2011-2012 > Academic research: refereed > Refereed conference paperVersion of RecordPublishe

Huang, H

Lo, VC

Wu, RPH

2011-2012 > Academic research: refereed > Refereed conference paperVersion of RecordPublishedVoR allowe

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Critical Behavior of the Random-Bond Clock Model Raymond P. H. Wu, Veng-cheong Lo, and Haitao Huang Department of Applied Physics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Abstract. The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the 8-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean 1   and different values of variance: from 2 0.1   to 2 3.0  . An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. Our results show that, a small amount of disorder (small ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type. Keywords: random-bond clock model, Kosterlitz-Thouless transition, Gaussian distribution PACS: 75.10.Hk, 75.10.Nr, 75.30.Hx, 75.40.Mg INTRODUCTION Unlike the phase transitions manifested in most spin models, Kosterlitz-Thouless (KT) transition is a specific phase transition observed in the critical behavior of the superfluid systems and it can be described by the two-dimensional XY model [1-3]. On the other hand, the q-state clock model is a discrete version of the XY model. It is expected to have various critical behavior under different q values. Extensive studies [4-10] on the clock model had shown that, for 4q  , the phase transition is Ising-like, and for 6q  , it is XY-like. The presence of defects interrupts the periodic structure of crystalline materials and the systems become disordered when the quantity of interruptions is large. It can be visualized by a random distribution of coupling coefficients between neighboring spins. In statistical point of view, it is natural to consider a Gaussian distribution for the coupling coefficients. It is our motivation to study the effect of disorder on the phase transition in the clock model. Since the critical behavior for the q-state clock model does not change appreciably on varying q values when q is large. For simplicity, we can use the random-bond 8-state clock model ( 8q  ) to study the KT transition. 2nd International Advances in Applied Physics and Materials Science CongressAIP Conf. Proc. 1476, 253-256 (2012); doi: 10.1063/1.4751605©   2012 American Institute of Physics 978-0-7354-1085-5/$30.00253MODEL We consider a system of spins on a square lattice with size N L L  . In the q-state clock model, the spins are confined on a plane with q different states each of which is specified by a phase angle  	2n n q  , where 0,1,2,..., 1n q  . The spin state at each site can be defined by the spin vector  	sin ,cosn n n S . The Hamiltonian is given by  ij i jijH J   S S , (1) For simplicity, the summation is restricted to the nearest neighbors only. The coupling coefficients ijJ  are assumed to follow the Gaussian distribution  2221 ( )( ) exp22JP J     , (2) where 1   is the mean and 2  is the variance of the distribution. The bond-randomness is reflected by the parameter  , in particular, 2 0   represents a pure system with no disorder and with constant coupling coefficient 1J  . The energy per spin E of the system is given by  1 ij i jijE JN   S S , (3) where    denoted the ensemble average of the quantities. The magnetization per spin m  is given by  1 1sin , cosi ii iN N     m , (4) and its magnitude is denoted by m. Furthermore, the susceptibility per spin   can be obtained from the fluctuations of m and is given by   	2 2BN m mk T       . (5) METHODS The helicity modulus per spin   is a measure of the resistance to an infinitesimal spin twist   across the system along one direction. It is given by  212NE sT        , (6) where sin( )( )ij i j ijijs J N     e x  and ije  is a unit vector pointing from lattice site i to j. The abrupt jump in the helicity modulus from the finite value (2 ) B ck T  to zero at the critical temperature cT  in the thermodynamic limit at the critical temperature is the key feature of the KT transition [2]. The fourth-order helicity modulus can be expressed as 254 22 2 2 2 2 24 3 21 3 3 4 3 32 4 4N NE E E s s E sN T T T T T    !      !   !   !   . (7) Supposed 4  is negative at the transition, then   cannot approach zero continuously but must make a discontinuous jump toward zero at the transition instead [11]. Thus, we can identify the KT transition. RESULTS AND DISCUSSION The magnetization and susceptibility against temperature for the random-bond 8-state clock model with various   are given in Fig. 1 In the pure case ( 2 0  ), the system undergoes two transitions at two critical temperatures instead of a single one. There exists an intermediate phase called the KT phase (or the massless phase) between the low-temperature ordered phase and the high-temperature disordered phase. The results for the susceptibility show the double-peak feature of the KT transition. As   increases, the critical temperature of the system decreases and the distance between two peaks in the susceptibility graph decreases. The results show the shrinkage of the intermediate phase as the amount of disorder increases.   (a)                                                                              (b) FIGURE 1.  (a) The magnetization (b) The susceptibility against temperature for the random-bond 8-state clock model with lattice size 128 128N    and various  : 0, 0.1, 0.5, 1.0, 2.0, and 3.0.  In order to demonstrate the disappearance of the KT transition under a large amount of disorder, we have calculated the fourth-order helicity modulus of the systems and the results are given in Fig. 2. For 2 0   and 2 0.5  , the fourth-order helicity modulus at the critical temperature are clearly negative, implying the system undergoes the KT transition. However, despite the noisy nature, the depth of the trough of the fourth-order helicity modulus reduces as   increases. For 2 2.0  , the trough is no longer observable and the helicity modulus does not jump discontinuously. Hence, we conclude that the transition is no longer of the KT-type and this is consistent with our above results.  255 FIGURE 2.  The fourth-order helicity modulus against temperature for the random-bond 8-state clock model with lattice size 128 128N    and various  : 0, 0.5, and 2.0.  Our results show that, a small amount of disorder (small  ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type. ACKNOWLEDGMENTS This work is financially supported by the research grant from The Hong Kong Polytechnic University under research grant No. RPRK. REFERENCES 1. J. M. Kosterlitz, and D. J. Thouless, J. Phys. C 6, 1181-1203 (1973). 2. J. M. Kosterlitz, J. Phys. C 7, 1046-1060 (1974). 3. D. R. Nelson, and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201-1205 (1977). 4. S. Miyashita, H. Nishimori, A. Kuroda, and M. Suzuki, Prog. Theor. Phys. 60, 1669-1685 (1978). 5. J. Tobochnik, Phys. Rev. B 26, 6201-6027 (1982). 6. M. S. S. Challa, and D. P. Landau, Phys. Rev. B 33, 437-443 (1986). 7. A. Yamagata, and I. Ono, J. Phys. A 24, 265-275 (1991). 8. Y. Tomita, and Y. Okabe, Phys. Rev. B 65, 184405 (2002). 9. S. K. Baek, P. Minnhagen, and B. J. Kim, Phys. Rev. E 81, 063101 (2010). 10. S. K. Baek, and P. Minnhagen, Phys. Rev. E 82, 031102 (2010). 11. P. Minnhagen, and B. J. Kim, Phys. Rev. B 67, 172509 (2003). 256Copyright of AIP Conference Proceedings is the property of American Institute of Physics and its content maynot be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express writtenpermission. However, users may print, download, or email articles for individual use.

Critical behavior of the random-bond clock model

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