Goldstone singularities and critical behaviour in isotropic systems

Abstract

The critical behaviour along the coexistence curve of isotropic ferromagnets is characterized by two diverging correlation lengths since longitudinal and transverse fluctuations of the order parameter become critical. Therefore, a critical theory valid for the whole phase diagram has to incorporate the crossover between a region characterized by one correlation length to another region where two different correlation lengths dominate. Treating this crossover problem entirely within the framework of the trajectory integral method. The author has calculated the equation of state and the susceptibilities. It is shown that coexistence behaviour is governed by a coexistence fixed point which is characterized by the vanishing interaction among the Goldstone modes. Taking the coupling of the critical transverse modes to the longitudinal modes into account, it is shown that coexistence behaviour is characterized by a Fisher renormalization. This central result sheds new light on the symmetry-broken phase in isotropic systems. The asymptotic forms of the equation of state and the susceptibilities at the coexistence curve are expressed by the specific heat exponent alpha t=1/2 for d>or=3 and for all spin dimensions. These results are in perfect agreement with the nonlinear sigma -model. The crossover from coexistence curve behaviour to ordinary critical behaviour is calculated for the equation of state and for the susceptibilities to O( in )( in =4-d).