fractional chiral hinge insulator
TRANSCRIPT
Fractional Chiral Hinge Insulator
N. Regnault
1√2| Ecole Normale Superieure Paris and CNRS 〉 + 1√
2| Princeton University〉
Benasque - February 2021
ÉCOLE NORMALES U P É R I E U R E
Acknowledgements
AnnaHackenbroich
(MPQ-Garching)
Ana Hudomal(Leeds)
Norbert Schuch(Vienna)
B. Andrei Bernevig(Princeton)
Reference: arXiv:2010.09728
Toward 3D topological order
2D TO: FQHE Higher ordertopological insulators
An intermediate step(3d but chiral hinge
modes)
3D TOToric code, fractons, ...
Any microscopicelctronic model?
Building a model state for a FCHI
Chiral Hinge Insulator: 2nd order TI in 3D [Schindler et al. 2018]
xy
z
Nx
Ny
Nz
PBC×PBC×PBC
0 20 40 60Nz
2πkz
−4
−2
0
2
4
E
4
3
2
21
4
31
2
3
24
∆1M
−∆22
−i∆22
z
x
y
Chiral Hinge Insulator: 2nd order TI in 3D [Schindler et al. 2018]
xy
z
Nx
Ny
Nz
OBC×PBC×PBC
0 20 40 60Nz
2πkz
−4
−2
0
2
4
E
4
3
2
21
4
31
2
3
24
∆1M
−∆22
−i∆22
z
x
y
Gapped vertical surfaces
Chiral Hinge Insulator: 2nd order TI in 3D [Schindler et al. 2018]
xy
z
Nx
Ny
Nz
OBC×OBC×PBC
0 20 40 60Nz
2πkz
−4
−2
0
2
4
E
4
3
2
21
4
31
2
3
24
∆1M
−∆22
−i∆22
z
x
y
Gapped vertical surfaces.
Gapless chiral & antichiral hinge modes.
Hinge modes = IQHE edge modes (BUTsurfaces 6= IQHE bulk).
Protected by the C4T symmetry (nbr hingesmod 2).
Fractional Chern insulator from Gutzwiller projection
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
−t
t
i∆
Two copies of CI with spin s =↑, ↓On-site repulsion U
∑x nx ,↑nx ,↓
FCI phase if U � t,∆ (Laughlin 1/2)
Model wave function from limit U →∞:
Double-occupancy forbidden, 1 particle per site:Spin s only remaining degree of freedom
Ground state obtained by Gutzwiller projection
PG =∏x
(1− nx ,↑nx ,↓)
Zhang, T. Grover, and A. Vishwanath PRB (2011)
Topological sectors by using PBC/APBC for the CIsInteracting model wave function: Gutzwiller projection of two spin
copies!
Fractional chiral hinge insulator (FCHI)
|ψs〉 ground state of CHI at half-filling with spin s =↑, ↓FCHI model wave function: Gutzwiller projection
|ΨFCHI〉 = PG [|ψ↑〉 ⊗ |ψ↓〉]
Properties accessible in Monte Carlo simulations to characterize ourmodel state:
Second Renyi entanglement entropy: S (2) = − ln(TrAρ
2A)
Spin fluctuations: Var(MA) = TrA(M2AρA)− (TrA(MAρA))2
Overlaps between different “topological sectors”
Monte Carlo simulations for S (2)
Partition into A & complement B, want S(2)A :
Spin configuration |v〉 = |vA, vB〉Swap operator
SWAP(|vA, vB〉 ⊗
∣∣v ′A, v ′B⟩) =∣∣v ′A, vB⟩⊗ ∣∣vA, v ′B⟩
Entanglement entropy
〈Ψ⊗Ψ|SWAP |Ψ⊗Ψ〉〈Ψ⊗Ψ|Ψ⊗Ψ〉 = e−S
(2)A
Double-layer simulation
Measured value decays exponentially with |∂A|More than 2 million CPU hours
Probing the FCHI
Luttinger liquids: 1D critical quantum system
Entanglement for subregion of size L scales as ln L.
Chiral conformal field theory for finite size N
(Second Renyi) Entropy:
S(2)crit(L;N) =
c
8ln
[N
πsin
(πL
N
)]Expect c = 1 forLuttinger liquid!
Fluctuations of U(1) current:
Var(MA) =K
2π2ln
[N
πsin
(πL
N
)]K 6= 1 indicatesfractionalization!
Study scaling of S (2) and Var(MA) with L
Disentangling the edge & hinge from the bulk contribution
Edge states in 2D: Hinge states in 3D:
ANx,A,Ny
Nx,A
Ny
Nx
x
y
z
Nx
Ny
Nz
ANx,Ny ,Nz,A
S(2)(Nx,A) = α2D + 2 × S(2)crit(Nx,A;Nx )
α2D bulk area law (constant)
S(2)(Nz,A) = α3D + 4 × S(2)crit(Nz,A;Nz )
α3D bulk area law (constant)
Monte-Carlo results
FCI:
FCHI:
Both have c = 1 andK = 1/2:
Fractionalized hingestates
5 10 15Nx,A
3.6
3.7
3.8
S(2
)ANx,A,Ny
(b)
c = 1± 0.1,α = 3.29± 0.05
30× 3
5 10Nx,A
0.94
0.96
0.98
1.00
1.02
1.04
Var
( MANx,A,Ny)
(c)
K = 0.5± 0.03,α′ = 0.95± 0.01
20× 4
5 10Nz,A
4.5
5.0
5.5
S(2
)ANx,Ny,Nz,A
(a)
c = 1.19± 0.07,α = 3.84± 0.05
c = 1.03± 0.14,α = 4.87± 0.12
2× 2× 20
3× 2× 20
5 10Nz,A
1.0
1.1
1.2
1.3
Var
( MANx,Ny,Nz,A
)
(b)
K = 0.49± 0.02,α′ = 0.86± 0.01
K = 0.49± 0.03,α′ = 1.11± 0.01
2× 2× 20
3× 2× 20
Topological entanglement entropy (TEE) γ
Area law for entanglement entropy:SA = α|∂A| − γKitaev-Preskill cut.
−γ = SABC−SAB−SBC−SAC+SA+SB+SC
Periodic boundary conditions in x
γPBC = −0.009± 0.102 ≈ 0
Same value if taken along z ,
γPBC ≈ 0 for two system sizes → not alayered construction.
Open boundary conditions in x
γOBC = 0.32± 0.16 ≈ ln(√
2) = 0.35
D
AB
C
x
yz
ACB
D
γ2DFCHI = (γOBC − γPBC )/2 ≈ ln(
√2)/2
Unconventional surface phase
γ2DFCHI = ln(
√2)/2
But: Strictly 2D topological phase has γ = 0 or γ ≥ ln(√
2)
Surfaces host unconventional topological phase that cannot bedescribed by 2D TQFT
FCI
FCHI
Signature from the bulk
What about the TEE in the bulk?
γPBC = −0.009± 0.102 ≈ 0
Same value if taken along z or x
γPBC ≈ 0 for two system sizes → not a layeredconstruction.
... but the cut is macroscopic in one directionZhang, Grover, Vishwanath PRB (2011).
Topological degeneracy?
Can play with PBC/APBC forthe CHI to generate 8 states(full PBC) or 4 states (OBCin x)
Should be isotropic → size isa killer.
Conclusion
Fractional CHI model wave function obtained from Gutzwillerprojection.
Fractionalized chiral hinge modes similar to edge modes of a fractionalChern insulator.
Gapped vertical surfaces hosting a topological phase that cannot berealized in 2D.
Outlook:3D is hard. Beyond Monte-Carlo?What is the nature of the phase?Fate of the Dirac cones in the interacting model?
Anisotropic limit OBC×PBC×PBC
OBC×PBC×PBC: 4 states → 2 linearly independent states.
PBC×PBC×PBC: 8 states → 4 linearly independent states.