Transverse Magnetic Field Coupling Effects on the Topology of Phase Diagrams of 2-D Ising Model

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1 2015 IJRET Volume 1 Issue 1 Prnt IN : Onlne IN : Themed ecton: Engneerng Technology Transverse Magnetc Feld Couplng Effects on the Topology of Phase Dagrams of 2-D Isng Model Bayor Jude mons *1, Baohua Teng 2, Lngl Wang 3 * 1,2,3 chool of Physcal Electroncs, Unversty of Electronc cence Technology of Chna, Chengdu, chuan, P.R. Chna ABTRACT The response behavor of the topology of varous phase dagrams correspondng to dfferent physcochemcal systems has been studed. Our study reles on usng the 2-D transverse feld sng model consderng only dstnct nearest neghbor par-exchange nteracton. We calculate phase dagrams n the polaraton-temperature phase space wth the ad of the Mathcad computer software. The theory s based on the smple temperature dependence of the potental parameters n the TIM wthn the framework of the mean feld theory approxmaton. We developed the more complex reentrant phase dagram that s common wth some bnary fluds, collods, protens many more systems. And by parameter modfcatons we obtaned some common exotc phase dagrams that correspondng to those obtaned both expermentally theoretcally. A qualtatve analyss of the topologcal nfluence of the transverse magnetc feld couplng s dscussed. Keywords: exchange nteracton, order-dsorder, phase dagram, topology, transverse magnetc feld, transverse Isng model (TIM). I. INTRODUCTION tudes of the phase dagrams assocated wth mcroscopc cooperatve nteracton wthn some physco-chemcal systems have offered useful nsght to the understng of the macroscopc phenomena that are characterstc of them. Indeed a wde number of systems dsplay phase transtons when the temperature or some other potental parameter s altered. Phase transtons are commonly observable n daly lfe as when water freees to form ce or n experments where a bar magnet loses ts magnetsm at hgh temperatures. But more complex phase transtons can be observed n optcal lattces n the more complex Mott nsulatorsuperflud phase transton [1]. Ths ntrgung phenomena has therefore courted lots of research work to underst ts propertes. Hence lots of research work on phase transtons at fnte temperature [2-8] has been carred out. The Lau-Gnsburg theory, the renormalaton group the symmetry breakng concepts [9-12] have been nstrumental n explanng phase transton propertes n systems. In ths paper we present theoretcal results dscussons on the unque nfluence of transverse magnetc feld on the phase transton propertes systems. Our focus s based on the 2-D sng model wth the transverse feld couplngs to the energy of ndvdual molecules at the lattce stes wth only nearest neghbour exchange nteracton consderatons. Qute often when questons are posed about the phase transton propertes of any system t stems from how phase changes are affected by the exchange nteracton energy the effect on systems n a ero external magnetc feld. However the magnetc feld nfluence s very crtcal on the Hamltonan energy of a system from the pont of vew of the TIM, thus affects the mcroscopc property changes the topologes of phase dagrams. everal researchers have carred out studes on the effect of the transverse feld on the spn usng several dfferent IJRET Receved: 1 Jan 2015 Accepted: 12 Jan 2015 January-February 2015 [(1)1: ] 117

2 approaches. For nstance, Jang et al. [13] have carred out the study of the spn-1 TIM on a honeycomb lattce wth a longtudnal crystal feld dscovered the presence of trcrtcal ponts whereby phase transtons change from second-order to frst order. By the use of the effectve feld theory (EFT) probablty dstrbuton technque, Htoutou et al.[14] dd an nvestgaton on the nfluence of the magnetc feld on the phase dagrams of a ste dluted spn-1 TIM on a square lattce. In all these studes the authors have reported the strong nfluence on the re-entrant behavour of the transverse feld. The longtudnal magnetc feld effect on the phase transtons n spn-3/2 spn-2 TIM has also been studed by way of both honeycomb square lattces usng the EFT wth correlatons[15,16]. Recently, Yussuf et al [17], studed the phase dagrams of the spn-1 TIM wth a longtudnal crystal feld n the presence of a longtudnal magnetc feld on a honeycomb lattce wthn the framework of the IEFT approxmaton. Our paper s concerned wth the nfluence of the transverse magnetc feld on the topology of the phase dagrams obtaned for the re-entrant behavour exhbted by some bnary flud mxtures, collods, protens etc. By alterng the mcroscopc potental feld parameters for our model (TIM), we agan obtan some phase dagrams that correspond to dfferent systems commonly obtaned n experments n theoretcal calculatons. These phase dagrams, thus reveal the ntrcate effect of the transverse feld couplng force on systems. II. METHOD AND MATERIAL In our model (TIM) we consder a two-dmensonal square lattce whch has N lattce stes. At the lattce stes we have the atoms or molecules of magnetc systems whch can ether have a spn up or spn down set of orentatons. In the case of bnary fluds these stes can be occuped by ether atoms or molecules of the mxture of two dfferent fluds. Whatever the case maybe of magnetc systems or bnary flud systems, the model features are the same therefore conssts of a lattce of lattce ste varables wth two characterstc propertes: (1) Each lattce ste ndependently takes on ether the value +1 or -1; (2) Interacton s between only pars of nearestneghborng spns. The sng model has a venerable tradton of beng effcent n the calculatons of the phase dagrams for complex fluds [18-22] wth reentrant phase behavour. The Hamltonan of the 2-D model wth the transverse feld gven as follows [18-22]: H J Where x x j, j (1) are the x- -components of a pseudospn-1/2 operator at ste n the lattce,, j the sum over only dstnct nearest-neghbourng pars. Ω s the transverse feld J s the exchange nteracton constant between nearest neghbour spns. Wthn the framework of the mean-feld theory, the component of the pseudospn can be wrtten as 2 tanh 2k T (2) Where And The order parameter of our model s the ensemble average of the pseudo-spn s. Mathematcally, any sngularty or loss n analytcty of ths order parameter at any fnte temperature means a change n phase. Ths fnte temperature then defnes the crtcal transton temperature for the system. Bascally ths order parameter of the system whch s the ensemble average of the pseudo-spn system from order ( J 0 0 descrbes the transton of the 0) to dsorder ( =0) state [22] by our graphs. For convenence sake we shall subsequently use n place of graphs for the phase dagrams. n labellng the It s known from expermental research carred out wth complex fluds, that temperature changes n systems may have a drect correlaton wth the feld potental parameters of the exchange nteracton the transverse feld [23-24]. Thus Camp Krvne [25] obtaned closed-loop shaped phase dagrams descrbed the reentrant phase behavor of complex fluds from ths concept. B Internatonal Journal of centfc Research n cence, Engneerng Technology (jsrset.com) 118

3 Dwellng on ths concept as well, mons et al [22] usng transverse Isng model assumng that the effectve exchange spn effectve transverse feld parameters J respectvely depends drectly on temperature. The proceeded further to obtan the followng temperature-dependent relatons: J n T J0 T0 T m 0 T0 (3) where T 0 are arbtrary constant. Here the parameters J 0, 0 ktare B reduced by B 0 notated stll as J 0, 0 t. kt, smply are We proceed further to obtan the phase dagrams by solvng eqn (2) wth the substtuton of eqns (1) (3). We therefore calculate graphs wth polaraton-as a functon of J 0,n,Ω 0 m versus effectve temperature. system s completely susceptble to beng n the dsordered state. As temperature s ncreased, for ntermedate temperatures, the system has the lkelhood of becomng ordered. For these ntermedate temperatures, the bgger the value of Ω 0, the smaller the ordered phase regon. Hence the polaraton temperature range for whch the ordered phase s stable becomes smaller dmnshes as Ω 0 s ncreased. At hgher temperatures the system once agan becomes dsordered for all polaraton ranges. One notceable trend about ths fgure s the egg-shaped profle. As Ω 0 s decreased ths egg-shape profle s beng enhanced. As Ω 0 s ncreased however, the profle s dstorted from egg-shape appears as oval shape untl ncompletely dmnshes. III. REULT AND DICUION The Phase dagrams In ths secton we obtan the phase dagrams n polaraton-temperature phase space. Graphs are plotted wth the polaratons as a functon of varous parameters of (J 0, n, Ω 0 m) vs. the effectve temperature t. Here J 0 s the effectve exchange nteracton, n s the temperature exponent for J 0, Ω 0 s the effectve transverse magnetc feld couplng whle m s the temperature exponent for the transverse feld. Ω 0. By varyng the effectve transverse feld parameter Ω 0, we can observe the effect t has on the phase transton dagram. Ths secton therefore examnes the nfluence of the transverse magnetc feld parameter Ω 0 over the topologes of systems of constant J 0, n m whle varyng Ω 0 wthn each fgure. The trends that evolve from the polaraton-temperature graphs gve us a qualtatve understng of the couplng behavour of Ω 0. Fgure 1: Phase transton dagram showng the polaraton aganst temperature t for varous effectve transverse feld parameter Ω 0 for a system where J 0 =1.6, n=1.6, m=2.0. Fg. 2 shows common features as fg. 1. They all show the phenomenon of re-entrant behavour. For fg. 2 the model s that for systems wth J 0 =1.1, n=1.6 m=2.0. When J 0 was reduced, t was observed that Ω 0 had to be correspondngly reduced n order to mmc the re-entrant behavour. All other propertes assocated wth fg. 1 are same wth fg.2. And t corroborates the nfluence of Ω 0 beng an nveterate modfer of the shape profle by skewng t n towards egg shaped when t s smaller. Fgure 1 gve the polaraton-temperature phase dagram for systems wth J 0 =1.6, n=1.6 m=2.0. Ths system shows the phenomenon of the re-entrant behavour assocated wth some complex fluds, bnary mxtures, collods, protens etc. For these systems, at a narrow range of low temperature close to ero, the Internatonal Journal of centfc Research n cence, Engneerng Technology (jsrset.com) 119

4 progressvely decreases. At hgh temperatures, the system s n the dsordered state. As Ω 0 s progressvely decreased, the regon for the ordered phase ncreases whle that for dsordered phase decreases. Also as Ω 0 s decreased one can observe appearance of the shape of jet-plane nose protuberance fashonng out. Fgure 2: Phase transton dagram showng the polaraton aganst temperature t for varous effectve transverse feld parameter Ω0 for a system where J0=1.1, n=1.6, m=2.0. Fg. 3 shows the phase dagrams for systems wth J 0 =0.9, n=1.6 m=1.0. These graphs have the U shaped profle that s commonly symptomatc wth ferromagnetc to paramagnetc Cure temperature phase transtons. Thus at lower temperatures, the system s n the ordered state. As temperature ncreases, the system s stable n the dsordered state. It can be observed that the smaller the value of Ω 0 the smaller the regon of the ordered phase. Also the range of polaratons for transtons can occur s larger when Ω 0 s smaller. As Ω 0 gets bgger, the regon for the dsordered state gets smaller whle ordered phase regon gets larger. Fgure 3: Phase transton dagram showng the polaraton aganst temperature t for varous effectve transverse feld parameter Ω 0 for a system where J 0 =0.9, n=1.6, m=1.0. Fg. 4 shows the phase dagrams for systems wth J 0 =1.6, n=0.8 m=2.0. These are commonly known as the reverse U-shapes. Here also we have the possblty two dstnct phase transtons. At lower temperatures, the system s stable n the ordered phase. However as temperature s gradually ncreased, the range polaraton of transtons to the dsordered phase also Fgure 4: Phase transton dagram showng the polaraton aganst temperature t for varous effectve transverse feld parameter Ω 0 for a system where J 0 =1.6, n=0.8, m=2.0. IV. CONCLUION Ths paper studes the couplng effect of the transverse feld parameter Ω0 on the Hamltonan of the Isng model. It develops a theory on smple temperature dependent relatonshp between the potental feld parameters n the model ths s used to calculate phase dagrams n the polaraton-temperature phase space. The topology of these dagrams then reveal a qualtatve depcton of the nfluence of the transverse feld couplng effect by smple parameter modfcaton usng the Mathcad software. Frst of all the graphs obtaned were categored accordng to the varous shapes of phase dagrams that have been observed for systems both theoretcally expermentally. Our calculaton was able to model the more complex phase dagram of the reentrant egg-shaped closed loop phase behavor n the frst category, other phase dagrams were obtaned by parameter modfcatons as well. uch as f the frst category systems. Other exotc shapes such as U shape the reverse U-shape were obtaned analyed. Our results bascally show that there s a drect nfluence of the transverse magnetc feld on all systems that exhbt phase transtons. Frst of all, for all these systems, as the exchange nteracton energy s decreased, Internatonal Journal of centfc Research n cence, Engneerng Technology (jsrset.com) 120

5 there s a correspondng decrease n the transverse feld couplng energy n order for system to showcase phase transtons between phases. Also the transverse feld has a sharp nfluence on the egg-shaped protuberance commonly found n systems wth reappearng phases as well as the jet-plane nose shape of some systems. It s observed that the regon for order or dsorder n systems dramatcally changes when the transverse magnetc feld couplng parameter s altered. As a result, the range of polaraton temperatures for whch phase transtons occur s also affected wth changes n the transverse feld couplng parameter nfluence. Fnally, as the transverse magnetc feld nfluence s ncreased, the systems begn to lose the property of havng any phase transtons so only sngle phases (of ether ordered or dsordered) subssts. Thus ths study has succeeded n explanng some trends observable on phase dagrams. Ths s mportant because t gves another angle to the mportant nfluence of the transverse feld parameter, whch s an mportant component of the Hamltonan of the Isng model. Thus any cursory consderaton on any expermental or theoretcal phase dagram, wll reveal the bengn nfluence of the transverse feld couplng constant so a qualtatve descrpton n these terms can be made. We recommend that future work be carred on the combned effect of the temperature exponents m n. [12] G. Pars, tatstcal Feld theory (Addson-Wesley, Readng, Massachusetts, 1988) [13] X.F. Jang, J.L. L, J.L. Zhong, C.Z. Yang, Phys. Rev. B 47 (1993) 827. [14] K. Htoutou, A. Oubelkacem, A. Anane, M. aber, J. Magn. Magn. Mater. 288(2005) 259. [15] W. Jang, L.Q. Guo, G. We, A. Du, Physca B 307 (2001) 15. [16] W. Jang, G. We, Z.H. Xn Phys. tatus old B 225 pp215. [17] Y. Yuksel H. Polat Journal of Magnetsm Magnetc Materals 322. pp [18] C. L. Wang, W. L. Zhong, P. L. Zhang. J. Phys.: Condens. Matter, 3 (1992) [19] T. Kaneyosh. Physca A, 293 (2001) 200 [20] J. M. Wesselnowa. old tate Comm., 121 (2002) 489. [21] A. aber,. Lo Russo, G. Matte, A Matton. JMMM, 251 (2002) 129 [22] B. J. mons, B. H. Teng,. Zhou, L. Zhou, X. Chen, M. Wu H. Fu. Chem. Phys. Lett., (2014) 121 [23]. H. Chen, J. Rouch, F. cortno, P. Tartagla Chen. J. Phys.: Condens. Matter, 6 (1994) [24] L. A. Daves, G. Jackson, L. F. Rull. Phys. Rev. Lett., 82 (1999) [25] X. Camp H. Krvne. Europhys. Lett., 66 (4) (2004) 527. V. REFERENCE [1] M. Grener, O. Mel, T. Esslnger, T. W. Hansch, I. Bloch, Nature (London) 415, 39 (2002). [2] H. E. tanley, Introducton to Phase Transtons Crtcal Phenomena (Oxford Unversty Press, Oxford, 1987). [3] P. Chakn T. Lubensky, Prncples of Condensed Matter Physcs (Cambrdge Unversty Press, Cambrdge, 1995). [4] N. Goldenfeld, Lectures on Phase transtons renormalaton group (Addson-Wesley, Readng, Massacusetts, 1992). [5] J. Cardy, calng Renormalaton n tatstcal Physcs (Cambrdge Unversty Press, Cambrdge, 1996). [6]. K. Ma, Modern theory of crtcal phenomena (Benjamn, Readng, Massachusetts, 1976). [7] G. Mussardo, tatstcal Feld Theory (Oxford Unversty Press, Oxford, 2010). [8] H. Nshmor G. Ort, Elements of Phase Transtons Crtcal Phenomena (Oxford Unversty Press, Oxford, 2010) [9] D. J. Amt, Feld theory, renormalaton group crtcal phenomena (World centfc, ngapore, 1984). [10] J. Jnn-Zustn, Quantum feld theory crtcal phenomena (Clarendon Press, Oxford, 1989). [11] K. G. Wlson J. B. Kogut, Physcs Reports 12C, 75 (1974). Internatonal Journal of centfc Research n cence, Engneerng Technology (jsrset.com) 121