# gnuplot-file # # name kagome is from japanese mats # (Journal of Nonlinear Math. Physics, v7, p57-72 (2000), S.M.Sergeev, # On Exact Solution of a Classical 3D Integrable Model) # # Gap: # Leung93-PRB47 expect gap=1/4 # Nakamura96-PRB53 delta-chain-gap=0.22 # alte Diag-Extrapol=PRB42_8436zeng90.pdf N<=21 #set term dumb # E = A0 + A3*L^-3 # GAP = D4*L^-2 + D5*L^-3 # N*GAP = D4 + D5*L^-1 # Mag = M0 + M1*L^-1 # ps=1 ps=0 set pointsize 1.5 # 14-32 a4=0 a45=0 # # # # x=L e0(x) =a0 + a3 *x**3 + 0*a4 *x**4 e05(x)=a05 + a35*x**3 + 0*a45*x**4 # g(x)= d4 + d5 *x g1(x)= d41 + d51*x m(x)= m0 + m1*x fit e0(x) "-" u ($1**(-1/2.)):(2*$2) via a0,a3 #12 -0.22686980 -0.21092529 0.18416013 % ll= 4, 4, 4 #18 -0.22356308 -0.21565843 0.11563650 % ll= 4, 7, 9 24 -0.22062428 -0.21629452 0.08884961 % ll= 7, 7,12 36 -0.219188 -0.2169078 0.059125 # ll=12,12,12 Mar04 e fit e05(x) "-" u ($1**(-1/2.)):(2*$2) via a05,a35 21 -0.21838761 -0.21175340 0.09650050 % ll= 7, 7, 7 27 -0.21813898 -0.21316163 0.07456556 % ll= 9, 9, 9 e fit g(x) "-" u ($1**(-2/2.)):((2*($3-$2))*$1) via d4,d5 12 -0.22686980 -0.21092529 0.18416013 % ll= 4, 4, 4 18 -0.22356308 -0.21565843 0.11563650 % ll= 4, 7, 9 24 -0.22062428 -0.21629452 0.08884961 % ll= 7, 7,12 #36 -0.219188 -0.2169078 0.059125 # ll=12,12,12 Mar04 e fit g1(x) "-" u ($1**(-2/2.)):((2*($3-$2))*$1) via d41,d51 #15 -0.21963813 -0.20567811 0.13131993 % ll= 3, 3, 7 21 -0.21838761 -0.21175340 0.09650050 % ll= 7, 7, 7 #21 -0.22053100 -0.21270744 0.10124700 % ll= 3,13,19 27 -0.21813898 -0.21316163 0.07456556 % ll= 9, 9, 9 e fit m(x) "-" u ($1**(-1/2.)):(($4)**.5) via m0,m1 #12 -0.22686980 -0.21092529 0.18416013 % ll= 4, 4, 4 #18 -0.22356308 -0.21565843 0.11563650 % ll= 4, 7, 9 #24 -0.22062428 -0.21629452 0.08884961 % ll= 7, 7,12 36 -0.219188 -0.2169078 0.059125 # ll=12,12,12 Mar04 # 9 -0.22050010 -0.18398543 0.205259 # ll= 3, 3, 3 #15 -0.21963813 -0.20567811 0.13131993 % ll= 3, 3, 7 21 -0.21838761 -0.21175340 0.0965005 # ll= 7, 7, 7 #21 -0.22053100 -0.21270744 0.10124700 % ll= 3,13,19 27 -0.21813898 -0.21316163 0.07456556 % ll= 9, 9, 9 e # fit taken by betts if(ps==1)set term postscript eps enhanced mono dashed "Helvetica" 20; set out "tmp/t8_erg.eps" set key right top yy=-2*0.236 #-0.597 set label 5 "N:" at 0.003,yy center set label 1 " 9" at 9**(-3/2.),yy center set label 2 "12" at 12**(-3/2.),yy center set label 3 "18" at 18**(-3/2.),yy center set label 4 "36" at 36**(-3/2.),yy center set xlabel "L^{-3}" set ylabel "E/N" plot [0:0.04] [-2*0.24:-2*0.21] e0(x**(1/3.)) t "" w l lt 1,e05(x**(1/3.)) t "" w l lt 1,\ "kagome.gpl" i 1 u ($1**(-3/2.)):(2*$2) t "even" w p pt 4,\ "kagome.gpl" i 2 u ($1**(-3/2.)):(2*$2) t "odd" w p pt 6,\ "kagome.gpl" i 5 u ($1**(-3/2.)):(2*$2) smooth cspline t "l^2=3" w l lt 3,\ "kagome.gpl" i 4 u ($1**(-3/2.)):(2*$2) smooth cspline t "l^2=4" w l lt 4,\ "kagome.gpl" i 3 u ($1**(-3/2.)):(2*$2) t "" w p pt 2 if(ps==0)pause -1 if(ps==1)set term postscript eps enhanced mono dashed "Helvetica" 20; set out "tmp/t8_gap.eps" set key left top yy=2*1.2 yy=0.05 set label 5 "N:" at 48**(-2/2.),yy center set label 1 " 9" at 9**(-2/2.),yy center set label 2 "12" at 12**(-2/2.),yy center set label 3 "18" at 18**(-2/2.),yy center set label 4 "36" at 36**(-2/2.),yy center set arrow from 0.01,0.25 to 0,0.25 set label 6 "Leung93" at 0.006,0.27 set xlabel "N^{-1}" set ylabel "{/Symbol D}" plot [.0:0.14] [:] g(x) t "" w l 1 1,g1(x) t "" w l 1 1,\ "kagome.gpl" i 1 u ($1**(-1)):(2*($3-$2)*$1) t "even" w p pt 4,\ "kagome.gpl" i 2 u ($1**(-1)):(2*($3-$2)*$1) t "odd" w p pt 6,\ "kagome.gpl" i 4 u ($1**(-1)):(2*($3-$2)*$1) smooth cspline t "l^2=4" w l lt 2,\ "kagome.gpl" i 5 u ($1**(-1)):(2*($3-$2)*$1) smooth cspline t "l^2=3" w l lt 3,\ "kagome.gpl" i 6 u ($1**(-1)):(2*($3-$2)*$1) smooth cspline t "" w l lt 3,\ "kagome.gpl" i 3 u ($1**(-1)):(2*($3-$2)*$1) t "" w p pt 2 set nolabel set noarrow if (ps==0) pause -1 if(ps==1) set term postscript eps enhanced mono dashed "Helvetica" 20; set out "tmp/t8_mag.eps" yy=0.22 yy=0.03 set key left top set label 5 "N:" at 48**(-1/2.),yy center set label 1 " 9" at 9**(-1/2.),yy center set label 2 "12" at 12**(-1/2.),yy center set label 3 "18" at 18**(-1/2.),yy center set label 4 "36" at 36**(-1/2.),yy center set xlabel "L^{-1}" set ylabel "m^+" plot [0.:0.4] [0.:0.5] m(x) t "",\ "kagome.gpl" i 1 u ($1**(-1/2.)):(($4)**.5) t "even" w p pt 4,\ "kagome.gpl" i 2 u ($1**(-1/2.)):(($4)**.5) t "odd" w p pt 6,\ "kagome.gpl" i 3 u ($1**(-1/2.)):(($4)**.5) t "" w p pt 2 if(ps==0)pause -1 exit # ################################################################# #i1 even, l^2_min>1 ll=... see triangle.dat (N=12..18 ea=1 tested) # SMag = \sum_{i,j}|S_iS_j|/(N^2) (0,60deg) # N=9*n for ABC-sublattice structure (localized magnons) # E/Nw base (1,0) (.5,.866) entartet ea>1 SMag=sum||/N^2 til8 12 -0.22686980 -0.21092529 0.18416013 % ll= 4, 4, 4 sym=12N/3 p6m EA=1 nonTri,Bi q00+i1c6=3,c3=0 q01-iX zeng90 e=2002 gap=0.3826682=Leung93 18 -0.22356308 -0.21565843 0.11563650 % ll= 4, 7, 9 sym= 8N/3 p2+ nonTri,nonBi q01-i0s1 q00+i0sX reel zeng90=fehlerE0=E2 3002=3012=6021=6031=> ri gap=0.2845674=Leung93 18 -0.22401341 -0.21651020 0.11723189 % ll= 3, 9,12 sym= 4N/3 pmm Tri,nonBi q00-i0x1y1 q00+i0x0y0 reel 3022=6011=6041 24 -0.22062428 -0.21629452 0.08884961 % ll= 7, 7,12 sym=12N/3 cmm nonTri,nonBi 24b q000+ 4012 24 -0.22316515 -0.21866958 0.09081985 % ll= 3,16,19 sym= 4N/3 pmm nonTri,nonBi 24c q000+ 4032=40-12 24 -0.22416453 -0.21866649 0.08673526 % ll= 4,12,16 sym=16N/3 p2+ nonTri,Bi 24 q20+ 4002=4022 ri? 30 -0.21923865 -0.21669106 0.06882069 # ll= 7,12,13 sym= 2N/3 p2 ea=2 nonTri,nonBi e=2105=2205 e0=? wf-dbl-q000-+ e0_000-=-13.15142626 e0_000+=-13.00146341 wf=3h states_between_Singlet_Triplet=88? 30 -0.22336206 -0.21970017 0.06946109 # ll= 4,19,21 sym=32N/3 p2+ nonTri,nonBi e=2005=2305 e0=??? e0000-=-13.35766837 e100=-13.38514400 e1=-13.18201000 EA(mmax,...)=1,11,35,... 30 -0.22294732 -0.22014342 0.06751563 # ll= 3,25,28 sym= 4N/3 p2+ nonTri,Bi e=11010 e0_105=-13.3768393 e0_010=-13.3285383 e0_000+=-13.20860491 e0_000-=-13.05504304 146min k(elow,d=4;5)=000,015; 36 -0.219188 -0.2169078 0.059125 # ll=12,12,12 sym=12N/3 p6m Tri,Bi q000+ e=6022 waldtmann++98 n1=32e6 =2206 E6= 0.28995925 EA(mmax..7/9)=1,13,55,71,8 (1,13,11+2*8+2*10+8,15+2*10+2*12+12,4+0+2*2+0) # # # e=6,0,2,2 mcf=2,2,-2,4 (0,60deg), mcf(0,120deg)=4,2,2,4 E1=-15.6175226=q0306- 36 -0.21954066 -0.21717572 0.05947829 # ll= 9,12, sym= 4N/3 1,4,0,12 E6= 0.30482374 n1=95e6 E000+=-15.80692775,-15.78624247 EA(mmax..7/9)=1,13,55,71,8 4isut.i185=22h ABC # ^^^Jul05^^^ ^^^^^^^^^^ not_in_fit # C.Zeng,V.Elser, PRB 42 p8436 (1990) # Numerical studies of antiferromagnetism on Kagome net # errata: table 2, N=18 E0,GAP, N=9 GAP, 36 -0.21964353 -0.21707582 0.0593460 # ll= 9,13, sym= 2N/3 1,3,0,12 nonABC E6= 0.32133375 n1=189e6 EA(mmax..7/9)=1,13,55,71,5 (k=-1 0*12 0*2 n1=378e6 8GB nzx=38.02 nzxmax=49 13*1leonardo.SH=33m) # E0(k=0+)=-15.81433399 E/2N=-0.21964353 13leonardo SH=33min # E0(k=6-)=-15.62945915 E/2N=-0.21707582 12leonardo SH=36min 4GB/2cpus #36 # ll= 7,19 sym= 2N/3 2,1,0,6 nonABC E6= 0.32569304 =2306=12012 EA(mmax..7/9)=1,13,55,71,7 # altix08: ?? ???? oldGS=wrong newGS(k!=0)! Apr08 see 36e #36 -0.22262229 -0.21662657 0.06311458 # ll= 4,28 sym=64N/3 2,0,0,6 nonABC k=0!!! E6=0 =2406=15012 k=0: -16.02880520 -15.52502099 sp=69h/3 -15.5971129=q_1_3_0_0-=-0.21662657 -15.4986320=q_0_3_0_0+ checked all a_b_0_0 # gap to big??? #36 n1=189e6 4GB # ll= 3,36,39 sym= 4N/3 2,5,0,6 E6= 0.00000000 0.14205687 =11012 ABC #42 ll=12,13,19 uc=14 7,0,2,2 nonABC sym=2N/3=28 l1=538e9/2 mem=2.7TB =3,-4,2,2 # #48 ll=16,16,16 uc=16 4,0,0,4 nonABC #54 ll=12,21,27 uc=18 6,0,0,3(60deg) 15,0,5,1???(120deg) ABC EA(mmax..)=1,19,136,430,685,119,4,.. #60 uc=20 nonABC #66 uc=22 nonABC #72 ll=21,21,36 uc=24 6,0,1,4 ABC EA(mmax..)=1,25,253,1329,... # EA(mmax..)=1,25,253,1329(=227+2*207+2*239+210)... Jul2004 # ToDo: spins v2.21 a0 gives wrong results (degeneracy is to high) # next_sym: 16,16,16(48), 19,19,19(57), 21,21,21(63), 25,25,25(75), ! # homogene triangles: n^2+3*m^2=ll = (0,1,4,9,16,25,...)+(0,3,12,27,...) # = 0,1,3,4,7,9,12,13,16,19,21,25,27,28,... # OR 1/4*((2n+1))^2+3/4*((2m+1))^2=ll = (1,9,25,49,...)/4+(3,27,75,...)/4 # = 1,3,7,13,19,21,... => ll*3=N_odd #24 ToDo #27 EA(mmax..7/9)=1,10, 26, 13 ABC + p6 (a0 used) 3Magnons #30b EA(mmax.. )=1,11, 35, 18, 10, 1 nonABC E(mmax)=N/2 E(m=7/9-2)=0 #33 EA(mmax.. )=1,12, 45 nonABC #36 EA(mmax.. )=1,13, 55, 71, 8 ABC (a4=a0 until mmax-3 proofed) 4Magnons #39 EA(mmax.. )=1,14, nonABC #42 EA(mmax.. )=1,15, nonABC #45 EA(mmax..7/9)=1,16, 91,201,110,4,.. ABC e=1,4,0,15 #45 EA(mmax..7/9)=1,16, 91,201, 97,0,.. nonABC e=1,3,0,15 #48 EA(mmax.. )=1,17, nonABC #54 EA(mmax..7/9)=1,19, 136,430,685,119, 4, .., .. ABC 6Magnons #57 EA(mmax.. )=1,20, nonABC but p6 #63 EA(mmax..7/9)=1,22, ABC p6 7Magnons #108 EA(mmax.. )=1,37, 595,..(by richter) ABC? #192 EA(mmax.. )=1,65,1953,..(by richter) ABC? # EA(2magnons)=(N/3)(N/3-7)/2+4*(N/3)/2-1? # N EA(mmax.. )=1,N/3+1,(N/3)(N/3-3)/2+1,... (if N%9=0) # # states between E0(S=1) and E0(S=0) (NST), degeneracy of groundstate (dE0): ToDo # N 18 21 24 27 30 30b 30c 33 36 # dE0 1 8 4 2 2 - 1 1 (*2 for odd N, Sud) # NST 14 48 127? 88 27 ? 40? >.. (*2 for odd N, Sud) #i2 odd ----------- test SMag+EA ---- ???? # SMag = \sum_{i,j}|S_iS_j|/(N^2) # base (1,0) (.5,.866) entartet ea>1 M=|mean| til8 # erst SiSj-mitteln, dann Betrag, wegen Werte nahe 0 # only N=9*n have ABC-SubLattice structure (for localized magnons) # l1=(nu+nd)!/(nu!nd!)/sym mem=(14..22)*l1 + hnz*6 # N E0/(2N) e=(0,60deg) 9 -0.22050010 -0.18398543 0.20525932 # ll= 3, 3, 3 sym=24N/3 p6m ea=4 zeng90 1 1 -2 1 15 -0.21963813 -0.20567811 0.13132004 # ll= 3, 3, 7 sym= 4N/3 cmm ea=1 zeng90 2 1 -3 1 e=5011=5021=5031 gap=0.4188006=Leung93 21 -0.21838761 -0.21175340 0.0965005 # ll= 7, 7, 7 sym=48N/3 p6 ea=8 zeng90 2 1 -1 3 e=7021=7041 gap=0.2786368=Leung93 21 -0.22053100 -0.21270744 0.10124700 # ll= 3,13,19 sym= 4N/3 cmm ea=2 e=7011=7031=7051 80m 27 -0.21813898 -0.21316163 0.07456556 # ll= 9, 9, 9 sym=24N/3 p6m ea=8 q00 k=-11 (0,6,3)/12 ea=3*7h 100m e=3003 lecheminant++97 SMag=3ZMag gemittelt ueber k=0,6,3,3 hoffentlich ok!? E0= -11.77950499 gap=0.2687769=Leung93 27 -0.21840734 -0.21330709 0.07579240 # ll= 7, 9,13 sym= 2N/3 p2 ea=1 e=3103=1209=1309 q000? am-flt e0_00=-11.78792604 e1_00=-11.51858296 am=214m 27 -0.22121262 -0.21550079 0.07760345 # ll= 3,21,27 sym= 4N/3 cmm ea= e=1109=1409 e0q10=-11.94548158 e0_01=-11.89135948 e0_00=-11.84915925 e1q40=-11.63704276 e1_00=-11.59626663 am=68m 33 -0.21833629 -0.2148597 0.06411854 # ll= 7,13 sym= 2N/3 p2 ea=1 nonABC e=1,3,0,11 l1=53e6 E-6=-1.02019593 ??? -14.41019505 -14.18074 == e=12011 33 -0.22153778 -0.21768851 0.06451766 # ll= 3 e=1,1,0,11 sym= 4N/3 p2 ea= nonABC l1=27e6 E-6=-1.35904977 E0_08=-14.6214932 E0_00=-14.31348115 th=59h k=-2 22k E1_02=-14.3674418 E1_11=-14.3154006=q_11_0(only_reel_checked)=E1/nw=-0.21690001 8wf.all=40h 39 -0.21820760 -0.21535 # ll=13,13,13 e=1,3,0,13 sym= 6N/3 p6m ea= nonABC l1=884e6*13=12GB+217GB hnz=41.00*n1 E0=-17.02019287 (korrekt sym?) E1=-16.7976? #45 l1=68.6e9 mem=960GB+18.5TB # ll= 9,21,39 e=1,4,0,15 sym= 4N/3 ABC EA(mmax..)=1,16,91,201,110,4,.. sym_k= -3 15k #45 # ll=13,13,25 e=1,3,0,15 sym= 4N/3 cmm nonABC EA(mmax..)=1,16,91,201, 97,0,.. sym_k= -4 15k #51 # ll=13,19,21 e=1,6,0,17 sym= 2N/3 p2 EA(mmax..)=1 nonABC #57 # ll=19,19,19 e=1,7,0,19 sym= 6N/3 p6m nonABC #63 # ll=21,21,21 e=1,4,0,21 sym= 6N/3 p6m ABC #69 #i3 p6m+p6 12 -0.22686980 -0.21092529 0.18416013 % ll= 4, 4, 4 36 -0.219188 - 0.059125 % ll=12,12,12 # 9 -0.22050010 -0.18398543 0.205259 # ll= 3, 3, 3 21 -0.21838761 -0.21175340 0.09650050 # ll= 7, 7, 7 27 -0.21813898 -0.21316163 0.07456556 # ll= 9, 9, 9 39 -0.218208 -0.21535 # ll=13,13,13 ToDo #i4 even, ll=4 12 -0.22686980 -0.21092529 0.18416013 % ll= 4, 4, 4 18 -0.22356308 -0.21565843 0.11563650 % ll= 4, 7, 9 #24 -0.22416453 -0.21866649 0.08673526 % ll= 4,12,16 24 -0.22416453 -0.21866649 0.08673526 % ll= 4,12,16 30 -0.22336206 -0.21970017 0.06946109 # ll= 4,19,21 #36 -0.22262229 -0.21662657 0.06311458 # ll= 4,28 gap to big? #i5 odd, ll=3 9 -0.22050010 -0.18398543 0.205259 # ll= 3, 3, 3 15 -0.21963813 -0.20567811 0.13132004 # ll= 3, 3, 7 21 -0.22053100 -0.21270744 0.10124700 # ll= 3,13,19 27 -0.22121262 -0.21550079 0.0771 # ll= 3,21,27 33 -0.22153778 -0.21768851 0.06451766 # ll= 3 e=1,1,0,11 E0_q8 E1_q2 #i6 even, ll=3 18 -0.22401341 -0.21651020 0.11723189 % ll= 3, 9,12 24 -0.22316515 -0.21866958 0.09081985 % ll= 3,16,19 30 -0.22294732 -0.22014342 0.06751563 # ll= 3,25,28