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We study the dynamic critical exponent from effective and microscopictheories. We employ a simple TDGL model, or model A in the classification ofHohenberg and Halperin, as an effective theory and the imaginary time formalismof the finite-temperature filed theory as a microscopic theory. Taking an O(N)scalar model as an example and carrying out the 1/N expansion up to the NLO inthe 1PI and 2PI effective actions, we compare the low-energy and low-momentumbehavior of the response function in the effective theory and of the retardedGreen's function in the microscopic theory. At the NLO of the 1PI 1/N expansionthe low-energy and low-momentum behavior of the two-point function is very muchdifferent in the microscopic and effective theories: in the field theory it isdominated by the propagating mode while in model A it is dominated by thediffusive mode. Also, in the microscopic theory the dynamic critical exponent,z, depends on whether the kinematics is relativistic or nonrelativistic. Incontrast, at the NLO of the 2PI 1/N expansion the microscopic and effectivetheories are equivalent. They satisfy exactly the same Kadanoff-Baym equation.Also, whether the kinematics is relativistic or nonrelativistic in themicroscopic theory becomes irrelevant. This implies that the diffusive modewith z = 2 + O(1/N) is dominant at low energies and momenta even in themicroscopic theory at the NLO of the 2PI 1/N expansion, though we do notexplicitly solve the Kadanoff-Baym equation. We also try to improve thecalculation of the dynamic critical exponent of model A by incorporating thestatic 2PI NLO correlations. The obtained critical exponent is slightly smallerthan the previous result and its N dependence is also milder than the previousone.

Dynamic Critical Exponent from One- and Two-Particle Irreducible 1/N Expansions of Effective and Microscopic Theories