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The dynamic critical exponent $z$ is determined numerically for the$d$-dimensional XY model ($d=2, 3$, and 4) subject to relaxational dynamics andresistively shunted junction dynamics. We investigate both the equilibriumfluctuation and the relaxation behavior from nonequilibrium towardsequilibrium, using the finite-size scaling method. The resulting values of $z$are shown to depend on the boundary conditions used, the periodic boundarycondition, and fluctuating twist boundary condition (FTBC), which implies thatthe different treatments of the boundary in some cases give rise to differentcritical dynamics. It is also found that the equilibrium scaling and theapproach to equilibrium scaling for the the same boundary condition do notalways give the same value of $z$. The FTBC in conjunction with the finite-sizescaling of the linear resistance for both type of dynamics yields values of $z$consistent with expectations for superfluids and superconductors: $z = 2$, 3/2,and 2 for $d=2$, 3, and 4, respectively.Comment: 21 pages, 16 figures, final versio

physical review. b, condensed matter

Dynamic critical exponent of two-, three-, and four-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>XY</mml:mi></mml:math>models with relaxational and resistively shunted junction dynamics

Dynamic critical exponent of two-, three-, and four-dimensionalXYmodels with relaxational and resistively shunted junction dynamics