, it seems to me at
all, as if he had not understood his work. As long as one does not go over to
the limit $n\to\infty$, it is trivial that for $H=0$ also $J=0$, and in fact
with parallel position of all magnets [footnote: One just applies Langvin’s
theory for the magnetic rod as a whole!] becomes $J$ prop.
$\mathfrak{Tg}$(n$\alpha$). In Ising’s linear chain parallel arrangement of
all spin vectors occurs if $e^{-2\epsilon/kT}\sim\frac{1}{n}$ holds, then
$T=\Theta\sim\frac{2\epsilon}{k\lg n}$; the Curie point $\Theta$ is thus
dependent on $n$ here and goes to zero as $n$ increases. For the wild spatial
model of Ising, however, would be $\Theta\sim\frac{2\epsilon n^{3/2}}{k\ln
n}$, $\Theta$ would go to infinity with increasing n, i.e. parallel
arrangement of all magnets would always exist. The fact that Ising conceived
this model as an argument against ferromagnetism seems to me a sign that he
has not understood his own work remotely. What do you think about it?”

Unfortunately Pauli’s answers are lost. Nowadays we know that the spatial
dimension is a relevant parameter for the existence and nonexistence of a
phase transition and not the numbers of nearest neighbours. Heisenberg argues
with his result obtained in Gaussian approximation. Ising’s result for the
chain is exact and holds in the thermodynamic limit this includes extensive
quantities which are proportional to $n$.

In a finite chain the probability that all spins are parallel is indeed finite
and given by

$P=\frac{1}{2}\Big{(}\frac{e^{\epsilon/k_{B}T}}{\cosh(\epsilon/k_{B}T)}\Big{)}^{n-1}$
(5)

which is below some temperature almost $1/2$ and leads to a ‘transition
temperature’ which goes to zero with $n\to\infty$. The ordering in finite
chains has been studied[82] in 2005. There the authors “have derived a
criterion that a finite Ising chain exhibits the ferromagnetic like behavior.
According to it, the transition time between the fully magnetized chain states
must exceed the measuring time, and the average number of domain walls must be
much less than 1. These conditions hold, i.e., a finite Ising chain does
display a ferromagnetic like order on the measuring time scale, if the
temperature is sufficiently small.”

##### Pauli at the Solvay conference in 1930

Figure 3:  Solvay conference 1930. Standing: E. Herzen, E. Henriot, J.
Verschaffelt, C. Manneback, A. Cotton, J. Errera, O. Stern, A. Piccard, W.
Gerlach, C. Darwin, P.A.M. Dirac, E. Bauer, P. Kapitsa, L. Brillouin, H. A.
Kramers, P. Debye, W. Pauli, J. Dorfman, J. H. Van Vleck, E. Fermi, W.
Heisenberg; seated in front: Th. De Donder, P. Zeeman, P. Weiss, A.
Sommerfeld, M. Curie, P. Langevin, A. Einstein, O. Richardson, B. Cabrera, N.
Bohr, W. J. De Haas. Pauli presented the Ising model in his talk and eleven
years later Kramers with Wannier proofed Pauli’s conjecture and calculated the
transition temperature for the two-dimensional Ising model. From Ref. [84].

Pauli was invited to present a review talk [83] at the 6th Solvay conference
in Brussels from October 20 to 25 with the title Les Théories Quantiques du
Magnétisme: L’électron Magnétique [The Quantum Theories of Magnetism: The
Magnetic Electron].

He expounded how the new development on the microscopic level in quantum
mechanics as the spin, the new formulation of by Heisenberg, Schrödinger and
Dirac and the exchange interaction contributed to the understanding of
magnetism and especially of ferromagnetism. With this he was able to formulate
the Hamiltonian function $\mathcal{H}$ for the Ising model (he considers the
chain)

${\mathcal{H}}=-A\sum_{k}(\sigma_{k},\sigma_{k+1})$ (6)

where the $\sigma_{k}$ is the spin matrix corresponding to the direction of an
possible external field and might be chosen to be in the $z$-direction. The
strength of the exchange interaction of two neighboring electrons indexed by
$k$ and $k+1$ is given by $A$. The sum is understood over all neighbours. This
has to be distinguished from the Heisenberg model

${\mathcal{H}}=-A\sum_{k}({\vec{\sigma}}_{k},{\vec{\sigma}}_{k+1})$ (7)

where the components of the vector contain all three spin matrices.

Pauli expressed his expectation that Ising’s model could lead to a phase
transition in higher dimensions. He stated in the discussion[83] of the
Hamiltonian (7): “There is in fact a very close relationship between the
problem of Ising and the one we have just treated” , and explains: “In Ising’s
calculation developed from the point of view of the old quantum mechanics, the
components of $\sigma_{i}$ that are perpendicular to the field are considered
to be zero, whereas in the new quantum theory these components do not commute
with the components in the direction of the field.” But he immediately
suspects for the Ising model (6): “Irrespective of this difference, it is
quite likely that an extension of the theory of Ising to the case of a lattice
of three dimensions would yield ferromagnetism even from the classical point
of view”.

###  9 Ising as a teacher from 1933 until the November pogrom in 1938

Ernst teaching in Caputh

©Herbert Sonnenfeld, Jüdisches Museum Berlin

  1. 1933

March 31 Dismission

  2. 1934

Teacher in Caputh

  3. 1937

Headmaster of the school

  4. 1938

November 10 Destruction of the school during the Pogrom

The Paradise Lasted A Year And A Half[56] for Ernst and Johanna until Hitler
came to power in March 1933 by the Enabling Act. The Enabling Act allowed
Hitler and his Cabinet to rule by emergency decree for four years. New
antisemitic laws followed which forced all Jewish scientists (but not only
them) to resign their position and/or leave Germany. On April 7, 1933, the Law
for the Restoration of the Professional Civil Service (Gesetz zur
Wiederherstellung des Berufsbeamtentums) came into force[10], which concerned
Ernst since he was dismissed from his teaching post in public schools. On May
1, 1933, the University of Hamburg took the Vow of allegiance of the
Professors of the German Universities and High-Schools to Adolf Hitler and the
National Socialistic State.

From one day to the next, all government jobs held by Jews were clearly at
risk. At the universities, all of them state institutions, waves of dismissals
began promptly[85] with all its consequences on the existing network of
physicists. These consequences for German and Austrian science and its future
were estimated[86] and it was indicated that the reduction in output in
departments with dismissals persisted at least until 1980.

Figure 4: Career path[87] of Nobel physicists starting from the birth place
(yellow) (a) for Stern until the escape 1933 and (b) for Pauli until the move
1940 to the United States ©Physics Today

Otto Stern resigning from his post at the University of Hamburg, found refuge
in the city of Pittsburgh becoming a professor of physics at the Carnegie
Institute of Technology. Wolfgang Pauli had already left the university in
1928, when he was appointed Professor of Theoretical Physics at ETH Zurich in
Switzerland. The German annexation of Austria in 1938 made him a German
citizen, which became a problem for him in 1939 after the outbreak of World
War II. In 1940 he tried in vain to obtain Swiss citizenship, which would have
allowed him to remain at the ETH. Pauli instead moved to the United States in
1940, where he was employed as a professor of theoretical physics at the
Institute for Advanced Study. Not until 1946, after the war, he became a
naturalized citizen of the United States and subsequently returned to Zurich.

Many physicist, who worked in the field of magnetism and had published on the
Ising model, had to emigrate, especially the young generation of post docs and
assistants. When J. J. Thomson, who discovered the electron, opened a meeting
of physicists in Oxford in 1936, he opened it with the words[88] “Heil Hitler!
Thanks to Hitler we have today with us Franz Simon, Kurt Mendelssohn [great
authorities in the field of low-temperature physics], Rudolf Peierls, and many
others”.

Leubuscher, who supervised Johanna’s thesis, half-Jewish lost in 1933 the
habilitation and emigrated to Great Brittain where she was in the network[89]
of the British Federation of University Women (like Lise Meitner, Marie
Jahoda, Charlotte Bühler). Johanna could stay at the College of Commerce
[Handelshochschule], not a state institution in Berlin, where she privately
worked for Prof. Franz Eulenburg. He was dismissed as a Jew not until 1935
because of the large amount of doctoral candidates he had to care for. He died
in 1943 under unexplained circumstances in the prison of the Gestapo.

Also Jewish pupils were dismissed from public schools and gathered in special
Jewish schools. One of such institution was the Landschulheim (boarding
school) Caputh, founded already in 1931 not only for Jewish children by
Gertrud Feiertag[90] with a special educational concept. The buildings had
been a summer recreational home, neighbored by the estate of Albert Einstein,
which later was included into the school buildings. Ernst Ising became after
his dismission in 1934 a teacher and in 1937 headmaster at Caputh (see the
picture at the beginning of the section).

In the following the political situation changed dramatically in Germany
reaching its culmination in the November Pogroms. The antisemitic and racist
Nuremberg Laws where enacted on September 15, 1935. The antisemitic atmosphere
finally exploded in violence. Also in Caputh was Heinz Bonnem, it is
reported[91]: “On the morning after the November pogroms, on November 10,
1938, the school was stormed by men from Caputh and destroyed to such an
extent that on November 15, the headmaster at the time, Dr. Ising sent the
message to the district school board in Potsdam, that resuming classes in the
foreseeable future is unthinkable. At that time, Heinz Bonnem was a student of
4b, as can be seen from a directory from 1938.” Similar can be found in the
memoirs of Sophie Friedlander and Gertrud Feiertag[90]: “One of the Nazis was
standing outside before the door of the house. With a motionless face, he
looked at his watch. As far as possible we will let everyone know, that we
want to gather together on our sports field in the forest. There Dr. Ising,
the headmaster, made lists and everyone, who resided in Berlin and could
accommodate children, took over a group, so that everyone had a temporary
home. Ernst Ising went back to Caputh and took two pictures of his ravaged
apartment. The country school home was not reopened after this destruction.”

Because of these two pictures Ernst Ising was arrested by the Gestapo but was
back after four hours. He could explain that he made them for the insurance
company and not for propaganda.

###  10 The impact and development of the Ising model until 1939

####  1 Nordheim, Bethe and Peierls

Friedrich Hund characterized in 1972 in his Geschichte der physikalischen
Begriffe [History of the physical terms[92] p.364] the understanding of
ferromagnetism in the mid-30ies as follows: “There are still gaps between the
three floors of the theory of ferromagnetism”, where he named the three floors
the qualitative quantum theory, the phenomenological - macroscopic equations,
and rules for technical magnetization [material constants]. He also refers to
Ising’s model however erroneously mentioned that Ising’s result were equvalent
to the results of Weiss’ theory. This goes back to Lothar Nordheims
contribution[93] in 1934 Quantentheorie des Magnetismus [Quantum Theory of
Magnetism) to Mueller-Pouillet’s Lehrbuch der Physik [Textbook of Physics]. By
the way, one of the rare publications which Ising was aware of where his paper
was cited. Nordheim (1899 - 1985), almost of the same age as Ising finished
his thesis in 1923 by Max Born in Göttingen and left Germany and immigrated to
the United States 1934. Ising might have known him from his time when he
started his physics studies in Göttingen.

Nordheim starts with a short review and a critic of the classical theory (p.
801). He notes: (1) that there is no explanation of the strength of the inner
field by a classical interaction (dipolar magnetic interaction is too weak);
(2) that the theory contains inner inconsistencies, since the susceptibility
should be zero and names the thesis of J. van Leeuwen. In consequence he
summarizes: “The whole complex of appearances of the magnetic properties of
matter is therefore to be addressed as a pure quantum effect.” (3) Later in
the paragraph on ferromagnetism he again emphasizes (p. 842) that there is no
classical explanation of the inner field and cites Yakov Frenkel who made this
point based on Yakov Dorfman’s experiments but did not work it out[94]. This
was done by Werner Heisenberg. (4) He then explains Heisenberg’s model and
Bloch’s calculation of the magnetization at low temperature[95]. An important
feature of the result was that a finite limiting value of the magnetization at
$T=0$ is obtained only in three dimensions. In this context he cites Ising’s
publication (p. 859) with the comment: “Incidentally, this [referring to
Bloch’s calculation] is in agreement with the result of the classical theory
that in a linear chain, if only the dipole interaction of the individual
magnetons is taken into account, no ferromagnetism is possible [citation of
Ising’s paper].” This is misleading since the Ising model is not based on the
dipole interaction but assumes a short-range interaction between the
magnetons. However it shows the importance of the space dimension for the
Ising and Heisenberg model. Felix Bloch[96] showed for the Heisenberg model
that the magnetization decreases from its maximum value as
$M=M_{0}(1-(T/T_{c})^{3/2})$. This decrease is a consequence of the existence
of spin waves in the ferromagnetic phase of the Heisenberg model. The three-
dimensionality of space turns out to be an essential condition for the
validity of this result.

As already noted, Hans Bethe, who was of Jewish ancestry, was also dismissed
from his position as an assistant professor at the University of Tübingen. He
accepted a temporary lectureship in Manchester, UK, before emigrating to the
US in 1935 to join the faculty of Cornell University. In Manchester Sir
Lawrence Bragg suggested him[97] a problem appearing in binary alloys in mean
field theory. This approximation called for alloys - Bragg-Williams
approximation - was completely independent from the geometrical configuration
of the two types of atoms in the alloy and an ordered phase existed even for
the one-dimensional case. Bethe tried to correct[98] this in 1935. In order to
do so he explicitly included in his mean-field calculation the configuration
of the neighboring atoms (in magnetic systems this would correspond to the
spin orientation, see Fig. 5(a)).

Figure 5:  Bethe-Peierls approximation for a two-dimensional Ising model: (a)
the neighboring spins $\sigma_{1}$ to $\sigma_{4}$ of $\sigma_{0}$ whose
configurations are taken into account by the Bethe’s approximation, (b) the
jumps of the specific heat in two dimensions in mean field theory (Bragg-
Williams) and the Bethe-Peierls approximation. (slightly changed taken from
Ref. [99])

In consequence the transition temperature became now dependent on the number
of nearest neighbours

$kT_{c}=\frac{A}{\log(z/(z-2))}$ (8)

but the jump in the specific heat remained (see Fig. 5(b)). The important
change was now that the phase transition temperature in the one-dimensional
case turned out to be zero. Bethe was aware of the analogy to the ferromagnet
(he explained this in the introduction) but he seemed to be not aware of
Ising’s publication from 1925. He didn’t cite Ising but remarked again at the
end of his paper when he pointed to the restriction of the ordering only in
two or more dimensions: “This should be noted in connection with
ferromagnetism which is restricted to three dimensions, due to a quantum
mechanical effect.”

The approximation is usually called Bethe-Peierls approximation since it seems
to originate from the close cooperation in Manchester between Bethe and
Peierls as the acknowledgement indicates. Indeed Peierls, who decided not to
return to Germany, worked on similar problems and published a paper on his
own[100] later when he got a position at Fowler’s institute in Cambridge. His
acquaintance with Hans Bethe was significant[101], a fellow student one year
his senior, knowing each other from the years in Munich in Sommerfeld’s group.
“They shared an interest in and a passion for physics that resulted in a
lifelong friendship that went far beyond the research-related acquaintance.”
In fact the formula above, Equ. (8), for the transition temperature has been
given by Peierls. He also does not mention Ising but turned to the Ising model
in the following paper[102].

The close connection of the mathematical treatment in statistical mechanics of
phase transitions in alloys and in ferromagnets led Peierls to study Ising’s
model[102]. He considered the problems to be mathematically equivalent and
cites studies of Bragg and Williams, Borelius, Fowler. Bethe and others.
However Heisenberg’s model was considered to be more realistic for
ferromagnets not knowing which properties of the model are important for the
aspects of the phase transition. For instance Peierls points to the difference
between the two models with the remark that Heisenberg’s model “depends not
only on the arrangement of the elementary magnets, but also on the speed with
which they exchange their places.” It was this difference (Peierls’
interpretation of the exchange interaction) which led him to the conclusion:
“The Ising model is therefore now only of mathematical interest.”

Figure 6:  Islands of different spin orientation for the planar Ising model
from Ref. [102].

Basis of his consideration was the result in the above mentioned
publication[100] where within the Bethe approximation the nonexistence of a
phase transition in one dimension was shown. “Since, however, the problem of
Ising’s model in more than one dimensions has led to a good deal of
controversy …it may be worth while to give its solution.” And indeed he could
end his paper with the statement: “Thus it follows rigorously that for
sufficiently low temperatures the Ising model in two dimensions shows
ferromagnetism and the same holds a fortiori also for the three-dimensional
model.” Although the argumentation turned out not to be strict the conclusion
was right as it was shown in 1964 by Griffith[103]. Peierls proof was also
extended[104] to higher dimensions. Peierl’s method proved a phase transition
but could not calculate the transition temperature, given later by Kramers and
Wannier (see Sect. 2). 1960 Domb[105] pointed out that the Bethe approximation
is exact on lattices that have no closed circuits of n-n bonds [Bethe-
lattice].

####  2 The Ising model in textbooks

##### Van Vleck 1932

The first citation of the Ising model in a textbook is found in Van Vleck’s
textbook[106] The Theory of Electric and Magnetic Susceptibilies published in
1932. Van Vleck had made his thesis in 1922 in the States supervised by Edwin
C. Kemble who was a pioneer of quantum mechanics in the States. Van Vleck is
considered as a Father of Modern Magnetism and got the Nobel Price in Physics
in 1977 for his fundamental theoretical investigations of the electronic
structure of magnetic and disordered systems together with P. W. Anderson and
N. F. Mott.

In the preface of his book he wrote: “The new quantum mechanics is perhaps
most noted for its triumphs in the field of spectroscopy, but its less
heralded successes in the theory of electric and magnetic susceptibilities
must be regarded as one of its great achievements.” He remembers[107]: “My
doctor’s thesis was the calculation of the binding energy of a certain model
of the helium atom, which Kemble and Niels Bohr suggested independently and
practically simultaneously, with Kramers making the corresponding calculation
in Copenhagen. The results did not agree with experiment for the old quantum
theory was not the real thing. However, when the true quantum mechanics was
discovered by Heisenberg and others in 1926, my background in the old quantum
theory and its correspondence principle was a great help in learning the new
mechanics, particularly the matrix form which is especially useful in the
theory of magnetism.”

In chapter XII §77 he discusses Heisenberg’s quantum mechanical theory of
ferromagnetism. Taking Heisenberg’s spin Hamiltonian in the classical limit
Van Vleck argues that one would yield the Weiss result and concludes
erroneously [page 331 footnote 19]: “In seeming contradiction, Ising found
that classically there was no ferromagnetism regardless of the crystalline
arrangement [citation of Ising’s paper]. This, however, was because Ising
arbitrarily took the coupling between elementary magnets to be proportional to
$\mu_{x_{1}}\mu_{x_{2}}$ rather than to the complete scalar product
$\mu_{1}\mu_{2}$ [$\mu$ represents the quantum mechanical spin vector]”. This
is different from Pauli’s comment at the Solvay conference, which Van Vleck
had attended. However Heisenberg’s model did show a phase transition in three
dimensions but Heisenberg used the Gaussian approximation (the same holds for
Ising’s model in mean field theory but this was never studied at this time).
Niss[108] considers this remark as so serious that the Ising model was
dismissed as a realistic model in the early 1930s.

Concerning Heisenberg’s model Van Vleck also mentions that a classical
analysis of this model would yield the Weiss result. The reason was that
Heisenberg in his paper made the Gaussian approximation as mentioned above and
therefore got the mean-field result. Some pages later [on p. 344] he noted:
“Even in absence of a magnetic field, there should be a discontinuity in
specific heat as one passes through the Curie point, a result first noted by
Weiss. In terms of Heisenberg theory, this is because…”. Again a result of the
approximations made.

##### Bitter 1937

After his Bachelor of Science degree in 1925 at Columbia University Francis
Bitter went to Europe for one year and studied in Berlin. Bitter recalled[109]
hearing Max Planck’s lecture on thermodynamics, attended the colloquium at
which Erwin Schrödinger introduced wave mechanics, and teached himself
electricity and magnetism from Abraham’s textbook, The Classical Theory of
Electricity and Magnetism (the English translation appeared later in 1937).
Returning back to the States he received his PhD in 1928 from Columbia
University. Rabi who visited Hamburg University for one year remembers[110]:
“During the exciting period 1925 - 28 when the world of physics was reborn
with the invention of quantum mechanics. It was a wonderful time to be a
graduate student with a lifetime before one for research and study and the
exciting task of remaking the old physics and bringing on the new.” Bitter
established The National Magnet Laboratory at MIT in Cambridge, Mass. in 1938,
worked at Westinghouse and later became university professor.

Figure 7:  The scaled energy and specific heat against the scaled temperature
for the Ising chain (adapted from Ref. [44])

In his[44] Introduction to Ferromagnetism in Chapter V Bitter presented
Ising’s treatment of the linear chain. He noted “The derivation given in this
section has been taken from Ising’s unpublished dissertation at the University
of Hamburg.” And indeed he presents the complete detailed calculation as
indicated in Chapter 5. Moreover he calculated in addition the thermodynamic
quantities as the energy and the specific heat $\sigma$ (see Fig. (7)), the
last is given by (in his notation)

$\frac{\sigma}{k_{B}}=\Big{[}\frac{\epsilon/(2k_{B}T)}{\cosh(\epsilon/(2k_{B}T))}\Big{]}^{2}.$

The appendix contains F. Zwicky’s updated article Cooperative Phenomena
published already in 1933. He points to general problems in understanding
phase transitions: (1) how a system of atoms can produce long-distance order,
(2) a sharp transition from one phase into another. However no conclusive
answers were given and the problem remained.

##### Becker, Döring 1939

Richard Becker was born in Hamburg and got his doctoral degree in 1909 under
August Weismann - a evolutionary biologist at the University of Freiburg.
After hearing lectures by Arnold Sommerfeld at the Ludwig Maximilian
University of Munich, Becker turned his professional interest to physics. He
also studied physics under Max Born at the Georg-August University of
Göttingen, Becker completed his Habilitation in 1922 under Planck and became a
Privatdozent at the University of Berlin. In 1926, he was appointed a full
professor at Technische Hochschule Berlin and head of the new physics
department there. In 1933 he was reassigned to Göttingen as a part of the
general anti-theoretical policies promoted under Deutsche Physik - his
position at Technische Hochschule Berlin was eliminated.

Werner Döring’s thesis was supervised by Becker in Berlin and Döring followed
him in 1939 as Privatdozent (assistant professor) at the University of
Göttingen. After the war he returned to Göttingen and became in 1977 professor
at the University of Hamburg.

The book[111] Ferromagnetismus (Ferromagnetism) started from lectures given by
Becker in Berlin in 1934/35. A remarkable document is the review of this book
by L. F. Bates in 1946 in Nature[112]. He says: “It is very unfortune that
this important work should have appeared at such an inopportune moment in the
world’s history, when it will be difficult for most of the English-speaking
peoples to obtain copies of it. Those of us who had the pleasure of meeting
Prof. R. Becker at the 1939 Conferences on Magnetism in Bristol [The
conference[113] Internal Strains in Solids July 11 - 13, 7 weeks before the
war broke out in 1939] and in Strasbourg must sympathize with him in no small
measure.” And Kittel[114] wrote: “In the decade of the 1940s physicists
everywhere were educated in the principles of ferromagnetism (and learned
German besides) from that excellent monograph Ferromagnetismus by R. Becker
and W. Döring, published in 1939. It was a decade in which many physicists
worked in ferromagnetism through the degaussing of ships in wartime; the
development of permanent magnet materials; … Becker and Döring was an exciting
and thorough book. There was a problem running through all of
Ferromagnetismus, like the cliche of Hamlet without the Prince of Denmark:
there was no domain theory, no theory of the origin, size or shape of
domains.”

In chapter 5 the Curie-Weiss theory is explained and criticized on the basis
of statistical mechanics. The authors explain the ordering through flocking
behavior of elementary magnets (Schwarmbildung) due to the short-range
interaction. In this way a cloudy structure (domains) of parallel elementary
magnets are created. One might guess that this modern picture (see the
chapters in this volume on flocking theory) goes back to Becker’s biological
education in Freiburg.

Figure 8: The scaled inverse susceptibility against scaled temperature for the
Ising chain (adapted fromRef. [111])

In this context also the Ising chain and its susceptibility is compared with
the result of Weiss theory (see Fig. (8)). They consider this model as “a
first step from the Langevin case of paramagnetism of free elemental magnets
in the direction of the case of ferromagnetism that actually interests us.”
The susceptibility below $\Theta$ in this approximation is not shown. It would
be a dashed line reflected on the vertical at $T=\Theta$, finite at $T=0$ and
zero at $T=\Theta$ with a slope twice as steep as above $\Theta$.

##### Extension to Condensation 1939

An important step was made by Felix Cernuchi and Henry Eyring[115, 116] (at
Princeton University) by extending the theory of alloys to liquids by
replacing one constituent of the alloy by a hole in the lattice model. This
allowed them to get phases with different density. Cernuchi was a post-doc and
student of Ralph H. Fowler; Eyring had a position as an instructor at
Princeton 1931 and became full professor 1938.

Such a lattice theory was seen as an application of the Ising model to
critical phenomena of liquids. In a Historical Note in the introduction to
their paper they explain the steps leading from Ising’s publication to their
theory. As important basis for developing statistically the model they cite
Bethe’s and Peierls’ paper discussed above. They conclude: “the existence of
two different phases at equilibrium is a direct consequence of the attractive
forces between the particles.” Their calculations are within the approximation
of Bethe and Peierls and they illustrate their result by a general phase
diagram for fluid systems.

####  3 The Van der Waals Centenary Congress in Amsterdam November 23 - 26,
1937

Already in November 1937, a conference took place in Amsterdam in honor of van
der Waals (Johannes Diderik van de Waals was born hundred years before, on
November 23, 1837 in Leiden). Phase transitions was a highly discussed topic.
Questions like how can the gas molecules “ know” when they have to coagulate
to form a liquid or solid, or could the partition function for a finite system
explain a sharp phase transition and is one partition function enough for two
phases, were discussed. Kramers (chairman of one session on phase transitions)
let vote the audience[117] with the result that no majority for one of the
opinions could be found. Kramers suggested[118, 119] the use of the
thermodynamic limit as an explanation of the sharp phase transition and the
existence of different phases. He named ferromagnetism as an example for that
behavior.

Kramers suggestion has been put on a rigorous basis later by Yang and Lee,
whose theory relies on the well-known mathematical fact that a sequence of
smooth functions can have a non smooth limit, provided that the convergence is
not uniform. Leo Kadanoff[120] explained in 2010: “Throughout, the behavior at
the phase transition is illuminated by an “extended singularity theorem”,
which says that a sharp phase transition only occurs in the presence of some
sort of infinity in the statistical system. The usual infinity is in the
system size. Apparently this result caused some confusion at a 1937 meeting
celebrating van der Waals, since mean field theory does not respect this
theorem. In contrast, renormalization theories can make use of the theorem.”

###  11 Escaping to Luxembourg

> Wer half mir
> Wider der Titanen Uebermuth?
> Wer rettete vom Tode mich, … Von Sklaverey?

> Who helped me
> Against the Titans’ insolence?
> Who rescued me from certain death,
> From slavery?

The lines written above are from Prometheus by Johann Wolfgang Goethe which
Ernst Ising could recite. He always had with him a little book with poems of
Goethe to read and correct if he lost some words. His son Tom Ising
remembers[121]: “My father liked acting and had a stage in the basement where
he and his friend, Heinz Wildhagen put on plays. Heinz spent his life as an
actor and theater owner. Later the actor Willie Busch became my father’s
dictation coach.”
The first quarter in Mersch ©Collection Julie Müller-Barthelemy 1939 January
27 Gestapo arrests Ernst Ising for a 4hour questioning
May 1 arrival in Mersch
September 1 Begin of World War II.
September 9 Birth of son Thomas 1940 May 10 Occupation of Luxemburg 1941
teaching at a Jewish school until October

After the short arrest of Ernst Ising Johanna and Ernst decided to ask for
emigration to the United States. Since the approval needed about two years and
their situation became more and more unsafe they organized their emigration to
Luxembourg for waiting to get the permission. On May 1, 1939 they arrived in a
hotel in Mersch[122] where they got one of the rooms (with 19 other families
in the hotel). Later, after the birth of the son Thomas on September 9, 1939,
they moved to a two room flat in Berschbach/Mersch.

On May 10, 1940 - Ernst Ising’s 40th birthday - the Germans occupied
Luxembourg and declared it part of the Greater German Reich. In consequence
German laws now also hold in Luxembourg with severe changes for their live. On
September 5, 1940, Gauleiter Gustav Simon introduced the “Ordinance on
Measures in the Field of Jewish Law”, the first of a whole series of anti-
Semitic “ordinances” that strictly regulated the lives of Jews and largely
restricted their freedom[123].

One month later in October 1940 there was a rumor that they had to leave
either to the occupied Belgium or France or to the unoccupied part of France.
Fortunately they could stay and in December Ernst found a job as a teacher for
Jewish children, which were similar to the year 1933 in Germany thrown out of
public schools and collected in a separate school set up in Luxembourg-Ville.

But time was running out. On July 1, 1941, the U.S. State Department
centralized all alien visa control in Washington, DC, so all applicants needed
to be approved by a review committee in Washington, and needed to submit
additional paperwork, including a second financial affidavit. At the same
time, Nazi Germany ordered the United States to shut down its consular offices
in all German-occupied territories. After July 1941, emigration from Nazi-
occupied territory was virtually impossible[124].

Johanna wrote[56]: “In 1941 the two-year waiting period was over, the money
for the trip from the Spanish border and for the boat passage was available,
almost all papers were ready. Only one more specification [affidavit] was
required of where our sponsors would lodge us in their house, how much pocket
money they would give us and more such details. We wrote, but before the
answer arrived the American consulates closed in June 1941 and no more visas
given out! It was a terrible disappointment and meant that we would to stay in
Luxembourg under the rule of the Nazis until the War was over.” Between 1938 -
1941, US law allowed only 27,370 immigration visas per year to be issued to
people born in Germany or Austria.

###  12 The international Conference on Magnetism in Strasbourg 1939

Figure 9:  Conference on Magnetism in Strasbourg May 21, 1939. Speakers were
33 Gorter, 37 Forrer, 38 Mott, 41 Casimir, 42 Foex, 43 Neel, 14 Kramers, 13
Sucksmith, 5 Krishnan, 7 P. Weiss, 8 Barnett, 9 Cabrera, 10 Bizetti for
further identification of the participants see the reference [129]. ©AIP
Emilio Segr$\grave{e}$ Visual Archives

While Ernst Ising prepared himself together with other emigrants by learning
English and Spanish for the new chapter of his life, 175 kilometers south in
Strasbourg the last international conference on magnetism took place from May
21 to 25, 1939. This was after the Solvay conference in 1930, the second major
international conference devoted to Magnetism and as Elliot [125] claims: “the
fundamental ideas which underpin our understanding of magnetic phenomena today
were largely in place. The Strasbourg conference of that year was able to look
back also to the triumphs of the classical era.” Indeed the development of the
international conferences on magnetism after the war showed this clearly.

S. J. Barnett[126] gave a complete report on the conference which was
published in Science as late as after the Second World War in 1946. There was
only a limited number of contributions [in total 18] allowed, and
attendance[127] was by invitation. The speakers had to submit their
contributions in advance, which were then distributed before the conference to
the participants. In addition the gathering was focused around six discussion
sections lasting three hours each[128]. However although the proceedings were
published in 1940 in France their distribution was long prevented by the
German occupation.

Kramers contribution is of special importance for the Ising problem although
no explicit results were presented. Therefore his talk[55] The Interaction
Between Magnetic Atoms In A Crystal was summarized by Weiss: “he takes the
point of view of a rigorous quantum-mechanical treatment… Mr. Kramers shows
the difficulties encountered by various previous theories in demonstrating the
existence of a Curie point and concludes: An exact solution that would prove
strictly the existence of a transition point has not been given for any
interaction model.”

Nevertheless some preliminary result were mentioned by Kramers, who had
already started to work on ferromagnetism[130] in 1934 where he observed[117]
“that the partition function can be obtained as the largest eigenvalue of a
matrix” and “that the model which he constructed does not show ferromagnetism
in one and two dimensions but does show it in three dimensions. This result
demonstrated that the ferromagnetic phase transition is possible in a
classical context and thereby settled an important issue.” Kramers continued
this investigations of the ferromagnetic transition in a paper[131] published
in 1936.

In his Strasbourg contribution Kramers gave a review of the present
understanding of ferromagnetism in solids. Although only approximate results
as in mean field approximation or in Gaussian approximation are available, he
announced some important unpublished steps for proceeding with theory and made
remarkable statements concerning the understanding phase transitions in
general.

Points made by Kramers were: (1) the importance of the thermodynamic limit,
(2) the description the two phases - high and low temperature phases - by one
partition function, (3) the importance of the dimension of the many-body
system for the existence of a phase transition, (4) the announcement of a new
way of calculation of the partition function which reproduced Ising’s result,
and (5) the existence of a special property relating the two magnetic phases.
But no further mathematical details were given.

Considering the results in one dimension for the Ising model and the
Heisenberg model he noted[55] (on p. 53): “The difficulties that prevent the
exact solution of an interaction problem for a two- or three-dimensional
network are great. However, we have the impression that this solution may be
found one day. We foresee that for the two- or three-dimensional network a
point of transition may well exist thanks to the infinite number of ways,
including one atom A can in the lattice be linked with another atom B by a
continuous chain of direct neighbour-to-neighbour interactions. Indeed, the
state of a solid below a transition point is a long-distance order state,
which we would like to see result from an elementary law of interaction, whose
direct effect would only be a small distance order.” (see item (3) below in
Sec. (1) on the Onsager solution)

He further remarked[55] (on p. 55): “The tendency to ferromagnetism is greater
in the model of Lenz-Ising than in that of Heisenberg. It seems likely to me
that the Lenz-Ising model is already ferromagnetic for a two-dimensional
network . The author [its himself] has developed a new method (not published)
which, for dimension one, leads to a result identical to that of Ising. For
two dimensions we find a Curie point, but unfortunately in this case it was
necessary to make in the calculation use of unsure simplifying assumptions and
that is why this proof of the existence of the Curie point cannot be
considered as sufficient.” D. Ter Haar writes[132] in Masters of Modern
Physics The Scientific Contributions of H. A. Kramers (on p. 6 in footnote 8)
that Dresden had informed him that according to Uhlenbeck: “Krames had already
found the transfer matrix method in 1937 and had mentioned it at the Van der
Waals Conference in 1937.” This might just refer to the method presented in
the 1934 and 1936 papers of Kramers which considered the Heisenberg model
rather than the Ising model. However the remark pointed to the fact that the
spatial dimension is an essential parameter for the existence or non-existence
of a phase transition. Indeed in 1966 N. D. Mermin and H. Wagner could
proof[133] the absence of ferromagnetism and antiferromagnetism in one- or
two-dimensional isotropic Heisenberg models.

Kramers also considered the new phenomenon of antiferromagnetism found by Neel
in the 30ies and speculated on the transition temperature for that phase:
“However, it is easy to see that the antiferromagnetic model of Lenz-Ising is,
from a formal point of view, identical to the simplest imaginable model of the
“order-disorder” problem in alloys.” (p. 56) Thus a kind of universality
appeared in connection with the Ising model.

It was already an ambivalent atmosphere at the conference[134]: “The Rector of
the university then closed the conference…in a final toast to the honour, and
preservation, of free scientific thought as it had just been exemplified at
the conference itself. It was a noble thought, grotesquely at odds with a
world collapsing around him.” Indeed not all conference participants were
amused[135]. As Stoner describes in his diary entry for May 24, 1939 the
evening’s conversations: “Mrs Becker was rather more obstreperous…they both
[also Gerlach] want Strasbourg…a German city”. And Bates remembers later[127]
“Most of us left Strasbourg with heavy forboding.”.

Three months later on September 1 the invasion of Poland and the Second World
War started.

###  13 Waiting time over- Consulate closed - Trembling in Mersch

Monastery Cinqfontaines (Fünfbrunnen) the only Luxembourgian internment camp
for Jews.

  1. 1941

October 6 all Jewish families in Luxembourg - Ville were deported to
concentration camps.

  2. 1942

In April moving to Mersch Mozart street (see below)

  3. 1942

January 20 Wannsee Conference

After the annexation of Luxembourg, numerous Jews tried to emigrate[123].
However, many elderly and sick people did not succeed in this. Out of concern
for their well-being, the Jewish consistory initiated the plan to establish a
home for the elderly in the Fünfbrunnen monastery. In the Jewish retirement
home, there was Jewish self-government under the control of the Gestapo.
Limited food allocations, a ban on going out and wearing the Star of David
characterized everyday life. The consistory tried to improve the situation of
the residents. For example, the ban on going out was relaxed. Attempts have
also been made to counteract the food shortage by growing vegetables or
bartering with local farmers. 23 deaths were recorded in the camp. The reasons
for this were the old age of the inmates, illnesses and the catastrophic
prison conditions. Since there were neither walls nor fences, Fünfbrunnen is
considered a “ghetto-like coercive community” and was similar to the so-called
Jewish houses or Jewish dormitories. All Jews who had to give up their
apartments by order of the Gestapo had to move in there. (see Land of
Memory[136] Kloster Fünfbrunnen). Hugo Heuman wrote in his diary[122]:
“Towards the middle of November 1941, the consistory was informed that all
Jews in the country, around 350 in number, were to be concentrated in
Fünfbrunnen, for which barracks were to be built.”

In fact the small village Fünfbrunnen nearby Troisvierges (Ulflingen) became
generally known as the only Luxembourgian internment camp[123] for Jews during
the Second World War. From 1941, the enormous cloister became the collecting
point for Jews. On August 11, 1941 the first transport with about 20 Jews
arrived in Fünfbrunnen. About one month later the deportations to the
concentration camps in the east began.

Fortunately due to the German laws Jewish husbands of German wifes where
exempted from the deportation and could stay with their family. A dramatic
change happened after the Wannsee-Conference[137] on January 20, 1942. In this
document (only one copy survived) the plan for the extinction of the European
Jews is documented (after a process of the decision making[138]). Until its
dissolution on April 6, 1943, around 300 Jews passed the camp. Fifty of them
gave Luxembourg as their place of birth. Most of them were German Jews who had
emigrated to Luxembourg after 1933.

From August 1942 Ernst worked at Fünfbrunnen where he had to educate three of
the interned children. He and his wife also had to care for the old people and
help them for the deportation. “Nothing happened to us”, Johanna wrote in her
momoires, “but the Nazi promised that those lazy Jews who lived in mixed
marriages and were not affected by the deportation rulings would be put to
work.”

###  14 The new approach to the Ising model

Although Kramers had announced his new method to calculate the Ising model in
May 1939 the result was published about two years later due to the development
of the war. The Netherlands where occupied by the Germans on May 10, 1940
leading successively to restrictions also at universities. Kramers had started
a collaboration with Wannier 1939 and worked in the following years with him
on the Ising problem in two dimensions.

Gregory Wannier born in Basel made his thesis under Stueckelberg at the
University of Basel in 1935 and afterwards went as exchange student for one
year to the United States to Princeton. Back to Europe in the year 1938 - 1939
he worked in Bristol with Neville Mott and also with Kramers.

In the year 1941 five authors[139, 140, 141, 142, 143, 144] Kramers, Wannier
(KW), Montroll (M), Lassettre and Howe (LH) presented a new method for
calculating the partition function of the two dimensional Ising model
according to the representation of the Hamiltonian given by Pauli at the
Solvay conference 1930. The papers and announcements were received in the year
1941 in the following order:

February 21 - 22

KW0: An abstract of a talk given at the 240th regular meeting of the American
Physical Society at Cambridge Mass. by Wannier under the title Statistics of
the Two-Dimensional Ferromagnet opens[145] the breakthrough in the struggle to
solve the Ising model. He announced: “Rigorous statistical study of the Curie
transition of ferromagnets is possible for a two-dimensional square net of
spins with coupling between neighbours.” Then he presented unpublished
calculations of strips of spins by Kramers, which show that the partition
function is given by the largest eigenvalue of a matrix. The existence of
ferromagnetism needs at least a two-dimensional geometry. The value of the
Curie temperature, where the partition function has a singularity is given and
noted that the specific heat has no jump but an infinite value there.

April 7

KW1: In footnote 1 they[139] indicate that it was entirely written by Wannier
due to the difficult communication during the wartime. In footnote 5 they
state that the elegant procedure handling the transfer matrix is due to
Montroll who applied it first to the theory of molecular chains. At the end of
the paper they generally thank Montroll for helpful discussions.

June 6

M1: In chapter 1 of part II General remarks Montroll[141] writes: “Wannier and
Kramers have studied ferromagnetic nets in detail by a method somewhat similar
to that described here. They first show that the partition function is the
largest characteristic value of a set of linear difference equations that are
derived by considering the change in the configuration of the lattice points
of the system by the addition of one more molecule. Their final characteristic
value problem [footnote 14] resembles ours rather closely. In footnote 14 the
hindrance in the communication with Wannier and Kramers on this matter is
noted: “Unfortunately the present state of world affairs has delayed
publication of this work.” The author is greatly indebted to Dr. Wannier for a
discussion of his unpublished work; indeed to a large degree it was this
discussion that inspired the writing of the present paper. Then in chapter 2
Calculation of the partition function for narrow strips he specifies in an
unnumbered footnote: “The results of this section are not new, having been
obtained by Wannier and Kramers; however, the approach, especially that in
c-e, is new. [a-b presented the Ising chain and the Ising ladder c-e the
extensions to 3 chains, an approximation for the ladder and the introduction
of an external field for the chain, thereby reproducing Ising’s result.]” The
paper ends by pointing to the future: “The author hopes to treat some of the
other nearest neighbor problems mentioned at the beginning of this paper at
some later date; and in conclusion he would like to thank Professor Onsager
for his interesting discussions and for the numerous suggestions made
throughout the course of this research.

June 7

KW2: This second part[140] presents results from calculation mostly in
approximation.

August 28

LH1: There is no reference[143] to KW or M. Regarding the work of Lassettre
and Howe Montroll remembers in the interview with Stephen Heims on September
26, 1983: “There is a fellow you may not think about, that’s John Howe, who
still exists, Lassiter [Lassettre] and Howe. I see him fairly regularly in La
Jolla, and I think, see, he was a little out of the network. All the other
individuals knew each other, saw each other, and sometimes it’s very hard to
think of who did what first, because it was such a net, whereas he was
completely separate.”

September 5

LH2: No citation[144] of the other papers noted here.

October 2

M2: In footnote 1, 2 Montroll[142] already cites M1 and the papers KW1 and
KW2.

1943

Kubo: In 1943 Kubo[146] published (his first paper) on this topic in Japanese.
Due to the wartime it was written without knowledge of the aforementioned
papers but based on Bethe’s work[98]. He starts to calculate the partition
function for a system of spins. He writes: “The difficulty of this sort of
approach lies in combinatory calculations of W [the Boltzmann weight], which
is elementary but becomes complicated in higher approximations and is almost
hopeless for rigorous treatments. … However, one wonders if an analytic method
can be devised for calculation of the asymptotic for of the expression (3)
[that is the partition function in the thermodynamic limit]. The author is not
able to conclude at present if this can be extended to two or three dimensions
and only hopes that the present note may give some hints for such
possibilities.” He then derives the magnetization starting from the partition
function for the system within an external field. Moreover Kubo proofs the
non-existence of a phase transition for such systems by showing that there is
no crossover of the largest eigenvalue of the ‘transfer matrix’ with smaller
eigenvalues.

Common to all the papers is that the occurence of a phase transition is
related to the degeneracy of the two largest eigenvalues of a certain matrix
of $2^{n}$ dimension on a $n\times n$ lattice. In the case of the magnetic
system this is the transition to the ferromagnetic phase, in the case of the
alloy it is the transition to a phase of regular separation. The case of
condensation of gasses is not mentioned. This cascade of papers shows how
science can progress in several ways at several places almost at the same
time. But on the other hand the success of networking is also demonstrated. It
is interesting that in his interview[147] Montroll pointed, when he spoke
about his connection with Kramers and the Ising model, to the importance of
the[148] “Vienna school of statistical mechanics”. The increase of this
network to a global dimension was caused by the spreading of physicists -
educated by professors of the physical institute of the university of Vienna -
all over the globe after the First and Second World War.

Niss[108] describes the part of the network concerning the Ising model in the
States in the following way: “Wannier taught Montroll statistical mechanics as
a graduate student in chemistry at the University of Pittsburgh. Montroll
subsequently spent the academic year 1940 - 1941 as a postdoctoral fellow at
Yale University with Onsager where he told Onsager about his and Wannier’s
ideas on the Ising problem. Kramers and Wannier were more interested than was
Montroll in the physical aspects of the Lenz-Ising model. Wannier had studied
the theory of transition points under Fowler in Cambridge and had attempted to
apply the Bethe method to the melting process just before he began
collaborating with Kramers in Utrecht in 1939.”

####  1 Transfer Matrix for the Ising chain

The partition function is given by the largest eigenvalue of a certain matrix,
called transfer matrix. This matrix on a $n\times n$ lattice is of $2^{n}$
dimension. For a linear chain the matrix is a two dimensional matrix and the
eigenvalues are easily found by solving a quadratic equation. For a ladder it
is a four dimensional matrix and the exact eigenvalues can be found only for
special cases e.g. when there is no external field.

Following the calculation of Kramers and Wannier[139] the partition function
$Z$ for the chain reads

$Z=\sum_{\sigma_{1},...\sigma_{n}}e^{-{\mathcal{H}}/k_{B}T}=\sum_{\sigma_{1},...\sigma_{n}}e^{\epsilon/2k_{B}T\sum_{i}\sigma_{i}\sigma_{i+1}+mH/k_{B}T\sum_{i}\sigma_{i}}=$
$=\sum_{\sigma_{1},...\sigma_{n}}V(\sigma_{1},\sigma_{2})V(\sigma_{2},\sigma_{3})...V(\sigma_{n},\sigma_{1})=TrV^{n}=\lambda_{1}^{n}+\lambda_{2}^{n},$

where the Hamiltonian Equ. (6) was used with $A=\epsilon/2$ and an external
magnetic field $H$ is taken into account coupled to the magnetic moment $m$ of
the electron spin. The important point was to recognize that the partition
function can be written as the product of the transfer matrix $V$

$V=\begin{pmatrix}e^{\beta+\alpha}&e^{-\beta}\\\
e^{-\beta}&e^{\beta-\alpha}\end{pmatrix},$

with $\alpha=mH/k_{B}T$ and $\beta=\epsilon/2k_{b}T$. It’s eigenvalues are

$\lambda_{1,2}=e^{\beta}\left(\cosh{\alpha}\pm\sqrt{\sinh^{2}\alpha+e^{-4\beta}}\right).$

Thus $Z$

$Z(T,H)=e^{n\beta}\left(\cosh{\alpha}+\sqrt{\sinh^{2}\alpha+e^{-4\beta}}\right)^{n},$

and the free energy $F$ is

$F(T,H)=n\Big{(}\beta+\ln\left(\cosh{\alpha}+\sqrt{\sinh^{2}\alpha+e^{-4\beta}}\right)\Big{)}.$

The magnetization $M$ is obtained by deriving the free energy

$M=\frac{\partial F(T,H)}{\partial
H}=mn\frac{\sinh\alpha}{\sqrt{\sinh^{2}\alpha+e^{-4\beta}}}.$

Thus Ising’s result is reproduced[139, 141, 146] (apart from the difference in
the temperature-dependent term due to the choice of the energy to be zero when
the two spins are parallel; see also the footnote 19 in M1[141]). Now in order
to proceed for the two-dimensional case in the same way, one has to recognize
that this is not possible including a magnetic field. However the idea is not
to calculate the magnetization but instead the specific heat in zero external
field. If there is a phase transition this is expected to be seen in a change
of the dependence of the specific heat at the transition temperature. In
Weiss’ theory this was a jump at the Curie temperature.

####  2 Duality (“the small miracle”)

A second important step was made by Kramers and Wannier (KW1[139]) only. They
found a hidden symmetry of the two-dimensional Ising model in zero external
field, which allowed them to proof the existence of a phase transition and to
determine the transition temperature. The symmetry is based on transformation
of the model interchanging links between spin and the spin variables defining
a new spin lattice (see e.g. the explanations by Krieger[151] p. 78).

In the case of the Ising chain in zero external magnetic field one may
introduce the new spin variables by replacing[149, 150] a two linked spins
$\sigma_{i}$ and $\sigma_{i+1}$ by a new link-spin

$\mu_{i}=\sigma_{i}\sigma_{i+1}\,.$

Then the partition function can be rewritten as

$Z(T)=2\sum_{\mu_{1},...\mu_{n}}e^{\epsilon/2k_{B}T\sum_{i}\mu_{i}},$

where the factor 2 appears because of the two possibilities for the boundary
spin (e.g. the first spin). This leads to the partition function of
independent link-spins and the result is

$Z(T)=2[2\cosh(\epsilon/(2k_{B}T))]^{n-1}.$

Thus the original spin model of interacting chain spins has been transformed
into a dual model of independent link-spins each in two possible energy states
- the model published in 1922 by Schottky. This is the reason for the result
obtained for the specific heat by Bitter in 1937 (see Sect. 2, also Bitter was
obviously unaware of Schottky’s paper) for the Ising chain to show the anomaly
already calculated by Schottky in 1922 (see Sect. 4). Unfortunately neither
Lenz nor Ising were interested in the specific heat, which could have been
calculated using Ising’s method.

This idea of duality then has been applied by Kramers and Wannier to the two-
dimensional model. It turns out that the dual model is the same as the
original model, thus the two-dimensional Ising model is self-dual (see e.g.
the explanations by Krieger[151] p. 78). The original and the dual lattice are
square lattices, where in the dual lattice spins are in the center of the
squares of the original lattice. With the definitions

$K=\frac{\epsilon}{2k_{B}T}\qquad\text{and}\qquad\sinh(2K^{*})=\frac{1}{\sinh(2K)}$

it can be shown that for the partition function the relation

$Z_{n}(K^{*})=2(\sinh(2K))^{n}Z_{n}(K),$

holds where $K^{*}\leq K_{c}$ and $K\geq K_{c}$ This self duality relates the
low-temperature phase region to the high-temperature region. Since they
coincide at the transition temperature the equation for $T_{c}$ can be
formulated as

$\sinh(2K_{c})=1,\qquad k_{B}T_{c}=2.27\epsilon.$

Applying these ideas to the three-dimensional Ising model shows that it is not
self dual and therefore no equation for $T_{c}$ results.

In the second paper of Kramers and Wannier (KW2[140]) it was shown that the
specific heat has to be infinite at the Curie point. No further information
could be given, but it was proven that the Curie-Weiss theory is not valid.

###  15 D-Day and rescue

> Ich dich ehren? Wofür?
> Hast du die Schmerzen gelindert
> Je des Beladenen?
> Hast du die Thränen gestillet
> Je des Geängsteten?
> Hat nicht mich zum Manne geschmiedet
> Die allmächtige Zeit
> Und das ewige Schicksal,
> Meine Herrn und deine?

> I honour thee, and why?
> Hast thou e’er lightened the sorrows
> Of the heavy laden?
> Hast thou e’er dried up the tears
> Of the anguish-stricken?
> Was I not fashioned to be a man
> By omnipotent Time,
> And by eternal Fate,
> Masters of me and thee?

From Prometheus by Johann Wolfgang Goethe

Mersch Mozart street

©Exhibition[122] Mersch 2007

  1. 1943

May 5 unfree labor at the Maginot Line

  2. 1944

D-Day Juni 6 - September 1, no railway service, Germans flee

  3. 1944

September 10 Liberation by the Allies

  4. 1947

April 9 left Rotterdam on board the freighter Lipscomb Lykes

  5. 1947

April 21 arrival in New York

Ernst Ising’ duty in Fünfbrunnen ended on April 6, 1943 when the last train
with Jews left Luxembourg without him. There is a statement in Johannas
memoirs which could bee seen in correspondence to Goethe’s Prometheus when she
says[56]: “And I was thankful that Ernest was not among them, but instead,
after having helped to make their trip as comfortable as possible, could
return to wife and child and that we could face for future, uncertain as it
was, together, side by side, encouraging and supporting each other. I was
grateful, but to whom? To God? I would have felt conceited to think that God
had selected me to be spared the horrors of deportation or separation while He
allowed those 80 down there to be carried to their almost certain death.”

As it was promised by the Nazis Ising was forced to work since May 5, 1943 in
Diedenhofen (south of Mersch) at the Maginot line. At 4 o’clock in the morning
he left for Diedenhofen and returned at 10 o’clock in the evening to Mersch.
They had to dismantle and load railroad tracks so they could be brought to the
Eastern Front and relocated there to aid the German advance.

New hope arose when on June 6, 1944, the D-Day, the allied forces landed in
the Normandy 18 years after the day Johanna met Ernst. Johanna started to note
their situation in a diary (included in her memoirs[56]) starting from August
26, 1944 since she was sure that it would not fall into the hands of the
Nazis. Due to the development on the new front in France on August 31, 1944
the work on the Maginot line was stopped. No trains were running the next day,
the Germans started to fly east and finally the Isings were freed on September
10, 1944.

The war was not over, but their lives were rescued. Ernst Ising had to look
for jobs earning money. He accepted a series of odd jobs: bookkeeping,
darkroom work, and even reconstruction work as hard as he had to do at the
Maginot line. In November 1946 the Isings could present themselves at the
American Consulate in Amsterdam and finally in April 1947 they were able to
leave for the United States!

Path of life for Ern(e)st Ising

  * •

Cologne

  * •

Bochum

  * •

Göttingen

  * •

Bonn

  * •

Hamburg

  * •

Berlin

  * •

Mersch

  * •

United States

###  16 On the steps to the solution

After Lars Onsager finished his studies in Trondheim in 1925 he worked on the
theory of electrolytes. His first publication was a revision of the Debye-
Hückel theory, which enabled him to get an assistant position at the ETH
Zürich at Debye’s institute for 1927 to 1928. There he got to know the German
Hans Falkenhagen who worked also at the institute. Both young scientists laid
the basis for their future career there. Onsager went back to the States and
after a semester at John Hopkins University Baltimore found a position at
Brown University Providence until 1933. Then due to economic depression the
university could not keep his position he had to look for a new one. He used
this situation for a visit in Europe to meet Falkenhagen who had in this year
a position[152] as extraordinarius professor at the university of Cologne.
Falkenhagen told W. Ebeling[153] how the visit happened: “The scientific
discussion was restricted to the short way from the railway station to our
house, since after I had introduced Onsager to my wife and her sister Gretl
Arledter who stayed with us, Onsager was lost for science. He took me aside
and asked wether my sister-in-law was still free.” Margarethe Arledter - born
in Maribor Slovenia former Austria - was the daughter[154] of Fritz Arledter a
well-known pioneer in the art of paper making in Cologne. After a few days on
September 7, 1933 Gretl and Lars got married and returned to the States. There
he got a position at the Yale University and contact to Montroll.

American universities during war were less busy, many students were off
fighting, and scientists were doing military research. In 1942 the United
States joined the anti-Hitler coalition. Many physicist were doing military
research, but as Hemmer[155] noted there were exceptions:“As an alien with an
Austrian wife, Onsager was not part of this (he became US citizen only in
1945) and consequently had time to prepare the next jewel in the crown; the
solution of the two dimensional Ising model.”

The first announcement of the existence of the solution was in a discussion
remark following a paper by Gregory Wannier at a meeting of the New York
Academy of Sciences on February 18, 1942 and presented on October 24, 1942 at
the Meeting of the New England Section held at Hartford, Connecticut. Onsager
contributed two papers, one of them - with the title Crystal Statistics \-
treated the exact solution of the Ising model in two dimensions. The three-
sentence abstract[156, 157] stated:
“The partition function for the Ising model of a two-dimensional ferromagnet
has been evaluated in closed form. The results of Kramers and Wannier
concerning the Curie point $T_{c}$ have been confirmed, including their
conjecture that the maximum of the specific heat varies linearly with the
logarithm of the size of the crystal. For an infinite crystal, the specific
heat near $T=T_{c}$ is proportional to $-\log|T-T_{c}|$.”

####  1 Onsager’s solution 1944 (“the big miracle”)

On October 4, 1943 (published February 13, 1944) Lars Onsager finally sent the
paper with his solution to Physical Review[158] whose abstract now started
with the sentence: “The partition function of a two-dimensional ferromagnetic
with scalar spins (Ising model) is computed rigorously for the case of
vanishing field.” In order to come to a solution (finding the largest
eigenvalue) it was necessary to understand the mathematical properties of the
transfer matrices. Onsager replied to a question how he found these
properties, that he had a lot of time during the war, so he began to
diagonalize the transfer matrix (a real tour de force as C. N . Yang said). He
started with an Ising ladder. Ising looked also on the ladder in his
thesis[47] in his attempt to get the magnetization in higher spatial
dimensions but included an external magnetic field which makes the problem
exactly unsolvable. Onsager then added a third chain and so on up to six
chains. The transfer matrix grows in this way from a $4\times 4$ matrix to a
$64\times 64$ matrix and finding that all the eigenvalues were of a special
form gave him the indication that the algebra of the problem was a product
algebra. Onsager could reduce the problem from studying instead of
$2^{n}\times 2^{n}$ matrices $(2n)\times(2n)$ matrices and then find the
eigenvalues for general $n$. Therefrom he could calculate the partition
function in the thermodynamical limit

$Z=\ln(2\cosh(2\beta
J))+\frac{1}{2\pi}\int_{0}^{\pi}d\phi\ln\frac{1}{2}(1+\sqrt{1-\kappa^{2}\sin^{2}\phi}),$
(9)

with $\beta=1/(k_{B}T)$ and $\kappa=2\sinh(2\beta J)/\cosh^{2}(2\beta J)$.

Figure 10: The change of the scaled specific heat for the isotropic to the
strongly anisotropic Ising plane. In the limit of one interaction going to
zero the result for the Ising chain is obtained. (adapted from Refr. [158])

From the solution he could calculate the specific heat[158, 159] showing its
logarithmic divergence at the critical temperature different from the jump
predicted by the mean-field theory

$\frac{C}{Nk_{B}}=\frac{2}{\pi}\big{(}\beta J\coth(2\beta
J\big{)}^{2}\big{[}2K(\kappa)-2E(\kappa)-(1-\kappa^{\prime})(\frac{\pi}{2}+\kappa^{\prime}K(\kappa))\big{]},$

with $K(\kappa)$ and $E(\kappa)$ a first and second class elliptic integral
respectively and $\kappa^{\prime 2}=1-\kappa^{2}$.

The transition temperature is given by the temperature where the specific heat
diverges logarithmically as conjectured by Kramers and Wannier

$\tanh\frac{2J}{k_{B}T_{c}}=\frac{1}{\sqrt{2}}.$ (10)