A microscopic calculation of dynamic critical exponents for displacive phase transitions

For anO(n)-isotropic lattice dynamicalQ⁴-model describing displacive phase transitions ind dimensions, we employ a microscopic 1/n-expansion in order to show that over-damped soft-phonon behavior emerges for frequencies smaller than those of the characteristic orderv_c=O(n^–x). This is concluded from the fact that the displacement propagatorD(q, v) assumes the time-dependent Ginzburg-Landau (TDGL) form with a damping coefficient=O(n^–x), whenv becomes smaller thanv_c. The exponentx is found to bex=4–d for 2<d<3,x=(d–1)/2 for 3<d<5, andx=2 ford>5. The dynamic critical exponents forv_c(q) and forD(0,v) are derived atT=T_c0 and toO(1/n). Their values are nontrivial for 2<d<4 and, within the TDGL-region, agree with the those appearing already for frequencies ofO(n⁰) in TDGL-models with nonconserved order parameter andO(n⁰)-damping coefficient. The latter case was studied by Halperin, Hohenberg, and Ma in 1972. Even in the TDGL-region, the energy conservation does not affect the dynamic exponents for largen(>2, since the specific heat is finite), but an energy diffusion singularity appears in theQ²-response function which is related to the basic quantity of the 1/n-method, the effective interactionU_eff. By an estimate of order we find that the damping coefficients resulting from the coupling between the relaxation modes contained inU_eff and the critical modes inD are of ordern^–w withw>x, such that the coupling between weakly damped critical modes is responsible for the crossover to the TDGL-behavior for largen. The exponentz=d/2, known to be generated by the coupling between order parameter and conservedO(n)-densities in TDGL-models, cannot be seen up to the order calculated. We also point out problems of a microscopic-expansion and comment upon differences between microscopic treatments for displacive transitions and those for the Bose condensation.