Abstract

Cubic replicated field theory is used to study the glassy phase of the short-range Ising spin glass just below the transition temperature, and for systems above, at, and slightly below the upper critical dimension six. The order parameter function is computed up to two-loop order. There are two, well-separated bands in the mass spectrum, just as in mean field theory. The small mass band acts as an infrared cutoff, whereas contributions from the large mass region can be computed perturbatively (d>6), or interpreted by the ϵ-expansion around the critical fixed point (d=6−ϵ). The one-loop calculation of the (momentum-dependent) longitudinal mass, and the whole replicon sector is also presented. The innocuous behavior of the replicon masses while crossing the upper critical dimension shows that the ultrametric replica symmetry broken phase remains stable below six dimensions.

Highlights

  • A spin glass is a prototype of complex systems, with its slow dynamics on macroscopic time scales, unusual equilibrium properties, and complicated phase space structure which breaks ergodicity

  • Without trying to overview this huge field, we only mention here the Janus Collaboration using the special purpose Janus computer, providing results about the spin glass phase in the physical threedimensional Edwards-Anderson model which are compatible with an ultrametrically organized replica symmetry broken (RSB) phase

  • Important examples for such discrepancies are the leading behavior of the order parameter function and momentum-dependent masses close to criticality. These details depend on the space dimension d which can be well illustrated by the breakpoint x1 of the order parameter function q(x), see Refs. [6,7,8,9]: x1 is proportional to τ ∼ (Tc −T )/Tc in mean field theory, and this behavior persists down to d = 8, with possibly a logarithm of τ at exactly eight dimensions

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Summary

INTRODUCTION

A spin glass is a prototype of complex systems, with its slow dynamics on macroscopic time scales, unusual equilibrium properties, and complicated phase space structure which breaks ergodicity. Without trying to overview this huge field, we only mention here the Janus Collaboration using the special purpose Janus computer, providing results about the spin glass phase in the physical threedimensional Edwards-Anderson model which are compatible with an ultrametrically organized replica symmetry broken (RSB) phase (see [5] for a recent list of references related to the Janus Collaboration). This ultrametric glassy phase emerged for the first time in the solution of the. Several results are summarized in listed and tabulated forms in the three appendices

THE REPLICATED CUBIC FIELD THEORY AND MASS RENORMALIZATION
THE FREE PROPAGATOR
The zero-loop term
The one-loop term
The one-loop results for the equation of state
The remaining two-loop term
CALCULATION OF THE CORRECTION TO THE ORDER PARAMETER FUNCTION
The calculation of x1
The Edwards-Anderson order parameter
THE STUDY OF THE MOMENTUM-DEPENDENT MASS
The longitudinal mass
DISCUSSION
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