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When More Becomes Different: A Historical Development Review of Phase Transitions and the Renormalization Group

Zen Revista, 10 PHYSICS

Published Sep 26, 2026• 0 citations
Publisher

Abstract

This article offers a historical development review of the theory of phase transitions and critical phenomena, the branch of physics whose development from Thomas Andrews's 1869 discovery of the critical point through Kenneth Wilson's renormalization group transformed the discipline's understanding of how collective behavior emerges from microscopic parts. The review reconstructs the development across six phases: the continuity surprise, Andrews's critical point and Johannes Diderik van der Waals's equation of state, whose 1873 dissertation unified the liquid and the gas and whose corresponding states law organized the fluids; the mean field era, Pierre Curie's magnetic measurements, Paul Weiss's molecular field hypothesis, and Lev Landau's order parameter, whose symmetry-based framework dominated the discipline for four decades; the exact solution, Wilhelm Lenz and Ernst Ising's model, Rudolf Peierls's argument, Lars Onsager's 1944 solution of the two-dimensional model, and Chen Ning Yang's spontaneous magnetization, whose eighth-root exponent destroyed the mean field's claims; the scaling synthesis, Edward Guggenheim's empirical corresponding states, Benjamin Widom's equation of state, and Leo Kadanoff's block spins, whose scaling laws connected the exponents; the renormalization group, Kenneth Wilson's iterative integration of fluctuations and Michael Fisher's epsilon expansion, whose fixed points explained universality itself; and the precision era, whose microgravity calorimetry and conformal bootstrap computations pushed the exponents to accuracies beyond experiment. Three synthetic claims are advanced. First, the theory's history is the defeat of mean field reasoning by exact mathematics and careful experiment, followed by the renormalization group's explanation of both the failure and the rare successes. Second, the universality discovery established that microscopic detail is irrelevant at criticality, the deepest expression of Philip Anderson's principle that more is different. Third, the renormalization group redefined physical theory itself, replacing single equations with methods that connect scales. An agenda is proposed spanning bootstrap precision, nonequilibrium criticality, and quantum matter.

DOI
10.5281/zenodo.22975317

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This article offers a historical development review of the theory of phase transitions and critical phenomena, the branch of physics whose development from Thomas Andrews's 1869 discovery of the critical point through Kenneth Wilson's renormalization group transformed the discipline's understanding of how collective behavior emerges from microscopic parts. The review reconstructs the development across six phases: the continuity surprise, Andrews's critical point and Johannes Diderik van der Waals's equation of state, whose 1873 dissertation unified the liquid and the gas and whose corresponding states law organized the fluids; the mean field era, Pierre Curie's magnetic measurements, Paul Weiss's molecular field hypothesis, and Lev Landau's order parameter, whose symmetry-based framework dominated the discipline for four decades; the exact solution, Wilhelm Lenz and Ernst Ising's model, Rudolf Peierls's argument, Lars Onsager's 1944 solution of the two-dimensional model, and Chen Ning Yang's spontaneous magnetization, whose eighth-root exponent destroyed the mean field's claims; the scaling synthesis, Edward Guggenheim's empirical corresponding states, Benjamin Widom's equation of state, and Leo Kadanoff's block spins, whose scaling laws connected the exponents; the renormalization group, Kenneth Wilson's iterative integration of fluctuations and Michael Fisher's epsilon expansion, whose fixed points explained universality itself; and the precision era, whose microgravity calorimetry and conformal bootstrap computations pushed the exponents to accuracies beyond experiment. Three synthetic claims are advanced. First, the theory's history is the defeat of mean field reasoning by exact mathematics and careful experiment, followed by the renormalization group's explanation of both the failure and the rare successes. Second, the universality discovery established that microscopic detail is irrelevant at criticality, the deepest expression of Philip Anderson's principle that more is different. Third, the renormalization group redefined physical theory itself, replacing single equations with methods that connect scales. An agenda is proposed spanning bootstrap precision, nonequilibrium criticality, and quantum matter.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
This article traces the development and study of phase transitions from late last century to the present day. We begin with a brief historical sketch and a description of the statistical mechanics of phase transitions. Particular attention is given to the modern era which began in 1944 with Onsager's celebrated solution of the two-dimensional Ising model. Points of development since Onsager, which are highlighted in this article, include the study of critical exponents, the scaling hypothesis, realisation of the universality of critical exponents and the recent renormalization group approach to critical phenomena. The basic idea of the renormalization group method, rather than the detailed application of the recipe to particular cases, is stressed and is discussed critically in some detail.
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
The nature and origins of renormalization group ideas in statistical physics and condensed matter theory are recounted informally, emphasizing those features of prime importance in these areas of science in contradistinction to quantum field theory, in particular: critical exponents and scaling, relevance, irrelevance and marginality, universality, and Wilson's crucial concept of flows and fixed points in a large space of Hamiltonians. Contents Foreword 1 Introduction 2 Whence came renormalization group theory? 3 Where stands the renormalization group? 4 Exponents, anomalous dimensions, scale invariance and scale dependence 5 The challenges posed by critical phenomena 6 Exponent relations, scaling, and irrelevance 7 Relevance, crossover, and marginality 8 The task for renormalization group theory 9 Kadanoff's scaling picture 10 Wilson's quest 11 The construction of renormalization group transformations: the epsilon expansion 12 Flows, fixed points, universality and scaling 13 Conclusions Acknowledgments Selected bibliography Appendix A Asymptotic behavior Appendix B Unitarity of the renormalization group Appendix C Nature of a semigroup Foreword It was a pleasure to participate in the Colloquium cosponsored by the Departments of Philosophy and of Physics at Boston University “On the Foundations of Quantum Field Theory.” In the full title, this was preceded by the phrase: “A Historical Examination and Philosophical Reflections,” which set the aims of the meeting. Naturally, the participants were mainly high-energy physicists, experts in field theories, and interested philosophers of science.