adiabatic connection of fractional chern insulators to ... · a. sterdyniak, n. regnault, & gm,...
TRANSCRIPT
![Page 1: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/1.jpg)
Adiabatic connection of Fractional Chern Insulators to Fractional Quantum Hall States
Gunnar MöllerCavendish Laboratory, University of Cambridge
Cavendish Laboratory
November 21, 2012, University of Leeds
A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012)
Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)
GM & N. R. Cooper, Phys. Rev. Lett. 103, 105303 (2009)
![Page 2: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/2.jpg)
Gunnar Möller University of Leeds, November 21st, 2012
• Motivation: physical realizations of lattices with high flux density
• Relationship between different realizations of topological band structures:
• Particle entanglement spectrum as a probe for quantum Hall states in lattices
Outline
‣ Adiabatic continuation from Fractional Chern Insulators to Fractional Quantum Hall states
![Page 3: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/3.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Magnetic flux through periodic potentials
• magnetic field in presence of crystal potentials in solid state systems
0 a n 1
![Page 4: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/4.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Magnetic flux through periodic potentials
• magnetic field in presence of crystal potentials in solid state systems
• cold atoms in `rotating’ optical lattices: 0 ∼ a n 1
0 a n 1
• Simulating Aharonov-Bohm flux by complexhopping in tight binding optical lattices 0 a n 1
• Simulating Aharonov-Bohm flux by Berry phases in real space
0 a n 1“Optical Flux Lattices”
Simulation of Landau-gauge (continuum) 0 ∼ L1/2
Cold A
toms
![Page 5: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/5.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Magnetic flux through periodic potentials
• magnetic field in presence of crystal potentials in solid state systems
• cold atoms in `rotating’ optical lattices: 0 ∼ a n 1
0 a n 1
• Simulating Aharonov-Bohm flux by complexhopping in tight binding optical lattices 0 a n 1
• Simulating Aharonov-Bohm flux by Berry phases in real space
0 a n 1“Optical Flux Lattices”
Simulation of Landau-gauge (continuum) 0 ∼ L1/2
Cold A
toms
![Page 6: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/6.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Magnetic flux through periodic potentials
• magnetic field in presence of crystal potentials in solid state systems
• cold atoms in `rotating’ optical lattices: 0 ∼ a n 1
0 a n 1
• Simulating Aharonov-Bohm flux by complexhopping in tight binding optical lattices 0 a n 1
• Simulating Aharonov-Bohm flux by Berry phases in real space
0 a n 1“Optical Flux Lattices”
Simulation of Landau-gauge (continuum) 0 ∼ L1/2
Cold A
toms
![Page 7: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/7.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Magnetic flux through periodic potentials
• magnetic field in presence of crystal potentials in solid state systems
• cold atoms in `rotating’ optical lattices: 0 ∼ a n 1
0 a n 1
• Simulating Aharonov-Bohm flux by complexhopping in tight binding optical lattices 0 a n 1
• Simulating Aharonov-Bohm flux by Berry phases in real space
0 a n 1“Optical Flux Lattices”
Simulation of Landau-gauge (continuum) 0 ∼ L1/2
• Chern bands: Berry flux in reciprocal space
0 a n 1
Cold A
toms
Solid State
Cold A
toms
![Page 8: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/8.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Magnetic flux through periodic potentials
• magnetic field in presence of crystal potentials in solid state systems
• cold atoms in `rotating’ optical lattices: 0 ∼ a n 1
0 a n 1
• Simulating Aharonov-Bohm flux by complexhopping in tight binding optical lattices 0 a n 1
• Simulating Aharonov-Bohm flux by Berry phases in real space
0 a n 1“Optical Flux Lattices”
Simulation of Landau-gauge (continuum) 0 ∼ L1/2
• Chern bands: Berry flux in reciprocal space
0 a n 1
Cold A
toms
Solid State
Cold A
toms
![Page 9: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/9.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms I: Optical Lattices with Complex Hopping
B
optical lattice + Raman lasers
H = −J
α,β
b†αbβe
iAαβ + h.c.+
1
2U
α
nα(nα − 1)− µ
α
nα
⇒ possibility to simulate Aharonov-Bohm effect of magnetic field by imprinting phases for hopping via Raman transitions
Aαβ = 2πnV
⇒ Bose-Hubbard with a magnetic field (! Lorentz force)
particle density vortex/flux density interaction nV U/JnV
J. Dalibard, et al. Rev. Mod. Phys. 83, 1523 (2011)
experimental realisation:
![Page 10: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/10.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms I: Optical Lattices with Complex Hopping
‘anti-magic’ optical lattice for Yb
Gerbier & Dalibard, NJP 2010
Staggered Flux ‘Rectified’ Flux
Realized by group of I. Bloch:Phys. Rev. Lett. 107, 255301 (2011). Experimental realization outstanding
Can tune flux density!
![Page 11: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/11.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms II: Berry phases of optically dressed states
spacially varying optical coupling of N internal states (consider N=2)
H =p2
2m1 +
2
−∆ ΩR(r)
ΩR(r) ∆
Yblocal spectrum En(r) and dressed states: |Ψr
J. Dalibard, F. Gerbier, G. Juzeliunas, P. Öhberg, RMP 2011
![Page 12: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/12.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms II: Berry phases of optically dressed states
spacially varying optical coupling of N internal states (consider N=2)
H =p2
2m1 +
2
−∆ ΩR(r)
ΩR(r) ∆
Yblocal spectrum En(r) and dressed states: |Ψr
adiabatic motion of atoms in space in the optical potential generates a Berry phase
J. Dalibard, F. Gerbier, G. Juzeliunas, P. Öhberg, RMP 2011
![Page 13: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/13.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms II: Berry phases of optically dressed states
for N=2, consider Bloch sphere of unit vector
spacially varying optical coupling of N internal states (consider N=2)
H =p2
2m1 +
2
−∆ ΩR(r)
ΩR(r) ∆
Yblocal spectrum En(r) and dressed states: |Ψr
adiabatic motion of atoms in space in the optical potential generates a Berry phase
nφ =1
8πijkµνni∂µnj∂νnk
n = Ψr|σ|Ψr
Berry flux generated:
Anφ d
2r =Ω
4π
Total flux quanta = # times Bloch vector wraps sphere
J. Dalibard, F. Gerbier, G. Juzeliunas, P. Öhberg, RMP 2011
![Page 14: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/14.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms II: Optical Flux Lattices
periodic optical Raman potentials are conveniently located by standing wave Lasersyielding an optical flux lattice of high flux density, here nϕ=1/2a2 (fixed)
(a) Local direction of unit Bloch vector (b) Local density of Berry Flux n nφ
Nigel Cooper, PRL (2011); N. Cooper & J. Dalibard, EPL (2011)
![Page 15: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/15.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Cold Atoms II: Optical Flux Lattices
periodic optical Raman potentials are conveniently located by standing wave Lasersyielding an optical flux lattice of high flux density, here nϕ=1/2a2 (fixed)
(a) Local direction of unit Bloch vector (b) Local density of Berry Flux n nφ
Nφ = 1
Nigel Cooper, PRL (2011); N. Cooper & J. Dalibard, EPL (2011)
![Page 16: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/16.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Fractionalization in Chern bands? Chern #1 Bands = FQHE ?
Proposition: correlated states reproduce the physics of FQHE
![Page 17: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/17.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Fractionalization in Chern bands? Chern #1 Bands = FQHE ?
Proposition: correlated states reproduce the physics of FQHE
![Page 18: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/18.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Fractionalization in Chern bands? Chern #1 Bands = FQHE ?
Proposition: correlated states reproduce the physics of FQHE
![Page 19: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/19.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Fractionalization in Chern bands? Chern #1 Bands = FQHE ?
Proposition: correlated states reproduce the physics of FQHE
![Page 20: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/20.jpg)
Gunnar Möller University of Leeds, November 21, 2012
H =− t1
rr
a†rar + h.c.
− t2
rr
a†rare
iφrr + h.c.
− t3
rr
a†rar + h.c.
+
U
2
r
nr(nr − 1)
Characteristic example: The Haldane Model
• tight binding model in real space on hexagonal lattice• with fine-tuned hopping parameters: obtain flat lower band, e.g. values [D. Sheng, PRL (2011)]
t1 = 1, t2 = 0.60, t2 = −0.58 and φ = 0.4π
F.D.M. Haldane, PRL (1988), Neupert et al. PRL (2011)
![Page 21: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/21.jpg)
Gunnar Möller University of Leeds, November 21, 2012
hαβ(k)unβ(k) = n(k)u
nα(k)
A(n,k) = −i
α
un∗α (k)∇ku
nα(k)
Bloch states
H =
k a†k,αhαβ(k)ak,β
• diagonalize Hamiltonian by Fourier transform
Topological (flat) bands in two-dimensions
• study Berry curvature in nth band:
Berry connection:
B(k) = ∇k ∧A(k)Berry curvature:
C = 12π
BZ d2kB(k)Chern number:
![Page 22: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/22.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Flux Lattices vs Chern Bands
= tight binding model in real space with topological flat bandsCBsreal space reciprocal space
real space reciprocal space
![Page 23: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/23.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Flux Lattices vs Chern Bands
= tight binding model in real space with topological flat bandsCBsreal space reciprocal space
real space reciprocal space
![Page 24: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/24.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Flux Lattices vs Chern Bands
= tight binding model in real space with topological flat bandsCBs
= tight binding model in reciprocal space with topological flat bands
real space reciprocal space
real space reciprocal space
N. R. Cooper, R. Moessner, PRL (2012) [arxiv:1208.4579]
Flux Lattice
absorption of photons = momentum + state transfers
![Page 25: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/25.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Flux Lattices vs Chern Bands
= tight binding model in real space with topological flat bandsCBs
= tight binding model in reciprocal space with topological flat bands
real space reciprocal space
real space reciprocal space
N. R. Cooper, R. Moessner, PRL (2012) [arxiv:1208.4579]
Flux Lattice
absorption of photons = momentum + state transfers
![Page 26: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/26.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Overview of underlying band structures
CBs
Lattices with homogeneous flux(Hofstadter bands)
Flux Lattices(vanishing overall flux)
![Page 27: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/27.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Overview of underlying band structures
CBs
Lattices with homogeneous flux(Hofstadter bands)
Flux Lattices
insertion of flux quanta through plaquettes [see Wu et al]
(vanishing overall flux)
![Page 28: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/28.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Overview of underlying band structures
CBs
Lattices with homogeneous flux(Hofstadter bands)
Flux Lattices
insertion of flux quanta through plaquettes [see Wu et al]
(vanishing overall flux)
![Page 29: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/29.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Overview of underlying band structures
CBs
Lattices with homogeneous flux(Hofstadter bands)
Flux Lattices
insertion of flux quanta through plaquettes [see Wu et al]
Magnetic Fields & Landau Levels
(vanishing overall flux)
![Page 30: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/30.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Overview of underlying band structures
CBs
Lattices with homogeneous flux(Hofstadter bands)
Flux Lattices
insertion of flux quanta through plaquettes [see Wu et al]
Magnetic Fields & Landau Levels
(vanishing overall flux)
+ Interactions
![Page 31: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/31.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interaction driven phases in these bands
CBs
Lattices withhomogeneous flux
Flux Latticesinsertion of flux quanta
through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
![Page 32: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/32.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interaction driven phases in these bands
CBs
Lattices withhomogeneous flux
Flux Latticesinsertion of flux quanta
through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
C>1
![Page 33: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/33.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interaction driven phases in these bands
CBs
Lattices withhomogeneous flux
Flux Latticesinsertion of flux quanta
through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
C>1
C>1
![Page 34: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/34.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Strongly correlated states from the Hofstadter spectrum
• Hofstadter spectrum provides bands of all Chern numbers [Avron et al.]
E
nφ
1
2
-1
-2
![Page 35: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/35.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Strongly correlated states from the Hofstadter spectrum
• Hofstadter spectrum provides bands of all Chern numbers [Avron et al.]
E
nφ
nφ
n
• Interactions stabilize fractional quantum Hall liquids in these bands!
• CF Theory: GM & N. R. Cooper, PRL (2009)
• Near rational flux density: LL’s with additional pseudospin indexR. Palmer & D. Jaksch PRL 2006L. Hormozi et al, PRL 2012
1
2
-1
-2
![Page 36: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/36.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interrelations between different interacting problems
CBs
Lattices withhomogeneous flux
Flux Lattices
insertion of flux quanta through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
C>1
C=1
C>1
![Page 37: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/37.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interrelations between different interacting problems
CBs
Lattices withhomogeneous flux
Flux Lattices
insertion of flux quanta through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
C>1
C=1
addition of oscillatorymagnetic field density
leaves LLL intact:should be adiabatic
C>1
![Page 38: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/38.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interrelations between different interacting problems
CBs
Lattices withhomogeneous flux
Flux Lattices
insertion of flux quanta through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
C>1
C=1
addition of oscillatorymagnetic field density
leaves LLL intact:should be adiabatic
adiabatically turning onlattice confinement
C>1
![Page 39: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/39.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interrelations between different interacting problems
CBs
Lattices withhomogeneous flux
Flux Lattices
insertion of flux quanta through plaquettes
Magnetic Fields & Landau Levels
FQHE
FCI
FQHE in lattices
FQHE ?
C>1
C=1
adiabatic continuation(Wannier or Bloch basis)
addition of oscillatorymagnetic field density
leaves LLL intact:should be adiabatic
adiabatically turning onlattice confinement
X.-L. Qi (2011), Wu et al (2012), Scaffidi & Möller PRL (2012),Liu & Bergholtz PRB (2013)
C>1
![Page 40: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/40.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Numerical evidence for “Fractional Chern Insulators”
• existence of a gap & groundstate degeneracy [checkerboard lattice]• chern number of groundstate manifold
[D. Sheng]
![Page 41: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/41.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Numerical evidence for “Fractional Chern Insulators”
• existence of a gap & groundstate degeneracy [checkerboard lattice]• chern number of groundstate manifold
[D. Sheng]
![Page 42: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/42.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Numerical evidence for “Fractional Chern Insulators”
• Finite size scaling of gap
“Fractional Chern Insulators (FCI)” [N. Regnault & A. Bernevig, PRX ’11]
• Particle Entanglement Spectra : count of excitations matches FQHE (here - Laughlin state)
• existence of a gap & groundstate degeneracy [checkerboard lattice]
• chern number of groundstate manifold [D. Sheng]
![Page 43: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/43.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Numerical evidence for “Fractional Chern Insulators”
• Strong numerical evidence for QHE physics, but no clear organising principle for different lattice models
• Finite size scaling of gap
“Fractional Chern Insulators (FCI)” [N. Regnault & A. Bernevig, PRX ’11]
• Particle Entanglement Spectra : count of excitations matches FQHE (here - Laughlin state)
• existence of a gap & groundstate degeneracy [checkerboard lattice]
• chern number of groundstate manifold [D. Sheng]
![Page 44: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/44.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Understanding Fractional Quantum Hall states
φm ∝ zme−|z|2/40
• Single particle states are analytic functions in symmetric gauge A =1
2r ∧ B
• Many particle states are still analytic functions - can write explicitly!
zj = xj + iyj
Ψν= 1m
=
i<j
(zi − zj)me−
i |zi|
2/40e.g. Laughlin:
![Page 45: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/45.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Understanding Fractional Quantum Hall states
φm ∝ zme−|z|2/40
• Single particle states are analytic functions in symmetric gauge A =1
2r ∧ B
• Many particle states are still analytic functions - can write explicitly!
zj = xj + iyj
Ψν= 1m
=
i<j
(zi − zj)me−
i |zi|
2/40e.g. Laughlin:
• Wavefunctions are nice! Can understand many features
" Quasiparticle excitations: charge / statistics" Incompressibility
" Correlations / You name the observable...
![Page 46: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/46.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Understanding Fractional Quantum Hall states
φm ∝ zme−|z|2/40
• Single particle states are analytic functions in symmetric gauge A =1
2r ∧ B
• Many particle states are still analytic functions - can write explicitly!
zj = xj + iyj
Ψν= 1m
=
i<j
(zi − zj)me−
i |zi|
2/40e.g. Laughlin:
• Wavefunctions are nice! Can understand many features
" Quasiparticle excitations: charge / statistics" Incompressibility
" Correlations / You name the observable...
Can we construct wavefunctions
forChern Insulators?
![Page 47: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/47.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Mapping from FQHE to FCI: Single Particle Orbitals
FCIFQHE
• Proposal by X.-L. Qi [PRL ’11]: Get FCI Wavefunctions by mapping single particle orbitals
![Page 48: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/48.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Mapping from FQHE to FCI: Single Particle Orbitals
FCIFQHE
• Proposal by X.-L. Qi [PRL ’11]: Get FCI Wavefunctions by mapping single particle orbitals
• Idea: use Wannier states which are localized in the x-direction• keep translational invariance in y (cannot create fully localized Wannier state if C>0!)
|W (x, ky) =
kx
f(x,ky)kx
|kx, ky
![Page 49: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/49.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Mapping from FQHE to FCI: Single Particle Orbitals
FCIFQHE
• Proposal by X.-L. Qi [PRL ’11]: Get FCI Wavefunctions by mapping single particle orbitals
• Idea: use Wannier states which are localized in the x-direction• keep translational invariance in y (cannot create fully localized Wannier state if C>0!)
• Qi’s Proposition: using a mapping between the LLL eigenstates (QHE) and localized Wannier states (FCI), we can establish an exact mapping between their many-particle wavefunctions
|W (x, ky) =
kx
f(x,ky)kx
|kx, ky
1:1
![Page 50: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/50.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Mapping from FQHE to FCI: Single Particle Orbitals
FCIFQHE
• Proposal by X.-L. Qi [PRL ’11]: Get FCI Wavefunctions by mapping single particle orbitals
• Idea: use Wannier states which are localized in the x-direction• keep translational invariance in y (cannot create fully localized Wannier state if C>0!)
• Qi’s Proposition: using a mapping between the LLL eigenstates (QHE) and localized Wannier states (FCI), we can establish an exact mapping between their many-particle wavefunctions
|W (x, ky) =
kx
f(x,ky)kx
|kx, ky
1:1
?
![Page 51: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/51.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Wannier states in Chern bands
H =
k a†k,αhαβ(k)ak,β
• Some formalism
α = 1 α = 2
α = 3
c†k,α =1√N
R
eik·(R+δα)c†R,α
hαβ(k)unβ(k) = n(k)u
nα(k)
• construction of a Wannier state at fixed ky
A(n,k) = −i
α
un∗α (k)∇ku
nα(k)
Hamiltonian
Eigenstates
Berry connection
|W (x, ky) =χ(ky)√
Lx
kx
e−i kx0 Ax(px,ky)dpx × eikx
θ(ky)2π × e−ikxx|kx, ky
Fourier transform
![Page 52: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/52.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Wannier states in Chern bands
H =
k a†k,αhαβ(k)ak,β
• Some formalism
α = 1 α = 2
α = 3
c†k,α =1√N
R
eik·(R+δα)c†R,α
hαβ(k)unβ(k) = n(k)u
nα(k)
• construction of a Wannier state at fixed ky in gauge with
A(n,k) = −i
α
un∗α (k)∇ku
nα(k)
Hamiltonian
Eigenstates
Berry connection
Fourier transform
|W (x, ky) =χ(ky)√
Lx
kx
e−i kx0 Ax(px,ky)dpx × eikx
θ(ky)2π × e−ikxx|kx, ky
‘Parallel transport’ of phase
Berry connection indicates changeof phase due to displacement in BZ
Ay = 0
![Page 53: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/53.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Wannier states in Chern bands
H =
k a†k,αhαβ(k)ak,β
• Some formalism
α = 1 α = 2
α = 3
c†k,α =1√N
R
eik·(R+δα)c†R,α
hαβ(k)unβ(k) = n(k)u
nα(k)
• construction of a Wannier state at fixed ky in gauge with
A(n,k) = −i
α
un∗α (k)∇ku
nα(k)
Hamiltonian
Eigenstates
Berry connection
Fourier transform
|W (x, ky) =χ(ky)√
Lx
kx
e−i kx0 Ax(px,ky)dpx × eikx
θ(ky)2π × e−ikxx|kx, ky
‘Parallel transport’ of phase ‘Polarization’
ensures periodicity of WF in ky # ky + 2!
Berry connection indicates changeof phase due to displacement in BZ
Ay = 0
![Page 54: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/54.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Wannier states in Chern bands
H =
k a†k,αhαβ(k)ak,β
• Some formalism
α = 1 α = 2
α = 3
c†k,α =1√N
R
eik·(R+δα)c†R,α
hαβ(k)unβ(k) = n(k)u
nα(k)
• construction of a Wannier state at fixed ky in gauge with
A(n,k) = −i
α
un∗α (k)∇ku
nα(k)
Hamiltonian
Eigenstates
Berry connection
Fourier transform
|W (x, ky) =χ(ky)√
Lx
kx
e−i kx0 Ax(px,ky)dpx × eikx
θ(ky)2π × e−ikxx|kx, ky
‘Parallel transport’ of phase ‘Polarization’
ensures periodicity of WF in ky # ky + 2!
Berry connection indicates changeof phase due to displacement in BZ
Ay = 0
ky-dependent phase factor, or `gauge’
More on gauge of Wannier orbitals: Y-L Wu, A. Bernevig, N. Regnault, PRB (2012)
![Page 55: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/55.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Wannier states in Chern bands
• construction of a Wannier state at fixed ky in gauge with
Fourier transform
|W (x, ky) =χ(ky)√
Lx
kx
e−i kx0 Ax(px,ky)dpx × eikx
θ(ky)2π × e−ikxx|kx, ky
‘Parallel transport’ of phase ‘Polarization’
ensures periodicity of WF in ky # ky + 2!
Berry connection indicates changeof phase due to displacement in BZ
Ay = 0
• or, more simply we can think of the Wannier states as the eigenstates of the position operator
Xcg|W (x, ky) = [x− θ(ky)/2π]|W (x, ky)
• role of polarization: displacement of centre of mass of the Wannier state
Xcg = limqx→0
1
i
∂
∂qxρqx
θ(ky) =
2π
0Ax(px, ky)dpx
![Page 56: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/56.jpg)
Gunnar Möller University of Leeds, November 21, 2012
H =− t1
rr
a†rar + h.c.
− t2
rr
a†rare
iφrr + h.c.
− t3
rr
a†rar + h.c.
+
U
2
r
nr(nr − 1)
An example: The Haldane Model
• tight binding model on hexagonal lattice• with fine-tuned hopping parameters: obtain flat lower band
t1 = 1, t2 = 0.60, t2 = −0.58 and φ = 0.4π
![Page 57: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/57.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Conventions for numerical evaluation
Real Space Reciprocal Space
G1 = 2πex/L1 sin(γ)
G2 = 2π[− cot(γ)ex + ey]/L2
v1 = sin(γ)ex + cos(γ)ey
v2 = ey
nx = 0 Lx − 1
Ly − 1
ny = 0
unβ(k+ LiGi) = un
β(k)• choose ‘periodic’ gauge with Bloch functions:
A few remarks:
Anx(q1, q2) = log [un∗
α (q1, q2)unα(q1 + 1, q2)]• use discretized Berry connection
kx
0Ax(px, ky)dpx →
q1(kx)
q1=0
Anx(q1, q2)
• discretize its integrals by the rectangle rule
![Page 58: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/58.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Qi’s Mapping
• Can introduce a canonical order of states with monotonously increasing position:
Ky = ky + 2πx = 2πj/Ly
j = ny + Lyx = 0, 1, ..., Nφ − 1
ky = 2πny/Ly
• Increase in position for ky # ky + 2! = Chern-number C, as
∂
∂kyXcg|x = − 1
2π
∂θ(ky)
∂ky=
2π
0B(px, ky)dpx
Y-L Wu, A. Bernevig, N. Regnault, PRB (2012)
More on Wannier states:
Z. Liu & E. Bergholtz, PRB 2013
![Page 59: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/59.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Case study: Bosons with contact interactions
Hint ∝
i<j
δ(ri − rj)
• Landau level momentum conserved:
• Linearized momentum
H =
j1,j2,j3,j4
Vj1j2;j3j4 c†j1c†j2 cj3 cj4
expand Hamiltonian in single-particle orbitals (finite size, periodic boundary conditions)
Vj1j2;j3j4 ∝ δj1+j2, j3+j4 Vj1j2;j3j4 ∝ δ mod Lxj1+j2, j3+j4
Ky = ky + 2πx = 2πj/Ly
kmaxy = Nφ − 1 Kmax
y = Lx × Ly − 1
T. Scaffidi, GM, PRL (2012)[arxiv:1207.3539]
![Page 60: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/60.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Case study: Bosons with contact interactions
Hint ∝
i<j
δ(ri − rj)
• Landau level momentum conserved:
• Linearized momentum
H =
j1,j2,j3,j4
Vj1j2;j3j4 c†j1c†j2 cj3 cj4
expand Hamiltonian in single-particle orbitals (finite size, periodic boundary conditions)
Vj1j2;j3j4 ∝ δj1+j2, j3+j4 Vj1j2;j3j4 ∝ δ mod Lxj1+j2, j3+j4
Different conservation laws # Problems live in different Hilbert spaces
Ky = ky + 2πx = 2πj/Ly
kmaxy = Nφ − 1 Kmax
y = Lx × Ly − 1
T. Scaffidi, GM, PRL (2012)[arxiv:1207.3539]
![Page 61: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/61.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Matrix elements in the Wannier basis
Hint ∝
i<j
δ(ri − rj)
Ky = ky + 2πx = 2πj/Ly
HFCI =
ky1,ky2,ky3,ky4x1,x2,x3,x4
ky1+ky2=ky3+ky4
c†W (ky1,x1)c†W (ky2,x2)
cW (ky3,x3)cW (ky4,x4)
kx1,kx2,kx3,kx4kx1+kx2=kx3+kx4
f∗(x1,ky1 )kx1
f∗(x2,ky2)kx2
f(x3,ky3)kx3
f(x4,ky4)kx4
a=A,B
u∗aα0(k1)u
∗aα0(k2)u
aα0(k3)u
aα0(k4)
Vj1j2;j3j4
![Page 62: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/62.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Case study: Bosons with contact interactions
-6
-5
-4
-3
-2
-1
0
-6
-5
-4
-3
-2
-1
0
FCIFQHE
• Magnitude of two-body matrix elements for delta interactions in the Haldane model
• System shown: two-body interactions for N = 6, Lx × Ly = 3× 4
Ktoty = 0 Ktot
y = 4 Ktoty = 8 Ktot
y = 0 Ktoty = 4 Ktot
y = 8
Kto
ty
=0
Kto
ty
=4
Kto
ty
=8
Kto
ty
=0
Kto
ty
=4
Kto
ty
=8
• Matrix elements differ in magnitude, but overall similarities are present• Different block-structure due to non-conservation of linearized momentum Ky
Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012) [arxiv:1207.3539]
• Lack of translational invariance of matrix elements in momentum space
V (ri − rj) ∝ δ(ri − rj)
lnVαα lnVαα
![Page 63: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/63.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Reduced translational invariance in Ky
• A closer look at some short range hopping processes
Ky
Ky
Ky
• for FCI: hopping amplitudes depend on position of centre of mass / Ky
Center of mass position
|V|2
0 2 4 6 8 100
0.5
1
K1y +K2
y
![Page 64: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/64.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Interpolating in the Wannier basis
• Can write both states in single Hilbert space with the same overall structure (indexed by Ky) and study the low-lying spectrum numerically (exact diagonalization)
• Can study adiabatic deformations from the FQHE to a fractionally filled Chern band
Th. Scaffidi & GM, Phys. Rev. Lett. (2012)
H(x) =∆FCI
∆FQHE
(1− x)HFQHE + xHFCI
FQHE of Bosons at
ν = 1/2
x = 0 x = 1
Laughlin state Same topological phase?
Half filled band of the (flattened) Haldane-model
B
• Here: look at half-filled band for bosons
![Page 65: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/65.jpg)
Gunnar Möller University of Leeds, November 21, 2012
|Ψ(x)|Φ|2 |PkGS |Ψ|2
0 0.2 0.4 0.6 0.8 1x0.7
0.8
0.9
1
N=6, 3 x 4N=6, 4 x 3N=8, 4 x 4N=10,4 x 5
0 0.2 0.4 0.6 0.8 1x0.7
0.8
0.9
1
Evaluating the accuracy of the Wannier states: Overlaps
Th. Scaffidi & GM, Phys. Rev. Lett. (2012) [arxiv:1207.3539]
Overlap Weight in GS sector
FCI FCIFQHEFQHE
Bosons at ν = 1/2
• Can write both states in single Hilbert space with the same overall structure (indexed by Ky) and study the low-lying spectrum numerically (exact diagonalization)
• Can study adiabatic deformations from the FQHE to a fractionally filled Chern band
H(x) =∆FCI
∆FQHE
(1− x)HFQHE + xHFCI
![Page 66: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/66.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Adiabatic continuation in the Wannier basis
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1
E
x
x2
0
0.025
0.05
0.075
0 1 2 3 4
ky
x2
0 5 10 15
0
0.025
0.05
0.075
Ky
1
FQHE FCI
• Spectrum for N=10:
Th. Scaffidi & GM, Phys. Rev. Lett. (2012) [arxiv:1207.3539]
![Page 67: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/67.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Adiabatic continuation in the Wannier basis
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1
E
x
x2
0
0.025
0.05
0.075
0 1 2 3 4
ky
x2
0 5 10 15
0
0.025
0.05
0.075
Ky
1
FQHE FCI FQHE FCI
• Spectrum for N=10: • Gap for different system sizes & aspect ratios:
Th. Scaffidi & GM, Phys. Rev. Lett. (2012) [arxiv:1207.3539]
![Page 68: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/68.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Adiabatic continuation in the Wannier basis
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.2 0.4 0.6 0.8 1
E
x
x2
0
0.025
0.05
0.075
0 1 2 3 4
ky
x2
0 5 10 15
0
0.025
0.05
0.075
Ky
1
• We confirm the Laughlin state is adiabatically connected to the groundstate of the half-filled topological flat band of the Haldane model
FQHE FCI FQHE FCI
• Spectrum for N=10: • Gap for different system sizes & aspect ratios:
• Clean extrapolation to the thermodynamic limit - (unlike overlaps)
Th. Scaffidi & GM, Phys. Rev. Lett. (2012) [arxiv:1207.3539]
![Page 69: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/69.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Entanglement spectra and quasiparticle excitations
|Ψ =
i
e−ξ,i/2|ΨA,i ⊗ |ΨB
,i
credit: A. Sterdyniak et al. PRL 2011
• Entanglement spectrum: arises from Schmidt decomposition of ground state into two groups A, B => Schmidt eigenvalues ξ plotted over quantum numbers for symmetries within each block
Dominant (universal) eigenvalues of PES yield count of excited states - and their wavefunctions - from groundstate wavefunction only!
Laughlin, N=8, NA=4
![Page 70: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/70.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Entanglement spectra and quasiparticle excitations
|Ψ =
i
e−ξ,i/2|ΨA,i ⊗ |ΨB
,i
credit: A. Sterdyniak et al. PRL 2011
• Entanglement spectrum: arises from Schmidt decomposition of ground state into two groups A, B => Schmidt eigenvalues ξ plotted over quantum numbers for symmetries within each block
Dominant (universal) eigenvalues of PES yield count of excited states - and their wavefunctions - from groundstate wavefunction only!
Laughlin, N=8, NA=4
![Page 71: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/71.jpg)
Gunnar Möller University of Leeds, November 21, 2012
FCI: Adiabatic continuation of the entanglement spectrum
|Ψ =
i
e−ξ,i/2|ΨA,i ⊗ |ΨB
,i
Dictionary:
Total #eigenvalues below entanglement gap = 804 + 800 + 800 + 800 + 800
Total #eigenvalues below entanglement gap = 4x(201 + 200 + 200 + 200 + 200)
Same number of states for all x
‘Infinite’ entanglementgap for pureLaughlin state
FQHE FCI
N = 10
NA = 5
Ktoty
x
ktotT
Th. Scaffidi & GM, Phys. Rev. Lett. (2012) [arxiv:1207.3539]
![Page 72: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/72.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Finite size behaviour of entanglement gap
• The entanglement gap remains open for all values of the interpolation parameter k• Finite size scaling behaviour encouraging, but analytic dependency on system size unknown
FQHE FCI
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
!-G
ap
"
N = 10,4!5N = 10,5!4
N = 8,4!4N = 6,3!4N = 6,4!3
![Page 73: Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012) Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)](https://reader034.vdocument.in/reader034/viewer/2022042916/5f54b30c0d6bba1e58488a7d/html5/thumbnails/73.jpg)
Gunnar Möller University of Leeds, November 21, 2012
Conclusions: FCI wavefunctions from Wannier states
• Wavefunctions of FCI’s in the Wannier basis are similar but not identical to FQH states in the Landau gauge
• We demonstrated the adiabatic continuity of the ground states at ν=1/2 using Qi’s mapping between Wannier basis and FQH eigenstates
• FCI wavefunctions not very accurate for the Haldane model (higher overlaps in models with N>2 sublattices)
higher overlaps also by explicit gauge fixing, see: Wu, Regnault, Bernevig, PRB (2012)
Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012) [arxiv:1207.3539]