Long-range spin glass in a field at zero temperature

TL;DR

This work tackles the zero-temperature spin-glass transition in a field by studying a one-dimensional long-range model as a proxy for higher dimensions. It develops and implements a novel Bethe $M$-layer loop expansion tailored to LR graphs, enabling a controlled $1/M$ expansion around a mean-field fixed point and the calculation of critical exponents. The authors derive an epsilon expansion around $\rho_c=5/4$, obtaining $\nu = 4 + O(\epsilon^2)$, $\omega = -4\epsilon + O(\epsilon^2)$, and anomalous dimensions with $\eta = 0$ and $\overline{\eta} = \epsilon + O(\epsilon^2)$, along with a fixed-point value $\lambda_c = 3\pi\Gamma\left(\tfrac{4\rho-5}{\rho-1}\right)/\Gamma\left(\tfrac{\rho}{\rho-1}\right)$. The work establishes a SR–LR correspondence above the upper critical dimension and provides analytical benchmarks for numerical simulations on LR lattices, aiding finite-size scaling analyses of spin glasses in a field. Overall, the paper demonstrates that the $M$-layer framework yields consistent, physically meaningful exponents for LR spin glasses and clarifies how topological loops influence critical behavior in these systems.

Abstract

We compute the critical exponents of the zero-temperature spin glass transition in a field on a one-dimensional long-range model, a proxy for higher-dimensional systems. Our approach is based on a novel loop expansion within the Bethe $M$-layer formalism, whose adaptation to this specific case is detailed here. The resulting estimates provide crucial benchmarks for numerical simulations that can access larger system sizes in one dimension, thus offering a key test of the theory of spin glasses in a field.

Figures (4)

• Figure 1: Numerical evaluation of the probability of edge occurrence $P(r)$. Once 1000 independent LRRGs are generated with the Monte Carlo procedure of Ref. Martin-Mayor_2012, $P(r)$ is evaluated as the number of present edges at distance $r$ normalized by the number of possible ones. The parameters considered here are the same as the ones used in Ref. Martin-Mayor_2012: $z=6$ and $\rho=1.792$ for different sizes $N$.
• Figure 2: Diagrams considered for the evaluation of two-point correlation functions (both connected and disconnected) up to one-loop order. Diagram $\mathcal{G}_1$ contributes to $\mathcal{O}(1/M)$ while diagrams $\mathcal{G}_2$, $\mathcal{G}'$ and $\mathcal{G}"$ contribute to $\mathcal{O}(1/M^2)$.
• Figure 3: Diagrams considered for the evaluation of the three-point connected correlation function up to one-loop order. Diagram $\mathcal{G}_3$ contributes to $\mathcal{O}(1/M^2)$ while diagrams $\mathcal{G}"'$, $\mathcal{G}_4$ and $\mathcal{G}_5$ contribute to $\mathcal{O}(1/M^3)$.
• Figure 4: Numerical evaluations of the number of NBP on one-dimensional LRRGs with fixed connectivity $z=6$ and decaying exponent $\rho=1.792$. On both figures $10^6$ NBP, with a fixed length $L$, are performed on the 1000 LRRGs previously generated with the Monte Carlo procedure of Ref. Martin-Mayor_2012. Top: Fourier transform of the number of NBP with fixed length $L=18$ for different system sizes $N$, compared with the expected curve $k^{\rho-1}$. Bottom: Fourier transform of the number of NBP with fixed system size $N=16384$ for different lengths $L$, compared with the expected curve $k^{\rho-1}$. Notice that both curves could be obtained for different values of the connectivity $z$ and the decaying exponent $\rho$. While the same behavior is expected, decreasing $\rho$ towards $\rho=1$ increases both the equilibration time and the finite-size effects.