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This article comes from the official account of Wechat: back to Park (ID:fanpu2019). The authors: Liu Yihan, Zhang Yi, Su Guifeng (Department of Physics, School of Mathematics and Science, Shanghai normal University)

Commemorating the 200th anniversary of the discovery of critical phenomena by French physicist Charles Cagniard Delatu.

"A few years ago, Cagniad Delatu did an experiment that gave me the opportunity to endow me with new words. How should I name the combination of liquid and steam according to the law of continuity? Cagniad Delatu didn't give it a name; what should I call it?"

-- Faraday to Hu Weili [1]

First, what is the critical phenomenon? What is critical phenomenon (critical phenomena)? In fact, phase transition and critical phenomena are the same thing, separate terms, but a "misunderstanding" in the history of physics, thinking that the two are different physical phenomena. In order to describe the phase transition and critical phenomena concretely, we take the water that we are used to in daily life as an example to make a brief and intuitive explanation.

The so-called phase transition, quoting most textbooks, is the change of matter-such as water-from one form of aggregation to another [Note 1]. Human beings know that there are three phases of gas, liquid and solid in water, which can be traced back to the historical records of China and ancient Egypt about 4000 years ago, but it has been more than half a century since we really understood the phase transition. The late famous statistical physicist Leo Kadanoff,1937-2015 took icebergs floating in the sea as an example to illustrate the coexistence of different phases of water: "the sea is liquid water, it surrounds ice-solid water. The breeze blows the clouds, and the water vapor in the air comes into contact with water in both the solid phase and the liquid phase." (of course, strictly speaking, water is not the only chemical component of sea water. The original text of this paragraph is shown in figure 1. ) [2]

To study the phase transition of a substance, a basic task is to determine its "phase diagram", that is, to find out what phase the substance is in at a given thermodynamic parameter-temperature T, pressure P and volume V for a "simple" thermodynamic system. And determine the boundaries between different phases. For example, figure 2 shows a phase diagram of water in the pressure-temperature plane, defining the solid (light blue), liquid (blue), and gas (ochre) three phases of water at different temperatures and pressures, as well as the boundaries between any two phases. The yellow dot near the middle of figure 2 is called the three-phase point (Triple point), which, as its name implies, is the intersection of the above three phases. Starting from the three-phase point, it "goes up" along the gas-liquid dividing line, which does not extend infinitely, but stops at the red dot in figure 2, which is the critical point (Critical point). For water, the corresponding thermodynamic parameters at the critical point are: critical pressure Pc = 22.09 MPa; critical temperature Tc = 374.14 °C (647.3 K), critical (specific) volume vc = 0.003155 m ³/ kg. The so-called "critical point" means that beyond this point, the difference between the gaseous state and the liquid state of water no longer exists, and it is no longer meaningful to ask whether the water is gaseous or liquid at this time. Therefore, with the critical point as the boundary, the area above (see the upper right corner of figure 2) is supercritical fluid (Supercritical fluid), where 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]

Fig. 2 pressure-temperature plane phase diagram of water [picture from network] although the critical point is only a point on the pressure-temperature phase diagram, the physical phenomena that occur near the critical point are very abundant-collectively referred to as "critical phenomena". A typical example is the so-called critical milklight (critical opalescence): when the thermodynamic parameters of a transparent gas or liquid approach the critical point, it becomes cloudy and gradually presents a milky white phenomenon. It is known from statistical physics that this is due to the large fluctuations near the critical point and the extremely strong scattering of light. This can be observed by laser scattering through phase separation at a critical point, as shown in the following video showing the critical emulsion phenomenon of an equal amount of aniline and cyclohexane mixture.

When the critical temperature is reached and the mixture changes from single-phase to two-phase (phase separation), the light spot on the screen will be disturbed. The spot "flickers" continuously until it spreads completely. Once the phase transition is completed and the two substances are completely separated, it will eventually form a single spot again. When the mixture is heated, the same reverse pattern can also be observed.

In addition, there are some other unique physical phenomena near the critical point, such as the specific heat of the system increases continuously in the process of approaching the critical point, and the specific heat coefficient and compression ratio tend to diverge (infinity) at the critical point.

It may be said that the discovery of critical phenomena begins with curiosity. In history, the French physicist Charles Cagniard Delatu (Charles Cagniard de la Tour,1777-1859) first discovered the critical phenomenon in his experiments in 1822. Many people may not realize that it has been two hundred years since his discovery. In the past two hundred years, earth-shaking changes have taken place in physics. The study of critical phenomena has become a mature field of modern condensed matter physics and complex system physics, and continues to bring new surprises.

In this paper, we will make a superficial review of the historical background of the discovery of critical phenomena. According to the famous statistical physicist Cyril Domb,1920-2012, this period of history can be classified as the "classical period" of the study of critical phenomena [4]. We will also briefly introduce some important advances in the study of critical phenomena in the classical period after Delatu's death.

Charles Cagniad de Delatu, who first discovered the critical phenomenon, was born on March 31, 1777 in Paris, France. As a student, he studied at the Paris Institute of Technology (l'Ecole Polytechnique) and the Institute of Engineering Geology (É coledu G é nie G é ographe). Since then, he has served as auditor of the Council of State and director of the Paris City Special Project. At the same time, he is a prolific scientist and inventor. Apart from the discovery of critical phenomena, from mechanics to acoustics to chemical biology, he has made important contributions in many different fields.

Delatu's academic research began in the fields of mechanics and thermodynamics. He invented a new type of heat engine in 1809. Between 1809 and 1815, he successively invented new hydraulic engines, new air pumps, and heat-driven winches. Delatu had been improving the design of these inventions until 1819. Since then, Delatu developed a strong interest in the physics of bird flight and human sound, began to study the mechanism of acoustics and sound production, and invested a lot of energy in this field. It is worth noting that it was this accidental turn of interest that led him to discover the critical phenomenon in the future. Between 1828 and 1831, Delatu began to study the crystallization process and the effect of acids on carbon, as well as the hardening of phosphorus, silicon and their crystals, and even mortar (mortar). Between 1832 and 1835, Delatu became interested in the application of Archimedes screw (Archimedean screw) principle in air pumps [5].

In 1835 Delatu began to turn to the study of alcohol fermentation. The work peaked between 1836 and 1838, when he found an active substance in beer yeast at the end of 1836. German physiologist Theodor Schwann,1810-1882 reached the same conclusion almost at the same time, but criticism from chemist Justus von Liebig,1803-1873 delayed this view for 20 years until French biologist Louis Pasteur,1822-1895 announced the discovery again.

Incidentally, reliable photographs or portraits of Delatu are still controversial. Some photos or portraits circulated in some documents and on the Internet often contradict each other. For example, what is common on the Internet, as shown in figure 3, is said to be a portrait of Delatu, and there is more reliable evidence that it is actually Prince Charles Edward of England.

Fig. 3 some portraits of the so-called Delatu circulated on the Internet are unreliable. 3. The discovery of critical phenomena and the early history of the invention of the steam engine in the late 17th and early 18th centuries aroused interest in fluid behavior at high temperature and high pressure. French physicist Denis Papin,1647-1712 invented the predecessor of the steam engine, the "Papin steamer" (Papin's digester, see figure 4 and model), while working as an assistant to Robert Boyle,1627-1691 at the Royal Society. In particular, he noticed that when heated under high pressure, the temperature of the water in the liquid phase is much higher than the usual boiling point, that is, the temperature of the boiling point increases with the increase of pressure. In the second half of the 18th century, the French chemist Antoine-Laurent de Lavoisier,1743-1794 proved that gas and steam are actually the same thing, the third state of matter other than solid and liquid. He also suggested that the gas could be liquefied at a sufficiently low temperature and a high enough pressure. This understanding led to the liquefaction of gaseous sulfur dioxide by cooling and compression in 1784 (Jean-Francois Clouet,1751-1801) [Note 7] and Monge (Gaspard Monge,1746-1818) [Note 8], the first successful gas liquefaction. Then the British physicist Michael Faraday,1791-1867 liquefied the gas through a series of successful experiments [7,8]. Hydrogen, oxygen, nitrogen and carbon monoxide, previously thought to be non-condensable gases-once called "permanent gases" (permanent gases)-were eventually liquefied in 1877.

Fig. 4 schematic diagram and model picture of Papan hot-pressure steamer (Papin's digester) Source: the existence of the critical point was found in the experiment of Papan hot-pressure steamer. In 1822, out of interest in acoustics, Delatu heated the flint ball in a steamer partially filled with liquid. When rotating the experimental device, the solid flint ball produces the splashing sound of water as it passes through the interface between the gas and liquid phases. Delatu noticed that when the temperature in the experiment was much higher than the boiling point of the liquid, the splash of the water stopped when it exceeded a certain temperature. This actually means the discovery of the supercritical fluid phase mentioned earlier (above the red critical point in figure 2). Since there is no gas-liquid boundary in this phase, there is no surface tension. Supercritical fluids can also dissolve substances like liquids or spread in solids like gases. At present, the research on supercritical fluid is still an important direction.

In two papers published in the Annals of Chemistry and Physics (Annales de Chimie et de Physique) [9], Delatu describes how he heats sealed alcohol glass tubes under high pressure (see the front page of the paper in figure 5). He observed that the liquid expanded to about twice its original volume and then turned into transparent steam, and the pipe looked empty. But when it cooled again, a "cloud" appeared in the glass tube. Now we have realized that this is actually the manifestation of the critical opacification phenomenon at the critical point. To give the reader an intuitive impression, figure 6 shows the critical milky light of ethane. Delatu also noted that increasing the pressure does not prevent the liquid from evaporating when the temperature is exceeded.

Fig. 5 the home page of Delatu's paper [9] in which he reported the discovery of critical phenomena.

Fig. 6 critical opacification of ethane (in the yellow circle) Source: https://handwiki.org in another subsequent paper [10], Delatu wants to prove that the existence of a specific limit temperature is a common phenomenon. The so-called limit temperature means that above this temperature, the liquid will evaporate no matter what the pressure is. Delatu reported the results of several material experiments in his paper. He determined the corresponding critical temperature by marking the disappearance of the meniscus of the liquid at 00:00 in surface tension. Delatu measured the critical temperatures of water, alcohol, ether and carbon disulfide, Tc, and found that there is indeed a specific temperature for each substance, where the liquid evaporates even without increasing pressure, and beyond that temperature all the liquid evaporates. The critical temperature of water measured by Delatu is about 362℃. Considering the historical conditions at that time, this is already a quite accurate result (the result of modern measurement is about 374 ℃). In his paper, he said that this "specific state" (é tat particulier): "always requires a very high temperature, almost independent of the capacity of the pipe" [10]. We now know that this "specific state" marks the end of the phase equilibrium curve, the critical point.

Many of his contemporaries did not realize the significance of his discovery and believed that the result was true only for the substances used in the Delatu experiment, not for a common phenomenon [11]. But Faraday showed his deep physical insight, and he realized the value of Delatu's work [12]. In 1844, Faraday wrote to Hu Weili (William Whewell,1794-1866): "A few years ago, Cagniad Delatu completed an experiment that gave me a chance to invent new words." Faraday then talked about the tipping point in the modern sense, "according to the law of continuity (law of continuity), how do I name the point where liquid and steam combine into one? Cagniad Delatu doesn't name it, so what should I call it?" (see note 11 above) Hu Weili suggested that it be called the vaporization point, or the non-liquefaction point of the liquid, or the Delatu state, and Faraday used the "Cagnia de la Tu state" (Cagniard de la Tour's state) and the "Cagniad de la Tu point" (Cagniard de la Tour point) in his later papers [13].

On July 5, 1859, Delatu died in Paris. However, his experimental findings ushered in the classical period of the study of critical phenomena and the subsequent journey of intellectual exploration.

The term "critical point" we use today was coined by British physical chemist Andrews [Note 10] (Thomas Andrews,1813-1885) in 1869, ten years after Delatu's death, who discovered "supercritical fluids" in the same year. His research results were published in the philosophical Journal of that year under the title "on the continuity of gaseous and liquid states of matter" (on the continuity of the gaseous and liquid states of matter) (see figure 7). In this famous paper, Andrews studied the pressure-volume curve of the two-phase coexistence line of carbon dioxide, liquid and gas. it is further clarified that what Delatu calls a "specific state"-that is, only at a certain temperature and pressure-a gas can be condensed into a liquid, or a liquid can evaporate into a gas. Above this is the supercritical phase, where the distinction between liquid and steam disappears (see figure 8).

In 1923, Dutch physicist J. H. van der Waals,1837-1923) explained the continuity between gas phase and liquid phase clearly in theory for the first time. Van der Waals showed in his doctoral thesis that intermolecular interaction can be introduced to generalize the law of ideal gas, and the equation of state of van der Waals gas named after him is obtained, which qualitatively explains Andrews' experimental results. At that time, famous physicists Maxwell and Boltzmann spoke highly of van der Waals's results [4]. Van der Waals's work in turn inspired his compatriot, Dutch physicist Heike Kamerlingh Onnes,1853-1926. The latter can estimate the critical point of permanent gas, which provides a theoretical basis for the final liquefaction of helium at low temperature-about 4K. Subsequently, the acquisition of low temperature led to the discovery of superconductivity. But the exploration history of low temperature 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 van der Waals equation of state are actually simple mean field (Mean Field) values, which do not accord with the measured critical exponents of thermodynamic systems. Belgian physicist Jules-Emil é Verschaffelt,1870-1955 first discovered this through experiments in 1896. He re-measured the rise of carbon dioxide in the capillary and analyzed the coexistence curve data combined with the new experimental data of coexistence density, and found that it was not consistent with the average field value. However, Wassafeldt's experimental results did not attract the attention of physicists at that time. In the 1930s, the famous Soviet physicist Lev Davidovich Landau,1908-1968 continued to develop the general framework of systematic mean field treatment of phase transition, that is, Landau continuous phase transition theory, which is a peak of phase transition phenomenological theory [18].

Fig. 7 the front page of Andrews' paper on the continuity of gaseous and liquid states of matter published in 1869, which starts with Delatu's experimental findings on critical phenomena [14]

Fig. 8 figure in Andrews' paper on the continuity of gaseous and liquid states of matter in 1869 [14]. The horizontal axis of the coordinate in the picture is the pressure and the longitudinal axis is the volume. It can be seen that with the increase of temperature, the density difference of the coexisting gas-liquid phase gradually tends to zero and disappears.

Fig. 9 readers, can you find Walshafeld in this famous group photo? (for answers, see note [12]) Photo Source: network in another research route on magnetism, French physicist Pierre Curie,1859-1906 found that demagnetization occurs when ferromagnetic materials exceed the critical temperature, which is often referred to as the "Curie point". In 1895, he noticed the similarity between gas-liquid phase transition and ferromagnetic phase transition, and put forward the important concept of "universality" of critical phenomena [13]. In order to understand the origin of magnetism, the German physicist Wilhelm Lenz,1888-1957 introduced a simple model, now commonly known as the "Ising model" [20]. In 1924, Lenz student Ising (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 and believed that there was no phase transition in the two-dimensional Ising model [21]. After the work of Peiers [22] and Klamers and Kramers & Wannier [23], Lars Onsager,1903-1976 analytically calculated the specific heat of the two-dimensional Ising model in the absence of an external magnetic field [24]-this work was so important that Dom called it the "Onsager Revolution" (Onsag Revolution) [4]. Onsag also gave the unproved spontaneous magnetization formula [25,26] [Note 14] in 1949, which was proved by Yang Zhenning (1922 -) in 1952 [27]. However, the exact solution of the three-dimensional Ising model has not been solved so far, which is always a great challenge for physicists. The history of the Ising model itself is enough to constitute a monograph. We will not repeat it, but only list some important developments in Table I. interested readers can further read the relevant article [36] and the literature cited in the article.

Table I Historical process of exact solution of Ising Model

Under the background of lack of exact solution of three-dimensional Ising model, people have to rely on numerical simulation. In his doctoral thesis in 1949, Dom proposed the development method of high and low temperature (cited in [4]). What is widely used today is the Monte Carlo method proposed by Maechopolis [Note 15] (Nicholas Metropolis,1915-1999) and Ulam [Note 16] (Stanislaw Ulam,1909-1986) in 1949.

In the 1960s, Cardanov and Fisher [Note 17] (Michael Fisher,1931-2021) realized that the general theoretical framework of phase transition must be based on the "scaling hypothesis" (scaling hypothesis). In particular, the scaling hypothesis leads to the "scaling relationship" (scaling relations) between various critical exponents close to the critical point. This idea opens the way for a complete theoretical description of critical phenomena through the "renormalization group" (renormalization group) method proposed by Kenneth G. Wilson,1936-2013 in 1971.

So far, our research and understanding of the critical phenomenon has reached a new height, but also a new starting point.

Annotation

[1] due to space constraints, we will not further discuss the classification of phase transitions here. The modern interpretation of (continuous) phase transition also includes the so-called symmetry breaking (symmetry breaking). For readers interested in the full picture of the development of phase transition and critical phenomena, please refer to the fascinating popular science reading structure, "marginal Miracle: phase transition and critical phenomena", Yu Li, Hao Berlin, Chen Xiaosong, Science Press (2005).

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

[3] because this paper focuses on the critical phenomenon of Cagnard's discovery, the author does not discuss the study of the later classical period in detail, and there will inevitably be omissions. I look forward to the opportunity to write another article in the future.

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

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

[6] it is actually an oil painting by an artist named M. Q. de La Tour around 1745 and is on display at the National Portrait Gallery of Scotland in Edinburgh. See https://www.britannica.com/ topic / Jacobite-British-history.

[7] Jean-Francois Clouet, a French chemist and metallurgist, promoted the transformation of French chemical research to specific problems and promoted the development of metallurgical industry.

[8] Gaspard Monge, French mathematician and physicist. The characteristic theories of descriptive geometry and partial differential equations are established, which promotes the development of spatial analytic geometry, differential geometry and pure geometry.

[9] William Whewell, a learned scholar in England in the 19th century, was also one of the most influential figures in British academic circles at that time. He wrote books on mechanics, mineralogy, geology, astronomy, political economy, architecture and many other disciplines, and left many works in the fields of philosophy of science, history of science and moral philosophy. such as Inductive History of Science (3 volumes, 1837), Inductive principles of Science (1840), History of Scientific thought (2 volumes, 1858), principles of Discovery (1860) and so on. He is a founding member and chairman of the British Association for the Advancement of Science, a member of the Royal Society, president of the Geological Society, and long-time dean of Trinity College, University of Cambridge. His influence was recognized by contemporary leading scientists such as John Herschel, Charles Darwin, Charles Lyle and Michael Faraday, who often sought philosophical and scientific advice from Hu Weili, even for help in academic terminology. Hu Weili invented the Faraday terms "anode", "cathode" and "ion". An interesting history is that in 1833, in response to the poet S.T. Coleridge's challenge, Hu Weili invented the word "scientist". The only terms previously used were "natural philosopher" (natural philosopher) and "scientific practitioner" (man of science).

[10] Thomas Andrews, British physical chemist, member of the Royal Society of London, member of the Royal Society of Edinburgh, former associate dean and professor of chemistry at Queen's College Belfast. Mainly engaged in the study of the critical state of matter.

[11] Jules-Emil é Verschaffelt, a Belgian physicist, studied under Heike Kamerlingh Onnes, a Dutch physicist and founder of cryogenic physics.

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

[13] in the opinion of the author (Zhang Yi), the Curie principle (Curie's Principle) put forward by Curie in 1894 pointed out the important position of symmetry in physics. Although this principle is still controversial, 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 speech, Onsager went to the blackboard to announce that he and Bruria Kaufman had solved the problem and wrote the formula on the blackboard. In 1949, Onsager reiterated his results at a conference on statistical mechanics in Florence, Italy. However, Kaufman and Onsager have never officially published their calculations.

[15] Nicholas Metropolis, Greek-born American physicist. It has an important contribution to the research of Monte Carlo method.

[16] Stanislaw Ulam, American mathematician and nuclear physicist. Taylor-Ulam configuration, which invented the design of hydrogen bomb, also contributed to number theory, set theory and so on.

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

[18] Kenneth Wilson, an American theoretical physicist, won the Nobel Prize in Physics in 1982 for establishing the theory of renormalization group transformation.

reference

[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 occasion to want a new word […] how am I to name this point at which the fluid and its vapour 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 L. P. Williams, The Selected Correspondence of Michael Faraday (Cambridge: Cambridge Univ. Press, 1971), Vol. 1427-428.

[2] L. P. 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.Murl L. Lavoisier, Recueil des m é moires de chemie (1792) 348; republished in uvres de Lavoisier, publi é es par les soins de son excellence le ministre de l'instruction publique et des Cultes (Paris: Impr. Imp é riale, 1862), t. II, 783803.

[7] M. Faraday and H. Davy, Phil. Trans. R. Soc. Lond. 13 (1823) 160-165; M. Faraday, ibid., 189198.

[8] M. Faraday, The Quarterly Journal of Science, vol. Xvi. (1824), pp. 229-240; reprinted in The Liquefaction of Gases. Papers by Michael Faraday, F.R.S. (1823-1845) with an Appendix consisting of Papers by Thomas Northmore in the Compression of Gases (1805-1806), Alembic Club Reprint No.12, pages 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; Supplementary, ibid., 178182.

[10] the original text is as follows: ". Cet é tat particulier exige toujours une temp é rature è s-é lev é e, presque ind é du tube de la it é 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 155177.

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

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

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

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

[18] L. D. Landau, Nature 137 (1936) 840841.

[19] P. Curie, Archives des Sciences physiques et naturelles, 3e p é riode, tome XXVI (1891) p. 13; reprinted in: Oeuvres de Pierre Curie, pp 214219, Paris: Gauthier-Villars (1908).

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

[21] E. Ising, Zeitschrift f ü r Physik 31 (1925) 253258 (entitled Report on the theory of ferromagnetism).

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

[23] H. A. Kramers and G. H. 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) 279287.

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

[27] C. N. Yang, Phys. Rev. 85 (1952) 808,816.

[28] P. W. Kastelyn, J. Math. Phys. 4 (1963) 287293.

[29] E. W. Montroll, R. B. Potts, and J. C. Ward, J. Math. Phys. 4 (1963) 308322.

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

[31] B. M. McCoy and T. T. Wu, Phys. Rev. 162 (1967) 436-475 (Part IV)

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

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

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

[35] B. M. McCoy, C. A. Tracy and T. T. Wu, Phys. Rev. Lett. 38 (1977) 793-796; B. M. McCoy and T. T. Wu, Phys. Rev. D 18 (1978) 1243-1252; B. M. McCoy and T. T. Wu, Phys. Rev. D 18 (1978) 1253-1258.

[36] for a wonderful introduction to the history of Ising models, see, for example, S.G. Brush, Reviews of Modern Physics 39 (1967) 883-893 (titled History of the Lenz-Ising Model), and M. Niss's trilogy: M. Niss, Arch. Hist. Exact Sci. 59 (2005) 267318 (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 historical review N. Metropolis, Los Alamos Science 15 (1987) 125.

[38] nowadays, renormalization group technology has been widely used in the theoretical study of various phase transitions and critical phenomena, and detailed discussions about renormalization groups can be found in almost every monograph on critical phenomena. Here we recommend its originator, K. G. Wilson. Summary: K. G. Wilson, Rev. Mod. Phys. 55 (1983) 583.

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