Fractional Topogical Insulators: numerical evidences N. Regnault Department of Physics, Princeton University, LPA, Ecole Normale Sup´ erieure Paris and CNRS
Acknowledgment Y.L. Wu (PhD, Princeton) C. Repellin (PhD, ENS) A. Sterdyniak (PhD, ENS) T. Liu (Master student, ENS) T. Hughues (University of Illinois) A.B. Bernevig (Princeton University)
Motivations : Topological insulators An insulator has a (large) gap separating a fully filled valence band and an empty conduction band Atomic insulator : solid Semiconductor : Si argon How to define equivalent insulators ? Find a continuous transformation from one Bloch Hamiltonian H 0 ( � k ) to another H 1 ( � k ) without closing the gap Vacuum is the same kind of insulator than solid argon with a gap 2 m e c 2 Are all insulators equivalent to the vacuum ? No
Motivations : Topological insulators What is topological order ? cannot be described by symmetry breaking (cannot use Ginzburg-Landau theory) some physical quantities are given by a “topological invariant” (think about the surface genus) a bulk gapped system (i.e. insulator) system feeling the topology (degenerate ground state, cannot be lifted by local measurement). a famous example : Quantum Hall Effect (QHE) TI theoretically predicted and experimentally observed in the past 5 years missed by decades of band theory 2D TI : 3D TI :
Motivations : FTI A rich physics emerge when turning on strong interaction in QHE What about Topological insulators ? strong no interaction interaction FQHE Time-reversal QHE (B field) breaking FCI Chern Insulator Time reversal FQSHE ? QSHE invariant 3D FTI ? 3D TI
Outline Fractional Quantum Hall Effect Fractional Chern Insulators Entanglement spectroscopy FTI with time reversal symmetry
Fractional Quantum Hall Effect
Landau level Cyclotron frequency : ω c = eB m Filling factor : ν = hn N eB = N φ R xx At ν = n , n completely filled levels and a energy gap � ω c R xy Integer filling : a ( Z ) topological B insulator with a perfectly flat band / perfectly flat Berry curvature ! Partial filling + interaction → N=2 h w c FQHE N=1 Lowest Landau level ( ν < 1) : h w c N=0 z m exp −| z | 2 / 4 l 2 � � N-body wavefunction : Ψ = P ( z 1 , ..., z N ) exp( − � | z i | 2 / 4)
The Laughlin wavefunction 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 wavefunction” 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 wavefunction” : quasihole 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 “Wilczek” approach : quasi-holes obey fractional statistics 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).
Fractional Chern Insulator
Interacting Chern insulators A Chern insulator is a zero magnetic field version of the QHE (Haldane, 88) Topological properties emerge from the band structure At least one band is a non-zero Chern number C , Hall conductance σ xy = e 2 h | C | Basic building block of 2D Z 2 topological insulator (half of it) Is there a zero magnetic field equivalent of the FQHE ? → Fractional Chern Insulator Here we will focus on the C = ± 1.
From CI to FCI To go from IQHE to FQHE, we need to : consider a single Landau level partially fill this level, ν = N / N Φ turn on repulsive interactions
From CI to FCI To go from IQHE CI to FQHE FCI, we need to : consider a single Landau level consider a single band partially fill this level, ν = N / N Φ partially fill this band, ν = N / N unit cells turn on repulsive interactions turn on repulsive interactions What QH features should we try to mimic to get a FCI ? Several proposals for a CI with nearly flat band that may lead to FCI But “nearly” flat band is not crucial for FCI like flat band is not crucial for FQHE (think about disorder)
Four (almost) flat band models 3 a 2 1 2 a 1 Kagome lattice, E. Tang et al. PRL (2011) Haldane model, Neupert et al. PRL (2011) 4 5 3 -t 2 b 2 6 1 2 t 2 t 2 b 1 t 1 -t 2 Checkerboard lattice, Ruby lattice, PRB (2011) K. Sun et al. PRL (2011).
The Kagome lattice model three atoms per unit cell, 3 a 2 spinless particles lattice can be realized in cold 1 2 a 1 atoms only nearest neighbor hopping e i ϕ three bands with Chern numbers C = 1, C = 0 and C = − 1 Jo et al. PRL (2012) e − i ϕ (1 + e − ik y ) e i ϕ (1 + e − ik x ) 0 e i ϕ (1 + e i ( k x − k y ) ) H ( k ) = − t 1 0 h . c . 0 k x = k . a 1 , k y = k . a 2
The flat band limit 15 δ ≪ E c ≪ ∆ ( E c being the interaction energy scale) 10 E(k x ,k y ) We can deform continuously the 5 C=-1 band structure to have a perfectly flat valence band 0 C=0 and project the system onto the -5 lowest band, similar to the C=1 projection onto the lowest 0 1 2 3 4 5 6 -10 0 1 2 3 4 5 6 k y Landau level k x nbr bands � H ( k ) = P n E n ( k ) n =1 E D nbr bands d � H FB ( k ) = − → n P n n =1 k
Two body interaction and the Kagome lattice Our goal : stabilize a Laughlin-like state at ν = 1 / 3. A key property : the Laughlin state is the unique densest state that screens the short range repulsive interaction. A nearest neighbor repulsion should mimic the FQH interaction. 3 We give the same energy 1 2 penalty when two part are sitting on neighboring sites (for fermions) or on the same site (for bosons). � On the checkerboard lattice : H F = : n i n j : U int Neupert et al. PRL 106, 236804 < i , j > (2011), Sheng et al. Nat. � H B = U : n i n i : int Comm. 2, 389 (2011), NR and i BAB, PRX (2011)
The ν = 1 / 3 filling factor An almost threefold degenerate ground state as you expect for the Laughlin state on a torus (here lattice with periodic BC) 0.07 0.06 0.05 E − E 0 0.04 0.03 N =8 0.02 N =10 0.01 N =12 N =12 ,NNN 0.00 0 5 10 15 20 25 30 35 k x + N x k y But 3fold degeneracy is not enough to prove that you have Laughlin-like physics there (a CDW would have the same counting).
Gap Many-body gap can actually increase with the number of particles due to aspect ratio issues. Finite size scaling not and not monotonic reliable because of aspect ratio in the thermodynamic limit. The 3-fold degeneracy at filling 1/3 in the continuum exists for any potential and is not a hallmark of the FQH state. On the lattice, 3-fold degeneracy at filling 1/3 means more than in the continuum, but still not much 0.07 0.06 0.05 0.04 ∆ E 0.03 N y =3 0.02 N y =4 N y =5 0.01 N y =6 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 1 /N
Quasihole excitations The form of the groundstate of the Chern insulator at filling 1/3 is not exactly Laughlin-like. However, the universal properties SHOULD be. The hallmark of FQH effect is the existence of fractional statistics quasiholes. In the continuum FQH, Quasiholes are zero modes of a model Hamiltonians - they are really groundstates but at lower filling. In our case, for generic Hamiltonian, we have a gap from a low energy manifold (quasihole states) to higher generic states. 0.3 0.28 0.26 N = 9, N x = 5 , N y = 6 0.24 E / t 1 The number of states below 0.22 0.2 the gap matches the one of 0.18 the FQHE ! 0.16 0 5 10 15 20 25 30 K x + N x * K y
The one dimensional limit : thin torus let’s take N x = 1, thin torus limit the groundstate is just the electrostatic solution (1 electron every 3 unit cells) a charge density wave and not a Laughlin state 1 1 0.1 0.1 E (arb. unit) E (arb. unit) 0.01 0.01 0.001 0.001 0.0001 0.0001 (a) FCI (a) FCI+2qh 1e-05 1e-05 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 18 k y k y Can we differentiate between a Laughlin state and a CDW ?
Entanglement spectroscopy
Entanglement spectrum - Li and Haldane, PRL (2008) example : system made of two spins 1 / 2 A B ρ = | Ψ � � Ψ | , reduced density matrix ρ A = Tr B ρ Entanglement spectrum : ξ = − log ( λ ) ( λ eigenvalues of ρ A ) as a function of S z � 1 1 3 2 ( |↑↓� + |↓↑� ) √ √ 4 |↑↓� + 4 |↓↑� |↑↑� 2 2 2 1.5 1.5 1.5 1 ξ 1 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 S z S z The counting (i.e the number of non zero eigenvalue) also provides informations about the entanglement
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