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1957 

 Ratio method
If you are expanding an exact power series, it is of interest to find the relation between the asymptotic behavior of the coefficients and the dominant singularity in the series (usually identified with the physical critical point). While the values of the coefficients themselves may be difficult to calculate, Domb and Sykes suggested that one can compare the ratios of successive terms and plot them versus 1/n. These ratios should vary linearly with 1/n, and one can estimate the limiting ratios by extrapolating to 1/n = 0, yielding an estimated value for the critical point.

Primary: C. Domb and M. Sykes, "On the susceptibility of a ferromagnet above the Curie point," Proc. Roy. Soc. 240 A (1957): 214-228.

Secondary: Domb 1996, 163. Fisher 1967, 681.
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Aug 1957 

 Logarithmic singularity in specific heat of liquid helium
Fairbank, Buckingham, and Kellers got to within a few microdegrees K of the lambda transition in superfluid helium 4 in the course of measuring the specific heat. Their results indicated that the specific heat tends logarithmically to infinity at the transition, whether from above or below. Neither the classical theory of liquids nor the recent theories of quantum liquids were much help on the shape of the singularities at the critical temperature. The realization that the Onsager solution for a completely different physical system (ferromagnets) was also the one theory predicting a logarithmic singularity in the specific heat was suggestive, at least to those theorists interested in locating commonalities in the critical points of diverse systems.

How did this experiment acquire "classic" status so quickly, given that it was not replicated for more than a decade, and the concurrent theories of critical exponents eventually leaned toward a more general power law at the expense of the logarithmic form when interpreting the data?

Primary: W. M. Fairbank, M. J. Buckingham, and C. F. Kellers, "Specific heat of liquid He4 near the lambda point," in J. R. Dillinger, ed., Low Temperature Physics & Chemistry: Proceedings of the Fifth International Conference (Madison, Wisconsin: University of Wisconsin Press), 50-52.

W. M. Fairbank and C. F. Kellers, "The lambda transition in liquid helium," in M. S. Green and J. V. Sengers, eds, Critical Phenomena: Proceedings of a conference held in Washington, D.C., April 1965 (National Bureau of Standards, 1966), 71-78.

Secondary: Ahlers 1980, 492. Brush 1983, 197. Domb 1996, 23. Heller 1967, 820.
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1958 - 1967

 Hybridizing Ehrenfest and Landau
Paul Ehrenfest's 1933 classification scheme for phase transitions, initially developed to incorporate the recent discovery of the "lambda transition" in liquid helium, dominated the conceptual vocabulary of the theory of phase transitions long after it was recognized that the anomalies it described were not consistent with the lambda transition. Landau and Lifshitz offered a slightly different classification scheme in the first edition of their 1938 textbook Statistical Physics, but the English version was not very influential (the initial print run of approximately 1000 copies had not been sold out by the mid- 1950s). This changed with the appearance of the second edition of their text in 1958. Jaeger (1998) argues that it was the hybridization of the schemes of Ehrenfest and of Landau and Lifshitz by the likes of Fisher (1967) that vastly increased the influence of their textbook, which advocated symmetry methods that carried over well into the theory of critical phenomena.

Primary: P. Ehrenfest, "Phasenumwandlungen im ueblichen und erweiterten Sinn, classifiziert nach den entsprechenden Singularitaeten des thermodynamischen Potentiales," Proc. Kon. Akad. Amsterdam 36 (1933): 153–157. L. Landau and E. Lifshits, Statisticheskaia fizika (Moscow- Leningrad: GTTL, 1938). L. Landau, and E. Lifshitz, Statistical Physics., trans. D. Shoenberg (Oxford: Clarendon, 1938). L. Landau and E. Lifshits, Statisticheskaia fizika (Moscow, 1951). L. Landau and E. Lifshitz, Statistical Physics, 2d. ed., trans. E. Peierls and R. Peierls (Reading MA: Addison- Wesley, 1958).

Secondary: Jaeger 1998, 69.
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1959 

 Antiferromagnetic susceptibility
The Onsager solution applies to a system with zero field, and it was not until the late 1950s that Fisher came upon a special class of antiferromagnetic models which yield an exact expression for the partition function in a finite field.

Primary: M. E. Fisher, "The susceptibility of the plane Ising model," Physica 25 (1959): 521-524. M. E. Fisher, "Lattice statistics in a magnetic field," Proc. Roy. Soc. A 254 (1960): 66-85; A 256 (1960): 502-513.

Secondary: Domb 1996, 182. Fisher 1967, 672.
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1959 

 Landau on Feynman diagrams
Lev Landau suggests an "algebra" of Feynman diagrams.

L.D. Landau, "Ob analiticheskikh svoistvakh vershinnakh chastei v kvantovoi teorii polia," ZhETF 37 (1959): 62; "On the analytic properties of vertex parts in quantum field theory"

See also: V.A. Kolkunov, L.B. Okun', and A.P. Rudik, "Osobye tochki nekotorykh feinmanovskikh diagramm," ZhETF 38 (1960): 877; A.P. Rudik and Iu.A. Simonov, "Novyi metod issledovaniia osobennostei diagramm Feinmana," ZhETF 45 (1963): 1016; B.N. Valuev, "Ob anomal'noi osobennosti i opredelenii amplitud nekotorykh protsessov," ZhETF 47 (1964): 649.
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1959 - 1960

 The Ginzburg criterion
As T approaches Tc one has to be more and more careful about specifying the magnitude of the statistical fluctuations which have been effectively "smoothed out" of the Landau mean field theory. Levaniuk and Ginzburg both provided quantitative criteria for the growth of the fluctuations and their consistency with Landau's assumptions.

Primary: A. P. Levaniuk, "K teorii rasseianiia sveta vblizi tochek fazovogo perekhoda vtorogo roda," ZhETF 36 (1959): 810- 818; "Contribution to the theory of light scattering near the second-order phase-transition points," Soviet Phys. JETP 9 (1969): 571-576.

V. L. Ginzburg, "Some remarks on phase transitions of the second kind and the microscopic theory of ferroelectric materials," Soviet Physics Solid State 2 (1961): 1824-1834; Fizika tverdogo tela 2 (1960): 2031.

Secondary: Domb 1996, 274.
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1960 

 Landau (unpublished) on fluctuation-driven phase transitions
Valery L. Pokrovsky recalls that Landau used a path integral over all configurations of the order parameter to formulate the general problem of fluctuation-driven phase transitions in 1960, but did not publish the incomplete results. Pokrovsky states that the integrand was the Gibbs- Boltzmann exponent of what was later called the Ginzburg- Landau-Wilson Hamiltonian. Further evidence (recollections, perhaps even an early draft) regarding the content of this work would be welcome for posting on HRST.

Valery L. Pokrovsky, "Notes on history of critical phenomena," History of Physics Newsletter (APS) 7 no. 3 (August 1998), available online at http://www.aps. org/FHP/pokrovsky.html.
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1960 

 Quasi-averages and spontaneous symmetry breaking
Bogoliubov formulates a method of quasi-averages for a quantum mechanical description of spontaneous symmetry breaking.

N.N. Bogoliubov, "On some problems of the theory of superconductivity," Physica 26, Suppl. (1960): S1-S16.
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1961 

 Padé approximants
Exact series expansions were most useful for high temperatures, where the coefficients were uniformly positive and the convergence behavior could be demonstrated fairly directly. For low temperature expansions it was more difficult to identify convergence patterns, since the sign of the coefficients was inconsistent. Baker revived Padé's method of approximating the ratio [L,M] of two polynomials of degree L and M so that a given expansion F(x) which contained many singularities (poles) could be represented as F(x) = [L,M] + terms of order (L+M+1) and higher. Baker showed that the Padé approximants could be used in many instances to continue a series expansion analytically to the critical point, thus yielding more accurate critical exponents. In particular, Baker analyzed spontaneous magnetization and found a value for ß roughly equal to 0.30.

Primary:

George A. Baker, Jr., "Application of the Padé approximant method to the investigation of some magnetic properties of the Ising model," Phys. Rev. 124 (1961): 768- 774.

Secondary:

Brush 1983, 253.

Domb 1996, 165.

Fisher 1967, 687.
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1961 - 1962

 Limiting validity of mean field theory
Baker is the first of several to demonstrate rigorously how to specify the limits of validity of mean field theory.

Primary: G. A. Baker, Jr., "One-dimensional order-disorder model which approaches a second-order phase transition," Phys. Rev. 122 (1961): 1477-1784.

G. A. Baker, Jr., "Certain general order-disorder models in the limit of long-range interactions," Phys. Rev. 126 (1962): 2071-2078.

Secondary: Fisher 1967, 666.
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1962 

 Specific heat of classical fluid: logarithmic divergence or finite discontinuity?

It was one thing for a quantum liquid like helium to exhibit divergences in properties like the specific heat near the critical temperature. Bagatskii, Voronel', and Gusak set out to measure the specific heat of a classical fluid (weakly-interacting argon) to obtain a more finely- resolved picture of the behavior very close to the critical point. Van der Waals' theory (and more generally, Landau's mean-field theory) yield finite discontinuities in the specific heat across the critical point, but the results obtained here suggested a much stronger divergence. With an eye on Fairbank, Buckingham, and Kellers' 1957 results for liquid helium, which had suggested a logarithmic singularity, Bagatskii, Voronel' and Gusak thought perhaps they were observing a similar logarithmic singularity. Voronel' and his colleagues performed similar experiments on nitrogen and oxygen which seemed to point to similar logarithmic singularities.

Was this an important experimental motive for rejecting incremental improvements in mean field theories as "good enough" for three dimensions?

Primary:

M. I. Bagatskii, A. V. Voronel', and V. G. Gusak, "Measurement of the specific heat cv of argon in the immediate vicinity of the critical point," Soviet Physics JETP 16 (1963): 517-518; "Izmerenie teploemkosti cv argona v neposredstvennoi blizosti k kriticheskoi tochke," ZhETF 43 (1962): 728-729.

A. V. Voronel', Yu. R. Chashkin, V. A. Popov, and V. G. Simkin, "Measurement of the specific heat cv of oxygen near the critical point," Soviet Physics JETP 18 (1964): 568-569; "Izmerenie teploemkosti cv kisloroda vblizi kriticheskoi tochki," ZhETF 45 (1963): 828-830.

A. V. Voronel' and M. Sh. Giterman, "Hydrostatic effect at the critical point of a binary alloy," Soviet Physics JETP 21 (1965): 958-; "Gidrostaticheskii effekt v kriticheskoi tochke binarnoi smesi," ZhETF 48 (1965): 1433-1436.

A. V. Voronel', V. G. Gorbunova, Yu. R. Chashkin, and V. V. Schekochikhina, "Specific heat of nitrogen at the critical point," Soviet Physics JETP 23 (1966): 597- 601; "Teploemkost' azota v okrestnosti kriticheskoi tochki," ZhETF 50 (1966): 897-904.

Secondary:

Ahlers 1980, 492.

Brush 1983, 254.

Domb 1996, 205.
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1962 

 Sublattice magnetization of antiferromagnetic MnF2
Heller and Benedek used nuclear magnetic resonance techniques to determine the critical exponent [beta] of MnF2 to unprecedented accuracy. This experiment provided important confirmation that antiferromagnets are like ferromagnets, which are in turn like fluids, thus giving impetus to the conviction that critical phenomena should encompass all these effects.

Primary: P. Heller and G. B. Benedek, "Nuclear magnetic resonance in MnF2 near the critical point," Phys. Rev. Letters 8 (1962): 428-432. P. Heller, "Nuclear-magnetic-resonance studies of critical phenomena in MnF2. I. Time-average properties," Phys. Rev. 146 (1966): 403-422.

Secondary: Brush 1983, 253. Domb 1996, 24. Heller 1967, 773.
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1962 

 Seminal textbooks
Two crucial early textbooks summarizing progress to date in the application of field theoretic approaches to many-body problems were published by Abrikosov, Gorkov and Dzyaloshinski and by Kadanoff and Baym.

A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzialoshinskii, Metody kvantovoi teorii polia v statisticheskoi fizike (Moscow: Fizmatgiz, 1962).

A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, trans. R. A. Silverman (Englewoods Cliffs, NJ: Prentice- Hall, 1963).

Leo P. Kadanoff and Gordon Baym, Quantum Statistical Mechanics (New York: W. A. Benjamin, 1962).
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Jul 17, 1962 - Jan 28, 1964

 Perpendicular susceptibility of planar lattice
Very few bulk thermodynamic properties could be calculated for planar lattices, but Fisher and Stephenson each obtained a result for the perpendicular susceptibility in zero field. Did this register at all among experimentalists?

Primary: M. E. Fisher, "Perpendicular susceptibility of the Ising model," J. Math. Phys. 4 (1963): 124-135. J. Stephenson, "Ising-model spin correlations on the triangular lattice," J. Math. Phys. 5 (1964): 1009-1024.

Secondary: Fisher 1967, 672.
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1963 

 Critical exponent inequalities
The rigor of the Onsager model is appealing, but highly constrained physically. Would it not be possible to prove relations between critical exponents for slightly more general systems? Essam and Fisher used a relatively simple droplet model to show that [alpha' + 2beta + gamma'] = 2. Rushbrooke then demonstrated that thermodynamic considerations alone could yield a rigorous inequality [alpha' + 2beta + gamma'] >= 2.

Primary: J. W. Essam and M. E. Fisher, "Pade approximant studies of the lattice gas and Ising ferromagnet below the critical point," J. Chem. Phys. 38 (1963): 802-812. G. S. Rushbrooke, "On the thermodynamics of the critical region for the Ising problem," J. Chem. Phys. 39 (1963): 842-843.

Secondary: Brush 1983, 254. Domb 1996, 25. Fisher 1967, 643.
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1963 

 The droplet model
In the early 1960s it remained a great challenge to construct a plausible connection between a respectable "microscopic" approach to a phase transition phenomenon and a simplified mathematical model that would say something about the relations between the critical exponents. Building on ideas introduced in the late 1930s in the wake of Joseph Mayer's theory of condensation, Essam and Fisher offered a heuristic droplet model that nonetheless yielded exact relations between the critical exponents [alpha]' + 2[beta] + [gamma]' = 2. Domb (1996) identifies the droplet model as a precursor to scaling theories.

Primary: J. W. Essam and M. E. Fisher, "Padé approximant studies of the lattice gas and Ising ferromagent below the critical point," J. Chem. Phys. 38 (1963): 802-812.

Secondary: Brush 1983, 253. Domb 1996, 215. Fisher 1967, 703.
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1965 

 S-matrix convergence without auxiliary regularization
The Han-Banach theorem regarding the expansion of a linear functional is used to show that it is possible to construct converging expressions for the elements of the S-matrix without using an auxiliary regularization.

B.M. Stepanov, Izv. Akad. Nauk SSSR. Ser. Mat. 29 (1965): 1037.
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1965 - 1966

 High-resolution work on magnetic systems
The time scale for magnetic measurements is fairly short compared to the time scale needed for fluid systems to come to equilibrium, and from a theoretical point of view magnetic systems were more amenable to modeling, but they still present important experimental obstacles to the study of critical phenomena. Even nominally pure pairs of samples can differ markedly in their critical points, and one must always find ways to subtract out the lattice and electronic contributions to the specific heat before arriving at the magnetic contribution. Teaney and his colleagues obtained results for MnF2 and RbMnF3 which were regarded as invaluable in establishing the shape and sharpness of the transition for magnetic systems with resolution sufficient to test theories of critical phenomena.

Primary: D. T. Teaney, "Specific-heat singularity in MnF2," Phys. Rev. Letters 14 (1965): 898-900. D. T. Teaney, V. L. Moruzzi, B. E. Argyle, "Critical point of the cubic antiferromagnet RbMnF3," J. Appl. Phys. 37 (1966): 1122-1123.

Secondary: Heller 1967, 790.
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Apr 1965 

 Experimental refutation of Orstein-Zernike
Chu chose two liquids whose refractive indices were much closer in value than was common in scattering experiments, with the challenging consequence that the scattering was then two orders of magnitude weaker. Sample, container, and surrounding thermostat liquid were chosen for compatible refractive indices so that the beam would not diverge within the sample cell. A standard Ornstein- Zernike-Debye plot of the results at several wavelengths implied a non-zero value for the critical exponent [eta] which could not be reconciled with classical Orstein- Zernike theory.

Primary: B. Chu, "Experiments on the Critical Opalescence of Binary Liquid Mixtures: Elastic Scattering," in M. Green and J. Sengers, eds., Critical Phenomena: Proceedings of a Conference (Washington, D.C.: National Bureau of Standards, 1966), 123-129.

Secondary: Heller 1967, 800.
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Apr 1965 

 NBS Conference on Critical Phenomena
In April 1965 two hundred scientists from eleven countries met in Washington for what participant Cyril Domb later dubbed "the founding conference of critical phenomena." This appears to have been the first time that such a diverse array of specialists gathered under the single rubric of critical phenomena.

You may view the contents of the conference in the Documents portion of the Physics of Scales site.

M. Green and J. Sengers, eds., Critical Phenomena: Proceedings of a Conference (Washington, D.C.: National Bureau of Standards, 1966), released as NBS Miscellaneous Document 273.
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