Conformal Field Theory of Composite Fermions in the QHE

Transcription

1 Conformal Field Theory of Composite Fermions in the QHE Andrea Cappelli (INFN and Physics Dept., Florence) Outline Introduction: wave functions, edge excitations and CFT CFT for Jain wfs: Hansson et al. results CFT for Jain wfs: W-infinity minimal models independent derivation of Jain wfs from symmetry arguments CFT suggests non-abelian statistics of quasi-holes

2 Quantum Hall Effect dim electron gas at low temperature T ~ 10 mk B y x and high magnetic field B ~ 10 Tesla J y E x Conductance tensor J i = ¾ ij E j ; ¾ ij = R 1 ij ; i; j = x; y Plateaux: ¾ xx = 0; R xx = 0 no Ohmic conduction gap ¾ xy = R xy 1 = e h º; º = 1( 10 8 ); ; 3; : : : 1 3 ; 5 ; : : : ; 5 ; High precision & universality Uniform density ground state: ½ o = eb hc º Incompressible fluid

3 Laughlin's quantum incompressible fluid Electrons form a droplet of fluid: incompressible = gap fluid = ½ ½ ½(x; y) = ½ o = const: x y q N º R # degenerate orbitals = # quantum fluxes D A = BA o ; o = hc e filling fraction: º = N D A = 1; ; : : : 1 3 ; 1 5 ; : : : density for quantum mech. º = 1 º = 1 3

4 Laughlin's wave function ª gs (z 1 ; z ; : : : ; z N ) = Y i<j (z i z j ) k+1 e P jz i j = º = 1 k+1 = 1; 1 3 ; 1 5 ; : : : º = 1 filled Landau level: obvious gap º = 1 3 non-perturbative gap due to Coulomb interaction effective theories! = eb mc À kt quasi-holes = vortices ª = Y i ( z i ) ª gs ª 1; = ( 1 ) 1 k+1 Y ( 1 z i ) ( z i ) ª gs i fractional charge Q = e k+1 & statistics µ ¼ = 1 k+1 Anyons vortices with long-range topological correlations

5 Conformal field theory of edge excitations The edge of the droplet can fluctuate: edge waves are massless t V ½ Fermi surface µ R / J edge ~ Fermi surface: linearize energy "(k) = v R (k k F ); k = 0; 1; : : : relativistic field theory in 1+1 dimensions, chiral (X.G.Wen '89) chiral compactified c=1 CFT (chiral Luttinger liquid)

6 CFT descriptions of QHE r z = re iµ ³ = e e iµ plane (bulk excit) cylinder (edge excit) µ (z 1 z ) anyon wavefunction h Á 1 Á i CF T (³ 1 ³ ) edge-excit. correlator wavefunctions: spectrum of anyons and braiding matrices edge correlators: physics of conduction experiments equivalence of descriptions: analytic continuation from the circle, use map CFT Chern-Simons theory in +1 dim general CFT is U(1) x neutral

7 Non-Abelian fractional statistics º = 5 described by Moore-Read Pfaffian state ~ Ising CFT x U(1) I Ã Ising fields: identity, Majorana = electron, spin = anyon fusion rules: Ã Ã = I electrons fuse into a bosonic bound state ¾ ¾ = I + Ã channels of fusion = conformal blocks h¾(0)¾(z)¾(1)¾(1)i = a 1 F 1 (z) + a F (z) ¾ state of 4 anyons is two-fold degenerate (Moore, Read '91) statistics of anyons ~ analytic continuation x matrix µ F1 F ze i¼ = µ µ F1 F 1 + (z 1)e i¼ = µ F1 (z) F µ µ F1 (z) F 0 z z 1 1 (all CFT redone for Q. Computation: M. Freedman, Kitaev, Nayak, Slingerland,...)

8 ª º= 1 p+1 = Y i<j z p ij Jain composite fermion Correspondence FQHE IQHE generalize to n filled Landau levels ª º= n np+1 = P LLL Y i<j z ij = Y i<j N N e = 1 º = p º = 1 B 1 º = p + 1 n Y z p ij ª º =n (¹z i ; z i ) i<j z p ij ª º=1; p even z ij = z i z j composite fermion: quasiparticle feeling the reduced B = B p ½ o 1 º = 1 n B many experimental confirmations + mean field theory (Lopez, Fradkin;...) ª º= n written directly in LLL using projection ¹z zi in np+1 ª º =n (Jain, Kamilla '97)

9 CFT for Jain: Hansson et al. ('07-'11) ª º= p+1 = A 4 N= Y w p+1 zn= 3 N= N= Y Y z p+1 ij (zi w j ) p 5 1 º = p + 1 result based on non-trivial algebraic identites recover Abelian two-component edge theory ª º= p+1 = A (@ z1 V + zn= V + V V but there is more: K = µ p + 1 p p p + 1 (Wen, Zee; Read,...) V = e ip p+ 1 ' e i 1 p Á charged neutral A : two fermions V + ; V one fermion descendant fields needed for non-vanishing result, yield correct shift Next: find improved CFT that complete the derivation

10 W-infinity symmetry Area-preserving diffeomorphisms of incompressible fluid Z d x ½(x) = N = ½ o A A = constant A A W-infinity symmetry can be implemented in the edge CFT CFT with higher currents characteristic of 1d fermions (+ bosonization) W k =: ¹F (@ z ) k F :; W 0 = ¹F F = J; W 1 = T =: J :; W =: J 3 :; representations completely known classification (V.Kac, A. Radul '9) c = n generically reproduce Abelian theories [U(1) n with K matrices but special representations for enhanced symmetry [U(1) \ SU(n) 1

11 W-infinity minimal models repres. with enhanced symmetry are degenerate and should be projected: W 1 minimal models (A.C., Trugenberger, Zemba '93-'99) [U(1) n! [ U(1) \ SU(n) 1! [ U(1) \ SU(n) 1 SU(n) = [ U(1) W n these edge theories reproduce Jain fillings, with usual K matrices for charge and statistics extra projection of SU(n) amounts to keeping edge excitations completely symmetric w.r.t. layer exchanges: single electron excitation reduced multiplicities of edge states non-abelian statistics of quasi-particles & electron (see later) º = n p n 1

12 Ex: c= minimal model [U(1) \ SU() 1! [ U(1) \ SU() 1 SU() = [ U(1) Vir c = ; 1 º = p + 1 Vir = SU() Casimir subalgebra take edge excitations symmetric w.r.t. two layers only neutral part is described by the Virasoro minimal model for c! 1 fields characterized by dimension h = k 4 i.e. total spin s = k ; NO electron has s = 1 s z identify two vertex operators by Dotsenko-Fateev screening operators V = e i p Á ; s z = 1 ; V» V + = Q + V ; Q + = J + 0 = I duj + (u) Q (Felder)

13 Derivation of Jain wf in Hansson et al. form use W-infinity minimal models to describe ground state wf 4-el. wf has two channels,, given by choices of ª = I C 1 J + 1 ª 1 ª = f0g + f1g I À J + V (z 1 )V (z )V (z 3 )V (z 4 ) C C C 1 C i z 1 z z 3 z 4 impose antisymm of electrons ª = 0 consider descendant with same charge: J + 0! J+ 1 = L 1J + 0 ; J + 1 z V + ª 0 = J + 1 V (z 1 ) J + 1 V (z ) V (z 3 ) V (z 4 ) + perm = ª Jain Underlying theory of Jain wf is W-infinity minimal model + Fermi statistics for electrons Indipendent, exact derivation of Jain state from symmetry principles universality, robustness, etc.

14 ª º= p+1 = A 4 N= Jain wf vs. Pfaffian wf Y w p+1 zn= 3 N= N= Y Y z p+1 ij (zi w j ) p 5 ; 1 º = p + 1 reminds of Paffian state in Abelian version (A.C, Georgiev, Todorov '01) N= Y ª Pfa = A 4 w M+ ij 3 N= N= Y Y z M+ ij (zi w j ) M 5 ; M odd; K = µ M + M M M + same vanishing behaviour: ª» z p 1 1 z 13 z 14 ; ª» (z 1 z 13 z 3 ) p 1 z 14 z 15 ; 1 º = p º = p p = 1 Jain is excited state of the M = 0 Pfaffian same pairing? fractional statistics? cf. Simon, Rezayi, Cooper '07; Regnault, Bernevig, Haldane '09

15 Pfaffian state & excitations 3 N= N= Y Y NY ª Pfa = S 4 w ij z ij 5 = zij Pf µ 1 z ij ; M = 0; projection = symmetric layers, Â 1 Â! Â 1 Weyl Majorana ground state & excitations are singlets distinguishable excit. (Abelian) identical excit. (non-abelian) 3-body pseudo-potential (Read, Rezayi '99) Jain state & excitations ground state: SU() singlet (up to short-distance deformation ) excitations: distinguishable Abelian also singlets ( irrep.) non-abelian W 1

16 Non-Abelian statistics of Jain quasiholes smallest q-hole, e.g. Q = 1 at, has neutral part : 5 º = s = 1 5 two components s z = 1 identified by the projection H +» J 0 + H i.e. q-holes in two layers are symmetrized f H H» H s=1 H + H» H s=0 = I HH» I + H s=1 fusion non-abelian statistics first non-trivial case is 4 q-holes: three independent states ª (1;34) = A zi DH + ( 1)H + ( )H ( 3)H ( 4) Y E V e (z i ) + (+ $ ) ª (13;4) = A zi h(+ + )i ; ª (14;3) = A zi h(+ +)i they trasform among themselves under monodromy multidimensional representation projection: permutation of different excit. exchange of identical excit.

17 Remarks k quasiholes have quantum dimension d k» k (not Rational CFT) other Jain q-holes: h(+ + ++)i, odd no., antisymm combinations, are projected out in the minimal theory. W 1 Jain quasi-particles & hierarchy (after Hansson et al.) also fit in edge is consistently non-abelian (long-distance physics) p k But W 1 energetics of projection not understood (in the bulk) entaglement spectrum does not seem to show projection

18 Conclusions CFT + W-infinity symmetry rederive Jain states independently of composite fermion picture Jain states are consistent & universal same CFT hints at non-abelian q-hole excitations open problems: investigate space of states and energy spectrum relation to Pfaffian and Gaffnian states and their CFTs experimental tests: thermopower, if measure can be extended to higher B puzzles in known experiments?