When Is A “wavefunction” Not A Wavefunction?
APS March Meeting, Baltimore March 20, 2013 N21.1When is a “wavefunction” not a wavefunction?A quantum-geometric interpretation of the Laughlin stateF. D.M. Haldane, Princeton University The FQHE is a fundamentally problem of interacting“guiding centers” with a non commutative geometry:what is the true meaning of Laughlin’s “wavefunction”? An interpretation of “z” in the Laughlin “wavefunction”as the intrisic geometry of flux attachment.Supported by DOE-SC0002140 and the W. M. Keck FoundationWednesday, March 20, 13
Quantized motion of three two-dimensional electrons in a strong magnetic fieldR. B. LaughlinPhys. Rev. B 27, 3383 – Published 15 March 1983 In 1983 Bob Laughlin’s numericaldiagonalization study of a systemof 3 interacting electrons (!) in thelowest Landau level led him to hisfamous wavefunction. Yi j(zizj )qYe1 2 zi zii It quickly became clear that this wasthe correct solution to the puzzle ofthe 1/3 FQHE and was the prototypemodel for the other FQHE statesWednesday, March 20, 13x iyz p2 B
The Laughlin state was convincing because itexplained why ν 1/3 was seen, but not ν 1/2. Its validity was subsequently established bynumerical exact diagonalization studies of theadiabatic evolution between a model system with a“short range pseudopotential” for which theLaughlin state was the exact ground state, anda realistic Coulomb interaction.FDMH and E. H. Rezayi, 1985Wednesday, March 20, 13
So it is known to work, but why? (In myopinion, this question was never satisfactorilyanswered)a common rationalization:“Laughlin’s wavefunctioncleverly lowers theCoulomb correlationenergy by placing its zeroesat the locations of theparticles”we will see that this is an empty statementWednesday, March 20, 13
A key aspect of the Laughlin wavefunction isthat is a holomorphic function (times anon-holomorphic Gaussian) This is the lowest-Landau level property: a a (z, z ) 0(annihilated by Landau-level lowering operator) (z, z ) f (z)eholomorphicfunctionWednesday, March 20, 131 2z zGaussiana† @12 z @z @1 z2@z†[a, a ] 1Schrödinger form ofLandau level(harmonic oscillator)ladder operators
Laughlin presented his state as a lowest-Landau-levelwavefunction, and this interpretation seems to havebeen uncritically accepted. But I will argue thatits holomorphic character has NOTHINGto do with the lowest Landau level (LLL)!some evidence against the LLL interpretation The Laughlin state also occurs in the second Landau levelThe Laughlin state occurs in models of Chern insulators with flatbands (no Landau levels)(In fact, the Laughlin state should not even beinterpreted as a Schrödinger wavefunction .)Wednesday, March 20, 13
When there is Landau quantization, the classical(with commuting components) displacement of anelectron splits up into a non-classical guidingcenter displacement plus the cyclotron-orbitradial vector:ab[r , r ] 0 R̃ rx ROantisymmetric symbolarea per London fluxquantum 2 2B[R̃a , R̃b ] i 2B abra R̃a R̃aguiding center of orbitab[R , R ] Wednesday, March 20, 132 abi B [R̃a , Rb ] 0independent
ab[R , R ] 2 abi B dicembre 1901 – Monaco di Baviera, 1º febbraio1976) è stato un fisico tedesco. Ottenne il PremioNobel per la Fisica nel 1932 ed è considerato unodei fondatori della meccanica quantistica. The remaining degrees of freedom afterIndice1 Meccanica quantistica2 Il lavoro durante la guerra3 Bibliografia3.1 Autobiografie3.2 Opere in italiano3.3 Articoli di stampa4 Curiosità5 Voci correlate6 Altri progetti7 Collegamenti esterniLandau quantization are the non-classicalguiding centers. They define aquantum geometry isomorphic to phasespace, and obey an uncertainty principleMeccanica quantistica They cannot be described by aQuando era studente, incontròGottinga nel 1922. Ciò permise lo sviluppo di unafruttuosa collaborazione tra i due.Werner Karl Hper laSchrödinger wavefunction! (only by aHeisenberg state)Heisenberg ebbe l'idea della, la prima formalmeccanica quantistica, nelprincipio di indeterminazioafferma che la misura simultanea di due variabili coniugate, comemoto oppure energia e tempo, non può essere compiuta senza un'inAssieme a Bohr, formulò l'della meRicevette il Premio Nobel per la fisica"per la creazione dquantistica, la cui applicazione, tra le altre cose, ha portato alla scoallotrope dell'idrogeno".The guiding centers are generic to any Landau levelIl lavoro durante la guerraWe must abandon the Schrödinger picture and anduse a Heisenberg description of the guiding center1 of 1La fissione nucleare vennescoperta in Germania nel 1939. HeisenbGermania durante la seconda guerra mondiale, lavorando sotto il reprogramma nucleare tedesco, ma i limiti della sua collaborazione sRivelò l'esistenza del programma a Bohr durante un colloquio a Cosettembre 1941. Dopo l'incontro, la lunga amicizia tra Bohr e Heisebruscamente. Bohr si unì in seguito al progetto Manhattan.Wednesday, March 20, 13Si è speculato sul fatto che Heisenberg avesse degli scrupoli moral
In Laughlin’s “symmetric gauge” scheme, the guidingcenters are described in a basis of circular statescentered at an arbitrary origin This circular shape is also the shape of the Landau orbits0m (z, zā†ā1 2z1 2z m)/z e@@z @ @zcircular basis guiding centerladder operators1 2z zLandau orbitladder operatorsaction on LLL states:Wednesday, March 20, 13
The Heisenberg form of Laughlin is therefore anunentangled product of a correlated guidingcenter state with a (trivial) harmonic oscillatorstate of the Landau orbits: qLi0 @Y i j†āi 0 iA āi 0 i 0at this point, we“purify” the Laughlinstate by removing itsLLL “baggage”Wednesday, March 20, 13†āj q1ai 0i0i 0
orbits are now gone: what The Landau†ādefines†[L, āi ] and ā ?†āixshape ofLandau orbitshape ofbasis of guidingcenter statesnot necessarilycongruent!Wednesday, March 20, 13L 12X iā†i āi āi ā†i 1 Xa b 2ḡab Ri Ri2 B iThe guiding center metric gab thatdefines the basis of guiding-centerladder operators is an INDEPENDENTvariational geometric parameter of theLaughlin state, that must be chosen tominimize the correlation energy
the guiding-center metric is a parameter ofthe pseudopotential Hamiltonian that can beused to define the Laughlin state:H(g) Zd2 q 2B XVm Lm (qg2 2B )e2 m1 2 22 qg B It is the shape of thecomposite boson formedby “flux attachment”Xe i R j)i q ·(Ri j2qgab g qa qb“Elementary droplet” or“composite boson” of 1/3 Laughlin state:central orbital filled, next two emptyarea-preserving zero-pointfluctuations of shape give q4 structure factorWednesday, March 20, 13
In generala a† x1@ā 2 z̄ @ z̄ †1 @ā 2 z̄@ z̄@12 z @z @1 z2@zguiding-center flux attachmentshape congruent toLandau orbitscongruent to z̄ constant z constant z̄ z z 22 1 “zeroes of wavefunction” only coincidewith locations of particles if β 0.Wednesday, March 20, 13
So where do the holomorphic functionscome from?coherent states of the guiding centersare non-orthogonal and overcomplete:ā z̄i z̄ z̄i00 )hz̄ z̄ i Sḡ ( r, rWednesday, March 20, 13complex positive-indefiniteHermitian overlap matrix thatdepends implicitly on the metric
The overlap defines a “quantum geometry” a “quantum distance” measure is defined by0 20 ) S( r, r ) 1d( r, r The non-null eigenfunctions of the overlap definean orthonormal basis:Z2 0d r0) S( r,rḡ22 B Wednesday, March 20, 131i psZ0( r) sd2 r2 2B( r)( r ) z̄( r )i
For the guiding-center coherent states, thenon-null eigenstates of S are degenerate, 1 z̄z̄with the form f (z̄ )e 2 Coherent-state0holomorphic!representation of Laughlin state:Y Z d2 ri Y @ i/(z̄i22 Bii j qz̄j )Ye 1z̄z̄ii2iLaughlin “wavefunction”, with replacementunchanged when composite boson has same shape as Landau orbit!Wednesday, March 20, 131A {z̄i }iz ! z̄ (z̄ z )
The metric is the true collective degree offreedom of the FQHE state: it adjusts tonon-uniform electric fields. Its curvaturefrom spatial nonuniformity provides anadditional gauge field similar to Berrycurvature in quantum Hall ferromagnetscutgives“Area” (perimeter)term in “momentumpolarization”Wednesday, March 20, 13relation toHall viscositythrough edgecurrentsproduced byentanglementcuts
V(x)fluid is compressed at edges es 0by creating Gaussian 0near edges:J Jegcurvature2 fluid densityfixed by flux densityx g Jg0 (x)001 (x)1 d2 12 dx2 (x)For larger s, fluid becomes more compressible(less distortion needed for a given density change)Wednesday, March 20, 13
R, ii p “flat bands”,i s.t. geth a similar i 1 On ChernP̂ Insulatorei(R1 , R2 , R3 ) ei' (R1 , R2 ) i'eR,i(R2 , R3 ) i'eR, i (R3 , R1 )R, iR, istructure by projecting lattice-site orbitals! We renormalizeeach band,projectedorbitalto unit weight: back tointoasingleandrenormalizingX †XX †ipchosens.t.h i 10 ,j h.c.][cecR0 ,j h.c.] MH tunit[cc tR,iR,iR,i12Rweight:R,iR,iR,hNNNihNNiBAHf /XR,i,R0 ,jVij (RR0 ) · c̄†R,i c̄†R0 ,j c̄R0 ,j c̄R,i! Resulting orbitals are non-orthogonal†and overcomplete. 0{c̄R,i , c̄R0 ,j } Sij R, RAnalogousto normalized,single-particlestates in the LLLThe overlapmatrixencodes coherentthe quantum(naturalbasis forindescribingFQHtoo,states)geometrythis casedifferencea showsa!a (Rr ) g (r)i 0 between topological and non-topological ! !g !aba bb !abandsdet(g) 1Wednesday, March 20, 13
summary The variable “z” in the Laughlin“wavefunction” depends on guiding centergeometry, not Landau orbit shape, which arechosen to be congruent in Laughlin’streatment The Laughlin “wavefunction” is really aguiding-center coherent-state amplitudeWednesday, March 20, 13
The Laughlin state also occurs in the second Landau level . Nobel per la Fisica nel 1932 ed è considerato uno dei fondatori della meccanica quantistica. Indice 1 Meccanica quantistica 2 Il lavoro durante la gu
6.1 Laughlin States 198 6.1.1 The View from the Wavefunction 198 6.1.2 The View from Chern-Simons Theory 200 6.1.3 The Chiral Boson 205 6.1.4 Electrons and Quasi-Holes 207 6.1.5 Tunnelling 212 6.2 The Bulk-Boundary Correspondence 214 6.2.1 Recovering the Laughlin Wavefunction 214 6.2.2 Wavefunction for Chern-Simons Theory 219
- Laughlin wavefunction. The initial ground-state wavefunction we introduced in the last lecture by Roman, had the form: 4(z) Y k l (z k z l)me P N i jzi 2 l2 B where l B q Be. By disturbing/ "creating a hole" in the electron density distribution at a point 2C, we get a new factor in the wavefunction: (z; ) YN i (z i ) Y k l (z k z l)me .
evidence for the veracity of the Laughlin wavefunction. While it’s certainly impressive, this isn’t the reason that the Laughlin wavefunction is interesting. Finding the ground state numerically is dicult and can only be done for a couple of dozen particles. While this may provide 99.99% overlap with the Laughlin wavefunction, by the time .
1-D BOX 1-D BOX DENSITIES. When drawing a wavefunction by inspection: 1. The wavefunction of the . n. th Energy level has . n-1 zero crossings 2. Higher kinetic energy means higher curvature and lower amplitude. 3. Exponential decay occurs when the Kinetic energy is “smaller”than the Potential energy.
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4.2 Molecular orbitals and atomic orbitals Consider a linear chain of four identical atoms representing a hypothetical molecule. Suppose that each atomic wavefunction is 1s wavefunction. This system of identical atoms has a center of symmetry C with respect to the center of the molecule (midway between the second and the third atom), and all m.
akuntansi musyarakah (sak no 106) Ayat tentang Musyarakah (Q.S. 39; 29) لًََّز ãَ åِاَ óِ îَخظَْ ó Þَْ ë Þٍجُزَِ ß ا äًَّ àَط لًَّجُرَ íَ åَ îظُِ Ûاَش
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