Scaling theory of the ordered phase of spin glasses

Abstract

The ordered phase of short-range spin glasses is described in terms of the scaling behaviour associated with a zero-temperature fixed point. The main ingredient of the theory is the exponent y which describes the growth with length scale L of the characteristic coupling at zero temperature, J(L) − JL^Y. The exponent y is estimated numerically for dimensions d=2,3. For Ising spin glasses we find y − -0.3 for d=2 and y − 0.2 for d=3, implying scaling to weak (strong) coupling for d=2(3), i.e. the “lower critical dimension” d_ℓ satisfies 2<d_ℓ<3. For d<dℓ y determines the divergence of the correlation length for T→O, while for d>d_{ℓ it} determines the large scale properties of the ordered phase, such as the long-distance behaviour of connected correlation functions, G(r) ∝ r^−y, and the singular response to a weak magnetic field, m_sing ∝ h^d/(d-2y), The decay of the connected correlation functions at large distances implies that the pure-state overlap distribution function P(q) is trivial, in contrast to the Sherrington-Kirkpatrick model. The dynamics of the system are also discussed, as is the extension to vector spin models.