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

TL;DR

This paper develops a perturbative field theory approach to analyze the critical exponents of the zero-temperature spin glass transition in finite dimensions below the upper critical dimension, revealing a new fixed point and critical exponents.

Contribution

It introduces a novel $1/M$ expansion around the Bethe lattice and constructs an $ ext{epsilon}$-expansion near the upper critical dimension to study the spin glass transition in a field.

Findings

Identifies a new zero-temperature fixed point for the transition.

Calculates critical exponents and correction-to-scaling exponent at first order.

Provides analytical and numerical results for non-standard diagrams.

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 $\epsilon$-expansion around the upper critical dimension $D_\text{uc}=8$, with $\epsilon=D_\text{uc}-D$. Following standard field theoretical methods, we can write a $\beta$ 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 $\epsilon$-expansion, the three independent critical exponents characterizing the transition, plus the correction-to-scaling exponent.