50 Years of the Renormalization Group

The following sections are included:

• Preface
• Contents

We present our personal histories with Michael Fisher. We describe how each one of us first came to Cornell University. We also discuss our many subsequent interactions and successful collaborations with him on various physics projects.

This chapter is about my memories of being a young postdoc from the UK coming to the center of activity on a very new and successful method for dealing with the hitherto impenetrable topic of phase transitions and critical phenomena, the Renormalization Group and the ϵ-expansion of Kenneth G. Wilson and Michael E. Fisher.

In the fall of 1966, I had known Michael Fisher for several years. I don’t remember where or when we first met. But we needed no introduction when he arrived at Cornell as the new Horace White Professor of Chemistry, Physics and Mathematics, and I came there on a one-year leave from Carnegie Tech. In the back of my mind is an image of Michael with Sorrel and young versions of their children arriving on campus not long after Elly and I and our even younger children came there. I was looking forward to working with Michael on a wide range of ideas in theories of phase transitions, and I wasn’t disappointed…

I first met Michael in the summer of 1961. I was in London for a few days, on my way to a conference in Vienna, visiting Oliver Penrose who was then at Imperial College. While there I went to Kings College to see Cyril Domb. Cyril took me to lunch in a kosher restaurant. After returning from lunch, Cyril introduced me to his group of young researchers. These included Martin Sykes and Michael Fisher, who told me something about their work. I cannot remember anymore of what Michael was working on, but I remember (or think I do) that his office was down a long corridor and had no windows. This must have been before Michael’s promotion to a Reader at Kings College…

I first heard of Michael Fisher in 1963 at the start of my postdoctoral year with Walter Kohn at La Jolla. I had been introduced to another young postdoc, Bob Griffiths. He told me that he was currently working on a proof that the free energy of a spin system exists. “That it what?” I asked. “That it exists,” said Bob firmly. “I’m using some ideas I got from Michael Fisher.” Well, I thought, this Fisher must be a man with deep philosophical interests — a sort of Plato of thermodynamics…

I had the good fortune and privilege of having Michael Fisher as my teacher, supervisor, mentor and friend. During my years as a scientist, teacher and supervisor of about 100 students and postdocs, I found myself innumerable times realizing that I am following or at least trying to follow Michael’s example. These pages are my attempt to convey recollections of my association with Michael, focusing on how he served as an example for me.

Unfortunately, I never had the privilege of collaborating directly with Michael Fisher on a research project, and we were never colleagues at the same institution. My interactions with him were therefore largely limited to encounters at scientific conferences and some brief visits to each other’s home institutes. Nevertheless, Michael was an important role model for me, and he had a major influence on my work. Beyond this, I came to consider him as a very good friend…

Working as a graduate student with Michael Fisher was one of the defining experiences of my life. It is hard to imagine how not only my career but also my character would be different if I had worked with someone else.

It is more than a little shocking to realize that I am older now than Michael was when I was in his group. To a beginning graduate student, the breadth and depth of his knowledge and understanding and his clarity of thought and expression were overwhelming. Above all, Michael was offended by sloppy thinking or incorrect statements. He would dress down anyone from a student to a seminar speaker for showing a plot without proper labels or with error bars that allowed for violations of thermodynamics. I remember an occasion when I had a meeting with him after a talk that ran late and wasted his time. In his annoyance, he decimated me and I felt utterly helpless — I simply could not think quickly enough and did not understand the physics well enough to defend myself. It was not easy to maintain confidence, let alone self-respect, under those conditions…

Fluctuation-dominated phase ordering refers to a steady state in which the magnitude of long-range order varies strongly owing to fluctuations and to the associated coarsening phenomena during the approach to steady state. Strong fluctuations can lead to a number of interesting phenomena, including a cusp singularity in the scaled correlation function, implying the breakdown of the Porod Law. First identified in a nonequilibrium system of passively sliding particles on a fluctuating surface, fluctuation-dominated order also occurs in several other systems, including an equilibrium Ising model with long-range interactions. This chapter discusses these systems and others where clustering effects are stronger.

Mixed-order phase transitions are transitions which have common features with both first-order and second-order transitions. I review some results obtained in the context of one of the prototypical models of mixed-order transitions, the one-dimensional Ising model with long-range coupling that decays as truncated inverse square distance between spins. The correspondence between this model and the Poland Scheraga model of DNA denaturation, a subject to which Michael Fisher made substantial contributions, is then outlined.

I summarize recent progress on obtaining rigorous upper bounds on superconducting transition temperature T_c in two dimensions independent of pairing mechanism or interaction strength. These results are derived by finding a general upper bound for the superfluid stiffness for a multi-band system with arbitrary interactions, with the only assumption that the external vector potential couples to the kinetic energy and not to the interactions. This bound is then combined with the universal relation between the superfluid stiffness and the Berezinskii–Kosterlitz– Thouless T_c in 2D. For parabolic dispersion, one obtains the simple result that k_B T_c ≤ E_F /8, which has been tested in recent experiments. More generally, the bounds are expressed in terms of the optical spectral weight and lead to stringent constraints for the T_c of low-density, strongly correlated superconductors. Results for T_c bounds for models of flat-band superconductors, where the kinetic energy vanishes and the vector potential must couple to interactions, are briefly summarized. Upper bounds on T_c in 3D remains an open problem, and I describe how questions of universality underlie the challenges in 3D.

Critical fluctuations in fluids and fluid mixtures yield a non-analytic asymptotic Ising-like critical thermodynamic behavior in terms of power laws with universal exponents. In polymer solutions, the amplitudes of these power laws depend on the degree of polymerization. Non-asymptotic behavior (upon the departure from the critical point) is particularly interesting in the case of polymer solutions, where it is governed by a competition between the correlation length of the critical fluctuations and the radius of gyration of the polymer molecules. If the correlation length is the dominant length scale, Ising-like critical behavior is observed. If, however, the radius of gyration exceeds the correlation length, tricritical behavior with mean-field critical exponents is observed. The Ising-like critical region shrinks with the increase of the polymer molecular weight. In the limit of an infinite degree of polymerization, the Ising-like critical region vanishes, yielding to theta-point tricriticality.

We discuss the role of series expansions in studies of critical phenomena. We review some general aspects of graph enumeration and counting and present the basis for using dimensionlity d as a continuous variable in series expansion studies. We discuss the series analysis method of differential approximants and partial differential approximants and emphasize the importance of embedding a particular model in a wider class of models to obtain better numerical estimates and a more comprehensive description of the critical behavior. We consider series studies of quenched random systems, especially classical and quantum Ising spin glasses. We argue that in classical spin glasses, Griffiths singularities hardly affect series extrapolations. In contrast, for quantum spin glasses, series expansions provide a powerful method for exploring Griffiths–McCoy singularities.

Continuous phase transitions in equilibrium statistical mechanics were successfully described 50 years ago with the development of the renormalization group framework. This framework was initially developed in the context of phase transitions whose universal properties are captured by the long wavelength (and long time) fluctuations of a Landau order parameter field. Subsequent developments include a straightforward generalization to a class of T = 0 phase transitions driven by quantum fluctuations. In the last 2 decades it has become clear that there is a vast landscape of quantum phase transitions where the physics is not always usefully (or sometimes cannot be) formulated in terms of fluctuations of a Landau order parameter field. A wide class of such phase transitions — dubbed deconfined quantum critical points — involve the emergence of fractionalized degrees of freedom coupled to emergent gauge fields. Here I review some salient aspects of these deconfined critical points.

Fisher and de Gennes showed that the cutoff of the order-parameter fluctuations in a liquid film of finite thickness near its critical point leads to a thickness-dependent free energy. This article presents a brief review of experimental studies of this Casimir-like effect in helium films near the superfluid transition and the tricritical point and in binary mixture liquid films near their critical points.

This chapter covers my personal remembrances in three Sections. Section 1 contains a brief personal perspective on Michael E. Fisher’s contributions to science. Section 2 tells how I came to work with Michael and describes events while I was under his supervision during my graduate years at Cornell University. Section 3 summarizes recent work, done in collaboration with Suraj Shankar, on thermalized buckling of isotopically compressed thin (perhaps atomically thin) sheets of materials, such as graphene or MoS₂. These investigations were inspired by Michael’s beautiful work on the effect of constraints at critical points, with fluctuations at all length scales, which leads to “Fisher renormalization” of critical exponents. Thin fluctuating sheets embedded in three dimensions, when they are tensionless as in a cantilever or “diving board” geometry, are automatically at a critical point everywhere in a low-temperature flat phase. However, when we consider thin sheets supported on multiple sides in various ways, Fisher’s ideas lead to the inequivalence of isotensional and isometric thermodynamic ensembles, which triggers dramatic differences in some of the critical exponents associated with the two types of boundary conditions. Readers not interested in my experiences as a student at Cornell University may only want to read Sections 1 and 3. Section 4 provides some concluding remarks.

I start with a review of my personal and scientific interactions with Michael E. Fisher, who was my post-doc mentor in 1972-1974. I then describe several recent renormalization group studies, which started during those years, and still raise some open issues. These include the magnets with dipole-dipole interactions, the puzzle of the bicritical points and the random field Ising model.

The Yang–Lee edge singularity is a prototypical example of the application of renormalization-group ideas to critical behavior and one to which Michael Fisher made several important contributions. Moreover, it has connections to several other problems such as the statistics of branched polymers, and its scaling limit in two dimensions provides a simple example of integrable field theory. This chapter aims to give a pedagogical introduction to these matters, with a few new ideas thrown in.

The remarkable technical contributions of Michael E. Fisher to statistical physics and the development of the renormalization group are widely known and deeply influential. But less well known is his early and profound appreciation of the way in which renormalization group created a revolution in our understanding of how physics — in fact, all science — is practiced and the concomitant adjustment that needs to be made to our conception of the purpose and philosophy of science. In this chapter, I attempt to redress this imbalance, with examples from Fisher’s writings and my own work. It is my hope that this tribute will help remove some of the confusion that surrounds the scientific usage of minimal models and renormalization group concepts, as well as their limitations, in the ongoing effort to understand emergence in complex systems.

A classical knot is described by a one-stroke trajectory of a string with entanglements. The replica method is a powerful tool in statistical mechanics for dealing with string-like objects like polymers or self-avoiding walks. We consider here the (N → 0) replica limit for Gaussian means of products of traces of N × N Hermitian matrices, which correspond to one-stroke graphs for knots. The Seifert surfaces of knots and links are thus related to a random matrix model. The zeros of Alexander polynomials on the unit circle are discussed for the case of n-vertices in analogy with the Yang–Lee edge singularity. The extension of one-matrix models to higher dimensional knots is considered, and also to the half-integral level k in a Chern–Simons gauge theory.

Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the ϕ⁴ field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the ϕ⁴ field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher ϵ-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art ϵ-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made.

Strong-randomness renormalization groups were first developed to treat various quantum critical ground states, especially in one-dimensional systems. After briefly examining some of the earlier works with these methods, the recent application of this approach to the many-body localization phase transition is reviewed.

We define a notion of zero-temperature entropy for impurities (line defects) in d + 1 space–time dimensions. We show that this entropy obeys a simple evolution equation under the renormalization group. We apply this result for impurities in magnets and in gauge theories. We find new critical impurities and phase transitions.

Unlike many of the authors who are contributing to this book, I was neither a student nor a postdoc with Michael Fisher, nor did I ever collaborate with him. Nevertheless, his impact on my work and career has been enormous. In my first encounter with him, however, his outstanding research and leadership in the general field of statistical physics took second place to his other great prowess as a flamenco guitarist…

We discuss the research performed by us in the relatively early days of the renormalization group, focusing on “beyond the bulk” properties of systems at and near bulk criticality. In all instances, the topics, along with specific problem addressed, originate in whole or in part from the work of Michael E. Fisher.

Our community has a deep and sophisticated understanding of phase transitions and their universal scaling functions. We outline and advocate an ambitious program to use this understanding as an anchor for describing the surrounding phases. We explain how to use normal-form theory to write universal scaling functions in systems where the renormalization-group flows cannot be linearized. We use the 2D Ising model to demonstrate how to calculate high-precision implementations of universal scaling functions and how to extend them into a complete description of the surrounding phases. We discuss prospects and challenges involved in extending these early successes to the many other systems where the RG has successfully described emergent scale invariance, making them invaluable tools for engineers, biologists, and social scientists studying complex systems.

We consider quantum and classical first-order transitions, at equilibrium and under out-of-equilibrium conditions, mainly focusing on quench and adiabatic protocols. For these phenomena, we review the finite-size scaling theory appropriate to describe the general features of the large-scale (and long-time for dynamic phenomena) behavior of finite-size systems.

Michael Fisher presented many important contributions to critical phenomena and distributed it widely in review articles. His paper with Kenneth Wilson on the expansion of critical exponents in 4 − ϵ dimensions gave the field of critical phenomena an enormous boost.

A few years later Alexander Polyakov presented his paper on the expansion around the lower critical dimension 2, which brought an important impetus for non-linear sigma-models. The Anderson localization transition due to disorder is described by such a model of supersymmetric matrices. It is applied to various dimensions and to ensembles with and without time-reversal invariance.

The low energy manifold for fermions at finite density is the Fermi surface. I describe renormalization group (RG) in which modes encountered on approaching the Fermi surface are systematically integrated out. The fixed point is described by strictly marginal coupling functions identified as the Landau parameters and marginally relevant coupling functions describing the BCS instability in various angular momentum channels.

I review a class of novel ordered states of “critical matter”, that exhibit strongly fluctuating universal power-law orders, controlled by an infrared attractive, non-Gaussian fixed point. I will illustrate how RG methods pioneered by Wilson and Fisher can be used to deduce critical phenomenology of such critical phases, resembling that of a critical point of second-order phase transitions, but requiring no fine-tuning.

An overview of recent advances in the theory of critical phenomena in d-dimensional weakly anisotropic systems is given. On the basis of a generalized shear transformation between anisotropic and isotropic systems, exact and approximate results are discussed for bulk and confined systems in two and three dimensions, where conformal-field theory and the minimal renormalization without ε-expansion play a crucial role. Stimulation for this research comes from the seminal work by V. Privman and M.E. Fisher in 1984 in which the principle of two-scale-factor universality for bulk systems has been extended to finite systems. Based on this principle for isotropic systems, we predict the validity of multiparameter universality with up to d(d + 1)/2 + 1 non-universal parameters in d-dimensional anisotropic bulk and confined systems with periodic boundary conditions (BC). The verification of multiparameter universality for confined anisotropic systems with realistic BC and the study of the intrinsic diversity of the critical behavior of magnetic materials, superconductors, liquid crystals, and solids with non-cubic symmetry are a major challenge to future research.

The following sections are included:

• Introduction
• Monte Carlo Simulations Related to Michael Fisher’s Fundamental Contributions
• Publications Co-authored by Michael Fisher Using MC Simulations
• Final Remarks and Acknowledgements
• References

I discuss the important work of Michael Fisher in developing the theory of finite-size scaling, which is now widely used in simulations of critical phenomena. A brief overview of the history of the subject and its formulation will be given. As an example of the power of FSS when combined with careful, large-scale Monte Carlo simulations, I briefly discuss the work which accurately determined the critical behavior of the three-dimensional Ising spin glass. Finite-size scaling breaks down above the upper critical dimension, which is d = 4 for most systems, because of a “dangerous irrelevant” variable, a term first coined by Michael. A brief discussion of how standard finite-size scaling has to be modified in this region will be given at the end.

Turning the divergent ϵ-expansion into a numerically sensible algorithm, relies on the knowledge of the behavior of the large order contributions. Two different pictures are known to compete there. The first one was based on Lipatov’s instantons, which is known to deal with the multiplicity of Feynman diagrams which grows factorially at high orders. However, this was challenged by ’t Hooft’s renormalons, which pointed out that renormalization could yield a similar growth through one single diagram. We study here a well-known model the O(N) model, in the large N-limit. The reason for returning to this familiar model, is that it deals with diagrams known to give renormalon effects. Through an explicit analytic result, we find no sign of a non-analyticity of perturbation theory due to these renormalons.

Numerous physical systems of different nature exhibit singular behavior in the vicinity of their critical points: Various thermodynamic and correlation functions acquire scaling forms with apparently universal critical exponents. The theory of phase transitions and critical phenomena is one of the most interesting areas of modern theoretical physics. The crucial breakthrough came with the application of the renormalization group and ε-expansion by K. Wilson, M. Fisher and others. Most conventional critical systems (liquid–vapour, magnets, superfluid transition) belong to the universality class of the O(n) symmetric ϕ⁴ model and its descendants. There, the application of the field-theoretic renormalization group allows for high-order calculations of the critical exponents within various regular schemes: 4 − ε-, 2 + ε-, 1/n- and τ-expansions with subsequent resummation of the resulting asymptotic series. Another direction of research is the application of renormalization group to more complex, dynamic and non-equilibrium systems such as growth phenomena, chemical reactions and turbulence. This chapter presents the results obtained by our informal international network (St. Petersburg, Dubna, Košice, Helsinki) and covers both directions.

The following sections are included:

• Introduction
• CFT Review
• Bootstrap Results
• Acknowledgements
• References

The first example of a quantum many body system not expected to have any quasiparticle excitations was the Wilson-Fisher conformal field theory. The absence of quasiparticles can be established in the compressible, metallic state of the Sachdev-Ye-Kitaev model of fermions with random interactions. The solvability of the latter model has enabled numerous computations of the non-quasiparticle dynamics of chaotic many-body states, such as those expected to describe quantum black holes. This chapter reviews thermodynamic properties of the SYK model, and describes how they have led to an understanding of the universal structure of the low energy density of states of charged black holes without low energy supersymmetry.

The cells of our body are compartmentalized by biomembranes and vesicles, which consist of molecular bilayers with a thickness of a few nanometers and represent two-dimensional fluids. Due to their fluidity, the bilayer membranes can easily remodel their composition, shape, and topology. Here, we focus on their topology and transformations between different topologies via fission and fusion processes. In general, these topological transformations can be characterized by changes in the Euler characteristic and in the topological genus. Fission processes proceed via the closure and cleavage of membrane necks as recently demonstrated for giant unilamellar vesicles (GUVs) and for unilamellar nanovesicles assembled in silico. Neck cleavage is controlled by constriction forces that compress the neck and provide a general physical mechanism for membrane fission. Fusion processes proceed via the adhesion of two membranes and by the formation of a fusion pore, which has the same shape as a membrane neck. In fact, when two membranes fuse, they undergo the same sequence of shapes as during fission but in reversed order. However, the ‘local surgery,’ necessary to form the fusion pore, involves alternative molecular pathways as reviewed here for tension-induced fusion. Multispherical vesicles, doped with membrane proteins that drive homotypic fusion, can be used to form high genus vesicles, which are accessible to experimental studies.

We cast a non-zero-temperature analysis of the jamming transition into the framework of a scaling ansatz. We show that four distinct regimes for scaling exponents of thermodynamic derivatives of the free energy, such as pressure, bulk and shear moduli, can be consolidated by introducing a universal scaling function with two branches. Both the original analysis and the scaling theory assume that the system always resides in a single basis in the energy landscape. The two branches are separated by a line T^* (Δϕ) in the T − Δϕ plane, where $\text{Δ}\varphi =\varphi -{\varphi }_c^{\text{Λ}}$ is the deviation of the packing fraction from its critical, jamming value, ${\varphi }_c^{\text{Λ}}$, for that basin. The branch for T < T^* (Δϕ) reduces at T = 0 to an earlier scaling ansatz that is restricted to T = 0, Δϕ ≥ 0, while the branch for T > T^* (Δϕ) reproduces exponents observed for thermal hard spheres. In contrast to the usual scenario for critical phenomena, the two branches are characterized by different exponents. We suggest that this unusual feature can be resolved by the existence of a dangerous irrelevant variable u, which can appear to modify exponents if the leading u = 0 term is sufficiently small in the regime described by one of the two branches of the scaling function.

I present my personal recollections of working together with Michael Fisher and what I learned from him. He stimulated the shift of my research interests toward the direction of theoretical biophysics. More specifically, together we started to investigate the mechanisms of motor proteins, biological molecular machines and cytoskeleton protein filaments. A very brief review of some of these topics is also presented.

It is a great pleasure and privilege to contribute to a volume of papers in honor of Professor Michael E. Fisher. Michael gave a colloquium in Oxford around 1978 and I was sufficiently enthralled to dare to talk to him afterward. Perhaps, I asked the right questions because I had the good fortune to have the opportunity to pursue a postdoc at Cornell. Michael was a superb postdoc advisor, he expected high standards and did so by example. It was an exciting time and a stimulating place to do science.

Later, in his career, Michael became interested in what are now termed active systems, in particular motor proteins. He was also much enthused by David Mermin’s work on topological defects. Here, I examine the two themes together and review recent works on nematic defects in living systems, particularly active, self-propelled defects.

In this chapter, we review recent advances in the theoretical, numerical, and experimental studies of critical Casimir forces in soft matter, with particular emphasis on their relevance for the structures of colloidal suspensions and on their dynamics. Distinct from other interactions which act in soft matter, such as electrostatic and van der Waals forces, critical Casimir forces are effective interactions characterized by the possibility to control reversibly their strength via minute temperature changes, while their attractive or repulsive character is conveniently determined via surface treatments or by structuring the involved surfaces. These features make critical Casimir forces excellent candidates for controlling the equilibrium and dynamical properties of individual colloids or colloidal dispersions as well as for possible applications in micro-mechanical systems. In the past 25 years, a number of theoretical and experimental studies have been devoted to investigate these forces primarily under thermal equilibrium conditions, while their dynamical and non-equilibrium behavior is a largely unexplored subject open for future investigations.

The following sections are included:

• Michael E. Fisher (1931-2021) Curriculum Vitae and Achievements
• Michael E. Fisher — Publications Until 2017
• Index