Ota, S B

Ota, Smita

IACS Institutional Repository

Indian J. Phys. 72A (5), 413-416 (1998)I J P  A— an international journalInhom ogeneity o f vortices in 2d classical X Y -m odel: a m icrocanonical M onte Carlo sim ulation studyS B Ota and Smita Ota Institute of Physics, Sachivalaya Marg, Bhubaneswar-751 005, IndiaAbstract : The extended 2d classical XY-model has been studied using microcanonical Monte Carlo simulations. Simulations have been carried out on 30 x 30 spin system on a square lattice. We find that the maximum inhomogeneity of vortex distribution occurs at a temperature which coincides with the position of specific heat peak in the Kosterlitz-Thouless (KT) case and in the coexistence region in the first order case. The inhomogeneity is found to be more in the KT case as compared to the first order one.Keywords : XY-model, vortices, microcanonicalPACS Nos. : 75.10 Hk, 02.70.Lq, 05.20.-y, 05.70.FhStudy of the two dimensional (2d) classical XY-model has unfolded several interesting physical properties of 2d system [1] and still demands further investigations. The 2d XY-model has been considered in the literature to understand the high temperature superconductors (HTSC), which is however not completely successful [2]. The experimental data can lead to interesting information on HTSC if one knows the right extension of the XY-model that explains them [3]. Thus, there is a need to understand the nature of the vortex-driven transition in 2d XY-model. The lack of long range order, the presence of topological defects called vortices, and the Kosterlitz-Thouless (KT) transition are some of its notable properties known as yet [4-21]. The two types of excitations that dominate at low temperatures are spin waves and vortices. The low- temperature phase has only bound vortex-antivortex pairs and the KT transition is associated with the unbinding of the vortex-antivortex pairs. Investigations have been carried out for. the possibility of a first order transition in this model, without disturbing the essential symmetry. Domany, Schick and Swendsen [22] suggested that by sufficiently reducing the width of the nearest neighbour interaction potential in this model a first order transition can be observed. The first order transition is understood to result as due to<£> 1 9 9 8 IA CS/ of vortices. In this paper, we report theMe classical 2d XY-system, which has sofarid  classical XY-model is given by :( 1)al meaning. For p2 = 1; the Hamiltonian reduces to the /iich admits the KT transition. By increasing the value of ide narrower and for p2 > 10 the transition becomes first-W e,spins 123,24]. \Vvcanonical MC simulations on a square lattice having 30 x 30 periodic boundary conditions and have calculated systemtemperature (7), vorticity, magnetization square and their respective standard deviations at each given total energy (£). We used 1 x 105 MCSS for equlibration and 1 x 105 MCSS lor averaging. The accuracy of the mean value of the physical quantities was estimated by performing block averages consisting of 5 x 103 MCSS each and then finding the standard deviation of block averages. The KT transition temperature occurs at 0.9 and the maximum of temperature dependence of specific heat occurs at 1.09.We analysed the Monte Carlo configurations evolved during the simulation to study the inhomogeneity of the vortices across the transition. To this end we estimated tfie average number (V (^r)) of vortices/antivortices at a distance r from any given vortex. From this we obtained the number of positive (negative) vortices (Vq+ (V^)) surrounding a positive vortex within a distance of V2a and l^ jla  (a is the lattice spacing). Forp2 = 1, n is seen that V^_, when determined over a distance of ^Jla , decreases with temperature initially upto 7=1.1 and then increases with further increase in temperature. The decrease of Vq_ with temperature is not observed, when determined over a larger distance of l^ lu  Whereas, shows a steady increase with increase in temperature. On the other hand, lor p2 = 50, both V ^  and V^ are seen to increase with increase in energy even when determined over a smaller distance of V2a [25].We next determined the net vortex charge (Vqt) within a distance of V2fl and 2y[2a from a vortex as functions of 7  (or E). Figure 1(a) shows the temperature dependence of VqJ for p2 = I, and comments on some features of this graph are in order. Firstly, Vqt goes through a maximum. Secondly, the magnitude of Vqt is reduced as the distance increases. Finally, the maximum occurs at 7 =  1.1, which corresponds to the specific heat peak. Similar graph is shown in Figure 1(b), forp2 = 50. The behaviour is qualitatively similar to the case for p2 -  1, except for the following differences. The magnitude of Vqf is comparatively smaller. The maximum of Vqt occurs in the coexistence region of the first-order transition. The vortices are therefore, not distributed homogeneously in the lattice. Examination of the configurations revealed the presence ofInhomogeneity o f vortices in 2d classical XY-model etc 415cortex clusters in the critical region, which has also been pointed out by Toboknik and Chester [13].Figure 1. (a) The temperature dependence of net voitex charge Vql in a radius of V2a (+) and 2yfla (x) for p2 = l . (b) The energy dependence of net vortex charge Vqt in a radius of V2a (+) and 2 Via (x) for p2 = 50. The data represents averages over 1 x 105 MCSS.We have not yet come across an explanation of the observed inhomogeneity of vortices. However, certain features can be understood as follows. In the low temperature (energy) insulating phase the vortices are bound tightly which results in a small value of V, In the high temperature (energy) Debye-Hiickel regime, Vqt is also small due to the presence of large number of free charges in the liquid phase. We speculate that clusters of vortices in the critical region are responsible for the peak in Vqt. The difference between p2 = 50 and 1 cases can be attributed td the change in interaction to m  ln(r fa) to r fa as p2 changes from 1 to 50 [3].In conclusion, we have studied the vortices in the classical 2d xV^odel undergoingKT and first-order transition. We have reported a subtle aspect of the vortices* that is the mhoniogeneily of vortex distribution.AcknowledgmentsSBO acknowledges discussions with Dr. A BaUmgartner. The authors thank Dr. V C Sahni lor helpful suggestions on the manuscript. So acknowledges the Council of Scientific & Industrial Research, India, for financial assistance.Referencesl11 For a review, see M N Barber Phys Rep, (Netherlands) 59 376 ( 1980)[^ 1 LJ de Jongh Solid State Commun. 70 955 (1989)M F Mila Phys. Rev, B47 442 (1993)I4] N D Mermin and H Wagner Phys. Rev. Lett. 17 1 133 (1966)W V L Berezinskii Sov. Phys. JETP 32 493 (1970)I6) J M Koatfctitz md D J Thoolesa J Phys. Cl L124 (1972); J. Phys. CIO 1181 (1973); J M Korterlnz'  Phys. C7 1046 (1974)72A(5)-ll416 S B Ota and Smita Ota[71 F J Wegner Z Phys. 206 465 (1967)[8] V L Berezinskii Sov. Phys. JETP 34 610 (1971)[9] J Zittartz Z  Phys. 23B 55,63 (1976)[10] J V Jose, L P Kadanoff, S Kirkpatrick and D J Nelson Phys. Rev. B16 1217 (1977)[11) C Kawabata and K Binder Solid State Commun. 22 705 (1977)[ 12] S Miyashita, H Nishimori, A Kuroda and M Suzuki Prog. Theor. Phys. 60 1669 (1978)[13] J Tobochnik and G V Chester Phys. Rev. B20 3761 (1979)[14] W J Shugard et al Phys. Rev. B2I 5209 (1980)[15] J E Van Himbergen and S Chakravorty Phys. Rev. B23 359 (1984)[16] H Betsuyaku Physica A106 3 11 (1981)[17] J F Fernandes, M F Ferreira and J Stankiewicz Phys. Rev. B34 292 (1986)[18] R Gupta, J Delapp, G G Batrouni, G C Fox. C F Baillie tind J Apostolakis Phys. Rev. Lett. 61 1996 (1988)[19] R Gupta and C F Baillie Phys. Rev. B45 2883 (1992)[20] U Wolff Nucl. Phys. B322 759 (1989)[21] J Kogut and J Polonyi Nucl. Phys. B265 [FS15] 3 13 (1986)[22] E Domany, M Schick and R FI Swendsen Phys. Rev. Lett. 52 1535 (1984)[231 M Creutz Phvs. Rev. Lett. 50 14 11 (1983)[24] S Ota, S B Ota and M F&hnlc J Phys. : Condens. Matter 4 5411 (1992)[25] The energy is used instead of temperature to present the points clearly in rhe coexistence region of the first order iransition.

Inhomogeneity of Vortices in 2D Classical XY-Model: A Microcanonical Monte Carlo Simulation Study

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/5323/1/Inhomogeneity%20of%20Vortices%20in%202D_By%20S%20B%20Ota.pdf

Inhomogeneity of Vortices in 2D Classical XY-Model: A Microcanonical Monte Carlo Simulation Study

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository