Fractional Topological Insulators A Bosonization Approach

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1 Journl of Modrn Physis Publishd Onlin Jnury 06 in Sis. rtionl Topoloil Insultors A Boniztion Approh D. Shmltzr Physis Dprtmnt City Coll of th City Univrsity of w Yor w Yor Y USA ivd Otobr 05; ptd 8 Jnury 06; publishd Jnury 06 Copyriht 06 by uthor nd Sintifi srh Publishin In. This wor is linsd undr th Crtiv Commons Attribution Intrntionl ins (CC BY). Abstrt A mtlli dis with stron spin orbit intrtion is invstitd. Th finit dis omtry introdus onfinin potntil. Du to th stron spin-orbit intrtion nd onfinin potntil th mtl dis is dsribd by n fftiv on-dimnsionl modl with hrmoni potntil. Th hrmoni potntil ivs ris to lssil turnin points. As rsult opn boundry onditions must b usd. W boniz th modl nd obtin hirl Bons for h spin on th d of th dis. Whn th fillin frtion is rdud to ν th ltron-ltron intrtions r studid by usin th Jordn Winr phs for omposit frmions whih iv ris to uttinr liquid. Whn th mtlli dis is in th proimity with suprondutor rtionl Topoloil Insultor is obtind. An primntl rliztion is proposd. W show tht by tunnin th hmil potntil w ontrol th lssil turnin points for whih rtionl Topoloil Insultor is rlizd. Kywords Spin-Orbit Chirl Bons Chins Mtlli Dis Topoloil Insultors. Introdution Th prsn of th spin-orbit intrtion in onfind omtris ivs ris to Topoloil Insultor (T.I.). ollowin rf. [] on mps th spin-orbit intrtion to spin dpndnt mnti fild B z. As rsult th non intrtin ltrons r mppd to two fftiv Quntum Hll problms for h spis of spin. Whn th ltron dnsity is tund to n intr ndu fillin ν (for h spin) th round stt is md up of two doupld spin spis whih form n intr Quntum Hll stt with opposit hirlitis. Whn is odd th systm is TI nd whn is vn w hv trivil insultor. Whn ν th prsn of ltron-ltron How to it this ppr: Shmltzr D. (06) rtionl Topoloil Insultors A Boniztion Approh. Journl of Modrn Physis

2 D. Shmltzr intrtion for h spis of spin ivs ris to rtionl Topoloil Insultor (.T.I.). Usin th proposl for th rtionl Quntum Hll whih is built on n rry of quntum wirs [] th uthors [] [4] hv shown tht by fin tunin th spin orbit intrtion for onfiurtion of oupld hins Topoloil Insultor (T.I.) mrs. Whn th fillin ftor is suh tht it orrsponds to omposit rmions rtionl Topoloil Insultor (.T.I.) hs bn introdud in rf. []. It hs bn shown tht for modl of oupld hins in th y dirtion th spin orbit intrtion n b ud wy rsultin in twistd boundry onditions for whih.t.i. ws obtind. Th purpos of this ppr is to dmonstt tht two-dimnsionl mtlli dis with spin-orbit intrtion nd ltron ltron intrtion ivs ris ithr to Topoloil Insultor or rtionl Topoloil whn th dis is in th proimity to suprondutor. W boniz [5] [6] th modl in th limit of stron spin orbit intrtions nd omtril onfinmnt. W find tht th d of th dis is quivlnt to on-dimnsionl modl with hrmoni potntil. W obtin hirl Boni modl [7] nd show tht T.I. mrs for th fillin ftor ν. or th fillin ftor ν w us th omposit ltrons mthod [8] [9] nd show tht th omposit Jordn Winr phs [0] ivs ris to n intrtin on-dimnsionl modl in th Boni form. W obtin uttinr liquid with th uttinr prmtr κ ν whih is th proimity to supr ondutor thn w obtin.t.i. An primntl vrifition is proposd. W show tht th.t.i. is obtind by tunnin th hmil potntil th intrtions nd th rdius of th dis. Th pln of th ppr is s follows. In Stion w prsnt th spin-orbit intrtions in mtls. In Stion. nd. w rviw th modl introdud in rf. []. W find it dvntous to us opn boundry onditions nd study th modl in th frmwor of Boniztion. In Stion. w introdu our nw modl. W onsidr mtlli dis with sron spin orbit intrtion nd onfinmnt. In Stion. w study th mtlli dis with stron spin orbit intrtions nd onfinmnt for th fillin ftor ν. Usin th omposit rmion mthod w obtin uttinr liquid whih in th proimity with suprondutor rprsnts.t.i. At th nd of this stion w onsidr th primntl rliztion of th modl. Stion 4 is dvotd to onlusions.. Th Spin Orbit in Two Dimnsions in th Prsn of Confinin Potntil Th Hmiltonin for two dimnsionl mtl in th prsn of prboli onfinin potntil is ivn by: ˆ µ H µ V ( ) m p E + y () γ Usin th onfinin potntil V ( y ) ( + y ) w obtin th ltri fild: E γ Ey γ y nd E z 0. W introdu fititious mnti fild µ B. As rsult th Hmiltonin in Eqution () is funtion of th spin orbit momntum nd ts th form: y H i z + i y + z m µ + V ( y) () is th ltti onstnt nd A y Ay r th u filds. 9

3 D. Shmltzr.. Th Emrin Topoloil Insultor for Two-Dimnsionl Modl Priodi in th y Dirtion with th illin tor ν for Systm of Coupld Chins In this stion w will rviw th modl introdud in rf. []. W find ssntil to modify th modl nd us opn boundry onditions. This modifition is importnt for voidin omplitions usd by th twist introdud by th spin-orbit intrtion. Th opn boundry onditions impos onstrint on th Boni filds (th riht nd lft Boni fild r not indpndnt). or th rminin prt w will boniz [7] th modl ivn in rf. [] usin th opn boundry onditions. W will us opn boundry onditions l for th mtli dis (s Stions.-.) Th mthodoloy for both modl will b sm thrfor w find it nssry to prsnt th dtils of th Boniztion mthod (for opn boundry onditions). Th modl onsidrd in rf. [] is s follows: In th y dirtion w hv hins with th tunnlin mtri lmnt t. Th onfinin potntil V( y ) obys for 0 < < V( y ) 0 nd for > V( y ). W will ssum opn boundry onditions in th dirtion. In th y dirtion th onfind potntil is fftivly zro for 0 < y < nd V( y ) for y > ( r th numbr of hins). W will us th onditions A y nd A y 0. H H + H 0 t γ y 0 d z µ n m γ H Ψ i + V Ψ t n+ n H t d Ψ Ψ + h.. H 0 is th on dimnsionl modl for h hin nd H t dsribs th tunnlin btwn th hins. Th V y nfors th opn boundry onditions. Ψ 0 Ψ 0 n (4) onfinin potntil Th opn boundry onditions void th twist. y y i i z z ; ( 0) ( ) 0 n Ψ Ψ Ψ Ψ Ψ Ψ As rsult th Hmiltonin is trnsformd H H0 + Ht γ y H0 dψ i z Ψ n γ dψ ( i ) µ Ψ n µ t d Ψ Ψ n+ +.. n ( yn yn ) z i + n n+ n H t h t dψ Ψ + h.. y n t w us th mppin n for n introdud by [] []. This prmtriztion rmovs th osilltin phs for rtin hnnls nn+ nd thrfor ps r opnd. () (5) 0

4 D. Shmltzr In th nt stp w boniz th modl for th fillin ftor nd th Boni phs is th spin orbit strnth). ν usin th ltroni dnsity n ϕ for h hin ( µ is th hmil potntil is th rmi momntum nd i dn i ± ϕ Ψ n n n + θ ( ) ( ) n θ p θ p iδ δ δ i ( θ ) n ϕn i 4 θ n nn i( θ + ϕ ( )) i 4θn n Ψ i i + r nti ommutin Klin ftors [0] [] []. Du to th boundry onditions th lft nd riht movrs r not indpndnt. Th Boni rprsnttion of. Th lft movrs r ivn by th rmion fild Ψ is ivn in trms of th riht movrs ( ). i i Ψ (7) W dfin nw hirl (riht movin) rmi fild Ω for > 0 nd Ω ( ) <. This implis tht th hirl rmioni fild Ω obys Ω ( ) Ω ( ). Ω is for 0 priodi in th domin < < (th domin 0 < < hs b nlrd to < < ). Usin th stp Ω. funtion θ [ ] w writ th rprsnttion of th hirl fild θ[ ] + ( ) θ[ ] W find: Ω (8) H H + H H H + H 0 t t t t 0 Ω n t 0 n+ n n t 0 n+ n n Ω H d i ; H t d + h.. ; H t d + h.. ; Th bul is pd nd only four hirl mods rmin plss ( ) ( ) Ω Ω Ω Ω (0) n n n n Th hirl d Hmiltonin is ivn by H hirl n H lft-d Hhirl n Hriht-d : H H + H ; hirl lft-d riht-d lft-d d Ω Ω + Ω n Ω H i i riht-d d Ω Ω + Ω Ω H i i Usin th proimity to suprondutor with th pirin fild stts r pd) without brin tim rvrsl symmtry. (6) (9) () w n p out th ds (th bul

5 D. Shmltzr H ˆ ˆ ˆ ˆ d i i Ω Ω + Ω Ω iδ ( ) ˆ ˆ iδ ˆ ˆ Ω Ω Ω Ω HC In th prsn of mnt whih brs rvrsl-symmtry th sptrum will l b pd out []... Th rtionl Topoloil Insultor for th illin tor ν t w will onsidr th modl t th fillin ftor ν. W us omposit rmions in on dimnsions nd Boniz th modl round (w mntion tht in on dimnsions w n Boniz round ny odd numbr of rmi momntum ). In this stion w will show how th mthod of omposit rmions wors in on dimnsions. Aordin to Eqution (6) omposit rmions is obtind whnvr n vn numbr of Jordn Winr d phss is tthd to rmion. If ± i nn i ϕn dsribs n ltrons omposit frmions ( ) d is obtind by modifyin th Jordn Winr phs to ± i n + nn i ϕn. As rsult on n + is obtind for obsrvs tht th Boni rprsnttion for th omposit frmions with ( ) ν. As rsult th Boniztion is invrint undr th frmi momntum nd fillin ftor mppin: nd ν ν. ollowin th stps ivn in Eqution (6) for th fillin ftor i dn i ; ± ϕ Ψ n n n + θ n θ p p ( ) i ( ) θn n δ δnn δ i ( θ ) i 4( ( n n ) ( ϕ θ + θ )) ; i ( θ ) i 4 ( n + ϕn θ + θ ) ; n i i Ψ + ; ; ; ptin th formultion ivn in Equtions (7)-(8) w hv i i Ψ Ω θ[ ] + ( ) θ[ ] ; ; ; ; ; ; () ν w find: In th nt stp w us th rltion nd obtin similr prssions to Equtions (8)-(). Th bul is pd nd only four hirl mods rmin plss ( ) ( ) n ; n ; n ; n ; () (4) Ω Ω Ω Ω (5) Th hirl d Hmiltonin is ivn by Hhirl n ; Hlft-d; Hhirl n ; Hriht-d; : H H + H ; hirl; lft-d; riht-d; lft-d; d Ω ; Ω + Ω ; ; Ω n ; riht-d d Ω Ω + Ω ; ; Ω ; H i i H ( i ) ( i ) (6)

6 D. Shmltzr Usin th rltion imposd by th opn boundry onditions with only on indpndnt Boni fild θ w hv: θ η θ η (7) W build from η nd η ( ) non-hirl Boni filds Θ Φ : η η η η Θ + Φ (8) v Hlft-d; d κ Φ n + Θ n κ ( ) v Hriht-d d κ ( n ) ( n ) Φ + Θ κ (9) v v0 κ ν This shows tht modl is uttinr liquid with th prmtr κ ν. Whn th hins r in th proimity with suprondutor w dd to th uttinr liquid Hmiltonin in Eqution (9) th pirin prt ivn in Eqution () (sond lin in Eqution ()). As rsult th modl of th oupld hins in proimity to suprondutor ivs ris to.t.i.. ollowin rf. [] th.t.i. is idntifid with th hlp of th Josphn priodiity whih msur th dnry of th round stt.. Th Mtlli Dis in th Prsn of th Spin Orbit Intrtion A liztion of Topoloil Insultor In this stion w prsnt our modl. it ws shown tht in stron mnti w n us th limit of lr mnti fild study th physis of ltrons in stron mnti filds [4]. w Usin th nloy with th stron mnti fild w propos to study th spin -orbit intrtion in th limit. As rsult on m dimnsionl modl in onfinin potntil mrs. or prboli potntil V( y ) with th ondition m w find onstrind Hmiltoni y h i + ( i ) µ + V ( y) m z y In th limit m w obtin V y z µ is rpld by on dimnsionl modl with prboli potntil... A liztion of Topoloil Insultor for th Mtlli Dis t illin tor ν In th sond quntizd formultion w find: H Ψ V y Ψ Du to th onstrind m modl. Th oordint y ts s th momntum. Th two dimnsionl prboli potntil V( y ) d z µ (0) + is rpld by on dimnsionl nd only rmins th oordint. W find: th potntil V( y ) ( y)

7 D. Shmltzr i H Ψ V y Ψ Ψ i + Ψ d z d µ µ () In th sond lin of Eqution () w hv usd th onstrint rltion whih mrs from th stron spin-orbit intrtion y This rsult is intrprtd s sond lss onstrind [5] [6] Th on dimnsionl fftiv modl ivn in Eqution () with th potntil llows to introdu sp µ µ dpndnt rmi momntum ˆ whr ˆ ± r th lssilly turnin points. W n mp this problm to th d of th dis. W introdu th nulr vribls for th d ( is th nulr vribl for th d whih is funtion of th oriinl oordint ). Th mppin btwn th sp dpndnt rmi momntum nd th nulr vribl is ivn by th funtion sin : Th turnin point ˆ µ sin for 0 ˆ sin for. ˆ ± uss th vnishin of th fild Ψ boundry onditions. As rsult w Boniz Ψ in trms of sinl movr. ± i dn ( ) i ϕ Ψ n n n + θ i ˆ d ( ) i ˆ d ( ). or this rn w must us opn Ψ Th rmi momntum is funtion of th hmil potntil µ instd of two rmi points ± th rmi momntum ( ) is dpndnt. Th vnishin points ( ) 0 iv ris to th fftiv d for th dis. is ivn by µ Du to th ft tht th rmi momntum is dpndnt w tht rmi vloity is l sp dpndnt ˆ () µ v ˆ In th nt stp w obtin th Boni rprsnttion for th mtlli dis. ( y> 0) ( y< 0) H H + H ( y> 0 ) ˆ ˆ H d y ; > 0 v i y ; > 0 y ; > 0 v i y ; > 0 ( y< 0 ) ˆ ˆ H d y ; < 0 v i y ; < 0 y ; < 0 v i y ; < 0 () ( y 0) H > rprsnts th Hmiltonin for th uppr hlf dis nd ( y 0) H < is th Hmiltonin for th lowr hlf. 4

8 D. Shmltzr Du to th turnin points w hv th rltions: ( ; 0 ) ( ; 0 ) ( ; 0 ) ( ; 0) y> y> y< y< (4) Usin th boundry onditions ivn in Eqution (4) w obtin for Eqution () th rprsnttion: ( y> 0 ) ˆ ˆ H d y ; > 0 v i y ; > 0 y ; > 0 v i y ; > 0 ˆ d ; 0 ; 0 ˆ y v i y > > ( y< 0 ) ˆ ˆ H d y ; < 0 v i y ; < 0 y ; < 0 v i y ; < 0 ˆ d ( y ; 0) v ( i ) ( y ; < 0) < ˆ t w mp th problm to th d of th dis. W find from th mppin th rltion d. Th trm v is rpld by th drivtiv on th boundry of th dis. d sin ˆ d v v d d ˆ df f ( ) ˆ dsin f ( ) ˆ d f ( ) ˆ 0 d 0 0 d W prss th Hmiltonin in Eqution (5) in trms of th hirl rmions on th boundry of th dis. W hv th mppin T ( ) ( ) T nd find: µ d ( )( ) ( ) + ( )( ) ( ) (7) H i i st w onsidr th proimity fft of suprondutor with th pirin fild ( ) iδ. As rsult of th pirin fild suprondutin p is opn on th ds. As rsults th Hmiltonin with th pirin fild ( ) iδ ivs ris to th Bonizd form of th T.I. Hmiltonin: (5) (6) µ H d i + i 8µ iδ + d ( ) ( ( ) ( ) + ( ) ( ) ) + HC.. (8).. Th Mtlli Dis in th Prsn of th Spin Orbit Intrtion A Composit rmion ormultion for.t.i. or prtiulr dnsitis th omposit frmions onstrution introdud by [8] n b usd. In on dimnsions th Jordn Winr onstrution llows to obtin omposit rmions. ptin th produr of sp dpndnt rmi momntum introdud in Stion. w find tht th turnin points dpnds on th hmil µ µ potntil. By hnin th hmil potntil to s > w obtin s µ ˆ µ. Th turnin points drss to. s s s Th onstrution of th omposit frmions lvs th position of th turnin point invrint. Th Jordn 5

9 D. Shmltzr Winr onstrution is bsd on th ft tht both Jordn Winr rprsnttions ± i dn ( ) ± i dn ( ) dsrib rmion. Th first rprsnttion rprsnts n intrtin rmion modl with th fillin ftor. Th sond on rprsnts non intrtin rmion modl with th fillin ftor. or th hmil potntil µ s ˆ µ. µ th omposit rmion with th momntum s. or s w obtin µ nd will oby th rltion ivin th sm turnin point or this s w rpt th formultion ivn in Eqution (). W rpl θ nd hirl bons θ θ. Du to th boundry onditions t th points ( ). W introdu θ ( ) η ( ) nd θ η ( ) ; ; i dn i ; ± ϕ Ψ n n n + θ + i ( θ ) i 4( ϕ θ + θ ) i 4 η + η( ) θ θ θ ϕ θ θ ; ; Ψ ϕ with th ˆ ± w us th rltions i ( θ + ϕ ) i 4 θ + θ i 4( η + η ) ˆ d ˆ d ; i ; i ; µ W mploy th mppin to th d of th dis. Whn w suprondutor is in th proimity of th dis th prin fild ( ) iδ will nrt p W introdu th filds: µ d H i + i 8µ iδ + d ( ) ( ( ) ( ) + ( ) ( ) ) + HC.. 9 η η η η Θ +Θ ( ) Θ + Φ Θ Φ +Φ Φ (9) (0) () 6

10 D. Shmltzr Θ msurs th hr dnsity whih is onjutd to Φ. W mp th Boni filds Θ nd Φ to th d of th dis: Θ Θ ( ) Φ Φ ( ). Th Boni form of th Hmiltonin in Eqution (9) rvls th uttinr liquid struturs with th intrtin prmtr κ ν. As rsult th hr stor rprsnts n.t.i.. v H µ d κ Φ + Θ κ ( ) ( ) 8µ + d ( ) os Θ( ) os Φ ( ) + δ 9 () v µ κ ν Comprin th rsults in Eqution () with th on ivn in Eqution (7) w noti tht κ nd th pirin oprtor is rpld by th symmtri form os Θ( ) os Φ ( ) + δ. As rsult th Josphn urrnt will b diffrnt for th two ss. Th us of th zro mod oprtors ivn in rf. [0] [7] n rvl th Josphn priodiity of th dnrt round stt. Whn th suprondutor is rpld by mnti systm p on th d of th dis vi spin-flippin bsttrin will ppr. In this s th Josphn hr urrnt will b rpld by Josphn spin urrnt []. Th primntl vrifition is don by msurin th Josphn urrnt btwn th mtlli dis nd th suprondutor whih will show diffrnt rsults for th T.I. nd th.t.i. Th primntl qustion is how to driv th dis to b ithr T.I. or.t.i. Our rsults show tht for th two ss w hv diffrnt turnin points µ for T.I. nd µ for.t.i. Th physil rdius of th 9 µ dis dtrmins wht stt n b obtind. Whn th rdius obys > th T.I. nd th.t.i. r possibl. W will hv ohrnt or mitur of th two phss. In ordr obsrv sinl phs w hv to µ µ hos th rdius to stisfy. or this s th phs with ν is not possibl ( th rin 9 rdius is shortr thn th turnin point ) from th othr-hnd th.t.i. with ν is possibl to obsrv. 4. Conlusions In th first prt of this ppr w hv prsntd th Boniztion for th modl introdud in rf. []. W hv found tht it is ssntil to us opn boundry onditions. This rsults r obtind by usin hirl Boniztion. Th rtionl s hs bn obtind with th hlp of th Jordn Winr trnsformtion for omposit rmions. In th sond prt w propos nw modl for rtionl Topoloil Insultor. W onsidr mtlli dis nd t dvnt of th stron spin orbit intrtion in th prsn of prboli potntil. W mp th problm to n on-dimnsionl modl with hrmoni potntil. On th d of th dis w find hirl frmion modl whih in th proimity to suprondutor ivs ris to rtionl Topoloil Insultor whn th rdius of th dis is tund to b lrr thn th frtionl turnin point. Th mppin to th on dimnsion llows showin tht th rtionl Topoloil Insultor mrs s n fftiv uttinr liquid modl for th fillin ftor ν. A possibl primntl rliztion of th modl is sustd bsd on tunnin of th hmil potntil nd th rdius of th dis. 7

11 D. Shmltzr frns [] vi M. nd Str A. (009) Physil viw ttrs [] Kn C.. Muhopdhyy. nd ubnsy T.C. (00) Physil viw ttrs [] Si E. Or Y. Str A. nd Hlpri B.I. (05) [4] Klinovj J. nd Tsrovny Y. (04) Physil viw B [5] Hldn.D.M. (98) Physil viw ttrs [6] uthr A. nd Emry V.J. (974) Physil viw ttrs 589. [7] lorni. nd Jiw. (887) Physil viw ttrs [8] Ji J.K. (007) Composit rmions. Cmbrid Univrsity Prss Cmbrid. [9] Bsu B. nd Bndyophdhyy P. [0] Shmltzr D. Kulov A. nd Mlrd M. (00) Journl of Physis: Condnsd Mttr [] Si E. nd Or Y. (04) Physil viw B [] uprt T. Sntos. yu S. Chmo C. nd Mudry C. (0) Physil viw B [] Ston M. (994) Boniztion. World Sintifi Publishin Co. Pt. td. Sinpor w Jrsy ondon nd Hon Kon. [4] I S. Krbli D. nd Sit B. (99) Physis ttrs B [5] Shmltzr D. (0) Journl of Physis: Condnsd Mttr [6] Winbr S. (0) turs on Quntum Mhnis. Cmbrid Univrsity Prss Cmbrid. [7] Boynovsy D. (989) Physil viw B