Notes on History of Critical Phenomena

by Valery L. Pokrovsky

Texas A&M University and

Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia

[These notes were provided after the 1998 March Meeting. They expand upon Pokrovsky's comments during the discussion period].

These notes are my immediate reaction to the presentation by Johanna Levelt Sengers, including the remarks of Michail Anisimov. To a lesser extent they also intended to add to the talk by Michael Fisher. They were not prepared beforehand and do not pretend to be exhaustive, being limited by the reserves of my memory. All three speakers were active participants in this history and they did a brilliant job. Nevertheless, some essential details were omitted. This is natural since history tends to become simplified in the course of time. It is also an effect of a scale transformation which people and events on one continent undergo in view of a person living on another one. Probably my view as a participant from USSR is distorted in a similar way.

The history of the theory of critical phenomena and phase transitions starts with Landau's work [1] of 1937, written just before his arrest. The main achievement of this work is not the mean-field approximation used for calculations, but the fundamental notion of spontaneous symmetry violation and the order parameter as a measure of this violation. It is impossible to exaggerate the impact which this idea had on practically all branches of physics and non-linear mechanics. Due to the concept of the order parameter, phase transition theory became a cross-disciplinary branch of science, much like the theory of oscillations. Landau gave simple prescriptions, how to describe order in terms of irreducible representations of the symmetry group. Here the pioneering work by E. M. Lifshitz [2] must be cited. He was the first to apply group theory to describe specific structural transitions. Since then group-theoretical analysis has become a powerful tool for the study of phase transitions.

The famous Onsager exact solution of the 2-d Ising model [3] shattered the quantitative aspect of Landau theory. It was based on a power series expansion of free energy with respect to the order parameter, whereas Onsager demonstrated a singularity in the free energy at the transition point. This discrepancy was resolved by A. P. Levanyuk [4] and V. L. Ginsburg [5] in 1959-60. They explained that mean field theory neglects fluctuations which grow rapidly near the transition point. Thus, mean field theory works well outside a small vicinity of the transition point and is invalidated by fluctuations within it. In this way the necessity to include fluctuations in phase transition theory was first recognized. Simultaneously Michael Fisher [6] approached the problem by attempting to generalize Onsager's results to non-exactly-solvable problems. By introducing critical exponents he made the decisive step to scaling.

Around 1960 Landau formulated the general problem of fluctuation-driven phase transitions via a calculation of the path integral over all configurations of the order parameter (unpublished). The integrand was the Gibbs-Boltzmann exponent of what was later called the Ginsburg-Landau-Wilson Hamiltonian. Despite serious efforts by Landau and his collaborators, the problem remained unsolved effectively until Wilson's work [7]. (Landau used to say that he spent more time working on this problem than on any other: an entire half-year. No doubt he kept it in mind afterward, but the tragic accident of January 1962 permanently interrupted his scientific work.)

I started to work on this problem in 1962, when I. M. Khalatnikov told me about the Landau's attempts. Together with Alexander Patashinksii, we formulated the field theory equations and conjectured correctly that the correlation functions of any order should obey scaling laws [8]. However, one of the two principal equations was erroneous, leading to incorrect critical exponents. This mistake was corrected 4 years later by A. M. Polyakov [9] and A. A. Migdal [10]. Their formulation used such physical requirements as causality and unitarity. It permitted, in principle, numerical calculations of the critical exponents. Unfortunately, the equations were too complicated to solve using computers of that time. Only with Wilson's renormalization group approach was the structure of the theory elucidated to the extent that standard methods could be employed.

Michael Fisher gave an excellent presentation of the scaling idea and its history. I would only mention the early work by A. N. Kolmogorov [11], who proposed a scaling approach for hydrodynamic turbulence. V. G. Vaks and A. I. Larkin [12] conjectured the universality hypothesis several years before Kadanoff and Wegner proved it [13]. According to this hypothesis, the critical behavior is determined by symmetry and how it is violated. All phase transitions may be divided into universality classes. An important contribution was made by B. D. Josephson [14], who first understood how to introduce the superfluid density and calculated its critical exponent.

Soon after publication of our work [8], Patashinskii and I recognized that something was wrong, and we tried to determine the consequences of scaling alone, without specifying the critical exponents. In this way we formulated our version of scaling [15], first presented at the International Symposium on Phase Transitions in Dubna, May 1965. The physical picture was that, for critical fluctuations the distribution of the order parameter remains invariant with temperature if the length scale and other observables are adjusted properly. This hypothesis is physically equivalent to L. P. Kadanoff's formulation [16], which was published 4 months later. In addition, in his work Kadanoff first formulated a program of elimination of short-range degrees of freedom by decimation of spin blocks, an embryo of the Wilson Renormalization Group, though still not a practical tool for calculations.

Among the works which prepared Wilson's revolution, a special place belongs to that of Larkin and Khmelnitskii [17]. Considering the Ising magnet with weak dipolar interaction in 3 dimensions, they showed that its critical behavior is the same as for the standard Ising magnet in 4 dimensions, where the mean field theory almost works. This choice of dimensionality enabled them to find the asymptotically exact solution, a direct predecessor of the Wilson-Fisher -expansion [18]. A. A. Migdal in 1970 [19] was the first to construct a fluctuation theory of the tricritical point in 3 dimensions.

A new symmetry of critical phenomena, the conformal symmetry, was discovered by Polyakov in 1970 [20]. It can be understood as a local scale transformation which does not violate local rotational invariance. This deeper level of scaling symmetry was especially fruitful in two dimensions. Many years later, in 1983, the three Sashas, Belavin, Polyakov and Zamolodchikov, constructed their famous Conformal Field Theory [21], which enabled them and others to determine all universality classes in two dimensions, and to calculate the complete algebra of fluctuating operators and their critical exponents. This work also had a deep impact on Field Theory. The permanent exchange of ideas between Statistical Physics and Field Theory is quite remarkable. It would not have existed without Landau's fundamental concept of the order parameter.

A new notion of topological excitations (vortices) and phase transitions driven by them was first introduced by V. L. Berezinskii in 1971 [22], two years before Kosterlitz and Thouless [23]. He did not find, however, their simple relation between the transition temperature and the transverse stiffness. As it was with the order parameter, the idea of topological excitations and phase transitions had a deep impact on field theory.

In conclusion, I cite two works which elucidated the nature of the order parameter in systems with macroscopic quantum coherence. The first is the famous work by N. N. Bogolyubov [24] on superfluidity in weakly interacting Bose gases. The second is the no-less-famous Ginsburg-Landau theory [25] in which the condensate wave-function was first introduced.

References

[1] L.D. Landau, ZhETF 7, 19 (1937); Phys. Zs. Sowjet. 11, 26 (1937).

[2] E.M. Lifshitz, ZhETF 11, 265 (1941).

[3] L. Onsager, Phys, Rev. 65, 117 (1944).

[4] A.I. Levanyuk, ZhETF 36, 810 (1959).

[5] V.L. Ginsburg, Fizika Tverdogo Tela (Solid State Physics, in Russian) 2, 2034 (1960).

[6] M.E. Fisher, Physica 25, 521 (1959).

[7] K.G. Wilson, Phys, Rev. Lett. 28, 548 (1972).

[8] A.Z. Patashinskii and V.L. Pokrovskii, ZhETF, 50, 439 (1964) [Sov. Phys. JETP 19, 677 (1964)].

[9] A.M. Polyakov, ZhETF 55, 1026 (1968) [Sov. Phys. JETP 28, 533 (1969)].

[10] A.A. Migdal, ZhETF 55, 1964 (1968) [Sov. Phys. JETP 28, 1036 (1969)].

[11] A.N. Kolmogorov, DAN SSSR, 30, 299; Ib. 31, 99 (1941).

[12] V.G. Vaks and A.I. Larkin, ZhETF 49, 975 (1965) [Sov. Phys. JETP 22, 678 (1966)].

[13] L.P. Kadanoff and F.J. Wegner, Phys. Rev. B4, 3989 (1971).

[14] B.D. Josephson, Phys. Lett. C2, 1113 (1969).

[15] A.Z. Patashinskii and V.L. Pokrovskii, ZhETF 50, 439 (1966) [Sov. Phys. JETP 23, 292 (1966)].

[16] L.P. Kadanoff, Physics 2, 263 (1966).

[17] A.I. Larkin and D.E. Khmel'nitskii, ZhETF 56, 2087 (1969) [Sov. Phys. JETP 29, 1123 (1969)].

[18] K.G. Wilson and M.E. Fisher, Phys. Rev. Lett. 28, 240 (1972).

[19] A.A. Migdal, ZhETF 59, 1015 (1970) [Sov. Phys. JETP Letters 12, 281 (1970)].

[20] A.M. Polyakov, Pis'ma v ZhETF 12, 538 (1970) [Sov. Phys. JETP Letters 12, 281 (1970)].

[21] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B240, 312 (1984).

[22] V.L. Berezinskii, ZhETF 61, 1144 (1971) [Sov. Phys. JETP 34, 610 (1972)].

[23] J.M. Kosterlitz and D. Thouless, J. Phys C 6, 1181 (1973).

[24] N.N. Bogolyubov, Izvestiya Akademii Nauk SSSR (Proceedings of Academy of Sciences of USSR, in Russian), ser. physics, 11, 77 (1947).

[25] V.L. Ginsburg and L.D. Landau, ZhETF 20, 1064 (1950) [English translation in The Collected Papers of L.D. Landau, Pergamon Press, Oxford, 1965].