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.