Introduction Graphene Bulk-edge correspondence Results
The Colors of Graphene:
Hofstadter Butterfly for the Honeycomb Latti...
Introduction Graphene Bulk-edge correspondence Results
The Hofstadter butterfly
Introduction Graphene Bulk-edge correspondence Results
The Hofstadter model (the square lattice)
The Hamiltonian
H =
4
j=1...
Introduction Graphene Bulk-edge correspondence Results
The Hofstadter butterfly
Φ/Φ0 = p/q
E
Introduction Graphene Bulk-edge correspondence Results
The Diophantine equation
Proposition [Thouless et al., 1982, Dana e...
Introduction Graphene Bulk-edge correspondence Results
The colored Hofstadter butterfly
σH -1-2-3-4 0 1 2 3 4 5 6 7 8 9 10-...
Introduction Graphene Bulk-edge correspondence Results
Lattice structure
BA
m, n
m, n + 1
m + 1, n
Coordinate system on
th...
Introduction Graphene Bulk-edge correspondence Results
The Hofstadter Hamiltonian
ψm,n =
ψA
m,n
ψB
m,n
∈ C2
ψ = (ψm,n)m,n ...
Introduction Graphene Bulk-edge correspondence Results
The Diophantine equation
The proof of [Thouless et al., 1982, Dana ...
Introduction Graphene Bulk-edge correspondence Results
The natural window condition
σH -1-2-3-4 0 1 2 3 4 5 6 7 8 9 10-10 ...
Introduction Graphene Bulk-edge correspondence Results
The natural window condition
σH -1-2-3-4 0 1 2 3 4 5 6 7 8 9 10-10 ...
Introduction Graphene Bulk-edge correspondence Results
The edge lattice
New physical space: the half plane lattice.
H = 2
...
Introduction Graphene Bulk-edge correspondence Results
The bulk-edge correspondence [Hatsugai, 1993]
Edge spectrum E(k) (r...
Introduction Graphene Bulk-edge correspondence Results
The transfer operator formalism
Define the transfer operators on the...
Introduction Graphene Bulk-edge correspondence Results
The detection of edge states
An edge state wave function must satis...
Introduction Graphene Bulk-edge correspondence Results
The detection of crossings
0
θ(t)
2π
-1
-2
-3
-4
-5
-6
π 2π
k
Evolu...
Introduction Graphene Bulk-edge correspondence Results
The detection of crossings
The result must still satisfy the Diopha...
Introduction Graphene Bulk-edge correspondence Results
Results
Statistics of results:



right 99.8%
correctable 0.1%...
Introduction Graphene Bulk-edge correspondence Results
The Conjecture
Conjecture: [A., Eckmann and Graf, 2014]
For the hex...
Introduction Graphene Bulk-edge correspondence Results
Introduction Graphene Bulk-edge correspondence Results
References
Agazzi, A., Eckmann, J.-P., and Graf, G. (2014).
The col...
1/5 1, 2,¨¨−2→3, −1, 0, 1, ¡2→−3, −2, −1
1/6 1, 2, 3,¨¨−2→4, −1, 0, 1, ¡2→−4, −3, −2, −1
1/7 1, 2, 3,¨¨−3→4,¨¨−2→5, −1, 0,...
The 2/3 Regularity
Errors: the natural window conditions
Errors: the bulk-edge correspondence
of 25

Pres_Zurich14

Published on: Mar 4, 2016
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Transcripts - Pres_Zurich14

  • 1. Introduction Graphene Bulk-edge correspondence Results The Colors of Graphene: Hofstadter Butterfly for the Honeycomb Lattice Andrea Agazzi, Gian Michele Graf, Jean-Pierre Eckmann Département de Physique Théorique Zurich, 14.10.2014
  • 2. Introduction Graphene Bulk-edge correspondence Results The Hofstadter butterfly
  • 3. Introduction Graphene Bulk-edge correspondence Results The Hofstadter model (the square lattice) The Hamiltonian H = 4 j=1 Tj . acting on H = 2 (Z2 ; C) (m, n) (m, n + 1) (m + 1, n) Explicitly Hψm,n = ψm−1,n + ψm+1,n e−2πimΦ/Φ0 ψm,n−1 + e2πimΦ/Φ0 ψm,n+1 Assumption: Rational magnetic field flux per unit cell Φ/Φ0 = p/q with p, q ∈ N .
  • 4. Introduction Graphene Bulk-edge correspondence Results The Hofstadter butterfly Φ/Φ0 = p/q E
  • 5. Introduction Graphene Bulk-edge correspondence Results The Diophantine equation Proposition [Thouless et al., 1982, Dana et al., 1985]: The Hall conductivity σH in the r-th spectral gap of H is the solution of the Diophantine equation r = σH · p + s · q where s ∈ Z. Remark (the window condition): It is natural to impose the uniqueness condition σH ∈ (−q/2, q/2) .
  • 6. Introduction Graphene Bulk-edge correspondence Results The colored Hofstadter butterfly σH -1-2-3-4 0 1 2 3 4 5 6 7 8 9 10-10 -9 -8 -7 -6 -5
  • 7. Introduction Graphene Bulk-edge correspondence Results Lattice structure BA m, n m, n + 1 m + 1, n Coordinate system on the honeycomb lattice structure. Hexagonal lattice structure Bipartite lattice Spinor-like wave function ψm,n = ψA m,n ψB m,n ∈ C2 ψ = (ψm,n)m,n ∈ H = 2 (Z2 ; C2 )
  • 8. Introduction Graphene Bulk-edge correspondence Results The Hofstadter Hamiltonian ψm,n = ψA m,n ψB m,n ∈ C2 ψ = (ψm,n)m,n ∈ H = 2 (Z2 ; C2 ) A B (m, n) (m − 1, n) (m, n − 1) The Hofstadter Hamiltonian (Hψ)m,n = ψB m,n + ψB m+1,n + e−2πimp/q ψB m,n+1 ψA m,n + ψA m−1,n + e2πimp/q ψA m,n−1
  • 9. Introduction Graphene Bulk-edge correspondence Results The Diophantine equation The proof of [Thouless et al., 1982, Dana et al., 1985] shows that also for the hexagonal lattice the value of σH in the r-th gap must satisfy r = σH · p + s · q . How about the window condition? Can the natural window condition still be applied?
  • 10. Introduction Graphene Bulk-edge correspondence Results The natural window condition σH -1-2-3-4 0 1 2 3 4 5 6 7 8 9 10-10 -9 -8 -7 -6 -5
  • 11. Introduction Graphene Bulk-edge correspondence Results The natural window condition σH -1-2-3-4 0 1 2 3 4 5 6 7 8 9 10-10 -9 -8 -7 -6 -5
  • 12. Introduction Graphene Bulk-edge correspondence Results The edge lattice New physical space: the half plane lattice. H = 2 (N × Z; C2 ) The new Hamiltonian H is the restriction of H to the Hilbert space H. Bloch decomposition in the unbroken symmetry direction gives H(k): ψB 1 (k) ψA 1 (k) (Hψ)m(k) = ψB m(k) + ψB m+1(k) + e−2πimp/q+ik ψB m(k) ψA m(k) + ψA m−1(k) + e2πimp/q−ik ψA m(k)
  • 13. Introduction Graphene Bulk-edge correspondence Results The bulk-edge correspondence [Hatsugai, 1993] Edge spectrum E(k) (red lines): E EF k 2π0 Theorem: # of (signed) crossings of the E(k) in k ∈ (0, 2π) = σH(r).
  • 14. Introduction Graphene Bulk-edge correspondence Results The transfer operator formalism Define the transfer operators on the half-plane at the energy E and wave-vector k as ψB m+1(k) ψA m(k) = T E m (k) ψB m(k) ψA m−1(k) . and a translation of q dimers by T E (k) = q m=1 T E m (k) = T E q (k) · · · T E 1 (k) . The latter operator allows us to describe the wave function on the whole lattice given its value at any two neighboring points in the lattice.
  • 15. Introduction Graphene Bulk-edge correspondence Results The detection of edge states An edge state wave function must satisfy: 1 Boundedness Be normalizable, i.e. be a contracting eigenvector of T E (k). 2 Edge Vanish to the left of the boundary i.e., ψB 1 (k) ψA 0 (k) ∼ 1 0 ψB 1 (k) ψA 1 (k) Lemma [A., Eckmann and Graf, 2014] Let (a(k), b(k)) be the contracting eigenvector of T E (k), then σH = 2π 0 dk 2πi ∂ ∂k log a(k) + ib(k) a(k) − ib(k) = θ(2π) − θ(0) 2π
  • 16. Introduction Graphene Bulk-edge correspondence Results The detection of crossings 0 θ(t) 2π -1 -2 -3 -4 -5 -6 π 2π k Evolution of the phase θ(k)/2π (on the y-axis) for two different discretizations as a function of k ∈ [0, 2π) (on the x-axis) for values of p/q = 8/19, r = 1. The red curve has a lower discretization than the blue one.
  • 17. Introduction Graphene Bulk-edge correspondence Results The detection of crossings The result must still satisfy the Diophantine equation r = σH · p + s · q . (1) Test: For each value of p, q, r insert the computed value of σH into (1). If s is “close to” a natural number, our guess can be corrected.
  • 18. Introduction Graphene Bulk-edge correspondence Results Results Statistics of results:    right 99.8% correctable 0.1% not correctable 0.1% of image pixels.
  • 19. Introduction Graphene Bulk-edge correspondence Results The Conjecture Conjecture: [A., Eckmann and Graf, 2014] For the hexagonal lattice, σH in the r-th gap must satisfy r = σH · p + s · q , where s ∈ Z under the relaxed condition σH ∈ (−q, q) . Question: can this ambiguity be solved algebraically?
  • 20. Introduction Graphene Bulk-edge correspondence Results
  • 21. Introduction Graphene Bulk-edge correspondence Results References Agazzi, A., Eckmann, J.-P., and Graf, G. (2014). The colored hofstadter butterfly for the honeycomb lattice. Journal of Statistical Physics, pages 1–10. Avila, J. C., Schulz-Baldes, H., and Villegas-Blas, C. (2013). Topological invariants of edge states for periodic two-dimensional models. Mathematical Physics, Analysis and Geometry, 16:137–170. Dana, I., Avron, Y., and Zak, J. (1985). Quantised Hall conductance in a perfect crystal. J. Phys. C, 18(22):L679. Hatsugai, Y. (1993). Edge states in the integer quantum hall effect and the riemann surface of the bloch function. Phys. Rev. B, 48:11851–11862. Thouless, D. J., Kohmoto, M., Nightingale, M. P., and den Nijs, M. (1982). Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49:405–408.
  • 22. 1/5 1, 2,¨¨−2→3, −1, 0, 1, ¡2→−3, −2, −1 1/6 1, 2, 3,¨¨−2→4, −1, 0, 1, ¡2→−4, −3, −2, −1 1/7 1, 2, 3,¨¨−3→4,¨¨−2→5, −1, 0, 1, ¡2→−5, ¡3→−4, −3, −2, −1 2/5 ¨¨−2→3, 1, −1, 2, 0, −2, 1, −1, ¡2→−3
  • 23. The 2/3 Regularity
  • 24. Errors: the natural window conditions
  • 25. Errors: the bulk-edge correspondence

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