Critical exponents of the spin glass transition in a field at zero temperature

Abstract

We analyze the spin glass transition in a field in finite dimension $D$ below the upper critical dimension directly at zero temperature using a recently introduced perturbative loop expansion around the Bethe lattice solution. The expansion is generated by the so-called $M$-layer construction, and it has $1/M$ as the associated small parameter. Computing analytically and numerically these non-standard diagrams at first order in the $1/M$ expansion, we construct an $ε$-expansion around the upper critical dimension $D_\text{uc}=8$, with $ε=D_\text{uc}-D$. Following standard field theoretical methods, we can write a $β$ function, finding a new zero-temperature fixed-point associated with the spin glass transition in a field in dimensions $D<8$. We are also able to compute, at first order in the $ε$-expansion, the three independent critical exponents characterizing the transition, plus the correction-to-scaling exponent.

Figures (2)

• Figure 1: Diagrams considered for the computation of the observables $\chi_2$ and $\chi_2^{dis}$ up to one-loop order. The diagram on the left gives the leading $\mathcal{O}(1/M)$ contribution, and the diagram on the right gives the first $\mathcal{O}(1/M^2)$ correction.
• Figure 2: Diagrams considered for the computation of the observable $\chi_3$ up to one-loop order. From left to right the diagrams give contributions of order $\mathcal{O}(1/M^2)$, $\mathcal{O}(1/M^3)$, $\mathcal{O}(1/M^3)$.