Supersymmetry in Disorder and Chaos (Random matrices, physics of compound nuclei, mathematics of random processes)

Transcription

1 Supesymmety n Dsoe an Chaos Ranom matces physcs of compoun nucle mathematcs of anom pocesses Lteatue: K.B. Efetov Supesymmety n Dsoe an Chaos Cambge Unvesty Pess Supesymmety an Tace Fomulae I.V. Lene J.P. Keatng D.E. Khmelntsk Kluwe NATO ASI Sees999 K.B. Efetov Aneson Localzaton an Supesymmety n 5 Yeas of Aneson Localzaton Wol Scentfc

2 Confeences an wokshops n Pas n Wokshop Supesymmety an Ranom Matces Instute en Poncae Apl 3-5 Wokshop Dsoee Quantum Systems Instute en Poncae May-July.

3 3 ' ' G A R U m n n n n A R G ' * ' U-anom <U>= n n n

4 Densty of states: Im G R Densty-ensty coelaton functon: R A K ' G ' G ' Non-lnea -moel Replca Wegne 979 Supematx Efetov 98 4

5 Ranom Matces The begnnng of the Ranom Matx Theoy RMT: Statstcal theoy of Complex Nucle E. Wgne 95 F. Dyson 96 The man assumpton: Matx elements mnof a amltonan of a complex system ae anom. The pobablty stbuton P: mn ae nepenent P Aexp m n a mn 7

6 Thee classes of unvesalty: othogonal untay an symplectc. Othogonal: tme evesal an cental nveson nvaance.. Untay: the tme evesal nvaance s boken. 3. Symplectc: the cental nveson nvaance s boken but the tme evesal one s not. R ot Level-level coelaton functon R unt R sympl sn x sn x sn xt x t x x x t sn x x x sn x x x x sn x x sn xt t t R x s the mean enegy level spacng 8

7 Dsoee systems: Schonge Equaton: ea c m U s s Aveagng ove mputes: E U U U U s a potental escbng scatteng by mputes the othes ae scattengs by magnetc an spn-obt mputes. U U ' ' No possblty to solve the equaton exactly fo an abtay soe. s so s the ensty of states s the mean fee tme 7

8 Dagammatc expansons fo the Geen functons G an subsequent aveages ove the anom potental Abkosov Gokov Dzyaloshnsk 96 G ' ' Expanson n Summaton of non-cossng agams: G R A p p / Whee ae the Wgne-Dyson fomulae an anom matx theoy? 8

9 Only sngulates mght help to fn somethng nontval f exste. Dffuson moes coopeons an ffusons Gokov Lakn an Khmelntsk 979 k Dk D s the classcal ffuson coeffcent As vegence fo = flms wes small metal patcles anom matces

10 Non-lnea supematx -moel fo escbng localzaton an not only effects Fo any coelaton functon O expesse n tems of the Geen functons Due to supesymmety Q F St[ D Q Q] 8

11 The man eas Gassmann antcommutng vaables : j Integals Beezn 96: All othe ntegals ae epettons of these two.

12 The most mpotant ntegals the bass of the metho exp * A * et A Not as fo conventonal et A complex numbes! Supevecto: S Supematx: a q Stq b a b - antcommutng S a b -conventonal St PP St P P St PP P St P P 3 3 P

13 3 D U G A R ] exp[ ' A A exp No weght enomnato! The bass of the metho. Scala pouct * * S S S * S * Tansposton T P P b a P T T T T T b a P All ules fo the supeobjects ae the same!

14 4 A possblty to aveage mmeately ove the anom potental U! D G A R ] [ 'exp ' U ' ' U U D G G A R ] 'exp[ ' ' ' / / The soe s avoe but an effectve nteacton appeas nstea. The next ea: Spontaneous beakng of the supesymmety exstence of Golstone moes.

15 A spontaneous aveage appeas! Q Q -s an 8x8 supematx A self-consstent soluton fo Q leas to Q A geneal stuctue fo Q: Q VV VV Degeneacy of the goun state gapless Golstone moes. The fee enegy functonal F can be obtane expanng n small gaents of Q the fequency s assume small. Ths s a way how one comes to a non-lnea supematx -moel. 5

16 F St[ D Q Q] 8 Physcal quanttes as ntegals ove the supematces B Qexp F[ Q] DQ Ang magnetc o spn-obt nteactons one changes the symmety of the supematces Q othogonal untay an symplectc. Depenng on the mensonalty geomety of the sample one can stuy ffeent poblems localzaton n wes an flms Aneson metal-nsulato tanston etc. Eveythng that can be wtten n tems of poucts of Geen functons can be expesse n tems of an ntegal ove the supematces wth the -moel

17 The explct stuctue of Q Q UQU U u v uv contan all Gassmann vaables All essental stuctue s n Q Q ^ cos ^ e sn ^ ^ e sn ^ cos ^ untay ensemble Mxtue of both compact an non-compact symmetes otatons: otatons on a sphee an hypebolo glue by the antcommutng vaables. 7

18 8 Explct fom ^ ^ Othogonal Untay ^ ^ Symplectc ^ ^ /

19 Ranom Matces Small Metal Patcles Zeo Dmensonalty. What s zeo mensonalty? In a fnte volume one comes to the space quantzaton of the ffuson moes: D / L D n E n s the Thouless Enegy n E n n D / L n 3... Zeo mensonalty D D / L In D only the moe wth n= s mpotant the - moel s zeo mensonal. 9

20 R The level-level coelaton functon Rx A R G y' y' G y' y' ' 6 V A R G y yg y y ' y s the spn vaable x V s the mean level spacng V s volume. R x Q Q exp F [ Q] Q F [ Q] St 4 Q Defnte ntegal ove the elements of Q D / L Eveythng s applcable f D / L s necessay. Ths s possble fo weak soe n thck wes D an 3D.

21 The ntegal ove the supematces Q n a moe human fom: exp Re x R oth exp Re x R sympl exp Re x R unt Calculaton of the ntegals gves the coesponng fomulae fo the Wgne-Dyson ensembles: poof of the elevance of the RMT fo soee metal patcles Efetov 98.

22 The begnnng of the Ranom Matx Theoy RMT: Statstcal theoy of Complex Nucle E. Wgne 95 F. Dyson 96 The man assumpton: Matx elements mn of a amltonan of a complex system ae anom. The pobablty stbuton P: mn ae nepenent P Aexp m n a mn

23 Thee classes of unvesalty: othogonal untay an symplectc. Othogonal: tme evesal an cental nveson nvaance.. Untay: the tme evesal nvaance s boken. 3. Symplectc: the cental nveson nvaance s boken but the tme evesal one s not. R ot Level-level coelaton functon R unt R sympl sn x sn x sn xt x t x x x t sn x x x sn x x x x sn x x sn xt t t R x s the mean enegy level spacng 3