{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT 261 72 "Series for the 1-triplet \+ hopping amplitudes and the 1-triplet dispersion" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 263 31 "in the Shastry-Sutherland model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }{TEXT 264 0 "" }{TEXT 265 17 "(Maple-Worksheet)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 " " {TEXT -1 85 "C.Knetter, E. M\374ller-Hartmann and G. Uhrig\nJ.Phys.: Condens. Matter 12, 9069 (2000)\n\n" }{TEXT 266 96 "\"Symmetries and \+ triplet dispersion in a modified Shastry-Sutherland model for SrCu_2(B O_3)_2\"\n\n\n" }{TEXT -1 32 "University of Cologne\nMay 2000\n\n" } {TEXT 267 7 "Summary" }{TEXT -1 646 "\nThe Hamiltonian H is given in E q. (1) defined on the lattice depicted in Fig. 1. The small expansion \+ parameter is x=J_2/J_1 (y=J_3/J_1 is treated as an additional paramete r.) For y=0 the model reduces to the standart Shastry-Sutherland model . In this file we give the series expansions for the 1-triplet hopping amplitudes and the 1-triplet dispersion up to and including order 15 \+ in x. We start by giving the splitted hopping amplitudes of Eq. (14). \+ From them the dispersion (27) is calculated at the end of the file. No te that there are two branches, omega1 and omega2, due to the A,B subl atice structure of the model.\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 21 "Some initia lizations:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "for rv1 from -4 to 4 do\n fo r rv2 from -4 to 4 do\n tm[rv1,rv2]:=0:\n dt[rv1,rv2]:0:\n od:\nod: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 120 "Non-zero 1-triplet hopping amplitudes \\bar(t_s) ( here tm[s_x,s_y]) and dt_s (here dt[s_x,s_y]) as defined in Eq. (14) :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "> " 0 "" {MPLTEXT 1 0 19423 "tm[-3,-1]:=-589/15120*(1/2*x-1/2*y)^14-946 34209/423360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[-3,1]:=-589/15120* (1/2*x-1/2*y)^14-94634209/423360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\nt m[-2,-2]:=439/400*(1/2*x-1/2*y)^12+5230567/648000*(1/2*x-1/2*y)^12*(1/ 2*x+1/2*y)+1388720251/38880000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-92203/ 12150*(1/2*x-1/2*y)^14+94793641507/777600000*(1/2*x-1/2*y)^12*(1/2*x+1 /2*y)^3-110126539561/1270080000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[-2 ,-1]:=-29/720*(1/2*x-1/2*y)^10-15451/86400*(1/2*x-1/2*y)^10*(1/2*x+1/2 *y)-723191/1728000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2-1463/4050*(1/2*x-1 /2*y)^12-32069747/62208000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-1817981/48 6000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+1687480243/4665600000*(1/2*x-1/2*y )^10*(1/2*x+1/2*y)^4-3549380399/155520000*(1/2*x-1/2*y)^12*(1/2*x+1/2* y)^2-669161/1088640*(1/2*x-1/2*y)^14+4339349430493/1119744000000*(1/2* x-1/2*y)^10*(1/2*x+1/2*y)^5-90789537337/874800000*(1/2*x-1/2*y)^12*(1/ 2*x+1/2*y)^3+519430907/1428840000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[ -2,0]:=-247/864*(1/2*x-1/2*y)^10-12887/10800*(1/2*x-1/2*y)^10*(1/2*x+1 /2*y)-18872993/7776000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2+224807/86400*( 1/2*x-1/2*y)^12-22568321/51840000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3+168 782567/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+66297183559/3499200000*( 1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4+22589328569/233280000*(1/2*x-1/2*y)^12 *(1/2*x+1/2*y)^2-662879267/27216000*(1/2*x-1/2*y)^14+33275139000527/33 5923200000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+4356263136659/13996800000* (1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-1440077053099/5715360000*(1/2*x-1/2*y )^14*(1/2*x+1/2*y):\ntm[-2,1]:=29/720*(1/2*x-1/2*y)^10+15451/86400*(1/ 2*x-1/2*y)^10*(1/2*x+1/2*y)+723191/1728000*(1/2*x-1/2*y)^10*(1/2*x+1/2 *y)^2+1463/4050*(1/2*x-1/2*y)^12+32069747/62208000*(1/2*x-1/2*y)^10*(1 /2*x+1/2*y)^3+1817981/486000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)-1687480243 /4665600000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4+3549380399/155520000*(1/2 *x-1/2*y)^12*(1/2*x+1/2*y)^2+669161/1088640*(1/2*x-1/2*y)^14-433934943 0493/1119744000000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+90789537337/874800 000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-519430907/1428840000*(1/2*x-1/2*y )^14*(1/2*x+1/2*y):\ntm[-2,2]:=439/400*(1/2*x-1/2*y)^12+5230567/648000 *(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+1388720251/38880000*(1/2*x-1/2*y)^12*( 1/2*x+1/2*y)^2-92203/12150*(1/2*x-1/2*y)^14+94793641507/777600000*(1/2 *x-1/2*y)^12*(1/2*x+1/2*y)^3-110126539561/1270080000*(1/2*x-1/2*y)^14* (1/2*x+1/2*y):\ntm[-1,-3]:=-589/15120*(1/2*x-1/2*y)^14-94634209/423360 000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[-1,-2]:=-29/720*(1/2*x-1/2*y)^ 10-15451/86400*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-723191/1728000*(1/2*x-1/ 2*y)^10*(1/2*x+1/2*y)^2-1463/4050*(1/2*x-1/2*y)^12-32069747/62208000*( 1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-1817981/486000*(1/2*x-1/2*y)^12*(1/2*x +1/2*y)+1687480243/4665600000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4-3549380 399/155520000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-669161/1088640*(1/2*x-1 /2*y)^14+4339349430493/1119744000000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5- 90789537337/874800000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3+519430907/14288 40000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[-1,-1]:=-895437134449/466560 000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-37387531/64800*(1/2*x-1/2*y)^10*( 1/2*x+1/2*y)^2-36046620480571/6998400000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y )^4-115/18*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^2-22/9*(1/2*x-1/2*y)^6*(1/2*x +1/2*y)-22798304507/54432000*(1/2*x-1/2*y)^14-125838744664349/22861440 000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y)-155237/2916*(1/2*x-1/2*y)^6*(1/2*x+ 1/2*y)^5+5757411497/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+12547/288*( 1/2*x-1/2*y)^8*(1/2*x+1/2*y)^2-18523/648*(1/2*x-1/2*y)^6*(1/2*x+1/2*y) ^4+3962472719/8957952*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^5+5327/432*(1/2*x- 1/2*y)^8*(1/2*x+1/2*y)+1039527945613/1289945088*(1/2*x-1/2*y)^8*(1/2*x +1/2*y)^7-4607/324*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^3-2479350307/7558272* (1/2*x-1/2*y)^6*(1/2*x+1/2*y)^9+12997/200*(1/2*x-1/2*y)^12+73003763425 /107495424*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^6+181360759/746496*(1/2*x-1/2 *y)^8*(1/2*x+1/2*y)^4-195786899/839808*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^8 -722225/7776*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^6+7063645/62208*(1/2*x-1/2* y)^8*(1/2*x+1/2*y)^3-1811/120*(1/2*x-1/2*y)^10-16237343/129600*(1/2*x- 1/2*y)^10*(1/2*x+1/2*y)-365603785262209/31104000000*(1/2*x-1/2*y)^10*( 1/2*x+1/2*y)^5+16/9*(1/2*x-1/2*y)^8-2/3*(1/2*x-1/2*y)^6-63915089/41990 4*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^7+486737806195081/27993600000*(1/2*x-1 /2*y)^12*(1/2*x+1/2*y)^3+74517570469/17280000*(1/2*x-1/2*y)^12*(1/2*x+ 1/2*y)^2:\ntm[-1,1]:=-895437134449/466560000*(1/2*x-1/2*y)^10*(1/2*x+1 /2*y)^3-37387531/64800*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2-36046620480571 /6998400000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4-115/18*(1/2*x-1/2*y)^6*(1 /2*x+1/2*y)^2-22/9*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)-22798304507/54432000* (1/2*x-1/2*y)^14-125838744664349/22861440000*(1/2*x-1/2*y)^14*(1/2*x+1 /2*y)-155237/2916*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^5+5757411497/7776000*( 1/2*x-1/2*y)^12*(1/2*x+1/2*y)+12547/288*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^ 2-18523/648*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^4+3962472719/8957952*(1/2*x- 1/2*y)^8*(1/2*x+1/2*y)^5+5327/432*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)+103952 7945613/1289945088*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^7-4607/324*(1/2*x-1/2 *y)^6*(1/2*x+1/2*y)^3-2479350307/7558272*(1/2*x-1/2*y)^6*(1/2*x+1/2*y) ^9+12997/200*(1/2*x-1/2*y)^12+73003763425/107495424*(1/2*x-1/2*y)^8*(1 /2*x+1/2*y)^6+181360759/746496*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^4-1957868 99/839808*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^8-722225/7776*(1/2*x-1/2*y)^6* (1/2*x+1/2*y)^6+7063645/62208*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^3-1811/120 *(1/2*x-1/2*y)^10-16237343/129600*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-36560 3785262209/31104000000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+16/9*(1/2*x-1/ 2*y)^8-2/3*(1/2*x-1/2*y)^6-63915089/419904*(1/2*x-1/2*y)^6*(1/2*x+1/2* y)^7+486737806195081/27993600000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3+7451 7570469/17280000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2:\ntm[-1,2]:=29/720*( 1/2*x-1/2*y)^10+15451/86400*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)+723191/1728 000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2+1463/4050*(1/2*x-1/2*y)^12+320697 47/62208000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3+1817981/486000*(1/2*x-1/2 *y)^12*(1/2*x+1/2*y)-1687480243/4665600000*(1/2*x-1/2*y)^10*(1/2*x+1/2 *y)^4+3549380399/155520000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2+669161/108 8640*(1/2*x-1/2*y)^14-4339349430493/1119744000000*(1/2*x-1/2*y)^10*(1/ 2*x+1/2*y)^5+90789537337/874800000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-51 9430907/1428840000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[-1,3]:=-589/151 20*(1/2*x-1/2*y)^14-94634209/423360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y): \ntm[0,-2]:=-247/864*(1/2*x-1/2*y)^10-12887/10800*(1/2*x-1/2*y)^10*(1/ 2*x+1/2*y)-18872993/7776000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2+224807/86 400*(1/2*x-1/2*y)^12-22568321/51840000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^ 3+168782567/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+66297183559/3499200 000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4+22589328569/233280000*(1/2*x-1/2* y)^12*(1/2*x+1/2*y)^2-662879267/27216000*(1/2*x-1/2*y)^14+332751390005 27/335923200000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+4356263136659/1399680 0000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-1440077053099/5715360000*(1/2*x- 1/2*y)^14*(1/2*x+1/2*y):\ntm[0,0]:=-3463578995819/116640000*(1/2*x-1/2 *y)^10*(1/2*x+1/2*y)^3-6416574971/777600*(1/2*x-1/2*y)^10*(1/2*x+1/2*y )^2-549342689806787/6998400000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4-20105/ 108*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^2-544/9*(1/2*x-1/2*y)^6*(1/2*x+1/2*y )-81804308413/27216000*(1/2*x-1/2*y)^14-10354199055569/228614400*(1/2* x-1/2*y)^14*(1/2*x+1/2*y)-3284687/5832*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^5 +14728059511/1944000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+2754491/2592*(1/2* x-1/2*y)^8*(1/2*x+1/2*y)^2-256913/432*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^4+ 30121919729/4478976*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^5+53443/216*(1/2*x-1 /2*y)^8*(1/2*x+1/2*y)-26501995751309/644972544*(1/2*x-1/2*y)^8*(1/2*x+ 1/2*y)^7-126887/324*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^3+58759695139/377913 6*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^9+2270989/4050*(1/2*x-1/2*y)^12-157632 52723/5971968*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^6+718685633/124416*(1/2*x- 1/2*y)^8*(1/2*x+1/2*y)^4+35166254743/5038848*(1/2*x-1/2*y)^6*(1/2*x+1/ 2*y)^8+7316509/46656*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^6+30837965/10368*(1 /2*x-1/2*y)^8*(1/2*x+1/2*y)^3+9*(1/2*x-1/2*y)^4*(1/2*x+1/2*y)-343707/2 56*(1/2*x-1/2*y)^4*(1/2*x+1/2*y)^9-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^2-4 *(1/2*x-1/2*y)^2*(1/2*x+1/2*y)-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^5-4*(1/ 2*x-1/2*y)^2*(1/2*x+1/2*y)^4-3942251/1024*(1/2*x-1/2*y)^4*(1/2*x+1/2*y )^11+1/8*(1/2*x-1/2*y)^4*(1/2*x+1/2*y)^4-4*(1/2*x-1/2*y)^2*(1/2*x+1/2* y)^6+69/4*(1/2*x-1/2*y)^4*(1/2*x+1/2*y)^3-4*(1/2*x-1/2*y)^2*(1/2*x+1/2 *y)^12-1185743/512*(1/2*x-1/2*y)^4*(1/2*x+1/2*y)^10-94431/128*(1/2*x-1 /2*y)^4*(1/2*x+1/2*y)^8-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^10-23883/64*(1 /2*x-1/2*y)^4*(1/2*x+1/2*y)^7-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^9-827/16 *(1/2*x-1/2*y)^4*(1/2*x+1/2*y)^5-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^7-150 1/10*(1/2*x-1/2*y)^10-49901809/32400*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-80 49957297655109/52488000000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+55/2*(1/2* x-1/2*y)^8-11*(1/2*x-1/2*y)^6+323429665/139968*(1/2*x-1/2*y)^6*(1/2*x+ 1/2*y)^7-4*(1/2*x-1/2*y)^2+2*(1/2*x-1/2*y)^4+33/2*(1/2*x-1/2*y)^4*(1/2 *x+1/2*y)^2-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^3-4*(1/2*x-1/2*y)^2*(1/2*x +1/2*y)^11-5231/32*(1/2*x-1/2*y)^4*(1/2*x+1/2*y)^6-4*(1/2*x-1/2*y)^2*( 1/2*x+1/2*y)^8-4*(1/2*x-1/2*y)^2*(1/2*x+1/2*y)^13+4055296823029/182250 00*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3+1962719377901/38880000*(1/2*x-1/2* y)^12*(1/2*x+1/2*y)^2:\ntm[0,2]:=-247/864*(1/2*x-1/2*y)^10-12887/10800 *(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-18872993/7776000*(1/2*x-1/2*y)^10*(1/2 *x+1/2*y)^2+224807/86400*(1/2*x-1/2*y)^12-22568321/51840000*(1/2*x-1/2 *y)^10*(1/2*x+1/2*y)^3+168782567/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y )+66297183559/3499200000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4+22589328569/ 233280000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-662879267/27216000*(1/2*x-1 /2*y)^14+33275139000527/335923200000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+ 4356263136659/13996800000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-14400770530 99/5715360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[1,-3]:=-589/15120*(1 /2*x-1/2*y)^14-94634209/423360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[ 1,-2]:=29/720*(1/2*x-1/2*y)^10+15451/86400*(1/2*x-1/2*y)^10*(1/2*x+1/2 *y)+723191/1728000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2+1463/4050*(1/2*x-1 /2*y)^12+32069747/62208000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3+1817981/48 6000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)-1687480243/4665600000*(1/2*x-1/2*y )^10*(1/2*x+1/2*y)^4+3549380399/155520000*(1/2*x-1/2*y)^12*(1/2*x+1/2* y)^2+669161/1088640*(1/2*x-1/2*y)^14-4339349430493/1119744000000*(1/2* x-1/2*y)^10*(1/2*x+1/2*y)^5+90789537337/874800000*(1/2*x-1/2*y)^12*(1/ 2*x+1/2*y)^3-519430907/1428840000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[ 1,-1]:=-895437134449/466560000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-373875 31/64800*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2-36046620480571/6998400000*(1 /2*x-1/2*y)^10*(1/2*x+1/2*y)^4-115/18*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^2- 22/9*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)-22798304507/54432000*(1/2*x-1/2*y)^ 14-125838744664349/22861440000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y)-155237/2 916*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^5+5757411497/7776000*(1/2*x-1/2*y)^1 2*(1/2*x+1/2*y)+12547/288*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^2-18523/648*(1 /2*x-1/2*y)^6*(1/2*x+1/2*y)^4+3962472719/8957952*(1/2*x-1/2*y)^8*(1/2* x+1/2*y)^5+5327/432*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)+1039527945613/128994 5088*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^7-4607/324*(1/2*x-1/2*y)^6*(1/2*x+1 /2*y)^3-2479350307/7558272*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^9+12997/200*( 1/2*x-1/2*y)^12+73003763425/107495424*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^6+ 181360759/746496*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^4-195786899/839808*(1/2 *x-1/2*y)^6*(1/2*x+1/2*y)^8-722225/7776*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^ 6+7063645/62208*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^3-1811/120*(1/2*x-1/2*y) ^10-16237343/129600*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-365603785262209/311 04000000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+16/9*(1/2*x-1/2*y)^8-2/3*(1/ 2*x-1/2*y)^6-63915089/419904*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^7+486737806 195081/27993600000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3+74517570469/172800 00*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2:\ntm[1,1]:=-895437134449/466560000 *(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-37387531/64800*(1/2*x-1/2*y)^10*(1/2 *x+1/2*y)^2-36046620480571/6998400000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4 -115/18*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^2-22/9*(1/2*x-1/2*y)^6*(1/2*x+1/ 2*y)-22798304507/54432000*(1/2*x-1/2*y)^14-125838744664349/22861440000 *(1/2*x-1/2*y)^14*(1/2*x+1/2*y)-155237/2916*(1/2*x-1/2*y)^6*(1/2*x+1/2 *y)^5+5757411497/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+12547/288*(1/2 *x-1/2*y)^8*(1/2*x+1/2*y)^2-18523/648*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^4+ 3962472719/8957952*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^5+5327/432*(1/2*x-1/2 *y)^8*(1/2*x+1/2*y)+1039527945613/1289945088*(1/2*x-1/2*y)^8*(1/2*x+1/ 2*y)^7-4607/324*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^3-2479350307/7558272*(1/ 2*x-1/2*y)^6*(1/2*x+1/2*y)^9+12997/200*(1/2*x-1/2*y)^12+73003763425/10 7495424*(1/2*x-1/2*y)^8*(1/2*x+1/2*y)^6+181360759/746496*(1/2*x-1/2*y) ^8*(1/2*x+1/2*y)^4-195786899/839808*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^8-72 2225/7776*(1/2*x-1/2*y)^6*(1/2*x+1/2*y)^6+7063645/62208*(1/2*x-1/2*y)^ 8*(1/2*x+1/2*y)^3-1811/120*(1/2*x-1/2*y)^10-16237343/129600*(1/2*x-1/2 *y)^10*(1/2*x+1/2*y)-365603785262209/31104000000*(1/2*x-1/2*y)^10*(1/2 *x+1/2*y)^5+16/9*(1/2*x-1/2*y)^8-2/3*(1/2*x-1/2*y)^6-63915089/419904*( 1/2*x-1/2*y)^6*(1/2*x+1/2*y)^7+486737806195081/27993600000*(1/2*x-1/2* y)^12*(1/2*x+1/2*y)^3+74517570469/17280000*(1/2*x-1/2*y)^12*(1/2*x+1/2 *y)^2:\ntm[1,2]:=-29/720*(1/2*x-1/2*y)^10-15451/86400*(1/2*x-1/2*y)^10 *(1/2*x+1/2*y)-723191/1728000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2-1463/40 50*(1/2*x-1/2*y)^12-32069747/62208000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3 -1817981/486000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+1687480243/4665600000*( 1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4-3549380399/155520000*(1/2*x-1/2*y)^12* (1/2*x+1/2*y)^2-669161/1088640*(1/2*x-1/2*y)^14+4339349430493/11197440 00000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5-90789537337/874800000*(1/2*x-1/ 2*y)^12*(1/2*x+1/2*y)^3+519430907/1428840000*(1/2*x-1/2*y)^14*(1/2*x+1 /2*y):\ntm[1,3]:=-589/15120*(1/2*x-1/2*y)^14-94634209/423360000*(1/2*x -1/2*y)^14*(1/2*x+1/2*y):\ntm[2,-2]:=439/400*(1/2*x-1/2*y)^12+5230567/ 648000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+1388720251/38880000*(1/2*x-1/2*y )^12*(1/2*x+1/2*y)^2-92203/12150*(1/2*x-1/2*y)^14+94793641507/77760000 0*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-110126539561/1270080000*(1/2*x-1/2* y)^14*(1/2*x+1/2*y):\ntm[2,-1]:=29/720*(1/2*x-1/2*y)^10+15451/86400*(1 /2*x-1/2*y)^10*(1/2*x+1/2*y)+723191/1728000*(1/2*x-1/2*y)^10*(1/2*x+1/ 2*y)^2+1463/4050*(1/2*x-1/2*y)^12+32069747/62208000*(1/2*x-1/2*y)^10*( 1/2*x+1/2*y)^3+1817981/486000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)-168748024 3/4665600000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4+3549380399/155520000*(1/ 2*x-1/2*y)^12*(1/2*x+1/2*y)^2+669161/1088640*(1/2*x-1/2*y)^14-43393494 30493/1119744000000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+90789537337/87480 0000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-519430907/1428840000*(1/2*x-1/2* y)^14*(1/2*x+1/2*y):\ntm[2,0]:=-247/864*(1/2*x-1/2*y)^10-12887/10800*( 1/2*x-1/2*y)^10*(1/2*x+1/2*y)-18872993/7776000*(1/2*x-1/2*y)^10*(1/2*x +1/2*y)^2+224807/86400*(1/2*x-1/2*y)^12-22568321/51840000*(1/2*x-1/2*y )^10*(1/2*x+1/2*y)^3+168782567/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+ 66297183559/3499200000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4+22589328569/23 3280000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-662879267/27216000*(1/2*x-1/2 *y)^14+33275139000527/335923200000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+43 56263136659/13996800000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-1440077053099 /5715360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[2,1]:=-29/720*(1/2*x-1 /2*y)^10-15451/86400*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-723191/1728000*(1/ 2*x-1/2*y)^10*(1/2*x+1/2*y)^2-1463/4050*(1/2*x-1/2*y)^12-32069747/6220 8000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-1817981/486000*(1/2*x-1/2*y)^12* (1/2*x+1/2*y)+1687480243/4665600000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^4-3 549380399/155520000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-669161/1088640*(1 /2*x-1/2*y)^14+4339349430493/1119744000000*(1/2*x-1/2*y)^10*(1/2*x+1/2 *y)^5-90789537337/874800000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3+519430907 /1428840000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[2,2]:=439/400*(1/2*x-1 /2*y)^12+5230567/648000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)+1388720251/3888 0000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-92203/12150*(1/2*x-1/2*y)^14+947 93641507/777600000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^3-110126539561/12700 80000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[3,-1]:=-589/15120*(1/2*x-1/2 *y)^14-94634209/423360000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ntm[3,1]:=-5 89/15120*(1/2*x-1/2*y)^14-94634209/423360000*(1/2*x-1/2*y)^14*(1/2*x+1 /2*y):\n\n\n\ndt[-3,-1]:=59/1680*(1/2*x-1/2*y)^14+272034373/1270080000 *(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ndt[-3,1]:=59/1680*(1/2*x-1/2*y)^14+2 72034373/1270080000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ndt[-2,0]:=137/864 *(1/2*x-1/2*y)^10+9259/10800*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)+24385351/7 776000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2-115969/86400*(1/2*x-1/2*y)^12+ 505199807/51840000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-103457317/7776000* (1/2*x-1/2*y)^12*(1/2*x+1/2*y)+24424685821/874800000*(1/2*x-1/2*y)^10* (1/2*x+1/2*y)^4-33494916437/466560000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2 +7387241/1209600*(1/2*x-1/2*y)^14+128666240927299/1679616000000*(1/2*x -1/2*y)^10*(1/2*x+1/2*y)^5-2043440995757/6998400000*(1/2*x-1/2*y)^12*( 1/2*x+1/2*y)^3+161057804303/2286144000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y): \ndt[-1,-3]:=-59/1680*(1/2*x-1/2*y)^14-272034373/1270080000*(1/2*x-1/2 *y)^14*(1/2*x+1/2*y):\ndt[-1,3]:=-59/1680*(1/2*x-1/2*y)^14-272034373/1 270080000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ndt[0,-2]:=-137/864*(1/2*x-1 /2*y)^10-9259/10800*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)-24385351/7776000*(1 /2*x-1/2*y)^10*(1/2*x+1/2*y)^2+115969/86400*(1/2*x-1/2*y)^12-505199807 /51840000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3+103457317/7776000*(1/2*x-1/ 2*y)^12*(1/2*x+1/2*y)-24424685821/874800000*(1/2*x-1/2*y)^10*(1/2*x+1/ 2*y)^4+33494916437/466560000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2-7387241/ 1209600*(1/2*x-1/2*y)^14-128666240927299/1679616000000*(1/2*x-1/2*y)^1 0*(1/2*x+1/2*y)^5+2043440995757/6998400000*(1/2*x-1/2*y)^12*(1/2*x+1/2 *y)^3-161057804303/2286144000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ndt[0,2] :=-137/864*(1/2*x-1/2*y)^10-9259/10800*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)- 24385351/7776000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2+115969/86400*(1/2*x- 1/2*y)^12-505199807/51840000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3+10345731 7/7776000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)-24424685821/874800000*(1/2*x- 1/2*y)^10*(1/2*x+1/2*y)^4+33494916437/466560000*(1/2*x-1/2*y)^12*(1/2* x+1/2*y)^2-7387241/1209600*(1/2*x-1/2*y)^14-128666240927299/1679616000 000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^5+2043440995757/6998400000*(1/2*x-1 /2*y)^12*(1/2*x+1/2*y)^3-161057804303/2286144000*(1/2*x-1/2*y)^14*(1/2 *x+1/2*y):\ndt[1,-3]:=-59/1680*(1/2*x-1/2*y)^14-272034373/1270080000*( 1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ndt[1,3]:=-59/1680*(1/2*x-1/2*y)^14-272 034373/1270080000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\ndt[2,0]:=137/864*(1 /2*x-1/2*y)^10+9259/10800*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)+24385351/7776 000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^2-115969/86400*(1/2*x-1/2*y)^12+505 199807/51840000*(1/2*x-1/2*y)^10*(1/2*x+1/2*y)^3-103457317/7776000*(1/ 2*x-1/2*y)^12*(1/2*x+1/2*y)+24424685821/874800000*(1/2*x-1/2*y)^10*(1/ 2*x+1/2*y)^4-33494916437/466560000*(1/2*x-1/2*y)^12*(1/2*x+1/2*y)^2+73 87241/1209600*(1/2*x-1/2*y)^14+128666240927299/1679616000000*(1/2*x-1/ 2*y)^10*(1/2*x+1/2*y)^5-2043440995757/6998400000*(1/2*x-1/2*y)^12*(1/2 *x+1/2*y)^3+161057804303/2286144000*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):\nd t[3,-1]:=59/1680*(1/2*x-1/2*y)^14+272034373/1270080000*(1/2*x-1/2*y)^1 4*(1/2*x+1/2*y):\ndt[3,1]:=59/1680*(1/2*x-1/2*y)^14+272034373/12700800 00*(1/2*x-1/2*y)^14*(1/2*x+1/2*y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 54 "Next we calculate a0, a1 and b as defined in Eq. (27):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "> " 0 " " {MPLTEXT 1 0 160 "a0:=simplify( tm[0,0]+2*( sum(sum(tm[i,j]*cos(k_x *i+k_y*j),i=0..0),j= 1..4) \n +sum(sum(tm[i,j ]*cos(k_x*i+k_y*j),i=1..4),j=-4..4) ) ):" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 178 "a1:=simplify( tm[0,0]+2*( sum(sum(tm[i,j]*cos(k_ x*i+k_y*j+Pi*(i+j)),i=0..0),j= 1..4) \n +sum( sum(tm[i,j]*cos(k_x*i+k_y*j+Pi*(i+j)),i=1..4),j=-4..4) ) ):" }}} {EXCHG {PARA 259 "> " 0 "" {MPLTEXT 1 0 160 "b:= simplify( dt[0,0]+2* ( sum(sum(dt[i,j]*cos(k_x*i+k_y*j),i=0..0),j= 1..4) \n \+ +sum(sum(dt[i,j]*cos(k_x*i+k_y*j),i=1..4),j=-4..4) ) ):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 45 "And finialy the dispersion given by Eq. (29):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "omega0: =(a0+a1)/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "DD:=(a0-a1)^ 2+4*b^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "omega1:=omega0+ sqrt(DD):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "omega2:=omega0 -sqrt(DD):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }