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, Cavendish Laboratory, University of Cambridge October 31, 2017 arXiv:1710.09350
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
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¨ oller & Cooper, 2015. ◮ Generalize the n φ → 0 continuum limit to the effective continuum limit at n φ → 1 / | C | (M¨ oller & Cooper, 2015). ◮ Investigate the stability (i.e. robustness in the effective continuum limit) of the many-body gap, ∆.
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Harper-Hofstadter Model We consider N spinless particles hopping on an N x × N y square lattice with a constant effective magnetic flux. E C = − 1 C =+1 +2 − 2 − 2 +2 C =+1 C = − 1 n φ magnetic translation-invariant phase interaction potential c † � e φ ij � P LB � � H = j c i + h.c. + P LB : ρ ( r i ) ρ ( r j ): t ij V ij i , j i < j lowest-band projection operator hopping parameter • bosons ⇒ on-site interactions • fermions ⇒ nearest-neighbour interactions
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¨ oller & Cooper, 2015): | kC | r + 1 ≡ r r ν = 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
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Scaling & Stability Aim to consider 2D isotropic limit ⇒ demand N x = N y . Nbr. Sites ◦ Note: 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 → ∞ .
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Harper-Hofstadter Model Approaching the Effective Continuum 4 | C | p − sgn ( C ) ≡ p p n φ = q , p ∈ N . 2 0 E -2 p lim p →∞ 2 p ± 1 lim q →∞ 1 q -4 0 0.2 0.4 0.8 1 0.6 n V (e.g. Hormozi et al. , 2012)
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 lim q →∞ ( q (2) ∆). q 2 ∆ ≈ 0 . 82 ∆ = 0 . 82 / q 2 70 L x =3, L y =8, MUC: p=227, q=35 x 13 4 8 . 24 60 50 (E-E 0 )/10 -7 3 q 2 ∆ / 10 − 1 8 . 22 ∆ / 10 − 6 40 2 8 . 2 30 ∆ 1 8 . 18 20 10 0 8 . 16 0 0 1 2 3 4 5 0 1 2 3 0 5 10 15 20 25 q − 2 / 10 − 6 q − 1 / 10 − 3 k x *L y + k y
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 lim q →∞ ( q (2) ∆). 4. Plot lim q →∞ ( q (2) ∆) against N . Read off lim N , q →∞ ( q (2) ∆).
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 / N c ⇒ N = ν N c : N c must be a multiple of s . 3. Isolated lowest Chern number C band at | C | p − sgn ( C ) ≡ p p n φ = q , p ∈ N . 4. Consider 2D systems ⇒ approximately unit aspect ratios: � � � 1 − N x � � � ≤ ǫ, for small ǫ. � � N y 5. Limited computation time: dim {H} < 10 7 .
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? 6 . 2 8 8 26 6 . 15 7 7 24 6 . 1 6 6 22 6 . 05 5 q 2 ∆ / 10 − 1 5 q ∆ / 10 − 1 q / 10 2 q / 10 2 6 . 2 10 10 6 4 20 4 26 9 9 6 . 1 8 24 8 5 . 95 3 3 N 18 N 7 7 22 5 . 9 2 2 6 6 6 16 0 0 . 05 0 . 1 0 0 . 05 0 . 1 5 . 85 1 1 q − 1 q − 1 14 5 . 8 0 0 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 N − 1 N − 1 (a) ν = 1 / 2 bosons (b) ν = 1 / 3 fermions Figure: Finite-size scaling of the gap for Laughlin states
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. N , q → ∞ limits commute! (if both limits can be taken) 6 . 2 8 8 26 6 . 15 7 7 24 6 . 1 6 6 22 6 . 05 5 q 2 ∆ / 10 − 1 5 q ∆ / 10 − 1 q / 10 2 q / 10 2 6 . 2 10 10 6 4 20 4 26 9 9 6 . 1 8 24 8 5 . 95 3 3 N 18 N 7 7 22 5 . 9 2 2 6 6 6 16 0 0 . 05 0 . 1 0 0 . 05 0 . 1 5 . 85 1 1 q − 1 q − 1 14 5 . 8 0 0 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 N − 1 N − 1 (a) ν = 1 / 2 bosons (b) ν = 1 / 3 fermions Figure: Finite-size scaling of the gap for Laughlin states
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Warm-up: | C | = 1 Band Bosons - Stability in the Continuum ν = 1 / 2 ν = 2 / 3 ν = 2 ν = 3 / 4 ν = 3 / 2 8 7 lim q →∞ ( q ∆) / 10 − 1 6 5 4 3 RR 2 1 0 0 0 . 04 0 . 08 0 . 12 0 . 16 0 . 2 N − 1 Figure: Finite-size scaling of the gap to the thermodynamic continuum limit at fixed aspect ratio
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Warm-up: | C | = 1 Band Bosons - Stability in the Continuum agrees with Bauer et al. � ν = 1 / 2 ν = 2 / 3 ν = 2 ν = 3 / 4 ν = 3 / 2 8 ν = 2 DMRG BIQHE: 7 lim q →∞ ( q ∆) / 10 − 1 more than just LLL involved 6 in stabilizing the state 5 (He et al. , 2017) 4 3 RR 2 ν = 2 competition expected 1 from continuum results 0 (Cooper & Rezayi, 2007) 0 0 . 04 0 . 08 0 . 12 0 . 16 0 . 2 N − 1 Figure: Finite-size scaling of the gap to the thermodynamic continuum limit at fixed aspect ratio
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Warm-up: | C | = 1 Band Bosons - Stability in the Continuum 9 L x =1, L y =7, MUC: p=848, q=77 x 11 8 7 ν = 1 / 2 (E-E 0 )/10 -4 6 x2 ν = 2 / 3 ν = 2 5 ν = 3 / 4 ν = 3 / 2 4 8 3 7 lim q →∞ ( q ∆) / 10 − 1 x2 2 6 1 ν = 2: � = BIQHE! 5 0 4 -1 0 1 2 3 4 5 6 7 8 k x *L y + k y 3 RR 2 L x =1, L y =8, MUC: p=969, q=88 x 11 1 8 0 ν = 3 / 2: indications for a 0 0 . 04 0 . 08 0 . 12 0 . 16 0 . 2 stable RR state (as in LLL) (E-E 0 )/10 -4 N − 1 6 Figure: Finite-size scaling of the gap to the 4 thermodynamic continuum limit at fixed 2 aspect ratio 0 0 2 4 6 8 k x *L y + k y
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Warm-up: | C | = 1 Band Bosons - Pair Correlation Functions no perfect correlation hole at ν > 1 / 2 Laughlin state N = 8 1.2 ν = 1 / 2 g ( r ) 0.8 ν = 2 / 3 ν = 2 0.4 0 ν = 3 / 4 ν = 3 / 2 40 27 0 13 27 40 013 y x finite-size effects no correlation hole N = 10 N = 20 1.2 1.2 g ( r ) 0.81 g ( r ) 1.1 0.41 0.93 0 0.78 45 30 30 20 0 15 30 45 015 0 10 20 30 010 y y x x charge density wave N = 9 N = 9 1.5 1.2 g ( r ) 1 g ( r ) 1.1 0.51 0.88 0 0.7 36 24 24 8 16 24 0 816 0 12 24 36 012 0 y y x x Figure: Pair correlation functions for the lowest-lying ground state in the ( k x , k y ) = (0 , 0) momentum sector
Introduction Theory Method Results for | C | = 1 , 2 , 3 Conclusion Warm-up: | C | = 1 Band Fermions - Stability in the Continuum ν = 1 / 3 ν = 2 / 5 ν = 2 / 3 ν = 3 / 7 ν = 3 / 5 30 lim q →∞ ( q 2 ∆) / 10 − 1 25 20 15 10 particle-hole symmetry! 5 0 0 0 . 02 0 . 04 0 . 06 0 . 08 0 . 1 0 . 12 0 . 14 0 . 16 0 . 18 N − 1 Figure: Finite-size scaling of the gap to the thermodynamic continuum limit at fixed aspect ratio
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