Spin waves in two and three dimensional magnetic materials. H. S. Wijesinhe, K. A. I. L. Wijewardena Gamalath

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1 Internatonal Letters of Chemstry, Physcs and Astronomy Onlne: ISSN: , Vol. 49, pp do: / ScPress Ltd., Swtzerland Spn waves n two and three dmensonal magnetc materals H. S. Wjesnhe, K. A. I. L. Wjewardena Gamalath Department of Physcs, Unversty of Colombo, Colombo3, Sr Lanka E-mal address: male@phys.cmb.ac.lk Keyword: Hesenberg model; spn waves; Suzuk Trotter decomposton method; smple cubc lattce; bcc lattce ABSTRACT The equatons of moton for the dynamc propertes of spn waves n three dmensons were obtaned usng Hesenberg model and solved for two and three dmensonal lattces analytcally up to an exponental operator representaton. The second order Suzuk Trotter decomposton method was extended to ncorporate second nearest nteracton parameters nto the numercal soluton. Computer based smulatons on systems n mcro canoncal ensembles n constant-energy states were used to check the applcablty of ths model for two dmensonal lattce as well as three dmensonal smple cubc and bcc lattces. In the magnon dsperson curves all or most of the spn wave components could be recognzed as peaks n the dynamc structure factor presentng the varaton of energy transfer wth respect to momentum transfer of spn waves. Second order Suzuk Trotter algorthm used conserved the energy. 1. INTRODUCTION Even though the spn s entrely a quantum mechancal concept, usng classcal models spn systems can be studed analytcally or numercally. Most classcal models lke sng model do not represent spn as three dmensonal vectors. But the classcal Hesenberg model uses the three dmensonal vector representatons for spns. Spn-dynamcs technque was used by Landau et.al [1] to study the crtcal and low-temperature magnetc exctatons usng smple classcal dynamcal spn model governed by equatons of moton. Accordng to the sotropc Hesenberg model, the total energy and net magnetzaton are good examples for conserved enttes. To nvestgate the exstence of spn waves n paramagnetc bcc ron, Tao et.al [2] used computer smulatons based on classcal Hesenberg model on a cubc lattce wth perodc boundary condtons and obtaned the equlbrum states by applyng Monte Carlo method. In most of the studes one level of nearest neghbours were coupled usng a sngle nteracton parameter or dmensons were lmted to two. To smulate a twodmensonal Hesenberg model wth ansotropy, Costa et.al [3] used Monte Carlo methods and molecular-dynamc technques. In the present work, a model based on Hesenberg model used by us to study the spn dynamcs of one dmensonal magnetc materal [4] was extended to observe the dynamc property of the formaton of spn waves n two dmensonal lattces as well as three dmensonal smple cubc and body centred cubc lattces of magnetc materals. The smulatons were carred out up to second nearest nteracton parameters. The systems were consdered to be n mcro canoncal ensembles n constant-energy states. To ntate the dynamc smulatons, the spn systems were brought nto thermal equlbrum wth heat baths of known temperature by usng Metropols algorthm whch s a manstay of Monte Carlo methods and statstcal smulatons. In the program a set of smulaton unts were used rather than SI unts where the Boltzmann constant was set to unty, nteracton parameter was absorbed nto smulaton tme, second nteracton parameter was defned relatvely to frst nteracton parameter, gyromagnetc factor was taken as unty and temperature was measured relatve to Cure temperature of each spn system. ScPress apples the CC-BY 4.0 lcense to works we publsh:

2 36 Volume THE DYNAMICS OF THE SPIN SYSTEM The tme evoluton of the probablty () t of a system n state at tme t s gven by the master equaton: d dt ( t) R( ) ( t) R( ) (1) where R( ) represents the transton rate from state to. The sum of the probabltes of all the states s one. For a canoncal ensemble, the equlbrum probabltes ρ μ ( ( t) ) at temperature T are gven by, 1 E / kt B e (2) Z where E μ s the energy assocated wth the state μ, k B s the Boltzmann constant and Z s the partton functon. Therefore the expectaton value of the thermodynamc varable Q s Q Q E Qe E e. (3) The condton for thermal equlbrum d dt 0 leads to an equaton that can be rewrtten n terms of selecton probablty g(μ ν) and acceptance rato A(μ ν). Assumng that there are M dscrete states n the phase space of the spn system and that the system can be n each of the possble states wth the same probablty (.e. a unform probablty) and thereby settng g(μ ν)=1/m, E E R g A A e R( ) g( ) A( ) A( ). (4) The largest of the acceptance ratos s made to be equal to one whle the others are adjusted to satsfy the constrant 1 f E E A E E (5) e otherwse By representng the classcal spn correspondng to lattce ste by a unt vector S 3, S 1, and spn-spn nteracton parameter J, the Classcal sotropc Hesenberg model can be descrbed by the Hamltonan, H J S S j. The dynamcs of the Hesenberg model s governed by the equaton of moton,, j ds dt S H (6) eff where γ s the gyromagnetc factor, and nearest neghbourng lattce stes NN() of th lattce ste, H eff s the effectve feld gven by summng over the k k H J S k x, y, z. Because eff jnn j

3 Internatonal Letters of Chemstry, Physcs and Astronomy Vol the effectve feld has unts of nteracton parameter J, the tme varable used n the spn dynamcs smulatons s also measured n unts of J [5]. By settng 1, equaton 6 can be rewrtten n matrx form ds dt z y 0 Heff H eff z x S Heff Heff 0 H eff S R S (7) y x Heff H 0 eff by a skew symmetrc matrx R and spn matrx S. All the set of dfferental equatons n ths equaton can be summarzed as an ordnary dfferental equaton, dy() t Ry ˆ () t (8) dt t S1 R1 0 where yt and Rˆ s a dagonal matrx. Assumng contnuous tme and SN t 0 R N ˆRt ˆR s a constant, the soluton to the above dfferental equaton takes the form y t Ce. However even f tme s dscrete and ˆR s a matrx, for dscrete tme steps τ, a soluton can be wrtten of ˆR evoluton operator e [6] as Rˆ y t e y() t (9) The exponentaton of the matrx operator R can be dentfed as another matrx operator gven by R n e R n!. For any skew-symmetrc matrx R, n0 R e s a rotaton matrx [7]. Therefore the evoluton operator n equaton 8 rotates each spn n yt () about the effectve felds by an amount proportonal to the feld magntude. The set of ordnary dfferental equatons derved n equaton 8 can be solved usng Suzuk Trotter decomposton method [8]. If the lattce s dvded nto four nterpenetratng sub lattces denoted by A, B C and D, then the effectve Hamltonan H eff ( X A, B, C, D) correspond to the effectve felds of the four sub lattces. For any spn, the X equatons of moton become, ds dt S H S H S H S H (10) eff A eff B eff C eff D If the correspondng skew symmetrc matrces are defned as Rˆ, ˆ ˆ and ˆ A RB RC R D, then Rˆ Rˆ ˆ ˆ ˆ A RB RC RD. By applyng the second order Suzuk Trotter decomposton to the evoluton operator and truncatng off the hgher order terms, the soluton for the equatons of moton can be wrtten as, Rˆ /2 ˆ /2 ˆ /2 ˆ ˆ /2 ˆ /2 ˆ D R C RB RA RB R C R D / 2 y t f ( ) y( t) e e e e e e e y( t) (11)

4 38 Volume 49 wth F F I ˆ. Snce Rˆ, Rˆ, Rˆ and Rˆ are skew-symmetrc matrces, they A B C D correspond to rotaton matrces. If a three dmensonal vector V s rotated about vector U by an angle α, then the rotated vector V ' can be computed usng the followng formula: ' V V cos U V sn 1 cos U U V (12) The space-dsplaced, tme-dsplaced spn correlaton functon and ts space-tme Fourer transform were used to study of spn dynamcs. Consderng a lattce space wth spn vectors correspondng to each lattce ste, the space-dsplaced tme-dsplaced spn correlaton functon s defned as [1]: C k ( r r ', t) S k ( t) S k (0) S k ( t) S k (0) (13) r r' r r' k wth k x, y, z. Here Sr () t s the k-component of spn vector S located at ste r at tme t n the lattce space. The angle brackets denote the average over Fgure1: Rotaton of a spn S n the sub the ensemble. The nformaton about nter-partcle lattce B correlatons and tme evoluton of the system s k k contaned n the dynamc structure factor S ( q, ). The statc structure factor S ( q) can be obtaned by ntegratng the dynamc structure factor over all the frequences [9]. Dynamc structure factor can be computed by applyng space-tme Fourer transformaton of the space-dsplaced, tmedsplaced spn-correlaton functon: k 1 qr r' t k S q, e e C r r ', tdt k x, y, z (14) 2 ' rr, The relaton of statc structure factor and dynamc structure factor gven n equaton 14 shows that the dynamc structure factor have to be a tme translaton nvarant and space translaton nvarant quantty. Therefore by takng only the real parts of the exponentals, the equaton 14 can be rewrtten as, t cutoff k 2 k S q, cost cos ( ') C ', tdt 2 q r r r r (15) 0 r, r' 3. THE SPIN DYNAMICS SIMULATION For an orgnal lattce wth each spn orented n z drecton, the Metropols algorthm was used to obtan thermal equlbrum by heat exchange wth a constant temperature envronment. In achevng thermal equlbrum wth a constant temperature envronment, for each lattce ste, the spn was randomzed usng sphercal polar coordnate system. To the new energy term, the Metropols algorthm was appled. In order to avod the ansotropy, the Marsagla s method [9] was adopted to generate random orentaton n spn n Monte Carlo Fgure 2: 1000 randomly generated spn random orentaton usng Marsagla s method

5 Internatonal Letters of Chemstry, Physcs and Astronomy Vol smulaton. Ths s shown n fgure 2. The Suzuk Trotter decomposton was used to solve dynamcs of the system. The space-dsplaced, tme-dsplaced spn-correlaton functon C k ( r r',) t was computed and stored as a functon of r r r ' and tme t. For a sngle smulaton run however, the spn system s n constant energy snce the smulaton was done n mcro canoncal ensemble. To get the average over canoncal ensemble, about 50 smulatons per each system were performed. A correlaton functon matrx and the smulaton count were created. After the computaton of the ntegrand of the dynamc structure factor n equaton 15, the trapezodal rule was used for the numercal ntegraton. By settng magnetzaton drecton as the postve z drecton, the x y transverse components n x and y drectons were dentfed as S ( q, ) S ( q, ). 4. SPIN DYNAMICS OF TWO DIMENSIONAL LATTICE A method was developed to mplement spn dynamc smulatons wth second nearest neghbours on two dmensonal and three dmensonal smple cubc lattces. The frst nearest nteractons of th spn were coupled to t va the usual nteracton parameter, whereas second nearest spn nteractons were coupled to the same spn wth a much smaller secondary nteracton parameter. A two dmensonal lattce space wth N L L stes was constructed as shown n fgure 3. The atoms are denoted as cubes n non-standard notatons to show the lattce decomposton of the computatonal algorthm clearly. For a two dmensonal lattce, there are four nearest neghbour stes per each ste. In the smulaton, neghbours of second ste were chosen as the frst and thrd ste etc. Crcular boundares were used for the two end stes, so that ste 1 possesses ste 2 and ste N as ts neghbours, whle ste N possess ste N 1 and ste 1. The decomposton of the sub lattce was done by assgnng each odd-numbered ste to sub lattce A and each even-numbered ste to sub lattce B. The resultng lattce defnton s shown n fgure 4. Each spn n sub lattce A (red), experence the effectve felds due to ts nearest neghbours whch are all formed from sub lattce B (orange). Therefore when performng a rotaton on sub lattce A, the effectve felds due to sub lattce B stay fxed and vce versa. On a two dmensonal lattce space, the way the frst and second nearest neghbours were defned s shown n fgure 5. To 1 2 L 1 2 L L+1 L L+1 L+2 2L L 2 Second-rank second nearest neghbours neghbours 1 2 L L+1 L+2 2L Fgure 3: Two dmensonal lattce stes L L+1 L+2 2L L 2 Fgure 4: Left: Decomposed two dmensonal sub lattces - lattce A (red), lattce B (orange); Rght: Nearest neghbours of two-dmensonal sub lattce A are from sub lattce B Fgure 5: Frst (blue) and second (yellow) nearest neghbours of two-dmensonal spn system wth two couplng constants frst nearest Frst-rank neghbours neghbours

6 40 Volume 49 ntegrate such a spn system, two sub lattces are nadequate. If only two sub lattces are used, evoluton operator cannot be appled on th spn snce wthn dscrete tme step, the evoluton operator tself changes snce t depends on secondnearest neghbours whch are on the same sub lattce that th spn s n. To elmnate ths problem, the sub lattce decomposton shown n fgure 6 was ntroduced. As a result of ths sub lattce defnton, all the frst-nearest Sub Lattce A Sub Lattce B Sub Lattce C Sub Lattce D neghbours and second- nearest neghbours of any spn S are due to other sub lattces and thereby gave a Fgure 6: New sub lattce defnton for two couplng constants successful decomposton. To smulate the two dmensonal spn systems the data were collected for systems wth sngle couplng constant and two couplng constants separately and for dfferent temperatures and at least 50 smulaton runs were made. These data are tabulated n table 1. For most systems, a sngle run produces few megabytes of data and all the data s not lsted. Table 1: Smulated two-dmensonal systems Number Temperature Couplng Constant Number of Integraton Integraton of Atoms (T C ) J1 J2 Smulatons Step Sze Steps A two-dmensonal lattce wth 100 stes was smulated at a temperature of 0.5T C. Monte Carlo updates smulatng equlbrum and energy fluctuatons are shown fgure 7. The Monte Carlo smulaton shows a more relatvely stable system than for the one dmensonal case for the same temperature. In the one dmensonal system smulaton used only 30 atoms whereas for two-dmensonal system 100 atoms were used [4]. Wth the ncrease n number of atoms n the smulaton, the system s Fgure 7: Monte-Carlo updates for 2-D spn system and ts Stablzaton for nearest neghbour nteractons L 10, J 1.0, T 0.5T C

7 Internatonal Letters of Chemstry, Physcs and Astronomy Vol thermal equlbrum becomes much stable and energy per spn shows fewer fluctuatons. To fnd out the effect of nearest neghbour couplng on energy per spn at thermal equlbrum, the same system was smulated consderng frst and second neghbour nteractons. The resultng energy fluctuaton curve s shown n fgure 8. For only the frst nearest nteractons, the equlbrum state of energy per spn s around 1.5. At the same temperature, when both frst and second nearest spn nteractons were ncluded n couplng, the system s n a much lower equlbrum state of energy per spn ( 2.1). The dynamc structure factors can be computed and compared of systems at the same temperature up to the frst and second nearest neghbour spn nteractons to see the dependence of nearest neghbour couplng and the spn wave frequences. Fgure 9 shows the resultng dynamc structure factors of twodmensonal systems wth Fgure 8: Monte-Carlo updates for 2-D spn system and ts Stablzaton up to second nearest neghbour nteractons L 10, J1 1.0, J2 0.3; T 0.5T same lattce sze smulated at C the same temperature and for the same wave vector. There s a consderable shft n energy transfer for the same spn wave component, when only frst nearest neghbour nteractons are coupled (black) compared wth the nteractons coupled up to the second (red) nearest neghbours. In the smulaton, the frequency shft was 1 Δ rad. s. For the same propagaton vector, the second neghbour coupled system transfers consderably greater amount of energy than Fgure 9: Comparson of energy transfers for spns wth frst the nearest neghbour coupled (black) and up to second (red) nearest neghbour couplngs system. In ths partcular case, correspondng to a momentum transfer of 3π/5 second-rank coupled system transfers almost twce the energy of what frst-rank system does. The system used to generate above graphs was a two dmensonal array of spns. Snce under these condtons only fve momentum transfers are possble, they were plotted n fgure 10 to fnd out f ths pattern contnues. These fgures establsh the expected dependence of energy transfers and spn nteractons. Wth the avalablty of more nteractons, the spn system tend to transfer more energy at the same temperature relatve to T C, suggestng that when more spns are avalable to nteract, ferromagnetc spn systems generate spn waves of hgher frequences.

8 42 Volume 49 Δω = rad s 1 Δω = rad s 1 q = π 5 q = 2π 5 Δω = rad s 1 q = 4π 5 Δω = rad s 1 q = π Fgure 10: Comparson of energy transfers for spns wth frst (black) and up to second (red) nearest neghbour couplngs correspondng to dfferent momentum transfers 5. SPIN DYNAMICS OF THREE DIMENSIONAL SIMPLE CUBIC LATTICE In the case of the three dmensonal lattces, the spn smulaton were performed for a smple cubc (sc) crystal structure and a bodycentred cubc (bcc) crystal structure up to second nearest spn nteractons. For Second-rank nearest three-dmensonal lattce wth smple neghbours neghbour cubc crystal structure, numberng Frst nearest Frst-rank lattce stes and decomposng nto sub neghbour neghbours lattces s straghtforward and smlar to the prevously dscussed twodmensonal stuatons. Fgure 11: Three dmensonal smple cubc lattce stes

9 Internatonal Letters of Chemstry, Physcs and Astronomy Vol In three dmensons, there are sx nearest neghbours per spn. For a spn n a gven sub lattce, all the neghbours come from the other sub lattce (Fgure 11). To ntegrate the Sub Lattce A smple cubc spn system, the Sub Lattce B decomposton nto eght sub lattces as shown n fgure 12 was proposed. For a spn S belongng to a partcular sub Sub Lattce C Sub Lattce D Sub Lattce E lattce, all the frst and second-rank nearest neghbours come from other sub lattces. Therefore evoluton operator can be decomposed to apply as a seres of operators on spn S. The systems of whch spn wave occurrence s Fgure 12: Sub lattce defnton of sc lattce wth two nvestgated are tabulated n table 2 and couplng constants for each system at least 50 smulaton runs were made. Table 2: Smulated three-dmensonal smple cubc systems Number Temperature Couplng constant Number of Integraton Integraton of atoms (T C ) J1 J2 smulatons step sze steps Usng MatLab a 66 6 system was smulated successfully. The spn energy varaton n 500 Monte-Carlo steps for a smple cubc lattce wth only nearest neghbour nteractons are shown n fgure 13. A three dmensonal system usually havng more spns, 216 spns n the above partcular case arrve at the stable equlbrum wth even less fluctuatons than two dmensonal lattces. The magnon dsperson curve showng all the spn components can be generated for ths system by computng the dynamc structure Sub Lattce F Sub Lattce G Sub Lattce H Fgure 13: Thermal equlbrum of a sc lattce system wth one couplng constant at 0.5 T C factor for the ndvdual momentum vectors and plottng them n the same graph. Ths s shown n fgure 14. The magnon dsperson curve wth frst-rank couplng suggests that energy gaps between magnons close up as ther momentums ncrease. In the above system the peaks correspond to rad.s, rad.s 1 and rad.s. Therefore, spn waves correspondng to consecutve momentums show smaller energy dfferences at hgher momentums.

10 44 Volume 49 A three-dmensonal smple cubc (sc) lattce wth a lattce decomposton shown n fgure 12 was smulated wth the second couplng constant 0.3 tmes that of the frst couplng constant. The spn energy of the sc system wth these two nteracton parameters for a Suzuk Trotter ntegraton step sze s shown n fgure 15. Accordng to the fgure, the energy conservaton can be mantaned for a sc spn system. However t requres extremely small step szes. For such extreme step szes, generatng data enough to vsualze or compute space-dsplaced, tmedsplaced spn correlaton functons s therefore extremely tedous. Fgure 14: Magnon dsperson curve of a sc lattce wth one couplng constant and 216 spns Monte Carlo Updates Suzuk Trotter Integraton Fgure 15: Energy conservaton of smple cubc system wth two nteracton parameters for a step sze of

11 Internatonal Letters of Chemstry, Physcs and Astronomy Vol SPIN DYNAMICS OF BODY CENTERED CUBIC LATTICE For a body centered cubc structure (bcc), the frst and second nearest neghbour defnton s slghtly complex. Frst, a numberng method should be appled to dentfy the spns. In dong so, the lattce s defned as two nterpenetratng smple cubc lke lattces as shown n fgure 16. For one lattce, the usual numberng system s appled whle for the other lattce, a smlar numberng system s + = Fgure 16: The bcc lattce represented as two nterpenetratng smple cubc lattces appled but startng from the largest Second-rank number n frst lattce as 1. From neghbours the Second structure n the fgure 16, two ranks of nearest nearest neghbours are dentfed. These neghbours are shown n fgure 17. For a bcc structure, the lattce decomposton s much easer than for a smple cubc structure. It can be done by just Fgure 17: Nearest neghbour defnton for a bcc decomposng the two smple cubc lke spn system wth two couplng constants lattces each nto two sub lattces where a sc lattce was decomposed wth nearest neghbours as shown n fgure 18 wth a sngle couplng constant. Wth proper lattce ste numberng, sub lattce of any gven spn can be dentfed by ts ndex beng odd or even just as n the two dmensonal lattces. For the bcc structure, therefore, only four sub lattces Nearest result n and there are fourteen nearest neghbours ncludng eght frst nearest neghbours neghbours and sx second nearest neghbours. The systems of whch spn Fgure 18: Smple cubc lattce wth only nearest neghbours wave occurrence s nvestgated are tabulated n table 3. For each system at least 50 smulaton runs were made. A bcc spn system defned on a lattce was smulated usng the decomposton method proposed n equaton 11 and snce there are two spns per prmtve bcc cell, there are 2000 spns n total. The energy curve s shown n the fgure 19. Table 3: Smulated Three-dmensonal systems (bcc) Frst-rank neghbours Frst nearest neghbours Number Temperature Couplng Constant Number of Integraton Integraton of Atoms (T C ) J1 J2 Smulatons Step Sze Steps

12 46 Volume 49 Monte Carlo Updates Suzuk Trotter Integraton Fgure 19: Spn energy varaton of bcc lattce up to second nearest neghbours for L 10, J1 1.0, J2 0.3; T 0.5T C wth 500 Monte-Carlo steps and 200 The Monte Carlo smulaton yelds a much stable equlbrum. Accordng to the fgure, Suzuk Trotter ntegraton preserves energy almost exactly as suggested by the flat curve to the rght sde of the plot. Therefore above fgure suggests that decomposton method works successfully. 7. CONCLUSIONS Ths study was focused at utlzng computatonal power to study complex dynamcs of magnetc systems, manly studyng the propagatng dsturbances n the orderng of spn orentatons. Second order Suzuk Trotter decomposton technque along wth evoluton operator soluton to spn equatons of moton was successful n numercally solvng two-dmensonal and three-dmensonal spn systems and can be successfully extended to nclude a second nteracton parameter for two and three dmensonal systems wth smple cubc and body centred cubc structures. Usng the developed MatLab code, many systems were smulated and analysed to dentfy spn waves and ther propertes n varous condtons. Energy conservaton of the algorthm was shown for each stuaton. For a smple cubc lattce wth two nteracton parameters, there was a sgnfcant energy drft. However, ths can be attrbuted to the ntegraton method used. Second order Suzuk Trotter algorthm s accurate only to the second order of step sze. However the algorthm used was conservng energy when extremely small ntegraton step szes were used for very short tme perods. In the magnon dsperson curves all or most of the spn wave components could be recognzed as peaks n the dynamc structure factor. They clearly presented the varaton of energy transfer wth respect to momentum transfer of spn waves. Classcal Hesenberg model produces extremely complex and chaotc moton for spn orentatons. Spn waves occur n one-dmensonal, two-dmensonal, and three-dmensonal spn systems below Cure temperature, and when they occur, they occur n a group and varous

13 Internatonal Letters of Chemstry, Physcs and Astronomy Vol components of them correspond to dfferent momentums and energes. As temperature s ncreased but kept below Cure temperature, frequency of spn wave components s decreased regardless of the propagaton vector. There s a strong relaton between spn-spn nteracton and energy transfer of spn waves. Wth more nteractons avalable spn waves can transfer more and more energy. References [1] K. Chen, D. P. Landau, Phy. Rev. B, 49 (5) (1994) [2] X. Tao, D. P. Landau, T. C. Schulthess, G. M. Stocks, Phys. Rev. Lett. 95(8) (2005) [3] J. E. Costa, B.V Costa, Phy. Rev. B, 54 (2) (1996) [4] H. S.Wjesnhe, K.A.I.L.Wjewardena Gamalath, I.L.C.P.A. 8(1) (2015) [5] A. H Morrs, The Physcal Prncples of Magnetsm, John Wley and Sons (New York, 1966) [6] S. H. Tsa, D. P. Landau, Comp. Sc. Eng. 10 (1) (2008) [7] D. P. Landau, A. Bunker, H. G. Evertz, M. Krech, S. H. Tsa, Prog. Theo. Phys. 138 (2000) [8] M.W. Spong, S. Hutchnson, M. Vdyasagar, Robot modelng and control (Wley, 2006) [9] N. Hatano, M. Suzuk, Quantum Annealng and Other Optmzaton Methods, (Sprnger, Berln, 2005) arxv:math-ph/ [10] C. Menott, M. Krämer, L. Ptaevsk, S. Strngar, Phys. Rev. A 67(5) (2003) [11] G. Marsagla, The Annals of Mathematcal Statstcs 43 (2) (1972) ( Receved 31 March 2015; accepted 07 Aprl 2015 )