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50 years of Yukhnovskii's critical point theory: its place in the constant flow of theoretical physics

Yu. Kozitsky

TL;DR

The paper surveys how Ihor Yukhnovskii's layer-by-layer integration in the space of collective variables provides an alternative yet complementary route to the $3$D Ising critical point, alongside the $\varepsilon$-expansion. By mapping spins to unbounded collective variables with a local quartic potential $P(\rho)= a_2 \rho^2 + a_4 \rho^4$ and performing momentum-shell decimations across a hierarchy of Brillouin zones, it derives a flow of the local parameters $(a_2,a_4)$ governed by averaged kernels $\bar{\alpha}_n$, effectively realizing a real-space RG procedure. The manuscript clarifies a deep connection between Yukhnovskii's method and Dyson's hierarchical model, recasting the approach as a structured RG with piecewise-constant $\alpha(k)$ and a lattice-hierarchy of block spins. It situates these insights within the broader development of quantum field theory and statistical physics, highlighting the conceptual unity behind seemingly distinct critical-point frameworks and underscoring the continued relevance of such hierarchical, layer-wise analyses for rigorous and semi-rigorous RG approaches.

Abstract

Half a century ago, Ihor Yukhnovskii elaborated a method of studying the critical point of the three-dimensional Ising model based on a layer-by-layer integration in the space of collective variables. His method was an alternative to that based on the $\varepsilon$-expansion for which K. G. Wilson was awarded the Nobel Prize in Physics in 1982. However, Yukhnovskii's technique, which yielded similar results, provided even deeper insight into the nature of this phenomenon. At that time, we, professor's students, saw only this aspect of his theory. Later, I realized that the mentioned Yukhnovskii's work naturally fits into a more general context of the turbulent development of quantum field theory and statistical physics in the last quarter of the twentieth century. The aim of the present article is to look at the main aspects and the impact of Yukhnovskii's theory from this perspective.

50 years of Yukhnovskii's critical point theory: its place in the constant flow of theoretical physics

TL;DR

The paper surveys how Ihor Yukhnovskii's layer-by-layer integration in the space of collective variables provides an alternative yet complementary route to the D Ising critical point, alongside the -expansion. By mapping spins to unbounded collective variables with a local quartic potential and performing momentum-shell decimations across a hierarchy of Brillouin zones, it derives a flow of the local parameters governed by averaged kernels , effectively realizing a real-space RG procedure. The manuscript clarifies a deep connection between Yukhnovskii's method and Dyson's hierarchical model, recasting the approach as a structured RG with piecewise-constant and a lattice-hierarchy of block spins. It situates these insights within the broader development of quantum field theory and statistical physics, highlighting the conceptual unity behind seemingly distinct critical-point frameworks and underscoring the continued relevance of such hierarchical, layer-wise analyses for rigorous and semi-rigorous RG approaches.

Abstract

Half a century ago, Ihor Yukhnovskii elaborated a method of studying the critical point of the three-dimensional Ising model based on a layer-by-layer integration in the space of collective variables. His method was an alternative to that based on the -expansion for which K. G. Wilson was awarded the Nobel Prize in Physics in 1982. However, Yukhnovskii's technique, which yielded similar results, provided even deeper insight into the nature of this phenomenon. At that time, we, professor's students, saw only this aspect of his theory. Later, I realized that the mentioned Yukhnovskii's work naturally fits into a more general context of the turbulent development of quantum field theory and statistical physics in the last quarter of the twentieth century. The aim of the present article is to look at the main aspects and the impact of Yukhnovskii's theory from this perspective.

Paper Structure

This paper contains 11 sections, 42 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic distribution
  3. Yukhnovskii's method and other models
  4. Posing
  5. Real space renormalization group
  6. Layer-by-layer integration
  7. Dyson's model and Yukhnovskii's method
  8. Dyson's hierarchical model
  9. The hierarchy of lattices
  10. Block spins
  11. Concluding remarks

Figures (4)

  • Figure 1: Diamond hierarchical lattice, in which bonds are replaced by diamonds.
  • Figure 2: Triangle hierarchical lattice, in which nodes are replaced by triangles.
  • Figure 3: (Colour online) Averaging in the Yukhnovskii method: (a) the new Brillouin zone $B'$ is cut out from the initial Brillouin zone $B$ with subsequent imposing periodicity on its boundaries; (b) the Fourier image $\alpha(k)$ of the interaction potential is set constant, i. e., $\alpha(k)= \hat{\alpha}_1$ on $B\setminus B'$.
  • Figure 4: (Colour online) Hierarchy of lattices corresponding to the Brillouin zones $B$, $B'$, $B"$.