The 200th anniversary of the critical phenomenon. Who first discovered this physical phenomenon?

Commemorates the bicentenary of the discovery of critical phenomena by French physicist Charles Cagniard de La Tour.

Written by Liu Yihan, Zhang Yi, Su Guifeng (Department of Physics, School of Mathematics and Physics, Shanghai Normal University)

"Some years ago, Cagnad de la Tour made an experiment which gave me the opportunity of inventing a new word... How shall I name the point where liquid and vapor become one, according to the law of continuity? Cagnad de la Tour did not give it a name; what shall I call it?"

——Faraday to Hu Weili[1]

1. What is critical phenomenon?

What are critical phenomena? In fact, phase change and critical phenomenon are the same thing. Separating them is just a "misunderstanding" in the history of physics, thinking that the two are different physical phenomena. In order to describe phase change and critical phenomenon concretely, we take water, which is common in our daily life, as an example to make a brief and intuitive explanation.

Phase transition, as most textbooks say, is the change of a substance, such as water, from one (aggregate) form to another (aggregate) form [Note 1]. Humans’ knowledge of the existence of water in three phases, gas, liquid, and solid, can be traced back to the historical records of China and ancient Egypt about 4,000 years ago, but the real understanding of phase transition has only been in the last half century or so. The late famous statistical physicist Leo Kadanoff [Note 2] (1937-2015) once used the example of an iceberg floating in the ocean to vividly illustrate the coexistence of different phases of water: "The ocean is liquid water, which surrounds ice, which is solid water. The breeze blows the clouds, and the water vapor in the air comes into contact with both solid and liquid water." (Of course, strictly speaking, seawater is not the only chemical component of water. For the original text of this paragraph, please refer to the text caption of Figure 1.) [2]

To study the phase change of a substance, a basic task is to determine its "phase diagram", that is, to find out what phase the substance is in under given thermodynamic parameters - usually temperature T, pressure P and volume V for "simple" thermodynamic systems - and determine the boundaries between different phases. For example, Figure 2 shows the phase diagram of water in the pressure-temperature (PT) plane, which clearly shows the three phases of water under different temperature and pressure conditions: solid (light blue area), liquid (blue area), and gas (ochre area), as well as the boundary between any two phases. The yellow dot near the middle of Figure 2 is called the triple point. As the name suggests, it is the intersection of the above three phases. Starting from the triple point, "going up" along the gas-liquid dividing line, it does not extend infinitely, but stops at the red dot in Figure 2, which is the critical point. For water, the thermodynamics corresponding to the critical point

The difference no longer exists, and it is no longer meaningful to ask whether water is gas or liquid at this time. Therefore, with the critical point as the boundary, the area above it (see the upper right corner of Figure 2) is a supercritical fluid, in which water will show more new characteristics.

Figure 1 L. Kadanoff: "Iceberg floating in the sea. This picture is intended to illustrate different phases of water. The sea is liquid water, which is then in contact with solid water in the form of ice. In the air above, breezes blow clouds through the air, which contains water vapor in contact with both the solid and the liquid forms of water. The change from one form to the next is termed a phase transition.[2]

Figure 2 Schematic diagram of the pressure-temperature (PT) plane phase diagram of water [Picture from the Internet]

Although the critical point is just a point on the pressure-temperature phase diagram, the physical phenomena occurring near the critical point are very rich - collectively referred to as "critical phenomena". A typical example is the so-called "critical opalescence": a transparent gas or liquid becomes turbid and gradually presents a milky white color when its thermodynamic parameters approach the critical point. From statistical physics, we know that this is due to the large fluctuations near the critical point, which causes extremely strong scattering of light. This can be observed by scattering laser light through phase separation at the critical point. The following video demonstrates the critical opalescence phenomenon of a mixture of equal amounts of aniline and cyclohexane.

As can be seen in the video, when the critical temperature is reached and the mixture changes from a single phase to two phases (phase separation), the light spot on the screen is disturbed. The light spot "flickers" until it completely diffuses. Once the phase change is complete and the two substances are completely separated, it eventually forms a single light spot again. The same but reversed pattern can also be observed when the mixture is heated.

Video 1 As the mixture cools at the transition temperature, the laser (as shown on the screen) flickers and diffuses until it becomes completely opaque (the video has been sped up 10 times). [3] [Go to the "Fanpu" official account to watch the video]

Video 2 Same as Video 1, side view (perpendicular to the direction of the laser, from left to right). Initially, a single laser beam passes through the mixture, and when it reaches the phase transition point, the beam diffuses significantly. (The video has been sped up 200 times). [3] [Go to the "Fanpu" official account to watch the video]

In addition, the system has some other unique physical phenomena near the critical point, such as the specific heat of the system continues to increase in the process of approaching the critical point, the specific heat coefficient and compressibility tend to "diverge" (infinity) at the critical point, etc.

It can be said that the discovery of critical phenomena began with curiosity. Historically, French physicist Charles Cagniard de la Tour (1777-1859) first discovered critical phenomena in his experiments in 1822. Many people may not realize that it has been two hundred years since his discovery! Physics has undergone tremendous changes in the past two hundred years. The study of critical phenomena has developed into a mature field of modern condensed matter physics and complex system physics, and it continues to bring new surprises.

In this article, we will briefly review the historical background of the discovery of critical phenomena. According to the classification of the famous statistical physicist Cyril Domb (1920-2012), this period of history can be classified as the "classical period" of critical phenomenon research [4]. We will also briefly introduce some important developments in critical phenomenon research in the classical period after Delatour's death [Note 3].

Biography of Didratu

Charles Cagniard de La Tour, who first discovered the critical phenomenon, was born in Paris, France on March 31, 1777. He studied at l'Ecole Polytechnique and École du Génie Géographe as a student. He later served as an auditor of the Council of State and director of special projects for the city of Paris. He was also a prolific scientist and inventor. In addition to discovering the critical phenomenon, he made important contributions in many different fields, from mechanics to acoustics to chemical biology.

Delatour's academic research began in the fields of mechanics and thermodynamics. In 1809, he invented a new heat engine. Between 1809 and 1815, he successively invented a new hydraulic engine, a new air pump, a heat-driven winch and many other devices. Before 1819, Delatour had been improving the design of these inventions. After that, Delatour became interested in the physics of bird flight and human vocalization, and began to study acoustics and the mechanism of sound generation, and invested a lot of energy in this field. It is worth noting that it was this accidental change of interest that led to his discovery of critical phenomena in the future. Between 1828 and 1831, Delatour began to study the crystallization process and the effect of acid on carbon, as well as phosphorus, silicon and its crystallization, and even the hardening of mortar. Between 1832 and 1835, Delatour became interested in the application of the Archimedean screw principle to air pumps[5].

In 1835, Delatour began to turn to the study of alcoholic fermentation. This work reached its peak between 1836 and 1838. At the end of 1836, he discovered that beer yeast contained an active substance. German physiologist Theodor Schwann (1810-1882) also independently came to the same conclusion almost at the same time, but the criticism of chemist Justus von Liebig (1803-1873) delayed this view for 20 years. It was not until 1857 that French biologist Louis Pasteur (1822-1895) announced this discovery again.

By the way, there is still controversy about the reliability of Delatour’s photos or portraits. Some photos or portraits circulating in certain documents and on the Internet often contradict each other. For example, as shown in Figure 3, which is commonly seen on the Internet, it is said to be a portrait of Delatour. There is relatively reliable evidence that it is actually the British Prince Charles Edward [Note 6].

Figure 3 Some so-called portraits of De La Tour circulating on the Internet are not reliable

Discovery and early history of the triple critical phenomenon

The invention of the steam engine in the late 17th and early 18th centuries sparked interest in the behavior of fluids under high temperatures and pressures. While working as an assistant to Robert Boyle (1627-1691) at the Royal Society of England, French physicist Denis Papin (1647-1712) invented the predecessor of the steam engine, the "Papin's digester" (see the schematic diagram and model in Figure 4). He also noted that when heated under high pressure, the temperature at which water remains in the liquid phase is much higher than the usual boiling point, that is, the boiling point increases with increasing pressure. In the second half of the 18th century, French chemist Antoine-Laurent de Lavoisier (1743-1794) proved that gas and steam are actually the same thing, a third state of matter besides solid and liquid. He also proposed that gas can be liquefied at sufficiently low temperatures and sufficiently high pressures. This understanding led to the first successful liquefaction of gas by Jean-Francois Clouet (1751-1801)[Note 7] and Gaspard Monge (1746-1818)[Note 8] in 1784 by cooling and compressing gaseous sulfur dioxide. This was followed by a series of successful experiments by British physicist Michael Faraday (1791-1867) to liquefy gases[7, 8]. Hydrogen, oxygen, nitrogen, and carbon monoxide, gases that were previously considered incondensable—once called “permanent gases”—were finally liquefied in 1877.

Figure 4 Schematic diagram and model of Papin's digester Image source: from the Internet

It was in the Papin hot press steamer experiment that De La Tour discovered the existence of the critical point. In 1822, out of interest in acoustics, De La Tour placed a flint ball in a steamer partially filled with liquid and heated it. When the experimental device was rotated, the solid flint ball produced a splashing sound of water because it passed through the interface between the gas and liquid phases. De La Tour noticed that when the temperature in the experiment was far above the boiling point of the liquid, the splashing sound of water stopped after exceeding a certain temperature. This actually means the discovery of the supercritical fluid phase mentioned above (the area above the red critical point in Figure 2). In this phase, since there is no gas-liquid phase boundary, there is no surface tension. Supercritical fluids can also dissolve substances like liquids, and can also diffuse in solids like gases. At present, the research on supercritical fluids is still an important direction.

In two papers published in Annales de Chimie et de Physique [9], Delatour described how he heated sealed glass tubes of alcohol under high pressure (see Figure 5 on the front page of Delatour’s paper). He observed that the liquid expanded to about twice its original volume and then turned into a transparent vapor, and the tube looked empty. But when it was cooled again, a “cloud” appeared inside the tube. We now recognize that this is actually a manifestation of the critical opalescence phenomenon at the critical point. To give the reader an intuitive impression, Figure 6 shows the critical opalescence of ethane. Delatour also noticed that above a certain temperature, increasing pressure does not prevent the evaporation of the liquid.

Figure 5. The first page of Delatour’s paper [9], in which he reports the discovery of critical phenomena.

Figure 6 Critical opalescence of ethane (inside the yellow circle in the middle picture) Image source: https://handwiki.org/

In a subsequent paper [10], Delatour wanted to prove that the existence of a certain limiting temperature is a universal phenomenon. The so-called limiting temperature is the temperature above which a liquid evaporates regardless of the pressure. In the paper, Delatour reported the results of experiments on several substances. He determined the corresponding critical temperature by the disappearance of the liquid meniscus when the surface tension is zero. Delatour measured the critical temperature Tc of water, alcohol, ether and carbon disulfide and found that for each substance there is indeed a certain temperature at which the liquid evaporates even without increasing pressure, and above this temperature, the liquid evaporates completely. Delatour measured the critical temperature of water to be about 362°C. Considering the historical conditions at the time, this is a fairly accurate result (modern measurements result in about 374°C). In his paper, he said that this "particular state" (état particulier): "always requires very high temperatures, almost independently of the capacity of the pipe" [10]. We now know that this "particular state" marks the end of the phase equilibrium curve, that is, the critical point.

Many of Delatour's contemporaries did not realize the significance of his discovery, believing that the result was specific to the materials used in Delatour's experiments and was not a general phenomenon.[11] However, Faraday showed great physical insight and recognized the value of Delatour's work.[12] In 1844, Faraday wrote to William Whewell (1794-1866)[Note 9]: "Some years ago, Cagniard de la Tour performed an experiment which gave me an opportunity of inventing a new word." Faraday then talked about the critical point in the modern sense: "According to the law of continuity, how should I name the point where liquid and vapor merge into one? Cagniard de la Tour did not name it, so what should I call it?" (See William Whewell, note 11 above) Whewell suggested calling it the vaporization point, or the non-liquefaction point of the liquid, or the Dela Tour state. Faraday used "Cagniard de la Tour's state" and "Cagniard de la Tour point" in his later papers[13].

Delatour died in Paris on July 5, 1859. However, his experimental discoveries initiated the classical period of critical phenomena research and subsequent intellectual explorations.

The term "critical point" that we use today was coined by the British physical chemist Thomas Andrews [Note 10] (1813-1885) in 1869, ten years after Delatour's death. In the same year, he discovered "supercritical fluid" and published his research results in the same year's "Philosophical Magazine" under the title "On the continuity of the gaseous and liquid states of matter" [14] (see Figure 7). In this famous paper, Andrews studied the pressure-volume curve of the coexistence line of carbon dioxide liquid and gas phases, and further explained what Delatour called "specific state" - that is, only under certain temperature and pressure - can gas condense into liquid, or liquid can evaporate into gas. Above this point is the supercritical phase, where the distinction between liquid and vapor disappears (see Figure 8).

In 1873, Dutch physicist JH van der Waals (1837-1923) first clearly explained the continuity between the gas phase and the liquid phase of matter theoretically. In his doctoral thesis, van der Waals showed[15] that the ideal gas law can be generalized by introducing intermolecular interactions, and obtained the equation of state of van der Waals gas named after him, which qualitatively explained Andrews' experimental results. Famous physicists Maxwell and Boltzmann at the time highly praised van der Waals' results[4]. Van der Waals' work in turn inspired his compatriot, Dutch physicist Heike Kamerlingh Onnes (1853-1926). The latter was able to estimate the critical point of permanent gases based on this, which provided a theoretical basis for the eventual liquefaction of helium at low temperatures, about 4K. The subsequent acquisition of low temperatures led to the discovery of superconductivity. But the history of low temperature exploration is another story[16].

The behavior of matter near the critical point can be characterized by a series of critical exponents. The "critical exponents" obtained from the van der Waals equation of state are actually simple mean field values, which do not correspond to the critical exponent values ​​of thermodynamic systems actually measured. Belgian physicist Jules-Emilé Verschaffelt (1870-1955) [Note 11] first discovered this through experiments in 1896 [17]. He remeasured the rise of carbon dioxide in the capillary and analyzed the coexistence curve data in combination with the new experimental value of the coexistence density, and found that it did not match the mean field value. However, Verschaffelt's experimental results did not attract the attention of physicists at the time. In the 1930s, the famous Soviet physicist Landau (Lev Davidovich Landau, 1908-1968) continued to develop a general framework for systematic mean field treatment of phase transitions, namely the Landau continuous phase transition theory, which is a peak of phase transition phenomenological theory [18].

Figure 7 The first page of Andrews’ 1869 paper On the Continuity of the Gaseous and Liquid States of Matter, which mentions Delatour’s experimental discovery of critical phenomena right away.[14]

Figure 8. A diagram from Andrews’ 1869 paper On the Continuity of the Gas and Liquid States of Matter[14]. In the figure, the horizontal axis is pressure and the vertical axis is volume. It can be seen that as the temperature increases, the density difference between the coexisting gas and liquid phases gradually approaches zero and eventually disappears.

Figure 9 Dear readers, can you find Woshafelt in this famous photo? (The answer is in the note [12] at the end of the article) Image source: Internet

In another research line on magnetism, French physicist Pierre Curie (1859-1906) discovered that ferromagnetic materials will demagnetize when the critical temperature is exceeded [19]. This critical temperature is usually called the "Curie point". In 1895, he noticed the similarities between the gas-liquid phase transition and the ferromagnetic phase transition, and proposed the important concept of the "universality" of critical phenomena [Note 13]. In order to understand the origin of magnetism, in 1920, German physicist Wilhelm Lenz (1888-1957) introduced a simple model - now commonly known as the "Ising model" [20]. In 1924, Lenz's student Ernst Ising (1900-1998) solved the one-dimensional case of the model in his doctoral thesis and found that there was no phase transition, but he mistakenly extended this conclusion to the two-dimensional case, believing that there was no phase transition in the two-dimensional Ising model [21]. After the work of Peiers [22] and Kramers & Wannier [23], Lars Onsager (1903-1976) finally calculated the specific heat of the two-dimensional Ising model in the absence of an external magnetic field in 1944 [24]. Onsager's work was so important that Dumm called it the "Onsager Revolution" [4]. Onsager also gave an unproven spontaneous magnetization formula in 1949 [25, 26] [Note 14], which was proved by Chen Ning Yang (1922-) in 1952 [27]. However, the exact solution of the three-dimensional Ising model has not yet been solved, which has always been a huge challenge for physicists. The history of the Ising model itself is enough to constitute the content of a monograph. We will not go into details here. We only list some important progress in Table I. Interested readers can read the relevant articles [36] and the references cited in this article.

Table I Historical progress of the exact solution of the Ising model

In the absence of an exact solution to the three-dimensional Ising model, people have to rely on numerical simulations. In his 1949 doctoral dissertation, Dumm proposed the high and low temperature expansion methods (see [4]). Today, the Monte Carlo method proposed by Nicholas Metropolis [Note 15] (1915-1999) and Stanislaw Ulam [Note 16] (1909-1986) in 1949 is widely used [37].

In the 1960s, Kadanoff and Fisher[Note 17] (Michael Fisher, 1931-2021) realized that the general theoretical framework of phase transitions must be based on the "scaling hypothesis", especially the "scaling relations" between various critical exponents describing the critical point were derived from the scaling hypothesis. This idea paved the way for a complete theoretical description of critical phenomena through the "renormalization group" method[38] proposed by Wilson[Note 18] (Kenneth G. Wilson, 1936-2013) in 1971.

At this point, our research and understanding of critical phenomena has reached a new height, and it is also a new starting point.

Notes

[1] Due to space limitations, we will not discuss further the classification of phase transitions here. Modern interpretations of (continuous) phase transitions also include so-called symmetry breaking. Readers interested in the full story of the development of phase transitions and critical phenomena are referred to the fascinating popular science book, Miracles at the Edge: Phase Transitions and Critical Phenomena, by Yu Lu, Hao Bolin, and Chen Xiaosong, Science Press (2005).

[2] Leo Kadanoff is a famous American statistical physicist and former president of the American Physical Society (APS). He has made outstanding contributions in the fields of statistical physics, chaos theory, and condensed matter physics.

[3] Since this article focuses on Cagniard’s discovery of critical phenomena, the author has not discussed in detail the subsequent research in the classical period. There will inevitably be omissions. I look forward to having the opportunity to write another article about this in the future.

[4] Theodor Schwann, a German physiologist, was one of the founders of the cell theory, the discoverer of the organic properties of yeast, the discoverer and researcher of pepsin, and coined the term “metabolism”.

[5] Justus von Liebig, a German chemist, is considered one of the founders of organic chemistry.

[6] It is actually an oil painting by a painter called MQ de La Tour from around 1745, and is on display at the Scottish National Portrait Gallery in Edinburgh, see
https://www.britannica.com/topic/Jacobite-British-history.

[7] Jean-Francois Clouet was a French chemist and metallurgist who promoted the shift of French chemical research towards specific problems and promoted the development of the metallurgical industry.

[8] Gaspard Monge, French mathematician and physicist. He founded descriptive geometry and the characteristic theory of partial differential equations, and promoted the development of spatial analytic geometry, differential geometry, and pure geometry.

[9] William Whewell was a 19th-century British scholar of many talents and one of the most influential figures in British academia at the time. He wrote books on many subjects, including mechanics, mineralogy, geology, astronomy, political economy, and architecture, and left many works in the fields of philosophy of science, history of science, and moral philosophy, such as History of Inductive Science (3 volumes, 1837), Principles of Inductive Science (1840), History of Scientific Thought (2 volumes, 1858), and Principles of Discovery (1860). He was one of the founding members and president of the British Association for the Advancement of Science, a member of the Royal Society, president of the Geological Society, and long-time master of Trinity College, Cambridge. His influence was recognized by his contemporaries, including John Herschel, Charles Darwin, Charles Lyell, and Michael Faraday, who often sought philosophical and scientific advice from Whewell, and even help with academic terminology. Woolly invented Faraday's terms "anode", "cathode" and "ion". An interesting historical note is that in 1833, in response to a challenge from the poet ST Coleridge, Woolly invented the word "scientist", previously using only the terms "natural philosopher" and "man of science".

[10] Thomas Andrews, a British physical chemist, is a member of the Royal Society of London and the Royal Society of Edinburgh. He was formerly the Vice-President and Professor of Chemistry at Queen’s College Belfast. His main research interests lie in the critical state of matter.

[11] Jules-Emilé Verschaffelt was a Belgian physicist who studied under the Dutch physicist Heike Kamerlingh Onnes, the founder of low-temperature physics.

[12] Back row, seventh from left (between Schrödinger and Pauli): Auguste Piccard, Emile Henriot, Paul Ehrenfest, Edouard Herzen, Theophile de Donder, Erwin Schrodinger, J.-E. Verschaffelt, Wolfgang Pauli, Werner Heisenberg, Ralph Fowler; Leon Brillouin.

[13] In my opinion, the Curie's Principle proposed by Curie in 1894 points out the important role of symmetry in physics. Although there is still controversy about this principle, its historical role seems to have been ignored for a long time.

[14] On August 23, 1948, L. Tisza gave a lecture on the Ising model at Cornell University. At the end of the lecture, Onsager walked to the blackboard and announced that he and Bruria Kaufman had solved the problem and wrote the formula on the blackboard. In 1949, Onsager reiterated his results at the statistical mechanics conference in Florence, Italy. However, Kaufman and Onsager never formally published their calculations.

[15] Nicholas Metropolis, Greek-American physicist, made important contributions to the study of Monte Carlo methods.

[16] Stanislaw Ulam, an American mathematician and nuclear physicist, invented the Teller–Ulam configuration for hydrogen bomb design and also made contributions to number theory and set theory.

[17] Michael Fisher, a British statistical physicist, was a member of the Royal Society of London and the American Physical Society (APS). He made important contributions to phase transitions and critical phenomena.

[18] Kenneth Wilson, an American theoretical physicist, won the 1982 Nobel Prize in Physics for his work on the theory of renormalization group transformations.

References

[1] The original text is as follows: M. Faraday to W. Whewell: “Cagniard de la Tour made an experiment some years ago which gave me the occasion to want a new word […] how am I to name this point at which the fluid and its vapor become one according to a law of continuity. Cagniard de la Tour has not named it; what shall I call it?”. Letter dated November 12, 1844, published in LP Williams, The Selected Correspondence of Michael Faraday (Cambridge: Cambridge Univ. Press, 1971), Vol. 1, 427-428.

[2] LP Kadanoff, arXiv: 0906.0653v2

[3] See:
https://www.doitpoms.ac.uk/tlplib/solid-solutions/demo.php

[4] C. Domb, The critical point: a historical introduction to the modern theory of critical phenomena, Taylor & Francis (London 1996).

[5] B. Berche, M. Henkel, and R. Kenna, J. Phys. Studies 13 (2009) 3201.

[6] A.-L. Lavoisier, Recueil des mémoires de chemie (1792) 348; republished in Œuvres de Lavoisier, publiées par les soins de son excellence le ministre de l'instruction publique et des Cultes (Paris: Impr. impériale, 1862), t. II, 783-803.

[7] M. Faraday and H. Davy, Phil. Trans. R. Soc. Lond. 113 (1823) 160-165; M. Faraday, ibid., 189-198.

[8] M. Faraday, The Quarterly Journal of Science, vol. 19-33, Pub. by William F. Clay, Edinburgh and Simpkin, Marshall, Hamilton, Kent & Co., London (1896).

[9] C. Cagniard de la Tour, Ann. Chim. Phys., 21 (1822) 127-132; Supplément, ibid., 178-182.

[10] The original text is as follows: "...cet état particulier exige toujours une température très-élevée, presque indépendante de la capacité du tube". C. Cagniard de la Tour, Ann. Chim. Phys., 22 (1823) 410-415.

[11] Y. Goudaroulis, Revue d'Histoire des Sciences 47 (1994) 353-379.

[12] M. Faraday, letter to W. Whewell, 9th November 1844. See also [11].

[13] M. Faraday, Philosophical Transactions for 1845, Vol. 135, pp 155-177.

[14] T. Andrews, Phil. Trans. Roy. Soc. London 159 (1869) 575-590.

[15] JD van der Waals, doctoral thesis, Leiden (1873); reprinted in On the continuity of gaseous and liquid states, ed. with an introductory essay by JS Rowlinson, North-Holland Amsterdam (1988).

[16] See R. Srinivasan, Resonance, Vol. 1, No. 12 (1996) p. 6.

[17] JE Verschaffelt, Verslagen 5 (1896) 94-103.

[18] LD Landau, Nature 137 (1936) 840-841.

[19] P. Curie, Archives des Sciences physiques et naturelles, 3e période, tome XXVI (1891) p.13; reprinted in: Oeuvres de Pierre Curie, pp 214-219, Paris: Gauthier-Villars (1908).

[20] W. Lenz, Physikalische Zeitschrift. 21 (1920) 613-615.

[21] E. Ising, Zeitschrift für Physik 31 (1925) 253–258 (entitled Report on the theory of ferromagnetism).

[22] R. Peierls, Proc. Cambridge Phil. Soc. 32 (1936) 477-481.

[23] HA Kramers and GH Wannier, Phys. Rev. 60 (1941) 252-262; Part II, ibid., 263-276.

[24] L. Onsager, Phys. Rev. 65 (1944) 117-149.

[25] L. Onsager, Nuovo Cim 6 (Suppl 2) (1949) 279–287.

[26] B. Kaufman, Phys. Rev. 76 (1949) 1232-1243; B. Kaufman and L. Onsager, ibid. (1949) 1244-1252.

[27] CN Yang, Phys. Rev. 85 (1952) 808-816.

[28] PW Kastelyn, J. Math. Phys. 4 (1963) 287-293.

[29] EW Montroll, RB Potts, and JC Ward, J. Math. Phys. 4 (1963) 308-322.

[30] TT Wu, Phys. Rev. 149 (1966) 380-401 (Part I); Phys. Rev. 155 (1967) 438 (Part II); H. Cheng and TT Wu, Phys. Rev. 164 (1967) 719-735 (Part III).

[31] BM McCoy and TT Wu, , Phys. Rev. 162 (1967) 436-475 (Part IV).

[32] BM McCoy and TT Wu, Phys. Rev. 176 (1968) 631-643; BM McCoy, Phys. Rev. Lett. 23 (1969) 383-386; BM McCoy, Phys. Rev. 188 (1969) 1014-1031.

[33] RB Griffiths, Phys. Rev. Lett. 23 (1969) 17-19.

[34] E. Barouch, BM McCoy and TT Wu, Phys. Rev. Lett. 31 (1973) 1409-1411; CA Tracy and BM McCoy, Phys. Rev. Lett. 31 (1973) 1500-1504; TT Wu, BM McCoy, CA Tracy and E. Barouch, Phys. Rev. B 13 (1976) 315-374.

[35] BM McCoy, CA Tracy and TT Wu, Phys. Rev. Lett. 38 (1977) 793-796; BM McCoy and TT Wu, Phys. Rev. D 18 (1978) 1243-1252; BM McCoy and TT Wu, Phys. Rev. D 18 (1978) 1253-1258.

[36] For an excellent introduction to the history of the Ising model, see, for example, S.G. Brush, Reviews of Modern Physics 39 (1967) 883-893 (entitled History of the Lenz-Ising Model); and M. Niss’s “trilogy”: M. Niss, , Arch. Hist. Exact Sci. 59 (2005) 267-318 (History of the Lenz-Ising Model 1920-1950); ibid. 63 (2009) 243 (1950-1965); ibid. 65 (2011) 625 (1965-1971).

[37] See N. Metropolis and S. Ulam, Journal of the American Statistical Association, 44 (1949) 335-341; see also the historical review by N. Metropolis, Los Alamos Science 15 (1987) 125.

[38] Nowadays, the renormalization group technique has been widely used in the theoretical study of various phase transitions and critical phenomena. A detailed discussion of the renormalization group can be found in almost every monograph on critical phenomena. We recommend the review by its originator KG Wilson: KG Wilson, Rev. Mod. Phys. 55 (1983) 583.

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