(*              GROUND STATE ENERGY AND  HOPPING AMPLITUDES                     *)
(*                FOR DIMERIZED & FRUSTRATED S=1/2 CHAIN                        *)
(*                        (MATHEMATICA-VERSION)                                 *)
(*                                                                              *)
(*                CHRISTIAN KNETTER  AND  GOETZ UHRIG                           *) 
(*                                                                              *)
(*                      UNIVERSITY OF COLOGNE                                   *)
(*                             JULY 1999                                        *)
(*                                                                              *)
(*                                                                              *)
(*  ....... Read this file in Mathematica with:  In[i] <<filename ...........   *)
(*                                                                              *)
(*                                                                              *) 
(*                                                                              *)
(*                              SUMMARY                                         *)
(*                            -----------                                       *)
(*              (paper: Eur. Phys. J. B 13, 209-225 (2000))                     *)
(*              (Equation numbers refer to that article!!!)                     *)
(*                                                                              *)
(*   The ground state energy as given below is identical to the one given in    *)
(*   Eq.(44) and refers to Hamiltonian (6). The results (plots) in Section 4    *)
(*   refer to Hamiltonian (4), which is connected to (6) by the substitutions   *)
(*   in Eqs.(5).                                                                *)
(*                                                                              *)
(*   The hopping amplitudes below are defined through Eq.(38) and partially     *)
(*   given in Appendix C. The dispersion                                        *)
(*   is given by \omega:=J*\sum_{i=0}^{10} [ai*Cos(i*k)].                       *)
(*                                                                              *)
(*                                                                              *)
(*   Both, ground state energy and hopping amplitudes, contain alpha            *)
(*   and lambdahat = 1/4 * lambda. ( lambda and alpha as they appear in        *)
(*   Hamiltonian (4).):                                                         *)

lambdahat=1/4*lambda



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(*                                                                             *)
(*              GROUND STATE ENERGY PER SPIN AS IN EQ.(44)                     *)
(*            ----------------------------------------------                   *)
(*                                                                             *)


epsilon0 = (1-2*alpha)^2*((-106469295871/373248000-107584683283/15552000*alpha^2-89796462557/5832000*alpha^3-160938279937/5832000*alpha^4+57686123141/972000*alpha^5+143920286959/972000*alpha^6-339171/2*alpha^7-82849717337/46656000*alpha+336527/16*alpha^8)*lambdahat^10+(-2354594813/24883200+8121212969/259200*alpha^5-14053262981/6220800*alpha^2-1591335559/345600*alpha^3-9560574943/1555200*alpha^4-453741/64*alpha^6-248391/32*alpha^7-7341879263/12441600*alpha)*lambdahat^9+(-8477587/12960*alpha^2-414359/12960+4002*alpha^5-4527/2*alpha^6-152558/81*alpha^3+2774357/1620*alpha^4-139801/648*alpha)*lambdahat^8+(-215995/864*alpha^2-173579/432*alpha^3-81557/6912-879/8*alpha^5-257909/3456*alpha+14865/16*alpha^4)*lambdahat^7+(-463/96-227/9*alpha+42*alpha^3+159/2*alpha^4-1307/12*alpha^2)*lambdahat^6+(-89/48-311/24*alpha-93/4*alpha^2+45/2*alpha^3)*lambdahat^5+(-27/4*alpha+3/4*alpha^2-13/16)*lambdahat^4+(-3/4-3/2*alpha)*lambdahat^3-3/4*lambdahat^2)

 



(*                                                                             *)
(*                                                                             *)
(*             EFFECTIVE HOPPING ELEMENTS AS IN APPENDIX C                     *)
(*           -----------------------------------------------                   *)
(*                                                                             *)
(*    (NOTE: 1) We do not substitute \bar{\alpha}=1-2\alpha here as in         *)
(*              Appendix C, but give the results in \alpha direct!!!!!         *)
(*           2) The amplitude a0 given below correspondes to a_0-E_0           *)
(*              in the paper!!!!)                                              *)
     







(*   a0=a_0-E_0 from Appendix C in the paper!!!!!!!!!!!                      *)
a0 = 1+(-16+48*alpha-48*alpha^2)*lambdahat^2+(-12+8*alpha+48*alpha^2-96*alpha^3)*lambdahat^3+(286*alpha^2-408*alpha^3-4*alpha^4-26*alpha-37/4)*lambdahat^4+(856*alpha^2+432*alpha^3-2504*alpha^4+1040*alpha^5-89*alpha-87/2)*lambdahat^5+(-37177/27*alpha^2+12184*alpha^3-287020/27*alpha^4-3056*alpha^5+3952*alpha^6-5431/9*alpha-6109/108)*lambdahat^6+(-4783603/648*alpha^2-6708173/324*alpha^3+16192579/162*alpha^4-8522435/81*alpha^5+42750*alpha^6-2172*alpha^7+680927/1296*alpha-171007/2592)*lambdahat^7+(333013541/9720*alpha^2-305681053/4860*alpha^3-1698043853/4860*alpha^4+891039551/1215*alpha^5-617483446/1215*alpha^6+276012*alpha^7-86594*alpha^8+28144121/19440*alpha-127561967/155520)*lambdahat^8+(790853593/64800*alpha^2+35414961073/48600*alpha^3+20016446569/194400*alpha^4-588393903319/97200*alpha^5+130134260969/18225*alpha^6-31605553363/18225*alpha^7+786391/4*alpha^8-652853/2*alpha^9-243167385961/9331200*alpha-22039105589/18662400)*lambdahat^9+(-178870309767433/279936000*alpha^2-6267321515107/5832000*alpha^3+387175658303851/34992000*alpha^4+136115216713153/8748000*alpha^5-268721615586503/2916000*alpha^6+3535421669809/40500*alpha^7-21714315290243/1458000*alpha^8+3142532962607/279936000*alpha-26468071/4*alpha^9+2130093/4*alpha^10+180705782417/373248000)*lambdahat^10-((-15+60*alpha-60*alpha^2)*lambdahat^2+(-15+30*alpha+60*alpha^2-120*alpha^3)*lambdahat^3+(-15-72*alpha+480*alpha^2-576*alpha^3+48*alpha^4)*lambdahat^4+(420*alpha^2+1224*alpha^3-3480*alpha^4+1680*alpha^5-123*alpha-63/2)*lambdahat^5+(-1805/3*alpha^2+65608/9*alpha^3-88460/9*alpha^4-3024*alpha^5+6000*alpha^6-1181/9*alpha-2885/36)*lambdahat^6+(-31585/108*alpha^2+35225/6*alpha^3+786593/27*alpha^4-2714626/27*alpha^5+74940*alpha^6-7032*alpha^7-116651/216*alpha-8909/48)*lambdahat^7+(4036777/3240*alpha^2-2523677/540*alpha^3+207800999/1620*alpha^4-77380483/405*alpha^5-9607598/45*alpha^6+453900*alpha^7-160014*alpha^8-11550853/6480*alpha-2692171/5760)*lambdahat^8+(-577320043/97200*alpha^2+3124167011/97200*alpha^3+12238091329/194400*alpha^4+69320855921/97200*alpha^5-34736133089/12150*alpha^6+15606465814/6075*alpha^7+426099/4*alpha^8-1124721/2*alpha^9-13461044767/3110400*alpha-2710532761/2073600)*lambdahat^9+(-92200177159/3888000*alpha^2+68595589901/729000*alpha^3+37892724091/729000*alpha^4+19394315399/9000*alpha^5-1321798825321/364500*alpha^6-870913207916/91125*alpha^7+692566166161/30375*alpha^8-13069305*alpha^9-289418494169/23328000*alpha-86855383237/23328000+1323654*alpha^10)*lambdahat^10)


a1 = 2*(-1+2*alpha)*lambdahat-4*lambdahat^2+2*(1-14*alpha+12*alpha^2-8*alpha^3)*lambdahat^3+2*(-48*alpha^2+40*alpha^3-40*alpha^4+18*alpha+5/2)*lambdahat^4+2*(1106/3*alpha^2-116*alpha^3-116*alpha^4-104*alpha^5-149/6*alpha-35/4)*lambdahat^5+2*(-4070/3*alpha^2+166768/27*alpha^3-54188/27*alpha^4-1936*alpha^5+192*alpha^6-10271/27*alpha+3121/108)*lambdahat^6+2*(-7691341/1296*alpha^2-17063945/648*alpha^3+25204769/324*alpha^4-5771243/162*alpha^5-8055*alpha^6+2698*alpha^7+2650495/2592*alpha+540083/5184)*lambdahat^7+2*(78427253/2592*alpha^2-62918825/1296*alpha^3-130172809/324*alpha^4+264102637/324*alpha^5-189862319/486*alpha^6+9189*alpha^7+8375*alpha^8+3393133/1728*alpha-14506793/62208)*lambdahat^8+2*(-201608063/46656*alpha^2+180753785879/291600*alpha^3+186294757253/1166400*alpha^4-675478924409/116640*alpha^5+146620899712/18225*alpha^6-118784295031/36450*alpha^7+2202761/8*alpha^8-84629/4*alpha^9-354693274169/18662400*alpha+21257692853/37324800)*lambdahat^9+2*(-76278932281757/139968000*alpha^2-23154357006457/17496000*alpha^3+33265783979473/3499200*alpha^4+65903332932527/4374000*alpha^5-39642163501451/486000*alpha^6+90114205222457/1093500*alpha^7-18410199928663/729000*alpha^8+716496246491/23328000*alpha+2526488220859/559872000+1139536*alpha^9-523491/2*alpha^10)*lambdahat^10


a2 = 2*(-1/2+2*alpha-2*alpha^2)*lambdahat^2+2*(-1+2*alpha+4*alpha^2-8*alpha^3)*lambdahat^3+2*(32*alpha^2-48*alpha^3-4*alpha^4+10*alpha-15/4)*lambdahat^4+2*(382*alpha^2-964/9*alpha^3-332*alpha^4+104*alpha^5-155/2*alpha-283/36)*lambdahat^5+2*(-3299/3*alpha^2+15244/3*alpha^3-35770/9*alpha^4-112*alpha^5+352*alpha^6-1906/9*alpha+79/8)*lambdahat^6+2*(-251599/108*alpha^2-886561/54*alpha^3+1308109/27*alpha^4-994354/27*alpha^5+9188*alpha^6-1416*alpha^7+144517/216*alpha-3937/432)*lambdahat^7+2*(165121433/7776*alpha^2+5799079/3888*alpha^3-1099700995/3888*alpha^4+453456559/972*alpha^5-36866657/162*alpha^6+46043*alpha^7-33079/2*alpha^8-11551447/15552*alpha-31313245/124416)*lambdahat^8+2*(-2290133257/58320*alpha^2+91857139/216*alpha^3+12334306181/19440*alpha^4-136656477257/29160*alpha^5+19433703263/3645*alpha^6-5554567022/3645*alpha^7-183371/2*alpha^8-12682477639/933120*alpha-34275*alpha^9+657898577/1866240)*lambdahat^9+2*(-947948976169/2799360*alpha^2-85480715791/58320*alpha^3+22076077105553/3499200*alpha^4+7387879055039/437400*alpha^5-15003453988537/218700*alpha^6+3530581491166/54675*alpha^7-2325478036667/145800*alpha^8+107974380769/3499200*alpha-2136754*alpha^9+361255*alpha^10+162775450919/111974400)*lambdahat^10


a3 = 2*(-1/2+3*alpha-6*alpha^2+4*alpha^3)*lambdahat^3+2*(-20/3*alpha^2-16*alpha^3+16*alpha^4+32/3*alpha-8/3)*lambdahat^4+2*(172*alpha^2-928/3*alpha^3+208*alpha^4-48*alpha^5-68/3*alpha-9/2)*lambdahat^5+2*(-4099/18*alpha^2+1052*alpha^3-1702*alpha^4+1488*alpha^5-648*alpha^6-172/9*alpha-337/72)*lambdahat^6+2*(3846461/2592*alpha^2-1287943/144*alpha^3+7034831/648*alpha^4-411389/324*alpha^5-1433/2*alpha^6-1821*alpha^7+805993/5184*alpha-142177/3456)*lambdahat^7+2*(11046281/1296*alpha^2+6978857/162*alpha^3-7791607/36*alpha^4+19280201/81*alpha^5-630635/27*alpha^6-57960*alpha^7+9020*alpha^8-695939/432*alpha-72667/648)*lambdahat^8+2*(-22598707469/466560*alpha^2+44906376283/233280*alpha^3+2918727266/3645*alpha^4-2529946145/729*alpha^5+36988487023/9720*alpha^6-15987631883/14580*alpha^7-1122873/4*alpha^8-1645616351/373248*alpha+812435221/3732480+194167/2*alpha^9)*lambdahat^9+2*(-1332707302459/15552000*alpha^2-5474328479741/4374000*alpha^3+46475505627701/17496000*alpha^4+10288172598181/729000*alpha^5-201804906037853/4374000*alpha^6+12412531608448/273375*alpha^7-34123734486217/2187000*alpha^8+574322*alpha^9+458531/2*alpha^10+71069365133/2916000*alpha-222160666897/559872000)*lambdahat^10


a4 = 2*(-5/8+5*alpha-15*alpha^2+20*alpha^3-10*alpha^4)*lambdahat^4+2*(-116/3*alpha^2-56/9*alpha^3+72*alpha^4-48*alpha^5+67/3*alpha-67/18)*lambdahat^5+2*(2863/72*alpha^2-17593/27*alpha^3+21727/18*alpha^4-802*alpha^5+146*alpha^6+4943/72*alpha-13373/864)*lambdahat^6+2*(1528795/648*alpha^2-2263015/324*alpha^3+905797/162*alpha^4+303839/81*alpha^5-6774*alpha^6+2204*alpha^7-128239/1296*alpha-117461/2592)*lambdahat^7+2*(3385051/1296*alpha^2+30011029/972*alpha^3-18161681/144*alpha^4+27387461/162*alpha^5-19797170/243*alpha^6+1088*alpha^7+11079/2*alpha^8-2901907/2592*alpha-2933293/41472)*lambdahat^8+2*(-970487819/46656*alpha^2+21036681319/583200*alpha^3+124119218131/291600*alpha^4-44200644059/29160*alpha^5+137913948079/72900*alpha^6-37935956923/36450*alpha^7+272890*alpha^8-474144803/777600*alpha-44280*alpha^9-424801577/4665600)*lambdahat^9+2*(12267364207/273375*alpha^2-4289567391133/8748000*alpha^3-44838655697/97200*alpha^4+39623640480551/4374000*alpha^5-44946037330991/2187000*alpha^6+9959752281943/546750*alpha^7-7379154102881/1093500*alpha^8+2689935/2*alpha^9-397142*alpha^10+650838603113/139968000*alpha-268426879699/279936000)*lambdahat^10


a5 = 2*(-35*alpha^2+70*alpha^3-70*alpha^4+28*alpha^5+35/4*alpha-7/8)*lambdahat^5+2*(-1444/9*alpha^2+3976/27*alpha^3+3412/27*alpha^4-320*alpha^5+160*alpha^6+1534/27*alpha-767/108)*lambdahat^6+2*(705599/2592*alpha^2-3300613/1296*alpha^3+1292735/216*alpha^4-2078951/324*alpha^5+6221/2*alpha^6-495*alpha^7+209945/1728*alpha-297881/10368)*lambdahat^7+2*(10081007/12960*alpha^2-43390373/19440*alpha^3-9866929/1620*alpha^4+47503027/1620*alpha^5-108415133/2430*alpha^6+31327*alpha^7-8707*alpha^8+6772217/25920*alpha-29668951/311040)*lambdahat^8+2*(23827219387/2332800*alpha^2-78396249667/1166400*alpha^3+199591837481/1166400*alpha^4-106245063161/583200*alpha^5+4470478103/145800*alpha^6+47030017/900*alpha^7+74327/8*alpha^8-98103/4*alpha^9+11623399937/18662400*alpha-12727147777/37324800)*lambdahat^9+2*(168698916647/2332800*alpha^2-21168427907/3499200*alpha^3-2535545374469/1458000*alpha^4+5469747909443/874800*alpha^5-316670001011/36450*alpha^6+929083689203/218700*alpha^7+544009821887/546750*alpha^8-1318761*alpha^9+191010*alpha^10-7035779179/1749600*alpha-158901820853/139968000)*lambdahat^10


a6 = 2*(-315/4*alpha^2+210*alpha^3-315*alpha^4+252*alpha^5-84*alpha^6+63/4*alpha-21/16)*lambdahat^6+2*(-77995/162*alpha^2+66655/81*alpha^3-24130/81*alpha^4-81068/81*alpha^5+1400*alpha^6-560*alpha^7+40415/324*alpha-8083/648)*lambdahat^7+2*(-256405/288*alpha^2-4114127/1296*alpha^3+21613103/1296*alpha^4-3333805/108*alpha^5+4550687/162*alpha^6-11921*alpha^7+3249/2*alpha^8+2667287/5184*alpha-311791/4608)*lambdahat^8+2*(171425603/23328*alpha^2-15492872807/291600*alpha^3+18596809327/145800*alpha^4-1601355457/14580*alpha^5-259499903/4050*alpha^6+1213526953/6075*alpha^7-142530*alpha^8+34304*alpha^9+1066674737/1166400*alpha-306024377/1166400)*lambdahat^9+2*(2701816994221/93312000*alpha^2+18706113929/17496000*alpha^3-1633293793873/2332800*alpha^4+23054062116947/8748000*alpha^5-12707972953183/2916000*alpha^6+3872357802241/1093500*alpha^7-4982043888779/4374000*alpha^8-1401083/12*alpha^9+1315703/12*alpha^10-77651621239/279936000*alpha-930758172391/1119744000)*lambdahat^10


a7 = 2*(-693/4*alpha^2+1155/2*alpha^3-1155*alpha^4+1386*alpha^5-924*alpha^6+264*alpha^7+231/8*alpha-33/16)*lambdahat^7+2*(-220247/162*alpha^2+809546/243*alpha^3-297610/81*alpha^4-47680/81*alpha^5+1476520/243*alpha^6-6048*alpha^7+2016*alpha^8+22535/81*alpha-22535/972)*lambdahat^8+2*(-26909287/7776*alpha^2-109739299/23328*alpha^3+28831627/576*alpha^4-1018222201/7776*alpha^5+511658777/2916*alpha^6-31001858/243*alpha^7+362443/8*alpha^8-21071/4*alpha^9+320762903/248832*alpha-208160129/1492992)*lambdahat^9+2*(-290110172207/279936000*alpha^2-398832929773/5832000*alpha^3+8294035326887/34992000*alpha^4-138628852163/546750*alpha^5-260905960459/972000*alpha^6+1138805211887/1093500*alpha^7-5228973207149/4374000*alpha^8+1280739/2*alpha^9-543077/4*alpha^10+605480691619/139968000*alpha-237610672123/373248000)*lambdahat^10


a8 = 2*(-3003/8*alpha^2+3003/2*alpha^3-15015/4*alpha^4+6006*alpha^5-6006*alpha^6+3432*alpha^7-858*alpha^8+429/8*alpha-429/128)*lambdahat^8+2*(-863303/243*alpha^2+8183966/729*alpha^3-14010346/729*alpha^4+3089380/243*alpha^5+9538480/729*alpha^6-24280352/729*alpha^7+25872*alpha^8-7392*alpha^9+3509345/5832*alpha-501335/11664)*lambdahat^9+2*(-64132879057/4478976*alpha^2+3820104731/279936*alpha^3+17480266441/186624*alpha^4-57832242295/139968*alpha^5+112814457731/139968*alpha^6-5174755147/5832*alpha^7+38627437847/69984*alpha^8-1363139/8*alpha^9+133825/8*alpha^10+1722341231/497664*alpha-5375265485/17915904)*lambdahat^10


a9 = 2*(-6435/8*alpha^2+15015/4*alpha^3-45045/4*alpha^4+45045/2*alpha^5-30030*alpha^6+25740*alpha^7-12870*alpha^8+2860*alpha^9+6435/64*alpha-715/128)*lambdahat^9+2*(-19564295/2187*alpha^2+75442511/2187*alpha^3-343436527/4374*alpha^4+211700636/2187*alpha^5-54079172/2187*alpha^6-239305840/2187*alpha^7+375069388/2187*alpha^8-109824*alpha^9+27456*alpha^10+11313629/8748*alpha-11313629/139968)*lambdahat^10


a10 = 2*(-109395/64*alpha^2+36465/4*alpha^3-255255/8*alpha^4+153153/2*alpha^5-255255/2*alpha^6+145860*alpha^7-109395*alpha^8+48620*alpha^9-9724*alpha^10+12155/64*alpha-2431/256)*lambdahat^10




(*                                                                             *)
(*                                                                             *)
(*      Dispersion \omega(k) as given by Eq.(39) in Mathematica-format         *)
(*     ----------------------------------------------------------------        *)
(*                                                                             *)
(*                                                                             *)
 
omega=J*(a0+a1*Cos[k]+a2*Cos[2*k]+a3*Cos[3*k]+a4*Cos[4*k]+a5*Cos[5*k]+a6*Cos[6*k]+a7*Cos[7*k]+a8*Cos[8*k]+a9*Cos[9*k]+a10*Cos[10*k])