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On the nature of the spin glass transition

Gesualdo Delfino

TL;DR

Addresses spin-glass transitions in finite dimensions by exploiting a 2D line of renormalization-group fixed points that imply a one-parameter continuous internal symmetry. The analysis uses the replica method in a conformal-scattering framework to obtain exact fixed points and show $N$-independence for $n\to 0$, connecting 2D criticality to a symmetry enhancement and explaining the absence of a finite-temperature transition. In $d>2$, the symmetry can be spontaneously broken, yielding a spin-glass phase with a continuous overlap akin to Parisi's order parameter in the infinite-range model, providing a unified symmetry-based view across dimensions. The work clarifies why Landau-Ginzburg descriptions are elusive for spin glasses and highlights the deep link between finite-dimensional criticality and mean-field theory.

Abstract

We recently showed that the two-dimensional Ising spin glass allows for a line of renormalization group fixed points. We observe that this exact result corresponds to enhancement to a one-generator continuous internal symmetry. This finally explains why no finite temperature transition to a spin glass phase is observed in two dimensions. In more than two dimensions, instead, the continuous symmetry can be broken spontaneously and yields a spin glass order parameter which, for fixed temperature and disorder strength, takes continuous values in an interval. Such a feature is shared by the order parameter of the known mean field solution of the model with infinite-range interactions, which corresponds to infinitely many dimensions.

On the nature of the spin glass transition

TL;DR

Addresses spin-glass transitions in finite dimensions by exploiting a 2D line of renormalization-group fixed points that imply a one-parameter continuous internal symmetry. The analysis uses the replica method in a conformal-scattering framework to obtain exact fixed points and show -independence for , connecting 2D criticality to a symmetry enhancement and explaining the absence of a finite-temperature transition. In , the symmetry can be spontaneously broken, yielding a spin-glass phase with a continuous overlap akin to Parisi's order parameter in the infinite-range model, providing a unified symmetry-based view across dimensions. The work clarifies why Landau-Ginzburg descriptions are elusive for spin glasses and highlights the deep link between finite-dimensional criticality and mean-field theory.

Abstract

We recently showed that the two-dimensional Ising spin glass allows for a line of renormalization group fixed points. We observe that this exact result corresponds to enhancement to a one-generator continuous internal symmetry. This finally explains why no finite temperature transition to a spin glass phase is observed in two dimensions. In more than two dimensions, instead, the continuous symmetry can be broken spontaneously and yields a spin glass order parameter which, for fixed temperature and disorder strength, takes continuous values in an interval. Such a feature is shared by the order parameter of the known mean field solution of the model with infinite-range interactions, which corresponds to infinitely many dimensions.

Paper Structure

This paper contains 7 sections, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Symmetries and criticality
  3. Ising spin glass
  4. Generalities
  5. d=2
  6. $d>2$
  7. Conclusion

Figures (2)

  • Figure 1: The phase diagram of the $\pm J$ random bond Ising model on the hypercubic lattice is symmetric under $p\to 1-p$. Left:$d=2$ with the indication of the magnetic (P, N, Z) and spin glass (G) renormalization group fixed points. Right:$d=3$.
  • Figure 2: Magnetic and overlap scattering amplitudes for the random bond Ising model. $a_i$ labels a particle excitation in the $i$-th replica of the $a$-th copy ($a \neq b$, $i \neq j$). Overlap amplitudes involve different copies. Time runs upwards.