Unitary engineering of two- and three-band Chern insulators

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:1312.6469[cond-mat] APS Meeting, 03-03-2014, 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 z is nothing but the spin coherent state of two-band spin Hamiltonian CP1 wave function corresponding to monopole vector potential 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 Chernnumber 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 C=1 Ex) 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 Cherninsulartor (b=0) 1) : Increasing Chernnumber of one monopole. 2) : Increasing Chernnumber of another monopole

12. Band Switching 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) Chernnumber 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, nnnhoppings through the center of the hexagon. Edge state Topological phase transition! Jo et al, PRL (2012)

15. Conclusion • 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!