Unitary engineering of two- and three-band Chern insulators Dept. of physics SungKyunKwan...
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Transcript of Unitary engineering of two- and three-band Chern insulators Dept. of physics SungKyunKwan...
Unitary engineering of two- and three-band Chern insulators
Dept. of physicsSungKyunKwan University, Korea
Soo-Yong Lee, Jin-Hong Park, Gyungchoon Go, Jung Hoon Han
arXiv:1312.6469[cond-mat]
APS Meeting, 03-03-2014, Denver
Contents
1. Introduction : Dirac monopole
2. Two-band Chern insulator 3. Three-band Chern insulator
4. Topological Band switching
5. Conclusion
Dirac MonopoleMaxwell eq. with magnetic monopole
Dirac quantization
Vector potential corresponding to monopole
Singularity at :
Dirac string
Ray, Nature (2014)
Fang, Science (2003)
Dirac MonopoleVector 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!!
Two-band Chern Insulator
Monopole charge = Chern number in Two-band Chern In-sulator
3-dim d-vector Pauli matrices
ex) Hall conductivity of the quantum Hall insulator
Two-band Chern InsulatorQ) How do we change the monopole charge?A) Unitary transformation !
Additional term by a certain unitary transformation can put in an extra sin-gular vector potential which generates a higher Chern number.
New Hamiltonian New wave ft.
New vector poten-tial
Two-band Chern Insulator
1) : Turning Chern number on and off.
2) : Increasing Chern number.
cf) Eigenvalues are always +d and –d and z is independent of the magnitude of d-vector
How to generate an arbitrary Chern number insulator?
1. Write down the unity Chern number model in the mo-mentum 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 model4. (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 C=2Ex)
Three-band Chern Insulator
Gell-mann matrices
8-dim n-vector
Three-band Chern Insulator
Chern number of each band
SU(3) Euler rotation
c, d disappear in Hamiltonian
Gell-mann matrices
8-dim n-vector
Three-band Chern Insulator
A pair of monopole charges = Combination of the two band Chern insulartor (b=0)
Two redundant U(1) gauges c and d (non-degenerate)
1) : Increasing Chern number of one monopole.
2) : Increasing Chern number of another monopole
Band SwitchingAnother class of the three-band model
3-dim d-vector
Chern number of each band(factor 2 difference from the two-band model, He et al. PRB 2012 Go et al. PRB 2013)
Ex) Kagome lattice model(Ohgushi, Murakami, Nagaosa PRB 1999)
Spin-1 matricescf) Eigenvalues are always +d,0,-d
Band SwitchingBand switching by basis change unitary transformation
Ex)
3-dim Reminder of 8-dim except 3-dim d vector space (orthogonal)
Generally,
cf)
Band Switching
Ex) In the Kagome lattice, the additional term represents the next(n), nn, nnn hoppings through the center of the hexagon.
Edge state
Eigenenergies :
Jo et al, PRL (2012)
only when
unitary Trans.
Topological phase transition!
Conclusion
• Monopole charge-changing operations become unitary transformations on the two-band Hamiltonian.
• For the three-band case, we propose a topology-engi-neering 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!