Stability of Fractional Chern Insulators in the Effective ...nqs2017.ws/Slides/2nd/Andrews.pdf ·...

Introduction Theory Method Results for |C | = 1, 2, 3 Conclusion

Stability of Fractional Chern Insulators in theEffective Continuum Limit of |C | > 1

Harper-Hofstadter Bands

Bartholomew Andrews & Gunnar Moller

TCM Group, Cavendish Laboratory, University of Cambridge

October 31, 2017

arXiv:1710.09350


Outline

Introduction

TheoryHarper-Hofstadter ModelComposite Fermion TheoryScaling of Energies

MethodEffective Continuum LimitThermodynamic Effective Continuum

Results for |C | = 1, 2, 3Many-body GapsCorrelation FunctionsParticle Entanglement Spectra

Conclusion


Introduction

I Fractional Chern Insulators (FCIs) generalize the FQHE tosystems with non-trivial Chern number, C .

I The Harper-Hofstadter model has provided some of thefirst examples of FCIs (Sørensen et al., 2005), and hosts afractal energy spectrum with any desired Chern number.

I Examine states of the composite fermion (CF) series predictedby Moller & Cooper, 2015.

I Generalize the nφ → 0 continuum limit to the effectivecontinuum limit at nφ → 1/|C | (Moller & Cooper, 2015).

I Investigate the stability (i.e. robustness in the effectivecontinuum limit) of the many-body gap, ∆.


Harper-Hofstadter Model

We consider N spinless particles hopping on an Nx × Ny squarelattice with a constant effective magnetic flux.

C=−1

C=−1

C=+1

C=+1

−2+2

+2−2

E

nφ

H =∑i ,j

[tij

hopping parameter

eφij

magnetic translation-invariant phase

c†j ci + h.c.]

+ PLB

lowest-band projection operator

∑i<j

Vij

interaction potential

:ρ(ri )ρ(rj):

PLB

• bosons ⇒ on-site interactions

• fermions ⇒ nearest-neighbour interactions


Composite Fermion Theory

Predicted filling fraction from CF theory on the lattice for awell-isolated lowest band (Moller & Cooper, 2015):

ν =r

|kC |r + 1≡ r

s, where r and s are co-prime

I C = Chern number of the band

I k = number of flux quanta attached to the particles

I |r | = number of bands filled in the CF spectrum

I sgn(r) = sgn(C ∗) for the CF band relative to C

I |s| = ground state degeneracy


Scaling & Stability

Aim to consider 2D isotropic limit ⇒ demand Nx = Ny .

◦ Note: Nbr. SitesNbr. MUCs = q is a measure of MUC size.

Scaling relations (Bauer et al., 2016):

∆ ∝ q−1 for bosons (contact interactions),

∆ ∝ q−2 for fermions (NN interactions).

Investigate stability (robustness of many-body gap)...

1. ...in the effective continuum limit: q →∞.

2. ...in the thermodynamic limit: N →∞.


Harper-Hofstadter ModelApproaching the Effective Continuum

0 0.2 0.4 0.6 0.8 1n

V

-4

-2

0

2

4E

limq→∞1q

limp→∞p

2p±1

nφ = p|C |p−sgn(C)

≡ pq, p ∈ N.

(e.g. Hormozi et al., 2012)


Basic Methode.g. N = 8 fermions in the |C | = 2 band at ν = 1/3 filling

1. Plot the many-body energy spectrum for a particular{C , r ,N} configuration and for a variety of MUC sizes, q.Identitify the ground states, predicted by CF theory.

2. Read off the many-body gap, ∆, for each energy spectrum.

3. Plot ∆ against q. Read off limq→∞(q(2)∆).

4. Plot limq→∞(q(2)∆) against N. Read off limN,q→∞(q(2)∆).

0

10

20

30

40

50

60

70

0 5 10 15 20 25

(E-E

0)/

10

-7

kx*Ly + ky

Lx=3, Ly=8, MUC: p=227, q=35 x 13

∆0

1

2

3

4

0 1 2 3 4 5

∆/

10−

6

q−2/10−6

∆ = 0.82/q2

8.16

8.18

8.2

8.22

8.24

0 1 2 3

q2∆/1

0−1

q−1/10−3

q2∆ ≈ 0.82


Basic Methode.g. N = 8 fermions in the |C | = 2 band at ν = 1/3 filling

1. Plot the many-body energy spectrum for a particular{C , r ,N} configuration and for a variety of MUC sizes, q.Identitify the ground states, predicted by CF theory.

2. Read off the many-body gap, ∆, for each energy spectrum.

3. Plot ∆ against q. Read off limq→∞(q(2)∆).

4. Plot limq→∞(q(2)∆) against N. Read off limN,q→∞(q(2)∆).


Constraints

1. We are only interested in filled CF levels:

N must be a multiple of r .

2. ν = N/Nc ⇒ N = νNc:

Nc must be a multiple of s.

3. Isolated lowest Chern number C band at

nφ =p

|C |p − sgn(C )≡ p

q, p ∈ N.

4. Consider 2D systems ⇒ approximately unit aspect ratios:∣∣∣∣1− Nx

Ny

∣∣∣∣ ≤ ε, for small ε.

5. Limited computation time:

dim{H} < 107.


Approaching the Thermodynamic Effective ContinuumQ: In which order should we take the N → ∞ and q → ∞ limits?

A: Doesn’t matter. We take the effective continuum limit first.

5.8

5.85

5.9

5.95

6

6.05

6.1

6.15

6.2

0 0.05 0.1 0.15 0.2 0.25

6

6.1

6.2

0 0.05 0.1

q∆/

10−

1

N−1

0

1

2

3

4

5

6

7

8

q/1

02

q−1

678910

N

N, q →∞ limits commute! (if both limits can be taken)

(a) ν = 1/2 bosons

14

16

18

20

22

24

26

0 0.05 0.1 0.15 0.2 0.25

22

24

26

0 0.05 0.1

q2∆/

10−

1

N−1

0

1

2

3

4

5

6

7

8

q/1

02

q−1

678910

N

(b) ν = 1/3 fermions

Figure: Finite-size scaling of the gap for Laughlin states


Approaching the Thermodynamic Effective ContinuumQ: In which order should we take the N → ∞ and q → ∞ limits?A: Doesn’t matter. We take the effective continuum limit first.

5.8

5.85

5.9

5.95

6

6.05

6.1

6.15

6.2

0 0.05 0.1 0.15 0.2 0.25

6

6.1

6.2

0 0.05 0.1

q∆/

10−

1

N−1

0

1

2

3

4

5

6

7

8

q/1

02

q−1

678910

N

N, q →∞ limits commute! (if both limits can be taken)

(a) ν = 1/2 bosons

14

16

18

20

22

24

26

0 0.05 0.1 0.15 0.2 0.25

22

24

26

0 0.05 0.1

q2∆/

10−

1

N−1

0

1

2

3

4

5

6

7

8

q/1

02

q−1

678910

N

(b) ν = 1/3 fermions

Figure: Finite-size scaling of the gap for Laughlin states


Warm-up: |C | = 1 BandBosons - Stability in the Continuum

0

1

2

3

4

5

6

7

8

0 0.04 0.08 0.12 0.16 0.2

limq→∞

(q∆

)/10−

1

N−1

ν = 1/2ν = 2/3ν = 3/4

ν = 2ν = 3/2

Figure: Finite-size scaling of the gap to the

thermodynamic continuum limit at fixed

aspect ratio

RR

0

1

2

3

4

5

6

7

8

9

-1 0 1 2 3 4 5 6 7 8

x2

x2

(E-E

0)/

10

-4

kx*Ly + ky

Lx=1, Ly=7, MUC: p=848, q=77 x 11

0

2

4

6

8

0 2 4 6 8

(E-E

0)/

10

-4

kx*Ly + ky

Lx=1, Ly=8, MUC: p=969, q=88 x 11

ν = 2: 6= BIQHE!

ν = 3/2: indications for astable RR state (as in LLL)



0

1

2

3

4

5

6

7

8

0 0.04 0.08 0.12 0.16 0.2

limq→∞

(q∆

)/10−

1

N−1

ν = 1/2ν = 2/3ν = 3/4

ν = 2ν = 3/2



aspect ratio

RR

agrees with Bauer et al. X

ν = 2 DMRG BIQHE:more than just LLL involved

in stabilizing the state(He et al., 2017)

ν = 2 competition expectedfrom continuum results

(Cooper & Rezayi, 2007)

0

1

2

3

4

5

6

7

8

9

-1 0 1 2 3 4 5 6 7 8

x2

x2

(E-E

0)/

10

-4

kx*Ly + ky

Lx=1, Ly=7, MUC: p=848, q=77 x 11

0

2

4

6

8

0 2 4 6 8

(E-E

0)/

10

-4

kx*Ly + ky

Lx=1, Ly=8, MUC: p=969, q=88 x 11

ν = 2: 6= BIQHE!




0

1

2

3

4

5

6

7

8

0 0.04 0.08 0.12 0.16 0.2

limq→∞

(q∆

)/10−

1

N−1

ν = 1/2ν = 2/3ν = 3/4

ν = 2ν = 3/2



aspect ratio

RR

0

1

2

3

4

5

6

7

8

9

-1 0 1 2 3 4 5 6 7 8

x2

x2

(E-E

0)/

10

-4

kx*Ly + ky

Lx=1, Ly=7, MUC: p=848, q=77 x 11

0

2

4

6

8

0 2 4 6 8

(E-E

0)/

10

-4

kx*Ly + ky

Lx=1, Ly=8, MUC: p=969, q=88 x 11

ν = 2: 6= BIQHE!



Warm-up: |C | = 1 BandBosons - Pair Correlation Functions

0 13 27 40 013

2740

00.40.81.2

xy

g(r

)

N = 8Laughlin state

ν = 1/2ν = 2/3ν = 3/4

ν = 2ν = 3/2

0 15 30 45 015

3045

00.410.81

1.2

xy

g(r

)

N = 10finite-size effects

0 10 20 30 010

2030

0.780.93

1.11.2

xy

g(r

)

N = 20 no correlation hole

no perfect correlation hole at ν > 1/2

0 12 24 36 012

2436

00.51

11.5

xy

g(r

)

N = 9charge density wave

0 8 16 24 08

1624

0.70.88

1.11.2

xy

g(r

)

N = 9

Figure: Pair correlation functions for the lowest-lying ground state

in the (kx , ky ) = (0, 0) momentum sector


Warm-up: |C | = 1 BandFermions - Stability in the Continuum

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

limq→∞

(q2∆

)/10−

1

N−1

ν = 1/3ν = 2/5ν = 3/7

ν = 2/3ν = 3/5

particle-hole symmetry!

Figure: Finite-size scaling of the gap to the thermodynamic continuumlimit at fixed aspect ratio


Warm-up: |C | = 1 BandFermions - Pair Correlation Functions

0 15 30 45 015

3045

00.420.83

1.3

xy

g(r

)

N = 9Laughlin state

ν = 1/3ν = 2/5ν = 3/7

ν = 2/3ν = 3/5

0 17 33 50 017

3350

-0.640

0.641.3

xy

g(r

)

N = 8

charge density wave/finite-size effects

0 18 36 54 018

3654

00.360.73

1.1

xy

g(r

)

N = 18

0 14 28 42 014

2842

-1012

xy

g(r

)

N = 9

0 15 30 45 015

3045

00.56

1.11.7

xy

g(r

)

N = 9

charge density wave




|C | = 2 BandBosons - Stability in the Effective Continuum

0

0.5

1

1.5

2

2.5

3

0 0.04 0.08 0.12 0.16 0.2

limq→∞

(q∆

)/10−

1

N−1

ν = 1/3ν = 2/5ν = 3/7

ν = 1ν = 2/3ν = 3/5

ν = 1/3: Laughlin-like state

ν = 1: BIQHE

in-line with sizedependence of BIQHEin optical flux lattices

(Sterdyniak et al., 2015)

Figure: Finite-size scaling of the gap to the thermodynamic effectivecontinuum limit at fixed aspect ratio


|C | = 2 BandBosons - Correlation Function

021

4263 0

21

42

63

0

0.47

0.94

1.4

x

y

g(r

)(0, 0)(0, 1)

(1, 0)(1, 1)

ν = 1/3

N = 7

(x mod |C |, y mod |C |)




|C | = 2 BandBosons - Correlation Function

021

4263 0

21

42

63

0

0.47

0.94

1.4

x

y

g(r

)(0, 0)(0, 1)

(1, 0)(1, 1)

|C |2 sheetsν = 1/3

N = 7

(x mod |C |, y mod |C |)




|C | = 2 BandBosons - Correlation Functions & Entanglement Spectra

0 21 42 63 021

4263

00.470.94

1.4

xy

g(r

)

ν = 1/3

N = 7

5

10

15

20

25

0 5 10 15 20

ξ

kx*Ly + ky

∆ξ = 14.25

|Ψ〉 =∑

k,n λk,n |ΨAk,n〉 ⊗ |Ψ

Bk,n〉

ξ ≡ − lnλ2(0, 0)(0, 1)

(1, 0)(1, 1)

0 23 47 70 023

4770

00.61.21.8

xy

g(r

)

ν = 2/5

N = 8

5

10

15

20

25

30

0 5 10 15 20

ξ

kx*Ly + ky

∆ξ = 4.09

(0, 0)(0, 1)

(1, 0)(1, 1)

Figure: Pair correlation functions for the lowest-lying ground state in the

(kx , ky ) = (0, 0) momentum sector, and correponding entanglement spectra


|C | = 2 BandFermions - Stability in the Effective Continuum

0123456789

0 0.03 0.06 0.09 0.12 0.15 0.18

limq→∞( q2

∆) /10

−1

N−1

ν = 1/5ν = 2/9ν = 3/13

ν = 1/3ν = 2/7ν = 3/11

d = 2

ν = 1/5 statepreviously established.

All others new!



|C | = 2 BandFermions - Correlation Functions & Entanglement Spectra

0 35 70 105035

70105

00.55

1.11.6

xy

g(r

)

ν = 1/5

N = 9

10

15

20

25

0 5 10 15 20 25 30 35

ξ

kx*Ly + ky

∆ξ = 13.76

(0, 0)(0, 1)

(1, 0)(1, 1)

0 36 73 109037

73110

00.53

1.11.6

xy

g(r

)

ν = 3/11

N = 9

5

10

15

20

25

0 5 10 15 20

ξ

kx*Ly + ky

∆ξ = 10.84

(0, 0)(0, 1)

(1, 0)(1, 1)


(kx , ky ) = (0, 0) momentum sector, and correponding entanglement spectra


|C | = 3 BandBosons - Stability in the Effective Continuum

0

0.4

0.8

1.2

1.6

2

0 0.04 0.08 0.12 0.16 0.2

limq→∞

(q∆

)/10−

1

N−1

ν = 1/4ν = 2/7ν = 3/10

ν = 1/2ν = 2/5ν = 3/8



|C | = 3 BandBosons - Correlation Functions

(0, 0)(0, 1)(0, 2)

(1, 0)(1, 1)(1, 2)

(2, 0)(2, 1)(2, 2)

021

4263 0

21

42

63

0

0.47

0.94

1.4

x

y

g(r

)

ν = 1/4

N = 7

026

5278 0

26

52

78

0.47

0.94

1.4

x

yg

(r)

ν = 1/2

N = 9


(kx , ky ) = (0, 0) momentum sector


Stability in the Effective ContinuumSummary

0

2

4

6

8

0 0.05 0.1 0.15 0.2

|C | = 1

|C | = 2

|C | = 3

limq→∞

(q∆

)/10−

1

N−1

01234567

0 0.20.40.60.8 1lim

q→∞

(q∆

)/10−

1|C |−1

N = 6

Figure: Finite-size scaling of the

gap at fixed aspect ratio for

bosonic Laughlin states

(a) bosons

|C | r ν limN,q→∞(q∆)

1 1 1/2 0.64± 0.012 1 1/3 0.27± 0.0053 1 1/4 0.13± 0.01

−1 1/2 0.18± 0.07

(b) fermions

|C | r ν limN,q→∞(q2∆)

1 1 1/3 2.56± 0.02−2 2/3 2.56± 0.02

2 1 1/5 0.46± 0.02−1 1/3 0.65± 0.16

Table: States with (effective) continuum limits that could beextrapolated to the thermodynamic limit


Conclusion

I Scaling to the effective continuum limit at fixed aspect ratioconverges faster than scaling at fixed flux density.

I Vast majority of finite-size spectra produce the ground statedegeneracy predicted by CF theory.

I Laughlin-like states with ν = 1/(|kC |+ 1) are the mostrobust, and yield a clear gap in the effective continuum limit.

I Instability may be caused by competing topological phases,charge density waves, or finite-size effects.

I Stable FCIs found with clear entanglement gaps in |C | > 1bands - largest gaps seen for |C | = 2 fermions.

I Pair-correlations are smooth functions modulated by |C | sitesalong both axes, giving rise to the appearance of |C |2 sheets.

ご清聴ありがとうございました！


Supplementary Slides

1. Approaching the Effective Continuum

2. Warm-up: |C | = 1 Band: Bosons & Fermions - Scaling of theGap with MUC Size

3. |C | = 2 Band: Bosons - Rectangular Geometries

4. |C | = 3 Band: Bosons - Rectangular Geometries


Approaching the Effective ContinuumQ: Should we fix nφ or fix aspect ratio?

A: Fix aspect ratio

0

0.5

1

1.5

2

-3 -2 -1

q∆/

10−

1

− log q

N = 9 at ν = 3/5

N = 8 at ν = 2/5

N = 6 at ν = 3/7

hollow symbols ⇒ fixed nφfilled symbols ⇒ fixed aspect ratio

(a) bosons

5

5.5

6

6.5

7

7.5

8

8.5

-3 -2

q2∆/1

0−1

− log q

N = 6 at ν = 1/3

scaling at fixed aspect ratio is more robust!

(b) fermions

Figure: Finite-size scaling of the gap in the |C | = 2 band


Approaching the Effective ContinuumQ: Should we fix nφ or fix aspect ratio?A: Fix aspect ratio

0

0.5

1

1.5

2

-3 -2 -1

q∆/

10−

1

− log q

N = 9 at ν = 3/5

N = 8 at ν = 2/5

N = 6 at ν = 3/7

hollow symbols ⇒ fixed nφfilled symbols ⇒ fixed aspect ratio

(a) bosons

5

5.5

6

6.5

7

7.5

8

8.5

-3 -2

q2∆/1

0−1

− log q

N = 6 at ν = 1/3

scaling at fixed aspect ratio is more robust!

(b) fermions

Figure: Finite-size scaling of the gap in the |C | = 2 band


Warm-up: |C | = 1 BandBosons & Fermions - Scaling of the Gap with MUC Size

5.4

5.7

6

6.3

−2 −1

q∆/

10−

1

− log q

limq→∞

(q∆) = 0.62

Bosonic Laughlin (ν = 1/2)0.8

1

1.2

1.4

−3 −2q

2∆/

2

− log q

agrees with Bauer et al. X

limq→∞

(q2∆) = 2.60

Fermionic Laughlin (ν = 1/3)

Figure: Finite-size scaling of q(2)∆ to a constant value in the continuumlimit for 8-particle Laughlin states


|C | = 2 BandBosons - Rectangular Geometries

5.9

6

6.1

6.2

6.3

0 0.75 1.5

q∆/

10−

2

q−1/10−3

q∆ ≈ 0.06

Figure: Magnitude of the gap for the

12-particle state at ν = 2/3

0

5

10

15

20

25

0 5 10 15

(E-E

0)/

10

-5

kx*Ly + ky

Lx=2, Ly=9, MUC: p=384, q=59 x 13

0

5

10

15

20

0 5 10 15

(E-E

0)/

10

-5

kx*Ly + ky

Lx=1, Ly=18, MUC: p=437, q=125 x 7



0.0

0.1

0.3

0.4

0.6

0.8

-3 -2

q∆/

10−

1

− log q

q∆ ≈ 0.03





0.0

0.1

0.3

0.4

0.6

0.8

-3 -2

q∆/

10−

1

− log q

q∆ ≈ 0.03



0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16

x2 x2

(E-E

0)/

10

-5

kx*Ly + ky

Lx=4, Ly=4, MUC: p=120, q=19 x 19

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16

(E-E

0)/

10

-5

kx*Ly + ky

Lx=1, Ly=16, MUC: p=133, q=80 x 5



0.0

0.1

0.3

0.4

0.6

0.8

-3 -2

q∆/

10−

1

− log q

q∆ ≈ 0.03



6

8

10

12

14

0 2 4 6 8 10 12 14 16

ξ

kx*Ly + ky

6

8

10

12

14

0 2 4 6 8 10 12 14 16

ξ

kx*Ly + ky

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