1. Unitary engineering of two- and three-band Chern insulators
Dept. of physics
SungKyunKwan University, Korea
Soo-Yong Lee, Jin-Hong Park, Gyungchoon Go, Jung Hoon Han
arXiv: [cond-mat]
APS Meeting, , Denver

2. Contents Introduction : Dirac monopole Two-band Chern insulator
3. Three-band Chern insulator
4. Topological Band switching
5. Conclusion

3. Dirac Monopole Maxwell eq. with magnetic monopole Fang, Science (2003)
Dirac quantization
Ray, Nature (2014)
Vector potential corresponding to monopole
Singularity at :
Dirac string

4. Dirac Monopole Vector potential corresponding to wave function z
CP1 wave function corresponding to monopole vector potential
z is nothing but the spin coherent state of two-band spin Hamiltonian
Dirac monopole always appears in the general two-band spin model!!

5. Two-band Chern Insulator
3-dim d-vector
Pauli matrices
Monopole charge = Chern number in Two-band Chern Insulator
ex) Hall conductivity of the quantum Hall insulator

6. Two-band Chern Insulator
Q) How do we change the monopole charge?
A) Unitary transformation !
New Hamiltonian
New wave ft.
New vector potential
Additional term by a certain unitary transformation can put in an extra singular vector potential which generates a higher Chern number.

7. Two-band Chern Insulator
cf) Eigenvalues are always +d and –d and z is independent of the magnitude of d-vector
1) : Turning Chern number on and off.
2) : Increasing Chern number.

8. How to generate an arbitrary Chern number insulator?
Write down the unity Chern number model in the momentum space. (ex : Haldane model, BHZ model etc.)
2. Apply the unitary transformation to change angle and
3. New d-vector gives a higher Chern number model
4. (When we apply the Fourier transformation, we get a real space model. To avoid non-valid hopping, multi-orbital character could be sometimes introduced. ex) p-orbital, t2g-orbital
Ex)
C=1
C=2

9. Three-band Chern Insulator
8-dim n-vector
Gell-mann matrices

10. Three-band Chern Insulator
8-dim n-vector
Gell-mann matrices
SU(3) Euler rotation
c, d disappear in Hamiltonian
Chern number of each band

11. Three-band Chern Insulator
Two redundant U(1) gauges c and d (non-degenerate)
A pair of monopole charges = Combination of the two band Chern insulartor (b=0)
1) : Increasing Chern number of one monopole.
2) : Increasing Chern number of another monopole

12. Band Switching 3-dim d-vector Spin-1 matrices
Another class of the three-band model
3-dim d-vector
Spin-1 matrices
cf) Eigenvalues are always +d,0,-d
Ex) Kagome lattice model(Ohgushi, Murakami, Nagaosa PRB 1999)
Chern number of each band
(factor 2 difference from the two-band model, He et al. PRB 2012 Go et al. PRB 2013)

13. Band Switching Band switching by basis change unitary transformation
Ex)
cf)
3-dim
Reminder of 8-dim except 3-dim d vector space (orthogonal)
Generally,

14. Band Switching Eigenenergies : only when unitary Trans.
Ex) In the Kagome lattice, the additional term represents the next(n), nn, nnn hoppings through the center of the hexagon.
Edge state
Topological phase transition!
Jo et al, PRL (2012)

15. Conclusion Thank you for your attention!
Monopole charge-changing operations become unitary transformations on the two-band Hamiltonian.
For the three-band case, we propose a topology-engineering scheme based on the manipulation of a pair of magnetic monopole charges.
Band-switching is proposed as a way to control the topological ordering of the three-band Hamiltonian.
Thank you for your attention!