Bosonization: mapping of electron models onto a model describing collective excitations (charge, spin excitations, diffusion modes, etc).

Transcription

1 Exact bosonzaton fo nteactng femons n abtay dmensons. (New oute to numecal and analytcal calculatons) K.B. Efetov C. Pen H. Mee 1 1 RUB Bochum Gemany CEA Saclay Fance 1 Phys. Rev. Lett (9) A detaled dscusson: Phys. Rev. B (1) Numecal ealzaton: Evand Kandelak (RUB Bochum) Bosonzaton: mang of electon models onto a model descbng collectve exctatons (chage sn exctatons dffuson modes etc).

2 Ogn of the wod: 1D electon systems. A geneal femonc Hamltonan q q q V H ; Femonc antcommutaton ealtons. Howeve thee ae also bosonc vaables. k k ex

3 Can one efomulate the model n tems of the bosonc vaables? The man dea: wtng the electonc oeatos ex( ) ex dx A smle Hamltonan H N dx H [ K ] -Densty fluctuatons oeato K-comessblty N-aveage densty Imotance of long wave length exctatons! as Bosonzaton: fo Tomonaga- Luttnge model (long ange nteacton) (Luttnge Tomonaga (196?)) The most geneal fom conjectued by K.E. & A. Lakn (1975) Mcoscoc theoy Haldane (198)

4 Fomal elacement of electon Geen functons by oagatos fo collectve exctatons! Equvalent eesentaton Vey often dect exansons wth electonc Geen functons ae not effcent (nfaed dvegences hgh enegy cutoffs esectng symmetes). Sn exctatons Tansfomaton fom electons to collectve exctatons: Bosonzaton

5 Why should one bosonze the electonc systems? A. Geneal nteest to descton of low temeatue behavo. Man contbuton comes fo the collectve exctatons and t may be moe convenent to have the coesondng bosonc felds. B. Monte Calo smulatons ae dffcult fo the femons. The comutaton tme gows exonentally wth the nvese temeatue 1/T nteacton V and the sze of the system N. 5

6 A Dffcultes n Monte Calo smulatons fo femonc systems: negatve sgn oblem exonental gowth of the comutaton tme wth the sze of the system. T[ Aex( H)] T( H) Negatve Sgn Poblem A Standad MC ocedue Random choce of A Femonc sgn oblem ases when one of A A A sgn / sgn / sgn Asgn Exonentally nceasng tme due to the cancellaton oblem n sgn! Asgn sgn ex( cn) A ex( cn)

7 Can one bosonze n hghe dmensons? Eale attemts: A. Luthe 1979: Secal fom of Fem suface (squae cube etc.). Almost 1D. F.D.M. Haldane 199: Patchng of the Fem suface no aound cone scatteng Futhe develoment of the atchng dea: A. Houghton & Maston 1993; A.H. Casto Neto & E. Fadkn 1994; P. Koetz & Schonhamme 1996; Khveshchenko R. Hlubna T.M. Rce 1994 et al; C. Castellan D Casto W. Metzne 1994 Man assumton of all these woks: long ange nteacton

8 I.L. Alene & K. B. Efetov 6 Method of quasclasscal Geen functons sulemented by ntegaton ove suevectos Logathmc contbutons to secfc heat and suscetblty ae found. Good ageement wth known esults n 1D. No estcton on the ange of nteacton but not a full account of effects of the Fem suface cuvatue n d>1. In all the aoaches only low enegy exctatons wee consdeed: no chance fo usng n numecs. Pesent wok: Exact mang of femon models onto bosonc ones. New ossbltes fo both analytcal and numecal comutatons. Wanng: Exact n the themodynamc lmt! Snglng out slow modes can stll be convenent fo analytcal calculatons

9 Statng Hamltonan H nt H H H ; ; c c c c t H c c c c V H ; nt 1 The nteacton tunnelng and dmensonalty ae abtay! Small smlfcaton of the fomulas V V V c c c c V H () nt V / ) / ex( T H T Z

10 Hubbad-Statonovch tansfomaton wth a eal feld ) ( 1/T ) ( ) ( nt ex d H T H T Z c c ) ex( ) ex( ) ( nt nt H H H H Decoulng of the quatc nteacton D d V d c c T d H T nt 1 ex ex ex ex ex H c H c

11 Femons n an extenal feld Z 1 Z f V ex d D f t f Z f d ex T ln / ( ) Anothe eesentaton fo Z f det 1T ex Z f d Bass fo femonc MC Unleasant featue of Z f []: non-localty n tme Futhe tansfomatons ae desable! 11

12 Devaton of the model (man stes): 1 ; ; ex f du d G G Z Z ; G u ; G -femonc Geen functon n the extenal feld. Bounday condtons G G G ; G u ; ; G G s the Geen functon of the deal Fem gas

13 A Refomulatng the theoy n tems of bosonc felds A Bosonc eodc bounday condtons: G G Then the atton functon Z s A A Z b ex 1 Z A z dud z { u} n G -Foue tansfom of the Fem dstbuton n() n 1 e

14 14 (Almost) Fnal equaton fo A un z A u } { u z Fst check assumng that s small d d d k k d d e k A T A / / / / / k n k n k u k A k k Left hand sde of the equaton fo A: smlaty wth the Boltzmann equaton.

15 15 d d d d d k k k n k n V k d T Z Z / / / / 1 ln ex The soluton n the man aoxmaton. Random hase aoxmaton: Fst ode of exanson n the nteacton between collectve modes. Chances to constuct a feld theoy fo the bosonc nteactng exctatons! Wanng: uncetanty at k Howeve ts contbuton s small n the themodynamc lmt!

16 Analytcal calculatons: BRST (Becch Rouet Stoa Tyutn) -ossblty of ntegaton ove the auxlay feld befoe dong aoxmatons How to calculate B A f s the soluton of the equaton F A? A well known tck: A Ba Fa det da A F B a Next ste: Fa C ex ff a F det a ex F df / a dd -Gassmann vaables

17 Descton wth a suesymmetc acton and suefelds Intoducng new Gassmann vaables * and suefelds T * * R a z f z z z R u * s antcommutng but bosonc(!) The atton functon Z as the functonal ntegal ove Z Z ex S ss S sb

18 Z Z ex S[ ] D Fnal suefeld theoy (stll exact). S S B S S I Z -atton functon of the deal Fem gas S s the bae acton (fully suesymmetc) S * * * dr

19 * 1 1 * 1 drdr R R R V S B drdr R R V S I n R n R u R The nteacton tems.

20 S The tems and ae I nvaant unde the tansfomaton of the felds : * * * n * S -antcommutng vaables What to calculate? (Almost) suesymmety tansfomaton. Logathmc contbutons exst n any dmensons and they can be studed by RG. Reducton of the exact model to a low enegy one s convenent. Vaety of henomena fo e.g. cuates and othe stongly coelated systems can be attacked n ths way

21 Second ode etubaton theoy n both femonc and bosonc eesentatons. Geen functons. n 1 1 n n n n Femonc dagam (c) fom bosonc b.1 b. c. Necessty to exclude some states. 1 k k n n

22 u Monte Calo fo the bosonc model A z un The functons ( ) and ( ) ae taken at the same tme! z { u} Tycal Hubbad-Statonovch feld Z b [ ] Z f Howeve a good ageement s exected fo hyscal Z

23 Patton functon Z 1 Zb V ex D Z Bounday condtons b ex Z A z 1 A A dud Puely bosonc oblem! Lnea (almost seaable) eal equaton fo A Howeve: The soluton fo A s not unque and one should select one of them (not necessaly coesondng to the ognal one)

24 Regulazaton Remove all solutons of the homogeneous equaton (seudonveson). M u B un M A B Tkhonov egulazaton T T M M A M B Wtng a ostve eal Z b that dffes fom the ognal femonc one but the fnal esults ae exected to convege fo lage systems

25 Essental educton of the comutaton tme s stll ossble. 1. The sze of the matces to be nveted s N ( N -numbe of stes -numbe of tme slces) s N t s N t Howeve ths can be educed wokng n N sace. s N t The lmt can be taken fom the begnnng.. Hybd Monte-Calo can be effcently mlemented (s not fully used yet)

26 Howeve exact calculaton s stll ossble fo any sze of the system usng e-weghtng. A Z f Z f A Z Z f b Z f Zb Z b Zb Hubbad model: Aveagng wth ostve nstead of! Z f Z b t 1 V 4 =band wdth

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38 1 Consdeable femonc sgn oblem ( 1 1 ) n the egon nea

39 A consdeable educton of the comutaton tme s exected fom usng the hybd Monte-Calo ocedue (n the ocess of debuggng now)

40 Conclusons. The model of nteactng femons can be bosonzed n any dmenson fo any easonable nteacton. Can the bosonzaton be the key to the ultmate soluton of the sgn oblem? Can one solve non-tval models (e.g. models fo hgh temeatue cuates o fo quantum hase tanstons) usng the suesymetc model fo collectve exctatons? To be answeed