Entanglement Spectroscopy and its application to the fractional - PowerPoint PPT Presentation

Entanglement Spectroscopy and its application to the fractional quantum Hall phases N. Regnault Ecole Normale Sup´ erieure Paris and CNRS, Department of Physics, Princeton Global Scholar É C O L E N O R M A L E S U P É R I E U R E

Acknowledgment A. Sterdyniak (MPQ, Germany) B. Estienne (Universit´ e Pierre et Marie Curie, France) Z. Papic (University of Leeds) F.D.M. Haldane (Princeton University) R. Thomale (W¨ urzburg, Germany) M. Haque (Maynooth, Ireland) A.B. Bernevig (Princeton University)

Topological phases What is topological order ? phases that can’t be described by a broken symmetry. No local order parameter. At least one physical (i.e. measurable) quantity related to a topological invariant (like the surface genus). A system with a gapped bulk and gapped or gapless surface (or edge) modes. Simplest example : the integer quantum Hall effect (quantized Hall conductance). Since 2005, the revolution of topological insulators 2D TI : 3D TI :

Making things harder : strong interactions R xx Most celebrate example : the R xy B fractional quantum Hall effect. Alliance of a non-trivial band structure (Landau levels) and strong interactions. An exotic place : emergent fractional charges with fractional statistics or non-abelian. No classification of fractional phases (as opposed to the non-interacting case). Non-perturbative problem → variational methods and numerical simulations. No local order parameter → which phase is emerging ?

Outline Entanglement Spectrum Fractional Quantum Hall Effect FQHE and Entanglement Spectrum ES and Fractional Chern Insulators

Entanglement Spectrum

Entanglement spectrum (Li and Haldane 2008) Start from a quantum state | Ψ � 1D: A Create a bipartition of the system into B A and B 2D: Reduced density matrix ρ A = Tr B | Ψ � � Ψ | A B L Entanglement Hamiltonian : ρ A = e − H ent The eigenvalues of H ent are the entanglement energies { ξ i } . Lower entanglement energies ≃ higher weights in ρ A . If O = O A + O B and , the ξ i can be labeled by the O A quantum numbers. Entanglement entropy S A = − Tr A [ ρ A ln ρ A ], area law for gapped systems (i.e. S A ∝ L d − 1 ).

Entanglement spectrum Example : system made of two spins 1 / 2 A B Entanglement spectrum : ξ as a function of S z , A ( z projection of the spin A ) � 1 1 3 2 ( |↑↓� + |↓↑� ) 4 |↑↓� + 4 |↓↑� √ √ |↑↑� 2 2 2 1.5 1.5 1.5 1 1 ξ 1 ξ ξ 0.5 0.5 0.5 0 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 S z,A S z,A S z,A The counting (i.e the number of non zero eigenvalue) also provides information about the entanglement

The AKLT spin chain A prototype of a gapped spin-1 chain. S j +1 + 1 � 2 � � S j · � � � S j · � � H AKLT = S j +1 3 j j The ground state of the AKLT Hamiltonian is the valence bond state. A B S=1/2 S=1/2 } } S=0 S=1 For an open chain, the two extreme unpaired spin- 1 2 correspond to the edge excitations (4-fold degenerate ground state)

The AKLT spin chain, the Li-Haldane conjecture 35 35 AKLT Heis. 2 ES for an open chain 30 30 1 ξ 25 25 with 8 sites and l A = 4. 0 20 20 0 1 ξ ξ S z,A 15 15 S z , A : z -projection of A 10 10 total spin. 5 5 D ξ 0 0 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 S z,A S z,A Reduced density matrix is 81 × 81 but only two non-zero eigenvalues for the AKLT model. The trace has introduced an artificial edge → a spin- 1 2 edge excitation. The ES mimics the edge spectrum of the model. Away from the model state : An entanglement gap ∆ ξ between a low (entanglement) energy structure related the model state and a high energy structure. ∆ ξ should stay finite at the thermodynamical limit if the two phases are in the same universality class.

Fractional Quantum Hall Effect

Fractional Quantum Hall effect Landau levels (spinless case) N=2 Cyclotron frequency : ω c = eB m , h w c Filling factor : ν = hn N N=1 eB = N Φ h w c Partial filling + interaction → FQHE N=0 R xx Lowest Landau level ( ν < 1) : z m exp � −| z | 2 / (4 l 2 � B ) R xy B N -body wave function : Ψ = P ( z 1 , ..., z N ) exp( − � | z i | 2 / (4 l 2 B )) What are the low energy properties ? Gapped bulk, Massless edge Strongly correlated systems, emergence of exotic phases :fractional charges, non-abelian braiding.

The Fractional QHE FQHE is a hard N -body problem : a single Landau level (the lowest one for ν < 1, no spin) the effective Hamiltonian is just the (projected) interaction ! � H = P LLL V ( � r i − � r j ) P LLL i < j (insert in V your favorite interaction plus screening effect, finite width, Landau level,...) Z Two major methods : L =+3 z L =+2 z variational method : find a wave functions describing L =+1 z L =0 low energy physics (symmetries, CFT, model...) z L =-1 N F z numerical calculation : exact diagonalizations on L =-2 z L =-3 z different geometries (sphere, plane, torus, ...), DMRG Nbr orb. ≃ N Φ

The Laughlin wave function A (very) good approximation of the ground state at ν = 1 3 | zi | 2 ( z i − z j ) 3 e − � � Ψ L ( z 1 , ... z N ) = i 4 l 2 i < j ρ x The Laughlin state is the unique (on genus zero surface) densest state that screens the short range (p-wave) repulsive interaction. Topological state : the degeneracy of the densest state depends on the surface genus (sphere, torus, ...)

The Laughlin wave function : quasihole excitations 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 Quasi-holes obey fractional statistics (fractional charge + flux) Adding quasiholes/flux quanta increases the size of the droplet For given number of particles and flux quanta, there is a specific number of qh states that one can write These numbers/degeneracies can be classified with respect some quantum number (angular momentum L z ) and are a fingerprint of the phase (related to the statistics of the excitations).

The torus geometry : topological degeneracy k = 2 y In the LLL, the one-body wf are : � 2 � Ly ( k y + kN φ )( x + iy ) e − x 2 k = 1 2 π − 1 2 π ( k y + kN φ ) 2 k = 3 y y � 2 e k ∈ Z e 2 Ly k = 0 The Laughlin ν = 1 / m is m -fold degenerate y k = y 0 1 2 3 on the torus. 2 1 1 0 N = 4 N=4 bosons F Number of orbitals is N φ . 0.1 0.08 Each orbital is labeled by its quantum 0.06 Energy number k y . 0.04 Invariant under the magnetic translations. 0.02 K T y = ( � i k y , i ) mod N Φ . 0 0 2 4 6 8 10 12 14 16 K T There is another quantum number (purely y Model interaction for the many-body) related the center of mass degeneracy. Laughlin state, N = 6 , N Φ = 18

The Haldane’s exclusion principle The number of quasihole states per momentum sector can be predicted by a generalization of the Pauli’s principle. For the Laughlin ν = 1 / m , no more than 1 particle in m consecutive orbitals (including periodic boundary conditions on the torus). Example Laughlin ν = 1 / 3 state with 9 flux quanta k = k = 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 y y 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 k = 0 1 2 3 4 5 6 7 8 k = 0 1 3 4 6 7 y y 2 5 8 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 Can be generalized to the Moore-Read or Read-Rezayi states (non-abelian excitations) : no more than k particles in k + 2 consecutive orbitals .

The Haldane’s exclusion principle Example : Finding back the 3-fold degeneracy of the Laughlin ν = 1 / 3 with N φ = 3 × N = 9 k = y 0 1 2 3 4 5 6 7 8 1 0 0 1 0 0 1 0 0 0.08 0.07 T K =0+3+6 mod 9=0 y 0.06 0.05 Energy k = 0 1 2 3 4 5 6 7 8 y 0.04 0 1 0 0 1 0 0 1 0 0.03 T K =1+4+7 mod 9=3 0.02 y Laughlin GS 0.01 0 k = 0 1 2 3 4 5 6 7 8 y 0 1 2 3 4 5 6 7 8 9 0 0 1 0 0 1 0 0 1 K T y T K =2+5+8 mod 9=6 y

The Laughlin wave function : edge excitations B B Non-interacting case (i.e. IQHE) One dimensional chiral mode (a) E = 0 (c) E = 2 t t with a linear dispersion relation energy E ≃ 2 π v L n The degeneracy of each momentum many-body energy level E t is (b) E = 1 (d) E = 2 t t given by the sequence 1 , 1 , 2 , 3 , 5 , 7 , ....

The Laughlin wave function : edge excitations B B Interacting case (Laughlin ν = 1 3 ) One dimensional chiral mode (b) E = 1 (d) E = 2 t t with a linear dispersion relation E ≃ 2 π v L n (a) E = 0 t The degeneracy of each energy many-body energy level E t is (c) E = 1 (e) E = 2 t t given by the sequence 1 , 1 , 2 , 3 , 5 , 7 , .... momentum

FQHE and Entanglement Spectrum

Orbital entanglement spectrum FQHE on a cylinder (Landau gauge) : orbitals are labeled by k y , rings at position 2 π k y L l 2 B Divide your orbitals into two groups A and B , keeping N orb , A orbitals : orbital cut ≃ real space cut (fuzzy cut) A B Laughlin state N = 12, half cut OES Laughlin N= 12, N A = 6 on a cylinder L= 15 60 50 K y 40 1 1 2 3 5 7 ... K =01234567.... y 30 ξ K = y 0 1 2 3 4 5 6 7 20 1 0 1 1 1 0 1 0 K y 10 } } 0 0 2 4 6 8 10 12 14 16 18 A B K y,A Fingerprint of the edge mode (edge mode counting) can be read from the ES. ES mimics the chiral edge mode spectrum. For FQH model states, nbr. levels is exp. lower than expected.