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Ernst Ising
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Ernst Ising (German: [ˈiːzɪŋ]; May 10, 1900 – May 11, 1998) was a German physicist, who is best remembered for the development of the Ising model. He was a professor of physics at Bradley University until his retirement in 1976.[1]

Key Information

Life

[edit]

Ernst Ising was born in Cologne in 1900. Ernst Ising's parents were the merchant Gustav Ising and his wife Thekla Löwe. After school, he studied physics and mathematics at the University of Göttingen and University of Hamburg. In 1922, he began researching ferromagnetism under the guidance of Wilhelm Lenz. He earned a Ph.D. in physics from the University of Hamburg in 1924 when he published his doctoral thesis (an excerpt or a summary of his doctoral thesis was published as an article in a scientific journal in 1925 and this has led many to believe that he published his full thesis in 1925[2][3][4]). His doctoral thesis studied a problem suggested by his teacher, Wilhelm Lenz. He investigated the special case of a linear chain of magnetic moments, which are only able to take two positions, "up" and "down", and which are coupled by interactions between nearest neighbors. Mainly through following studies by Rudolf Peierls, Hendrik Kramers, Gregory Wannier and Lars Onsager the model proved to be successful explaining phase transitions between ferromagnetic and paramagnetic states.[5][6]

After earning his doctorate, Ernst Ising worked for a short time in business before becoming a teacher, in Salem, Strausberg and Crossen, among other places. In 1930, he married the economist Dr. Johanna Ehmer (later known as Jane Ising).[7] As a young German–Jewish scientist, Ising was barred from teaching and researching when Hitler came to power in 1933. In 1934, he found a position, first as a teacher and then as headmaster, at a Jewish school in Caputh near Potsdam for Jewish students who had been thrown out of public schools. Ernst and his wife lived in Caputh near the famous summer residence of the Einstein family. In 1938, the school in Caputh was destroyed by the Nazis, and in 1939 the Isings fled to Luxembourg, where Ising earned money as a shepherd and railroad worker. After the German Wehrmacht occupied Luxembourg, Ernst Ising was forced to work for the army. In 1947, the Ising family emigrated to the United States. Though he became professor of physics at Bradley University in Peoria, Illinois, he never published again. Ising died at his home in Peoria in 1998, just one day after his 98th birthday.

Work

[edit]
Main article: Ising model

The Ising model is defined on a discrete collection of variables called spins, which can take on the value 1 or −1. The spins interact in pairs, with energy that has one value when the two spins are the same, and a second value when the two spins are different.

The energy of the Ising model is defined to be where the sum counts each pair of spins only once. Notice that the product of spins is either +1 if the two spins are the same (aligned), or −1 if they are different (anti-aligned). J is half the difference in energy between the two possibilities. Magnetic interactions seek to align spins relative to one another. Spins become randomized when thermal energy is greater than the strength of the interaction.

For each pair, if

, the interaction is called ferromagnetic;
, the interaction is called antiferromagnetic;
, the spins are noninteracting.

A ferromagnetic interaction tends to align spins, and an antiferromagnetic tends to antialign them.

The spins can be thought of as living on a graph, where each node has exactly one spin, and each edge connects two spins with a nonzero value of J. If all the J values are equal, it is convenient to measure energy in units of J. Then a model is completely specified by the graph and the sign of J.

The antiferromagnetic one-dimensional Ising model has the energy function where i runs over all the integers. This links each pair of nearest neighbors.

In his 1924 Ph.D. thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition. On the basis of this result, he incorrectly concluded that his model does not exhibit phase transition in any dimension.

It was only in 1949 that Ising knew the importance his model attained in scientific literature, 25 years after his Ph.D. thesis. Today[when?], each year, about 800 papers are published that use the model to address problems in such diverse fields as neural networks, protein folding, biological membranes and social behavior.[5][8] Analysis of Google Scholar results shows exponential increase in the occurrence of "Ising model" in paper titles, with a doubling period of approximately ten years, reaching 1800 occurrences in 2025.

The Ising model had significance as a historical step towards recurrent neural networks. Glauber in 1963 studied the Ising model evolving in time, as a process towards equilibrium (Glauber dynamics), adding in the component of time.[9] Shun'ichi Amari in 1972 proposed to modify the weights of an Ising model by Hebbian learning rule as a model of associative memory, adding in the component of learning.[10] This was popularized as the Hopfield network (1982).[11]

See also

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Notes

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Ernst Ising

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Ernst Ising (10 May 1900 – 11 May 1998) was a German-American physicist best known for developing the Ising model, a discrete mathematical framework in statistical mechanics that describes interacting magnetic spins on a lattice to explain ferromagnetism and phase transitions. Born in Cologne, Germany, to merchant Gustav Ising and Thekla Löwe, he studied mathematics and physics at the universities of Göttingen, Bonn, and Hamburg from 1919 to 1924, earning his doctorate from Hamburg under Wilhelm Lenz with a thesis that exactly solved the one-dimensional version of the model Lenz had proposed in 1920. Although Ising's analysis demonstrated no spontaneous magnetization or phase transition in one dimension at finite temperature—initially viewed as a limitation—the model's simplicity and solvability in higher dimensions, later advanced by Lars Onsager's 1944 exact solution for the two-dimensional case, established it as a cornerstone for studying critical phenomena, universality, and renormalization group theory in condensed matter physics. After brief industrial work at AEG's patent office and secondary school teaching in Germany and Luxembourg amid rising political instability, Ising emigrated to the United States in the early 1940s, obtaining citizenship in 1953 and serving as a physics professor at Bradley University in Peoria, Illinois, from 1948 until his retirement in 1976.

Early Life and Education

Family Background and Upbringing

Ernst Ising was born on May 10, 1900, in , , to Gustav Ising, a merchant, and his wife Thekla (née Löwe). Two years later, in 1902, the family moved to , an industrial center in the region, where Ising spent most of his childhood. He commenced formal schooling in in 1907, progressing through primary and secondary education in the local system amid the stable bourgeois environment of a merchant family. Ising's maternal lineage traced to a traditional Jewish family background, though biographical accounts indicate a secular upbringing without emphasis on religious observance.

Academic Training in Göttingen

Ising began his higher education at the in Easter 1919, focusing on mathematics and physics. This enrollment came shortly after his completion of the Gymnasium in in 1918 and a brief stint of military training amid the post-World War I context in Germany. The , established as a leading center for advanced studies in the natural sciences, offered Ising exposure to foundational coursework in these fields during his initial semesters. Although specific professors supervising Ising at are not documented in available records, the institution's curriculum emphasized analytical rigor and theoretical principles, aligning with the era's emphasis on . Ising's time there laid preliminary groundwork for his later specialization, but after approximately one year—marked by a semester-long absence—he continued his studies at the in 1920 before transferring to , where he pursued advanced research. His doctoral training ultimately occurred under Wilhelm Lenz in Hamburg, culminating in his 1924 thesis on .

Scientific Contributions

Formulation of the Ising Model

![{\displaystyle E=-\sum {ij}J{ij}S_{i}S_{j}\,}][float-right] The , proposed by Ernst Ising in his 1925 doctoral thesis under Wilhelm Lenz at the , represents through a lattice of interacting magnetic . Each lattice site hosts a spin variable Si=±1S_i = \pm 1, corresponding to the two possible orientations of an atomic along a quantization axis, reflecting early quantum mechanical constraints on spin directions. Interactions occur between nearest-neighbor , with the JijJ_{ij} determining the preference for alignment: positive JijJ_{ij} favors ferromagnetic ordering where neighboring tend to point in the same direction. The energy of the system, expressed via the Hamiltonian, is H=−∑⟨i,j⟩JijSiSj−h∑iSiH = -\sum_{\langle i,j \rangle} J_{ij} S_i S_j - h \sum_i S_i, where the first term captures pairwise interactions (summed over nearest neighbors ⟨i,j⟩\langle i,j \rangle), the second accounts for an external magnetic field hh, and the negative signs ensure lower energy for aligned spins in ferromagnetic cases (Jij>0J_{ij} > 0). Ising derived this form to model the emergence of spontaneous magnetization in materials without an external field, treating the lattice as a regular array—initially a one-dimensional chain—to enable exact computation of thermodynamic properties via summation over all spin configurations weighted by Boltzmann factors. For the one-dimensional case emphasized in Ising's work, the Hamiltonian simplifies to H=−J∑iSiSi+1−h∑iSiH = -J \sum_i S_i S_{i+1} - h \sum_i S_i, assuming uniform coupling J>0J > 0 along the . ![{\displaystyle E=\sum {i}S{i}S_{i+1}\,}][center]

Analysis and Limitations in One Dimension

In his 1925 doctoral , Ernst Ising formulated and exactly solved the one-dimensional version of the model, considering a linear of N spins Si=±1S_i = \pm 1 with nearest-neighbor interactions and no external . The Hamiltonian for this system is given by E=−J∑iSiSi+1E = -J \sum_i S_i S_{i+1}, where J>0J > 0 favors ferromagnetic alignment. Ising derived the partition function through a recursive method, effectively prefiguring the approach, yielding Z≈(2cosh⁡βJ)NZ \approx (2 \cosh \beta J)^N in the . From this, Ising computed the magnetization per spin as m=tanh⁡(βJ+βh)m = \tanh(\beta J + \beta h) in the presence of hh, but found that in zero field, m=0m = 0 for all finite temperatures T>0T > 0, with spontaneous alignment only emerging [at T](/page/AT&T) = 0. The free energy remains analytic across all temperatures, indicating the absence of or singularity associated with long-range order. This exact solvability highlighted the model's tractability in one dimension but revealed its inability to produce a finite-temperature ferromagnetic , as the thermal fluctuations disrupt correlations over long distances. The primary limitation of the one-dimensional analysis was its failure to capture the essential physics of real ferromagnetic materials, which exhibit below a critical Tc>0T_c > 0 in three dimensions. Ising himself noted this discrepancy, interpreting the result as evidence that the simplified lattice model with short-range interactions insufficiently modeled bulk ferromagnetism. Contemporaries, including , criticized the model on these grounds, using the lack of a to justify alternative quantum mechanical approaches based on exchange interactions rather than classical spins. Despite this, the exact solution provided a benchmark for understanding dimensionality effects, later generalized by theorems like Mermin-Wagner, which prohibit breaking in low dimensions for short-range interactions.

Broader Implications and Initial Overshadowing

The one-dimensional solution derived by Ising in his 1925 dissertation demonstrated the absence of or phase transitions at any finite temperature, which contemporaries interpreted as evidence that the model failed to capture the ferromagnetic ordering observed in real materials, predominantly three-dimensional systems. This negative result, coupled with Ising's own extrapolation suggesting limited applicability to higher dimensions, contributed to the model's initial neglect in the physics community during the late and , as researchers pursued alternative mean-field approaches like those of Weiss and Heisenberg that promised broader explanatory power for magnetism. Citations of Ising's work remained sparse, with the paper receiving minimal attention beyond immediate circles in German . Subsequent breakthroughs revitalized the model, beginning with Lars Onsager's exact solution for the two-dimensional case in 1944, which revealed a finite-temperature and deviating from mean-field predictions, thus highlighting the model's capacity to exhibit phenomena absent in one . This paved the way for its central role in elucidating , including universality classes of , as formalized in the and through scaling hypotheses and Kenneth Wilson's theory, which earned Wilson the 1982 for advancements directly inspired by Ising-like lattice models. The model's simplicity facilitated pioneering computational techniques, such as simulations introduced in 1953 for estimating partition functions in higher dimensions, enabling numerical exploration of inaccessible analytic regimes. Beyond , the Ising framework's implications extended to disordered systems via extensions like the model (Edwards-Anderson, 1975), influencing studies of frustration and optimization problems in , and to interdisciplinary applications modeling , neural networks, and biological , where binary states approximate decision-making or cellular automata behaviors under local interactions. Despite these developments, Ising's personal contributions were initially overshadowed, with the model's revival often attributed to later solvers like Onsager, and widespread recognition of Ising's foundational role emerging only in the postwar era through historical reviews and his own modest career trajectory outside mainstream research hubs.

Career in Germany

Early Teaching and Research Positions

Following his PhD in 1924 from the , Ising briefly entered industry, working from 1925 to 1926 at the patent office of Allgemeine Elektrizitätsgesellschaft (AEG) in . Dissatisfied with this role, he transitioned to education, beginning as a teacher at the Salem boarding school in near from 1926 to 1927. To qualify for higher civil service teaching positions, Ising pursued additional studies in and at the University of from 1928 to 1930, passing the required state examinations that year. On December 23, 1930, he married Johanna Ehmer and secured a position as Studienassessor—a probationary higher civil service role—at a high school in near , where he taught and physics. In the early , Ising was transferred as a to Crossen (now Krosno Odrzańskie, ), continuing in without advancing to university-level research or professorial roles. No formal research positions are recorded during this period; his career focused on classroom instruction amid limited opportunities for Jewish scholars in German academia.

Professional Disruptions under the Nazi Regime

Following the Nazi Party's assumption of power on January 30, 1933, Ernst Ising, classified as Jewish under Nazi racial laws, faced immediate professional repercussions as a employed as a high school teacher in public schools. On March 31, 1933, he was dismissed from his position pursuant to early implementations of anti-Jewish policies targeting , which were formalized in the Law for the Restoration of the Professional of , 1933. This explicitly excluded individuals of Jewish descent from state employment, effectively barring Ising from academic research and mainstream teaching roles. In response to his dismissal, Ising secured employment in 1934 as a teacher for Jewish children at the Jüdisches Landschulheim, a in Caputh near established for Jewish pupils expelled from public institutions. He served there until November 1938, when the school was destroyed during the pogroms of November 9–10, which targeted Jewish communal properties across as part of coordinated state-sponsored violence. This event further curtailed Jewish educational efforts, confining Ising's professional activities to increasingly precarious, segregated environments amid escalating persecution. Throughout the Nazi era, Ising's Jewish ancestry precluded any return to university-level physics research or public-sector roles, limiting him to informal or community-based teaching amid broader institutional exclusion of Jewish scientists. These disruptions halted his early scientific momentum, with no documented publications or advancements in during this period, reflecting the regime's systematic purge of Jewish professionals from intellectual life.

Post-War Transition and Emigration

Reconstruction Efforts in Lübeck

Following the Allied liberation of in , Ernst Ising continued to reside there with his family until their in , with no documented involvement in reconstruction activities in or any other part of . During the immediate post-war period from 1945 to , Ising's efforts centered on personal and familial stabilization amid economic hardship, rather than contributing to urban or institutional rebuilding in his native country. , where the family had fled in to escape Nazi , served as their base; Ising had previously supported them through menial labor such as farm work and railroad maintenance during the occupation. Ising's professional reconstruction was deferred until arrival in the United States, as opportunities in war-devastated were limited for someone of Jewish descent who had been barred from academia and under the Nazi since 1933. No primary accounts or biographical records indicate attempts to secure positions or engage in scientific, educational, or civic reconstruction in —a city heavily damaged by Allied bombing in 1942, which faced extensive post-war rebuilding of its historic architecture and infrastructure. Instead, the period involved preparatory steps for relocation, including a brief to in April 1946 for respite. By early 1947, Ising arranged passage on the freighter Lipscomb Lykes from , arriving in New York on April 12, marking the transition to a new career in American academia without interim engagement in European reconstruction. This decision reflected broader patterns among Jewish scientists, who often prioritized over reintegration into a politically unstable and resource-scarce .

Departure for the United States

Following the Allied liberation of in September 1944, where Ising had sought refuge since fleeing in 1939, he and his wife Johanna Ehmer Ising initiated plans to emigrate to the , motivated by the desire for stability after years of persecution and wartime displacement. The process was hindered by extensive post-war immigration bureaucracies, requiring over two years to secure the requisite visas and documentation after the European theater of concluded on May 8, 1945. In spring 1947, the couple departed from a European port aboard the American freighter Lipscomb Lykes, a vessel typically used for cargo transport between continents. They arrived in in April 1947, marking Ising's permanent relocation at age 46. This transatlantic crossing concluded a decade of enforced nomadism, during which Ising had worked intermittently as a teacher, laborer, and military conscript under occupation, with no return to academic research in .

Career in the United States

Initial Academic Roles

Upon emigrating to the in April 1947, Ernst Ising obtained his first academic position as an instructor in physics and mathematics at the State Teachers College (now ) in . He served in this role for one year, from 1947 to 1948, marking a significant transition from his disrupted career in postwar to adapting to American higher education. This position required Ising to teach primarily in English, a language he learned as an adult immigrant, representing a challenging shift from his prior experience in German secondary education. During this period, Ising focused on undergraduate instruction without notable research output, consistent with the teaching-oriented nature of the institution and his personal circumstances following displacement.

Long-Term Professorship at Bradley University

In 1948, Ernst Ising accepted a position as Professor of Physics at Bradley University in , marking the beginning of a 28-year tenure dedicated primarily to . Following a brief teaching role at Minot State Teachers College in , Ising shifted focus to classroom instruction at Bradley, where he emphasized foundational principles in courses such as general physics, , and . His approach prioritized clarity and empirical understanding over advanced research, reflecting his post-war circumstances and preference for stable academic service rather than publication. During his time at Bradley, Ising ceased original research output, with no further peer-reviewed publications after his arrival, as he devoted efforts to mentoring students and curriculum development in a teaching-oriented institution. In 1968, the university awarded him an honorary doctorate in recognition of his long-term contributions to physics education. His wife, Hanna (Americanized as Jane), also joined the faculty, supporting the household while Ising maintained a low-profile yet consistent presence in departmental affairs. Ising retired from Bradley University in 1976 at age 76, concluding a career phase defined by pedagogical stability amid earlier disruptions in . Post-retirement, he remained in Peoria, occasionally reflecting on his Ising model's unexpectedly enduring impact in , though his U.S. years centered on routine teaching rather than scientific acclaim.

Later Life and Recognition

Retirement and Personal Reflections

Ising retired from his professorship in physics at Bradley University in , in 1976 at the age of 76, following a new university regulation that mandated retirement at that age. After retiring, he and his wife, Jane (formerly ), undertook extensive travels, including trips to and . These activities reflected a shift toward personal pursuits following decades focused on teaching and family stability in the United States. Ising remained in Peoria until his death, maintaining an interest in and leisure. His son, Thomas Ising, recalled him as a dedicated who incorporated humor and hands-on experiments to engage students, though post-war priorities had centered on and rather than advancing his early research on . In retirement, Ising did not publish further scientific work, consistent with his career trajectory after emigrating, but he received late recognition for the , including an honorary doctorate from the in 1981. Ernst Ising died at his home in Peoria on May 11, 1998, one day after his 98th birthday, following a brief period in care. Personal accounts from family describe him as kind, gentle, and humorous, with enjoyments including reading histories and biographies, playing chess and bridge, listening to , and violin performance; in later years, he pursued with notable skill.

Posthumous Honors and Model's Enduring Impact

Following Ising's death on May 11, 1998, in , his legacy was honored through initiatives bearing his name. The University of Cologne's Physics Department awards the Ernst Ising Dissertation Prize annually to recognize exceptional doctoral theses, endowed with €4,000. The Ising Lectures, an annual series on statistical physics and related topics initiated in , , in 1997, have continued posthumously, reaching their 20th installment by 2017 and fostering discussions on the model's extensions and applications. The , defined by its Hamiltonian E=−∑ijJijSiSjE = -\sum_{ij} J_{ij} S_i S_j where Si=±1S_i = \pm 1 are spin variables and JijJ_{ij} denote interaction strengths, remains a cornerstone of . Exactly solvable in one dimension by Ising in —demonstrating no at finite temperature—and in two dimensions by in 1944, it elucidates , , and in interacting spin systems. Its captures real-world in materials like unary magnets, providing qualitative and quantitative benchmarks despite simplifications such as nearest-neighbor interactions and no external fields in the base case. Beyond physics, the model's framework has influenced diverse fields through generalizations. In , Monte Carlo simulations based on it simulate complex systems intractable analytically. Extensions appear in via energy-based models like restricted Boltzmann machines for and generative tasks. In social sciences, hierarchical variants model polarization and influence dynamics among agents with varying engagement levels. Biological applications include calcium release in cellular processes, where channel clusters exhibit Ising-like cooperative behavior near criticality. Quantum versions underpin studies of and quantum phase transitions, while time-varying interactions extend it to dynamic networks. By , marking its centennial, the model had inspired thousands of studies, affirming its interdisciplinary robustness.

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