Heisenberg Model. Sayed Mohammad Mahdi Sadrnezhaad. Supervisor: Prof. Abdollah Langari

Transcription

1 snbrg Modl Sad Mohammad Mahd Sadrnhaad Survsor: Prof. bdollah Langar bstract: n ths rsarch w tr to calculat analtcall gnvalus and gnvctors of fnt chan wth ½-sn artcls snbrg modl. W drov gnfuctons for closd chan. ftr that w rnormald th rsult for ntracton btwn chans for ro tmratur. Fnall accordng to slf smlart f onts wr found from ffctv rnormalaton nrg.. Calculaton of Egnnrg and Egnstats of Fnt Chan -. Lngth Chan hr ar stats for lngth on chan: amltonan of ths chan for on chan n XXY modl s: Now w orat th amltonan n th stats: so th oratonal form of amltonan s ١

2 Egnvalus and gnstats ar: : : : : -. Lngth Chan hr ar 8 stats: h amltonan s: caus th chan s on th amltonan oratonal matr s block dagonal: m m For m and m gnvalu ar and obvousl gnfunctons ar and. n othr hand for m and m ar smmtrc and th ar sam. So t s nough to calculat gnstats for m : So w wrt matr oraton: m ٢

3 So gnvalus and gnstats ar: : 8 : : Lngth Chan h chan wth artcls has stats and ts amltonan has block smmtr too. W want to fnd ground stat so t s nough to solvng quaton for m and m. lso ths two hav sac smmtr so w solv th quatons for m thn w tnd th rsult to m. h stats whch w hav to solv ar: --. On Chan amltonan for on chan s: ٣

4 ٤

5 So amltonan matr s: h Calculaton of Egnvalu: h amltonan quaton s h h h dt h dtrmnant s andd to dgr quaton: o solvng ths quaton w hav to us vctor form to fndng gnvctors: and Now w us th smmtr to dtrmn gnvctor s lmnts n trm of ach othr. a α a a a a α a a a a a a a a α a a 6 a a ٥

6 a a - - α a a 6 β a a a β a a a γ a 9 γ a ] - -] - -] - -] β γ β γ β α β α o valuat th lmnt th gnvalu quatons for rd and 8 th lmnts ar: a a 8 a a 9 a a a a a a 8 a a 9 ] ] α β α γ β γ So ] α γ mans th solutons hav bn satsf ths condtons: ] or α γ Four gnvalus ar arrvd from ] : ٦

7 ٧

8 h grah of ths four valus s: ut for α γ thr ar s soluton whch wll b arrvd from a s dgr ducaton: whch should b solvd b Mathmatca. h grah of all gnvalus n trm of ar scratchd n ths grah: ٨

9 --. Closd Chan For chan w rwrt th amltonan: and amltonan oratonal matr bcom: ٩

10 Calculat Egnvalu: n closd chan th rodc condton maks a rotatonal smmtr. W call th rotaton oraton. So all of stats of wll b catgor n two grou. h stats ar shown n nw form: n othr hand th amltonan and rotaton ar commutatv. Easl w can rach to answrs wth solvng amltonan for onl two stats and : h wth th knowldg of that momntum s nvarant of amltonan gnvalus and gnstats ar: h h Ψ Ψ h ± ١٠

11 ١١ h h ± h Ψ Ψ

12 ١٢ h h ± h 6 Ψ 6 Ψ

13 ١٣ h h ± h Ψ 8 Ψ

14 ١٤ h h ± h 9 9 Ψ Ψ

15 ١٥ h gnvalus for ths stats n ntrval of and - n ordr of lowr to hghr lvl ar wrttn: ε : 9 Ψ Ψ ε : Ψ ε : 7 Ψ Ψ ε : Ψ Ψ ε : 8 6 Ψ Ψ ε : Ψ h grah of ths nrg stats n trm of s

16 ١٦. Rnormalaton ssum a chan wth N artcl. W can dvd ths chan to N chan wth artcl. So th amltonan of sstm can b wrttn wth two art: N N whr s ntrnal amltonan of ach chan wth artcl and t s what w calculat n last art. s ntracton of th chan wth nt chan. n ths roct th sstm s consdrd n ro tmratur so th ground stat s what w ar lookng for. s an aromaton w consdr ach sub-chan a closd chan bcaus w know th analtcal solutons of ths modl. h ground stat of w hav dgnrat stat whch two of thm ar m and two of thm hav m whch m stats ar sam as m stats wth rvrs sns. n rnormalaton th amltonan of sstm wll b wrttn as ffctv amltonan for mag of sstm n gnstats of stat sac of sub-chans and t wll b don wth P orator: P So ffctv amltonan wll b: / / N ff N ff t ff P P

17 ١٧ So for ach sub-chan ffctv amltonan s t ff P P ε ε ε So ffctv amltonan for ntracton btwn sub-chans s t t ff P P P P For two nghbor sub-chan th ntracton amltonan s and th ffctv ntracton amltonan wll b: ] ff Now w calculat ach Paul mtrcs: ] ] ]

18 ] ] ] ] ] ] ] t P P ] ] ] ] } ١٨

19 P ] ] t ] ] ] 8 P ١٩

20 ] ] } ] ] ] 8 ] ] ] ٢٠

21 ] ] ] ] ] ] ] t P P ٢١

22 ] ] P t ] ] ] P ] ] 8 8 ] ٢٢

23 ٢٣ ] ] ] ] ] ] ] ] ] ] t P P

24 ٢٤ ] ] ] ] ] 8 ff ] g f whr ] 8 g f

25 Fd onts ar whr our sstm wll dcras to fnt chan sstm whr th ffctv amltonan s smlar to fnt chan amltonan. n th nt grah f and g ar scratch n trm of and th valu for fd onts ar drvn:.6 g f h valus for f onts ar. -. whch rov our ctaton that th f onts ar n th rgon of smmtr and ths rsult shows ths calculats ar corrct. ٢٥