Breakdown of some scaling law relations in expansion

G. Grinstein

The scaling laws relating the critical exponents characterizing the ferromagnetic phase transition are shown to be modified in the presence of quenched random magnetic fields. The hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$, for example, becomes $(d+{\ensuremath{\lambda}}_{u})\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$, where the index ${\ensuremath{\lambda}}_{u}$ is negative and is related to the range of the ferromagnetic exchange interactions. This breakdown of hyperscaling results from a singular dependence of the thermodynamic functions on the scaling field associated with an irrelevant operator.

Vladimir L. Kuzmin, Fedor M. Kuni

Ryuzo Abe, S. Hikami

The specific heat exponent α of classical n-vector model is studied up to the order of 1/n. It is shown that α has discontinuities at particular spatial dimensionalities, both for the short-range and the long-range interactions. The scaling law relations involving α break down at such dimensionalities.

Gérard G. Emch, Chuang Liu

J.A.C. Loodts, Jean-Louis Colot, R. Brout

Sangwoong Park, Bongsoo Kim, Sung Jong Lee

We find a breakdown of the critical dynamic scaling in the coarsening dynamics of an antiferromagnetic $XY$ model on the kagome lattice when the system is quenched from disordered states into the Kosterlitz-Thouless phases at low temperatures. There exist multiple growing length scales; the length scales of the average separation between fractional vortices are found to be not proportional to the length scales of the quasiordered domains. They are instead related through a nontrivial power-law relation. The length scale of the quasiordered domains (as determined from the optimal collapse of the correlation functions for the order parameter $\text{exp}[3i\ensuremath{\theta}(r)]$) does not follow a simple power-law growth but exhibits an anomalous growth with time-dependent effective growth exponent. The breakdown of the critical dynamic scaling is accompanied by unusual relaxation dynamics in the decay of the fractional $(3\ensuremath{\theta})$ vortices, where the decay of the vortex numbers is characterized by an exponential function of the logarithmic powers in time.