Order parameter for non-mean-field spin glasses

TL;DR

The paper tackles non-mean-field spin-glass models and the difficulty of identifying the infrared degrees of freedom due to frustration. It develops a renormalization-group scheme that relies on a minimal information principle and system symmetries, yielding a ground-state-projection order parameter that drives decimation and rescaling. Applying this to the Hierarchical Edwards-Anderson model, the authors obtain RG fixed distributions and predict the critical exponent $ν$ as a function of the interaction-range exponent $ς$, with strong agreement to Monte Carlo results in the non-mean-field regime. The work points to a broader framework for real-space RG in frustrated magnets and potential extensions to cubic lattices and spin-glass materials.

Abstract

We propose a novel renormalization group (RG) method for non mean-field models of spin glasses, which leads to the emergence of a novel order parameter. Unlike previous approaches where the RG procedure is based on a priori notions on the system, our analysis follows a minimality principle, where no a priori assumption is made. We apply our approach to a spin-glass model built on a hierarchical lattice. In the RG decimation procedure, a novel order parameter spontaneously emerges from the system symmetries, and self-similarity features of the RG transformation only. This order parameter is the projection of the spin configurations on the ground state of the system. Kadanoff's majority rule for ferromagnetic systems is replaced by a more complex scheme, which involves such novel order parameter. The ground state thus acts as a pattern which translates spin configurations from one length scale to another. The rescaling RG procedure is based on a minimal, information-theory approach and, combined with the decimation, it yields a complete RG transformation. Below the upper critical dimension, the predictions for the critical exponent $ν$, which describes the critical divergence of the correlation length, are in excellent agreement with numerical simulations from both this and previous studies. Overall, this study opens new avenues in the understanding of the critical ordering of realistic spin glasses, and it can be applied to spin-glass models on a cubic lattice and nearest-neighbor couplings which directly model spin-glass materials, such as AuFe, CuMn and other magnetic alloys.

Figures (2)

• Figure 1: Energy excitations for the hierarchical Edwards-Anderson model and block-spin decimation. $\textbf{A}$) Energy excitations: The four spins of model ${\cal M}$ are represented by dots, and the coupling between a spin pair by an arc connecting the dots. First- and second-level couplings are shown on bottom and top, and they correspond to the first two and last four terms in the right-hand side of \ref{['eq_91']}, respectively. Violated and non-violated couplings in the excited states are shown with red solid and blue dashed curves, respectively. Energy excitations are grouped according to the couplings that they violate: i) Excitation that violates all second-level couplings with respect to the ground state. ii) Excitations that violate one first-level and two second-level couplings. iii) Excitations that violate two first-level and two second-level couplings. $\textbf{B}$) Block-spin decimation. For a given sample of the disordered couplings $\textit{J}$ and $J'$, models ${\cal M}$ and ${\cal M}'$, on the left and right, are shown with their respective ground-state spin configurations, displayed in red and blue, respectively. Three spin configurations of ${\cal M}$ and ${\cal M}'$ are shown in black in panels $i$, $ii$ and $iii$, where only spins in the left half are shown for clarity. In each panel, the spin configuration of ${\cal M}$ and ${\cal M}'$ are related by the decimation relation \ref{['eq_dec']}. $i)$ Left spins of ${\cal M}$ are parallel to their ground state, and $\Phi_{\rm L} = +1$. As a result, the left spin of ${\cal M}'$ is parallel to its own ground state, and $\Phi'_{\rm L} = +1$. $ii)$ Same as $i$, for spins antiparallel to the ground states, with $\Phi_{\rm L} = \Phi'_{\rm L}=-1$. $iii)$ Left spins of ${\cal M}$ are neither parallel nor antiparallel to their ground state, i.e., $\Phi_{\rm L} = 0$. As a result, $\Phi'_{\rm L} = 0$, and the left spin in ${\cal M}'$ is either parallel or antiparallel to the ground state of ${\cal M}'$, with equal probability.
• Figure 2: Linearization of the RG transformation and critical exponent. $\textbf{A}$) Jacobian $\bm {\mathscr J}$ of the scaled RG transformation evaluated at the scaled critical fixed distribution ${\bm K}^k = \beta_{\rm c} \bm{L}_{\rm c}$, where $\bm L$ are the quantiles of the spin-coupling distribution, as a function of the values $F_i$, $F_j$ of the relative cumulative distribution function, for $\varsigma = 0.8$. Here the cumulative-distribution-function values $F_i$, $F_j$ serve as labels for the Jacobian rows and columns, respectively, and $i,j = 1,\cdots, N$, where $N$ is the number of bins of the discretization. Inset: eigenvalues ${\overline{\lambda}}_m$ of $\bm {\mathscr J}$ with norm smaller than unity (blue), the eigenvalue ${\overline{\lambda}_\star}$ with norm larger than unity (red), and the unit disk (black). $\textbf{B}$) Critical exponent $\nu$, which describes the divergence of the correlation length, as a function of the coupling-range exponent $\varsigma$. The values of $\nu$ obtained from the renormalization-group method are shown as red circles. Monte Carlo simulation predictions for $\nu$ from this work (black crosses) have been generated from the Hierarchical Edwards-Anderson model with power-law interaction decay for $\varsigma \leq 0.8$, and with fixed coordination number for $\varsigma > 0.8$---see \ref{['sec_mc']}. Monte Carlo predictions for $\nu$ from previous studies leuzzi2008dilutebanos2012correspondence, obtained from diluted one-dimensional Ising spin glasses with power-law interactions, are shown as brown squares and blue triangles. We also show the upper and lower critical dimensions, ${\varsigma_{\hbox{${\rm up}$} }}$ and ${\varsigma_{\hbox{${\rm low}$}}}$, respectively, and the exact value of $\nu$ at ${\varsigma_{\hbox{${\rm up}$} }}$ (green triangle).