Stability of Fractional Chern Insulators in the Effective Continuum - - PowerPoint PPT Presentation

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Stability of Fractional Chern Insulators in the Effective Continuum - - PowerPoint PPT Presentation

Jun 14, 2023 275 likes •708 views

Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Stability of Fractional Chern Insulators in the Effective Continuum Limit of | C | > 1 Harper-Hofstadter Bands Bartholomew Andrews & Gunnar M oller TCM Group,

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

Stability of Fractional Chern Insulators in the Effective Continuum Limit of |C| > 1 Harper-Hofstadter Bands

Bartholomew Andrews & Gunnar M¨

  • ller

TCM Group, Cavendish Laboratory, University of Cambridge

October 31, 2017 arXiv:1710.09350

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

Outline

Introduction Theory Harper-Hofstadter Model Composite Fermion Theory Scaling of Energies Method Effective Continuum Limit Thermodynamic Effective Continuum Results for |C| = 1, 2, 3 Many-body Gaps Correlation Functions Particle Entanglement Spectra Conclusion

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

Introduction

◮ Fractional Chern Insulators (FCIs) generalize the FQHE to

systems with non-trivial Chern number, C.

◮ The Harper-Hofstadter model has provided some of the

first examples of FCIs (Sørensen et al., 2005), and hosts a fractal energy spectrum with any desired Chern number.

◮ Examine states of the composite fermion (CF) series predicted

by M¨

  • ller & Cooper, 2015.

◮ Generalize the nφ → 0 continuum limit to the effective

continuum limit at nφ → 1/|C| (M¨

  • ller & Cooper, 2015).

◮ Investigate the stability (i.e. robustness in the effective

continuum limit) of the many-body gap, ∆.

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

Harper-Hofstadter Model

We consider N spinless particles hopping on an Nx × Ny square lattice 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
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SLIDE 5

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

Composite Fermion Theory

Predicted filling fraction from CF theory on the lattice for a well-isolated lowest band (M¨

  • ller & Cooper, 2015):

ν = r |kC|r + 1 ≡ r s , where r and s are co-prime

◮ C = Chern number of the band ◮ k = number of flux quanta attached to the particles ◮ |r| = number of bands filled in the CF spectrum ◮ sgn(r) = sgn(C ∗) for the CF band relative to C ◮ |s| = ground state degeneracy

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

Scaling & Stability

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

  • Note:
  • Nbr. Sites
  • Nbr. 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 → ∞.
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SLIDE 7

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

Harper-Hofstadter Model

Approaching the Effective Continuum

0.2 0.4 0.6 0.8 1

nV

  • 4
  • 2

2 4

E

limq→∞ 1

q

limp→∞

p 2p±1

nφ =

p |C|p−sgn(C) ≡ p q ,

p ∈ N. (e.g. Hormozi et al., 2012)

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SLIDE 8

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

Basic Method

e.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)∆).

10 20 30 40 50 60 70 5 10 15 20 25

(E-E0)/10-7 kx*Ly + ky

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

∆

1 2 3 4 1 2 3 4 5 ∆/10−6 q−2/10−6 ∆ = 0.82/q2 8.16 8.18 8.2 8.22 8.24 1 2 3 q2∆/10−1 q−1/10−3 q2∆ ≈ 0.82

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

Basic Method

e.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)∆).
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Introduction Theory Method Results for |C| = 1, 2, 3 Conclusion

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

  • ≤ ǫ,

for small ǫ.

  • 5. Limited computation time:

dim{H} < 107.

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SLIDE 11

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

Approaching the Thermodynamic Effective Continuum

Q: In which order should we take the N → ∞ and q → ∞ limits?

5.8 5.85 5.9 5.95 6 6.05 6.1 6.15 6.2 0.05 0.1 0.15 0.2 0.25 6 6.1 6.2 0.05 0.1 q∆/10−1 N−1 1 2 3 4 5 6 7 8 q/102 q−1 6 7 8 9 10 N

(a) ν = 1/2 bosons

14 16 18 20 22 24 26 0.05 0.1 0.15 0.2 0.25 22 24 26 0.05 0.1 q2∆/10−1 N−1 1 2 3 4 5 6 7 8 q/102 q−1 6 7 8 9 10 N

(b) ν = 1/3 fermions Figure: Finite-size scaling of the gap for Laughlin states

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

Approaching the Thermodynamic Effective Continuum

Q: 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 0.05 0.1 0.15 0.2 0.25 6 6.1 6.2 0.05 0.1 q∆/10−1 N−1 1 2 3 4 5 6 7 8 q/102 q−1 6 7 8 9 10 N

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

(a) ν = 1/2 bosons

14 16 18 20 22 24 26 0.05 0.1 0.15 0.2 0.25 22 24 26 0.05 0.1 q2∆/10−1 N−1 1 2 3 4 5 6 7 8 q/102 q−1 6 7 8 9 10 N

(b) ν = 1/3 fermions Figure: Finite-size scaling of the gap for Laughlin states

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SLIDE 13

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

Warm-up: |C| = 1 Band

Bosons - Stability in the Continuum

1 2 3 4 5 6 7 8 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/2 ν = 2/3 ν = 3/4 ν = 2 ν = 3/2

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

RR

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SLIDE 14

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

Warm-up: |C| = 1 Band

Bosons - Stability in the Continuum

1 2 3 4 5 6 7 8 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/2 ν = 2/3 ν = 3/4 ν = 2 ν = 3/2

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

RR

agrees with Bauer et al.

ν = 2 DMRG BIQHE: more than just LLL involved in stabilizing the state (He et al., 2017) ν = 2 competition expected from continuum results (Cooper & Rezayi, 2007)

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SLIDE 15

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

Warm-up: |C| = 1 Band

Bosons - Stability in the Continuum

1 2 3 4 5 6 7 8 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/2 ν = 2/3 ν = 3/4 ν = 2 ν = 3/2

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

RR

1 2 3 4 5 6 7 8 9

  • 1

1 2 3 4 5 6 7 8 x2 x2

(E-E0)/10-4 kx*Ly + ky

Lx=1, Ly=7, MUC: p=848, q=77 x 11 2 4 6 8 2 4 6 8

(E-E0)/10-4 kx*Ly + ky

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

ν = 2: = BIQHE! ν = 3/2: indications for a stable RR state (as in LLL)

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

Warm-up: |C| = 1 Band

Bosons - Pair Correlation Functions

0 13 27 40 013 27 40 0.4 0.8 1.2 x y g(r)

N = 8

Laughlin state ν = 1/2 ν = 2/3 ν = 3/4 ν = 2 ν = 3/2 0 15 30 45 015 30 45 0.41 0.81 1.2 x y g(r)

N = 10

finite-size effects 0 10 20 30 010 20 30 0.78 0.93 1.1 1.2 x y g(r)

N = 20

no correlation hole no perfect correlation hole at ν > 1/2 0 12 24 36 012 24 36 0.51 1 1.5 x y g(r)

N = 9

charge density wave 8 16 24 0 816 24 0.7 0.88 1.1 1.2 x y g(r)

N = 9

Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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SLIDE 17

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

Warm-up: |C| = 1 Band

Fermions - Stability in the Continuum 5 10 15 20 25 30 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 = 1/3 ν = 2/5 ν = 3/7 ν = 2/3 ν = 3/5

particle-hole symmetry!

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

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SLIDE 18

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

Warm-up: |C| = 1 Band

Fermions - Pair Correlation Functions

0 15 30 45 015 30 45 0.42 0.83 1.3 x y g(r)

N = 9

Laughlin state ν = 1/3 ν = 2/5 ν = 3/7 ν = 2/3 ν = 3/5 0 17 33 50 017 33 50

  • 0.64

0.64 1.3 x y g(r)

N = 8

charge density wave /finite-size effects 0 18 36 54 018 36 54 0.36 0.73 1.1 x y g(r)

N = 18

0 14 28 42 014 28 42

  • 1

1 2 x y g(r)

N = 9

0 15 30 45 015 30 45 0.56 1.1 1.7 x y g(r)

N = 9

charge density wave

Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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SLIDE 19

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

|C| = 2 Band

Bosons - Stability in the Effective Continuum 0.5 1 1.5 2 2.5 3 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/3 ν = 2/5 ν = 3/7 ν = 1 ν = 2/3 ν = 3/5

ν = 1/3: Laughlin-like state ν = 1: BIQHE in-line with size dependence of BIQHE in optical flux lattices (Sterdyniak et al., 2015)

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

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SLIDE 20

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

|C| = 2 Band

Bosons - Stability in the Effective Continuum 0.5 1 1.5 2 2.5 3 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/3 ν = 2/5 ν = 3/7 ν = 1 ν = 2/3 ν = 3/5

ν = 1/3: Laughlin-like state ν = 1: BIQHE in-line with size dependence of BIQHE in optical flux lattices (Sterdyniak et al., 2015)

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

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

|C| = 2 Band

Bosons - Correlation Function

21 42 63 0 21 42 63 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|) Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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SLIDE 22

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

|C| = 2 Band

Bosons - Correlation Function

21 42 63 0 21 42 63 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|) Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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SLIDE 23

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

|C| = 2 Band

Bosons - Correlation Function

21 42 63 0 21 42 63 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|) Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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SLIDE 24

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

|C| = 2 Band

Bosons - Correlation Function

21 42 63 0 21 42 63 0.47 0.94 1.4 x y g(r)

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

|C|2 sheets

ν = 1/3 N = 7 (x mod |C|, y mod |C|) Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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SLIDE 25

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

|C| = 2 Band

Bosons - Correlation Functions & Entanglement Spectra 0 21 42 63 021 42 63 0.47 0.94 1.4 x y g(r)

ν = 1/3 N = 7

5 10 15 20 25 5 10 15 20

ξ kx*Ly + ky ∆ξ = 14.25

|Ψ =

k,n λk,n |ΨA k,n ⊗ |ΨB k,n

ξ ≡ − ln λ2

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

0 23 47 70 023 47 70 0.6 1.2 1.8 x y g(r)

ν = 2/5 N = 8

5 10 15 20 25 30 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

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SLIDE 26

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

|C| = 2 Band

Fermions - Stability in the Effective Continuum 1 2 3 4 5 6 7 8 9 0.03 0.06 0.09 0.12 0.15 0.18 limq→∞

  • q2∆
  • /10−1

N−1 ν = 1/5 ν = 2/9 ν = 3/13 ν = 1/3 ν = 2/7 ν = 3/11

d = 2

ν = 1/5 state previously established. All others new!

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

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SLIDE 27

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

|C| = 2 Band

Fermions - Correlation Functions & Entanglement Spectra 0 35 70 105035 70 105 0.55 1.1 1.6 x y g(r)

ν = 1/5 N = 9

10 15 20 25 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 73 110 0.53 1.1 1.6 x y g(r)

ν = 3/11 N = 9

5 10 15 20 25 5 10 15 20

ξ kx*Ly + ky ∆ξ = 10.84 (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

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SLIDE 28

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

|C| = 3 Band

Bosons - Stability in the Effective Continuum 0.4 0.8 1.2 1.6 2 0.04 0.08 0.12 0.16 0.2 limq→∞(q∆)/10−1 N−1 ν = 1/4 ν = 2/7 ν = 3/10 ν = 1/2 ν = 2/5 ν = 3/8

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

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SLIDE 29

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

|C| = 3 Band

Bosons - Correlation Functions

(0, 0) (0, 1) (0, 2) (1, 0) (1, 1) (1, 2) (2, 0) (2, 1) (2, 2) 21 42 63 0 21 42 63 0.47 0.94 1.4 x y g(r) ν = 1/4 N = 7 26 52 78 0 26 52 78 0.47 0.94 1.4 x y g(r) ν = 1/2 N = 9

Figure: Pair correlation functions for the lowest-lying ground state in the (kx, ky) = (0, 0) momentum sector

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

Stability in the Effective Continuum

Summary

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 1 2 3 4 5 6 7 0 0.20.40.60.8 1 limq→∞(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 ± 0.01 2 1 1/3 0.27 ± 0.005 3 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 be extrapolated to the thermodynamic limit

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

Conclusion

◮ Scaling to the effective continuum limit at fixed aspect ratio

converges faster than scaling at fixed flux density.

◮ Vast majority of finite-size spectra produce the ground state

degeneracy predicted by CF theory.

◮ Laughlin-like states with ν = 1/(|kC| + 1) are the most

robust, and yield a clear gap in the effective continuum limit.

◮ Instability may be caused by competing topological phases,

charge density waves, or finite-size effects.

◮ Stable FCIs found with clear entanglement gaps in |C| > 1

bands - largest gaps seen for |C| = 2 fermions.

◮ Pair-correlations are smooth functions modulated by |C| sites

along 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！

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

Conclusion

◮ Scaling to the effective continuum limit at fixed aspect ratio

converges faster than scaling at fixed flux density.

◮ Vast majority of finite-size spectra produce the ground state

degeneracy predicted by CF theory.

◮ Laughlin-like states with ν = 1/(|kC| + 1) are the most

robust, and yield a clear gap in the effective continuum limit.

◮ Instability may be caused by competing topological phases,

charge density waves, or finite-size effects.

◮ Stable FCIs found with clear entanglement gaps in |C| > 1

bands - largest gaps seen for |C| = 2 fermions.

◮ Pair-correlations are smooth functions modulated by |C| sites

along 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！

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SLIDE 33

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

Supplementary Slides

  • 1. Approaching the Effective Continuum
  • 2. Warm-up: |C| = 1 Band: Bosons & Fermions - Scaling of the

Gap with MUC Size

  • 3. |C| = 2 Band: Bosons - Rectangular Geometries
  • 4. |C| = 3 Band: Bosons - Rectangular Geometries
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Introduction Theory Method Results for |C| = 1, 2, 3 Conclusion

Approaching the Effective Continuum

Q: Should we fix nφ or fix aspect ratio?

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 nφ filled symbols ⇒ fixed aspect ratio

(a) bosons

5 5.5 6 6.5 7 7.5 8 8.5

  • 3
  • 2

q2∆/10−1 − log q

N = 6 at ν = 1/3

(b) fermions Figure: Finite-size scaling of the gap in the |C| = 2 band

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SLIDE 35

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

Approaching the Effective Continuum

Q: Should we fix nφ or fix aspect ratio? A: Fix aspect ratio

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 nφ filled symbols ⇒ fixed aspect ratio

(a) bosons

5 5.5 6 6.5 7 7.5 8 8.5

  • 3
  • 2

q2∆/10−1 − 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

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SLIDE 36

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

Warm-up: |C| = 1 Band

Bosons & Fermions - Scaling of the Gap with MUC Size

5.4 5.7 6 6.3 −2 −1 q∆/10−1 − log q Bosonic Laughlin (ν = 1/2) 0.8 1 1.2 1.4 −3 −2 q2∆/2 − log q Fermionic Laughlin (ν = 1/3)

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

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SLIDE 37

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

Warm-up: |C| = 1 Band

Bosons & Fermions - Scaling of the Gap with MUC Size

5.4 5.7 6 6.3 −2 −1 q∆/10−1 − log q

lim

q→∞(q∆) = 0.62 Bosonic Laughlin (ν = 1/2) 0.8 1 1.2 1.4 −3 −2 q2∆/2 − log q

agrees with Bauer et al. lim

q→∞(q2∆) = 2.60 Fermionic Laughlin (ν = 1/3)

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

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SLIDE 38

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

|C| = 2 Band

Bosons - Rectangular Geometries 5.9 6 6.1 6.2 6.3 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

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SLIDE 39

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

|C| = 2 Band

Bosons - Rectangular Geometries 5.9 6 6.1 6.2 6.3 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

5 10 15 20 25 5 10 15

(E-E0)/10-5 kx*Ly + ky

Lx=2, Ly=9, MUC: p=384, q=59 x 13 5 10 15 20 5 10 15

(E-E0)/10-5 kx*Ly + ky

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

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SLIDE 40

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

|C| = 3 Band

Bosons - Rectangular Geometries 0.0 0.1 0.3 0.4 0.6 0.8

  • 3
  • 2

q∆/10−1 − log q q∆ ≈ 0.03

Figure: Magnitude of the gap for the 6-particle state at ν = 3/8

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SLIDE 41

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

|C| = 3 Band

Bosons - Rectangular Geometries 0.0 0.1 0.3 0.4 0.6 0.8

  • 3
  • 2

q∆/10−1 − log q q∆ ≈ 0.03

Figure: Magnitude of the gap for the 6-particle state at ν = 3/8

5 10 15 20 25 30 2 4 6 8 10 12 14 16 x2 x2

(E-E0)/10-5 kx*Ly + ky

Lx=4, Ly=4, MUC: p=120, q=19 x 19 5 10 15 20 25 30 2 4 6 8 10 12 14 16

(E-E0)/10-5 kx*Ly + ky

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

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SLIDE 42

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

|C| = 3 Band

Bosons - Rectangular Geometries 0.0 0.1 0.3 0.4 0.6 0.8

  • 3
  • 2

q∆/10−1 − log q q∆ ≈ 0.03

Figure: Magnitude of the gap for the 6-particle state at ν = 3/8

6 8 10 12 14 2 4 6 8 10 12 14 16

ξ kx*Ly + ky

6 8 10 12 14 2 4 6 8 10 12 14 16

ξ kx*Ly + ky