Incompressible Quantum Hall Fluid

1、incompressible quantum hall fluid after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al incompress

2、iblequantumhallfluid bo-yuhou anddan-taopeng instituteofmodernphysics,northwestuniversity xian,710069,p.r.china arx i v : h e p - t h /0210173v1 18 oct 2021 abstract afterreviewthequantumhalle ectonthefuzzytwo-spheres2andzhangandhus4-spheres4,theincompressiblequantumhall uidons2,s4andtorusarediscuss

3、edrespectively.next,thecorrespondinglaughlinwavefunctionsons2arealsogivenout.theadhmconstructionons4isdiscussed.wealsopointoutthatontorus,theincompressiblequantumhall uidisrelatedtotheintegrablegaudinmodelandthesolutioncanbegivenoutbytheyangbetheansatz. pacs:11.90.+t,11.25.-w keywords:quantumhalle e

4、ct,incomressiblequantumhall uid,noncommutativegeometry. 1introduction quantumhalle ects(qhe)havebeenanimportantresearchsubjectincondensedmatterphysicsandtheoreticalphysics.adecadebefore,laughlin1proposedthein-compressiblequantumhall uid(qhf)formulationforfractionalquantizationofthehalle ect(fqhf).so

5、onlaterhaldane2considereda2dimensionaleletrongasofnparticlesmovingonasphericalsurfaceofradiusrinaradial(diracmonopole)magnetic eldb= s email:byhouemail:dtpeng after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid

6、 on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al thepurposeofthispaperistogeneralizethedescriptionoftheqhfwhichhasbeenobtainedonthenoncommutative(n.c.)planer2andfuzzytwo-spheretozhangandhus4-sphereandtorus.inthenextsection,we rstrevie

7、wthehaldanesdiscriptionoftheqheon2-sphere,thenoncommutativealgebraandthemoyalstructureofthehilbertspaceisalsoshowninthissection.afterreviewzhangandhusgeneralizationoftheqheto4-sphere,wegiveoutthenoncommutativealgerawiththemoyalstructureonfuzzy4-sphereinsection3.insection4,byusingthemethodsuggestedby

8、susskind3andpolychronakos7,13,weconstructthen.c.chern-simonstheorytodescisbetheincompressibleqhfons2andgiveoutthealgebrastructureons2.thecorrespondingincompressibleqhftheoryforzhangandhus4-sphereisgiveninsection5.insection6,weconstructthedescriptionoftheincompressibleqhfontorusandrelateittotheintegr

9、ablegaudinmodelwhichcanbesolvedbythebethe-yangansatz.inthelastsection,weshortlygivesomesubjectswhichwillbeinvestigatedlater. 2 theoneparticlewavefunctionsinlllonthe rsthopf brations2 onthetwo-dimensionalspheres2=(s3su(2)/(s1u(1),thehamiltonianofasingleparticlewithchargeemovingaroundadiracmonopeleis

10、h= |2| 2 s , eb (1) heremisthee ectivemass,ristheraduisofs2andc= eb2 isthemagnetic eldstrenth andr = r |r| whichsatis esthecommutationrelations2 l,t=i t, + sr + s = (2) =l, tror (3) 2,l 2,l3and 2canbediagonalizedsimultaneouslyandtheircommontheoperatorss eigenfuntionsare j jm,s=dm,s(,), j|s|,m= j,j,(

11、4) j heredm,ssarethe niterotationmatrixelementsandtheindicesj,mandsarethe 2,l3ands respcetively.theeularangles=(,)eigenvaluesoftheoperatorsl after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and

12、 torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al equal=,=and=(,)whichhastheu(1)gaugefreedom.inlll(j=swithenergy1 2s! 2s! 2 )exp( i)= v v=sin 2 exp( i+i 21 21 (,), 2 =d 2 ,1 (1+|2)2 ,(7) andthecorrespondingsymplecticstructure(orthekahlerform)is =2i dd d

13、d, (8) wherek=ln(1+|2)isthekahlerpotential. thehilbertspacehnonthishopf brations2iscomposedbythen=2s+1oneparticlewavefunctionssm,(m= s,s)aroundthediracmonopoleg(s=ge).theoperatorsactingonthese2s+1statesarecovariant(2s+1)(2s+1)matrices,whichastheirreducibletensorialsetshouldbe(afternormalized) sjsj ,

14、0j2s,0mj,(9)(xm)m,m= mmmwhere(:)denotesthe3j-symbol.theseoperatorsconstitutetherightmodule14ofn.c.algebraanonfuzzyspheresimplywiththematrixproduct15: j1j2j j1j2j jj1j2 (xm)ln,(10)(xm)(xm)=1lm2mnsssm1m2m m j,m where:isthe6j-symbol.thismatrixproductisequivalenttothemoyalproducton fuzzysphere16.first,t

15、hecoherentstateswhichwerefoundbyperemolov17onthecosetspacej2=s2ingeodesicprojectioncoordinatesare s |)|,)=dm,(11) s(, )|s,m , m after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are di

16、scussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al whichsatisfy n 2j! 4 j1j2 thesymboloftheproductoftwooperatosxmandxequalsthe”starproduct”m12(moyarproduct)oftwosymbols,namely j1j2 dm1 dm2=( 1)j2 j1 m(2j+1) j,m j ddm(,)|(,)(,)|. (14) j1j2jm1m2m j1j2jsss jdm. (15) thi

17、sisaspecialcaseforthe productoftherightmoduleofanons214. 3 lllwavefunctionsasthesectionsofthe2ndhopft bundle onthe2ndhopfbundle( bration),zhangandhus4dimensionalfuzzyspheres4=(s7sp(4)/su(2)/(s3su(2),whichisequivalenttoso(5)/(so(3) so(3),thehamiltonianofoneparticleis h= 2 (1+r1 r2)(1+2r2)(2+r1+ r2)(3

18、+2r1)respectively.thesu(2)gaugepotentialvaluedbythesu(2)liealgebra is2 ii,ij=i ijkikandthecorrespondingcasimirisiii=i(i+1)whichspeci esthedimensionsofthesu(2)representationinthemonopolepotential.soforagiveni,fromthehamiltonian(16),wecanreadoutthatthedegeneracyoftheenergylevelisgivenbythedimensionali

19、tyofthecorrespondingirreduciblerepresentationd(r1,r2). 6 after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on

20、 $s2$ are al thegroundstate,whichisthelowestso(5)levellabeledby(r1=i=isn=1 p 2 ), p! ,p=m1+m2+m3+m4andk1=m3+m4,k2=2 m1+m2 ;ki,kiz(i=1,2)aretheeigenvaluesoftheangularmomentumofthe2 stable(keeps rinvariant)subgroupso(3) so(3)andi,izaretheeigenvaluesofthesu(2)isopin. thebasespaces4canbeparameterizedbyt

21、hefollowingcoordinatesystems 2 x1=sinsin cos( ), x3= sincos 2 2 cos(+), (19) x5=cos, where,0,)and,0,2).next,theorbitalcoordinatesxa(a=1,5),whichdescribethepointonthes4byxa=rxa,isrelatedthespinorcoordinates(=1,2,3,4)throughtherelation =1,xa=a=(a),(20) where(u1,u2)isanarbitrarytwocomponentscomplexvect

22、orsatisfyinguu=1andithasarotationsu(2)symmetrywhichmapstothesamepointxaons4.thecorrespondinggroupmanifoldiss3andthedirectionofthesu(2)isospinisspeci edbytheeularanglesi,i,i,whichdiscribethespinorsin: u1=cos i 2 ), u2=sin i 2 ). wherea,(a=1,5)arethe ve44diracmatriceswhichsatisfytheclifordalgebraa,b=2

23、ab.theexplicitsolutionof eq.(20)isgivenbyzhangandhu5: 3u1u11 =,(21)=(x4 ixii)4u2u2222(1+x5) (22) thegeometricconnectiongivesoutthesu(2)gaugepotentialaaoftheyangsmonopole de nedons4: i aa= 2 i andabisknownasthethoofttensor. after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang a

24、nd hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al byusingtheparameterizationofs4andthechoiceoftheisospincoordinates,theone-particlewavefunctioncanberewrittenas: k11,k12z;k

25、2,k2z;r2,=(sin) 1(1 cos)k1+ k1+1 (r,r) 1 2 , wherepnisthejacobipolynomialand k1i = k,m;i,iz|k2,k2z dm,k(,)duk1,k1z;k2,k2z;iz,iz(i,i,i)|i,iz .i,izz1z m,iz pr1+1+k2 k12(cos)uk1,k1z;k2,k2z;r2, (24) (25) theseone-particlewavefunctionsaretheyangssu(2)monopoleharmonics,i.e.the sphericalharmonicsonthecoset

26、spaceso(5)/su(2),whichislocallyisomorphictothespheres4s3. thehilbertspacehnonthissecondhopfbundle( bration)s4iscomposedbyalltheseone-particlewavefunctons.thecompletenessoftheseone-paticlewavefunctionsisensuredbythefollowingrelation: 1 dr1,r2=( pr1r20 2 ,q 2 ,p 2 ,0),(0,q r 2 t k1 j r 2 1 kj p p r 2

27、1 j k p n rr1r2jj1j2 rr1r2 p p2 after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al herer(r1,r2)

28、,j j1j2j1zj2z .thisalgebraisalsoaassociatealgbra. 4 twoincompressiblequantumhall uidons2 whenad0-braneentersad2-brane,itcandissolveintomagnetic ux,anditsdensityisequivalenttoamagnetic eldonthemembranewhilethedparticlecurrentsresultintheelectric eld.throughthet-duality,thed0-braneplaystheroleoftheele

29、ctonwhichistheendsofstringsond2-brane4,18,19,20,21.thisgiveacon gurationofthebranesandstrings-thequantumhallsolitonandthelowenergydynamicsdisplaythefractionalqhewhichcanbemodeledbyanoncommutativechern-simonstheory4.in3,susskindpropposedthenon-commutativechern-simonstheorytodescribetheareapreservingg

30、augetransformationoftheincompressibleelectrongasonaconstantmagneticb eld.heretheareapreservinggaugeis: xi=xi+ a jij 20 isthenoncommutativeparameter.andtheactionofthe noncommutativechern-simonstheoryis: s= k 3 a a a ,(30) wherekb=b andiscalledthe llingfraction.uptoatotaldivergent, thisnoncommutativec

31、hern-simonstheoryisequivalenttothematrixmodel: s= dt b 2 tr ab(xa+ia0,xa)xb+2a0 xa 2 + (i a0),(32) herexaisrepresentedbynnmatricesinmatrixtheory.isacomplexn-vector whichcomesfromthedropletboundarystates.thepotentialtermxa2 servesasaspatialregulatoranditbreaksthetranslationinvraianceoftheaction. inth

32、efuzzytwo-spheres2,theactionissimilarasthedropletinr2: s= dt b after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefuncti

33、ons on $s2$ are al hereisacomplexn-vectorwhichde nesthecp(n 1)modelmanifold22 andithasaleftglobalu(n)symmetryandarightlocalu(1)gaugesymmetrywhichcanbemapped (statisticalgauge eld)ford0 uidbytheseiberg-wittenmap.intothegauge elda thisaction,thetermeb 2s+1whichisthequantume ect. intheplanecase,theexpl

34、icitexpressionsofthelaughlinwavefunctionswasgivenin9.uponquantization,thegaussianconstraint(34)andthemomentmapequation(35)requirethephysicalstatestobesingletsofsu(n).thegroundstatebeingacompletelyantisymmetricsu(n)siguletwithandx hasthefollowingform: r2 wherex=12(x1ix2)whichsatisfytheconstrainequati

35、on(34)and|0 isannihilatedbyx+sands,wihletheexcitedstatescanbewrittenas |exc = n i=1 (trx )ci i1in i1(x )i2(x (n 1) |gr = i1in i1(x )i2(x (n 1) )ink|0 , (38) )ink|0 , (39) 1 onthefuzzytwo-spheres2,thekillingvectorsons2xd 1actonthe s state| hnwhichisequivalenttothesymboldm, s.thehamiltonianofthesystem

36、is 2 s 2)c(i h= (1+|j|2)2,(41) after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al andthesimplec

37、ticform = n j=1 2i jdjd (1 ij),x+=diag(z1,zn),and 2 theexcitedstate|exc astheeigenfunctionofthehamiltonian=tr(x )isexpressedbyjackpolynomial. i j 5 incompressiblequantumhall uidons4 onthefour-spheres4,wemaysetthebraneconstructionas(2,0)noncommutatityforelectronic uidorasiiasupergravityford0 uid. ont

38、hes4,thefourkillingvectorsalongthesurfacebecomethebasisi(i=1,4),thefundamentalsu(2)monopolewavefunctionsofc.n.yang23,24,whichiscon-formallyequivalenttotheinstantonon4-sphere.sos4hasahyperkahlersymplectic 1 metric.theleftglobalnnquarternionmatrix(n= 2 ,1 2222 , 111122,1221 , 121 , 2 11 , , 1 2 , 2 ,

39、, 1 after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al su(2)monopole(instanton).istheyangssu(2)

40、monopoleharmonicfunctionwith(p,q)=(1,0),here(p,q)isthesp(2)c2casimirwithp=r1+r2,q=r1 r2.inthehedgehoggauge,italsorepresentthe(anti)selfdualstationarysubgrouparoundtheradialdirection,e.g.sristhatalong r.inangularmomentumtheory,theconventional r rotationisdk,s.forthedetailofdfunctionpleaseconferto12.

41、nowthemodulispaceoftheqhfons4iss4 n/sn.next,wehavethehyperkahlersymplecticformforthisnoncommutatives4intermsofthesekillingvectors: = n i=1 dqidqi, (45) heretheqi(theji)dependsontheshiftofthe”center”ofn.c.solitonfromthenorth poleto(i,i,i,i). the”areapreserving”constrainequationwithquasiparticlesource

42、i,jturnstobex,x =(i+j+k)(1 n|v v|),(46) b1b2 wherex=q+b=x,b=,(i+j+k)areself-dualorbitalmomentum b2 b1 andwehavechooseni,jtogetanweylinvariant v|=1n(1,1). 6 . incompressiblequantumhallfluidontorus onthetorust,wechoosethefollowingcomovingframecoordinatesofelectrons: zi= 1 (xi+iyi),2 zi= 1 (xi iyi),2 (

43、i=1,2,n). (47) thenthemomentmapequationbecomes zj,zk = 1 after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$, the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on

44、 $s2$ are al sln(t)ontheellipticcurvetis e=( 1)(0) 1 (zjk) j k=j n (zji) i=j theweylre ectionisrealizedbyj kforejk(j=k).thed- atequationwithsourcethenis dz,=(1 n|v v|)2(u), inamorecommonbasis,leteijhave ejk,elm=ejmkl elkjm, .e0commuteswithe. b2 ,we=(0,0)(i)ije,where(i1,2)ab=a+1,2 zj (53) v|=(1,1).(5

45、4) thesl(n)bundle,thehamiltonianreducedbythemomentmapwithquasiparticleas asource,turnstobeadi erentialloperator(quantumlaxoperator)ofgaudinmodel lij=w(u)e(i)ij,lijij(55) =(0,0) where w(u)= (0)(u) after review the quantum hall effect on the fuzzy two-sphere $s2$ and zhang and hu's 4-sphere $s4$,

46、the incompressible quantum hall fluid on $s2$, $s4$ and torus are discussed respectively. next, the corresponding laughlin wavefunctions on $s2$ are al 7discussion inthispaper,wehavesucceededinconstructingthen.c.algebraofthes2ands4andgeneralizingthedescriptionsoftheincompressibleqhftothefuzzy4-spher

47、es4andtorust.onthetorus,wenoticethattheincompressibleqhfcanberelatedtotheintegrablegaudinmodelwhichcanbesolvedbythebethe-yangansatz. asseeninsection4,aftert-duality,thed0braneswhichenterind2branebecometheelectronswhichistheendsofthestringsond2brane.soforothernoncommutativemanifolds,itisinterestingto

48、relatethematrixalgebragiveninthispaperwithsomeapplicationsind-branedynamics. references 1ughlin,phys.rev.lett.50(1983)1395.2f.d.m.haldane,phys.rev.lett.51(1983)605. 3l.susskind,thequantumhallfluidandnon-commutativechern-simonstheory, hep-th/0101029.4b.a.berneving,j.brodie,l.susskindandn.toumbas,jhep

49、0102(2021)003, hep-th/0010105.5s.c.zhang,j.p.hu,science,294(2021)823.cond-mat/0110572. 6j.p.hu,s.c.zhang,collectiveexcitationsattheboundaryofa4dquantumhall droplet,cond-mat/0112432.7a.p.polychronakos,jhep0104(2021)011,hep-th/0103013. 8b.morariu,a.p.polychronakos,jhep0107(2021)006,hep-th/0106072.9s.hellerma