Extracting excitations from a fractional quantum Hall groundstate - PowerPoint PPT Presentation

Extracting excitations from a fractional quantum Hall groundstate N. Regnault Labortoire Pierre Aigrain, Ecole Normale Sup´ erieure, Paris 24/11/2010

Acknowledgment A. Sterdyniak, Z. Papic (PhD, ENS) A. Chandran (PhD, Princeton) R. Thomale (Postdoc Princeton) M. Hermanns (Postdoc Princeton) A.B. Bernevig (Princeton University) F.D.M Haldane (Princeton University)

Motivations : nu=4/11 paper, Fig.1 testing candidate wavefunctions for a 4 5 1.5 T ~ 35 mK 5 4 13 17 given fraction using numerical 11 13 7 1 2 simulations 2 11 3 1 7 1.0 5 3 2 R xx (k Ω ) 2 5 overlap can be misleading. At least one 3 3 0.5 known example where two different 3 10 8 6 states have large overlaps : Abelian 10 10 17 19 21 0.0 (Jain CF) vs non-abelian (Gaffnian). 6 7 8 9 10 11 12 13 14 MAGNETIC FIELD [T] is the groundstate enough to 10 characterize a FQH phase ? 8 6 new tools to probe the groundstate Energy 4 how deep are encoded the excitations 2 0 within the groundstate ? 0 5 10 15 20 L

Outline : 1. Orbital entanglement spectrum 2. Conformal limit 3. From the edge to the bulk 4. Probing the non-universal part of the OES 5. Conclusion

Orbital entanglement spectrum

Landau level without spin with spin 5<ν<6 N=2 hg µ b B 4<ν<5 h ω c 3<ν<4 N=1 hg µ b B 2<ν<3 h ω c 1<ν<2 N=0 hg µ b B 0<ν<1 Filling factor : ν = hn eB = N N φ Cyclotron frequency : ω c = eB m Lowest Landau level ( ν < 1) : z m exp � −| z | 2 / 4 l 2 � N-body wave function : Ψ = P ( z 1 , ..., z N ) exp( − � | z i | 2 / 4) the Hamiltonian is just the (projected) interaction ! � V ( � r i − � H = r j ) i < j (including screening effect, finite width, Landau level,...)

The Laughlin wave function A (very) good approximation of the ground state at ν = 1 3 | zi | 2 ( z i − z j ) 3 e − P � Ψ L ( z 1 , ... z N ) = i 4 l 2 i < j ρ x add one flux quantum at z 0 = one quasi-hole � Ψ qh ( z 1 , ... z N ) = ( z 0 − z i ) Ψ L ( z 1 , ... z N ) i ρ x Locally, create one quasi-hole with fractional charge + e 3

ν = 5 / 2 : the Moore-Read state R.L. Willett, L.N. Pfeiffer, K.W. West (PNAS 0812599106) � 1 � � ( z i − z j ) 2 Ψ pf ( z 1 , ..., z N ) = Pf z i − z j i < j add/remove one flux quanta − → create a pair of quasi-holes /quasi-electrons ( ± e / 4) non Abelian statistics !

Entanglement entropy for the FQHE look at the ground state | Ψ � cut the system into two parts A and B in orbital space ( ≃ real space, orbital partition) reduced density matrix ρ A = Tr B | Ψ � � Ψ | , block-diagonal wrt N A and L A z compute the entanglement entropy S A = − Tr A ( ρ A log ρ A ).

Entanglement entropy for the FQHE calculation directly done at the level of the Fock decomposition topological entanglement entropy : extract the γ from S A = cL − γ (Haque et al.). Only depends on the nature of the excitations. But : highly non-trivial looking at the entanglement spectrum : plot ξ = − log λ A vs L A z for fixed cut and N A Schmidt decomposition | Ψ � = � p exp( − ξ/ 2) | A , p � ⊗ | B , p � key idea : think about exp( − ξ ) as a Boltzmann weight, ξ as “energies” of a fictious Hamiltonian for N A particles

Entanglement spectrum (Li and Haldane) Laughlin N = 13 , l A = 36 (hemisphere cut), N A = 6 L A z angular momentum of A , ξ = − log λ A , λ A ’s are ρ A eigenvalues.

Entanglement spectrum a way to look at the Fock space decomposition “banana” shaped spectrum for pure CFT state (not only Jack polynomials) with a given maximum L A z “low energy” part : a signature of the state (edge mode degeneracy). i z 2 example Laughlin (1,1,2) : Ψ L , Ψ L × � i z i , Ψ L × � i and Ψ L × � i < j z i z j Probing physics of the edge from the ground state on a closed surface

Coulomb case and entanglement gap

Entanglement spectrum for the FQHE : some results probing non abelian statistics (Li, Haldane 2008) looking at (precursor of ) phase transition through closing entanglement gap (Zozulya, Haque, NR, 2009) differentiate states with large overlap but different excitations (from the ground state only !) (NR, Bernervig, Haldane 2009) non-trivial relation between ES and edge mode (Bernervig, NR 2009) when N → ∞ recover degenerate multiplets and linear (relativistic) dispersion relation for the edge mode (Thomale, Stedyniak, NR, Bernervig 2010) torus geometry, tower of edge modes ( L¨ auchli et al. 2010 )

Entanglement spectrum : beyond FQHE quantum Hall bilayers quantum spin systems superconductor topological insulators Bose-Einstein condensates SUSY lattice models

An application : probing statitics of excitations Write wavefunctions for localized excitations and move them ! e/4 A A B B N F N F N F N F e/2 e/4 In the Laughlin case (abelian excitations), the counting stays the same (1,1,2,...) | | | • • • (a) 010101010101010101010101 (b) 101010010101010101010101 (c) 101010101010010101010101 50 50 50 8.05 8 11 10 40 40 40 7.8 9 8 7.6 8 7 30 30 30 7.4 7.95 6 ξ ξ ξ 30 31 32 33 34 33 34 35 36 37 36 37 38 39 40 20 20 20 10 10 10 0 0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 A A A L z L z L z

An application : probing statistics of excitations In the Moore-Read case, the counting is able to detect if there is an even or odd number of excitations. 16 12 8 ξ 4 (a) 0202...02 (b) 1111...11 0 30 35 40 45 35 40 45 50 A A L z L z

Conformal limit

Different geometries, similar ES sphere cylinder N F N F disk R N F annulus N F Ψ = � µ c µ sl µ , c µ will one differ by some geometrical factors different eigenvalues of ρ A (shape of the ES) but the same number of non-zero eigenvalues (counting) The counting IS the important feature. For model states (CFT) , exponentially lower than expected