* Corresponding author. E-mail address: kim@tp.umu.se (B.J. Kim) Physica B 284 } 288 (2000) 413}414 Dynamic critical exponent of three-dimensional X> model Beom Jun Kim*, Lars Melwyn Jensen, Petter Minnhagen Department of Theoretical Physics, Umea> University, 901 87 Umea> , Sweden Abstract The dynamic critical behaviors are determined for the three-dimensional X> model with resistively shunted junction (RSJ) dynamics and time dependent Ginzburg}Landau (TDGL) dynamics. A short-time relaxation method is employed and turned into a standard "nite-size scaling from which a precise determination of the critical temperature as well as the dynamic critical exponent can be made. The RSJ and TDGL dynamics are shown to have di!erent dynamic critical behaviors. Comparisons with earlier works are made and discussed. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Scaling relations; Short-time relaxation; X> model Although there have been well-established consensus on thermodynamic properties of two- and three-dimen- sional (3D) X> models [1,2], there still remain unsolved questions on dynamic behaviors such as current}voltage characteristics, the form of complex conductivity, the #ux noise spectrum, and the dynamic critical exponent z (see Refs. [3,4] and references therein). The dynamics of 3D X> model also draws strong attention since it can be related to superconductors and super#uids. We use the resistively shunted junction (RSJ) and the time-dependent Ginzburg}Landau (TDGL) dynamics with the #uctuating twist boundary condition to investi- gate numerically the dynamic universality classes of these two dynamic versions of the 3D X> model (see Refs. [3,5] for details of numerical method used here). We compute z in two independent ways (z R and z t , respec- tively): from the scaling of resistance R, and from the short-time relaxation of the order parameter. From the "nite-size scaling using R(¹, ¸)"¸~z R o[(¹!¹ # )¸1@l] [6] RSJ and TDGL are found to have the same z R (+1.5) as well as the same critical temperature ¹ # +2.20 and the critical exponent l+0.67 [5]. In this paper, we focus on the short-time relaxation of tI (t),Ssign[+ i cos h i (t)]T [7,8] starting from the ordered state where all phase angles h i (t"0)"0. The advant- age of using this quantity is that the "nite-size scaling function has a simple form tI (t, ¹, ¸)"f [t¸~z t ,(¹!¹ # )¸1@l], (1) which implies that curves for di!erent system sizes ¸ should collapse onto a single curve at ¹"¹ # since tI (t, ¹ # , ¸)"f (t¸~zt ,0) with the scaling variable t¸~zt . From this method with ¹ # "2.20, z t "1.50(5) and 2.0(1) are found for RSJ and TDGL, respectively [5]. Furthermore, if the "rst argument of the scaling func- tion is "xed to a, i.e., t¸~z t "a, then tI for various ¸ should have a single crossing point at ¹"¹ # , as shown in insets of Fig. 1, where we used a"4.0 for RSJ (z t "1.5) and 0.5 for TDGL (z t "2.0). This intersection method yields ¹ # +2.200 for RSJ and 2.194 for TDGL, respectively, and the relatively larger discrepancy in the value of ¹ # for TDGL is due to the error from the discrete time step in numerical integration. From the obtained value of ¹ # and the known value of the critical exponent l"0.670, we then collapse all data points in insets of Fig. 1 to a single smooth curve by using tI "f [a,(¹!¹ # )¸1@l] with scaling variable (¹!¹ # )¸1@l, as shown in Fig. 1. In conclusion, RSJ and TDGL are found to have di!erent dynamic critical exponent in the short-time re- laxation method: z t "1.5 and 2.0, respectively. The equations of motion of TDGL are of purely relaxational 0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 9 8 1 - X