The Haldane Model - physics.rutgers.edu · Nicodemos Varnava The Haldane Model September 27, 2017 7...

The Haldane Model

Nicodemos Varnava The Haldane Model September 27, 2017 1 / 12

Why the Haldane Model?

First crystal model describing topological behavior.

Simple and intuitive tight binding model.Helped me understand what topological materials are all about.



First crystal model describing topological behavior.Simple and intuitive tight binding model.

Helped me understand what topological materials are all about.



First crystal model describing topological behavior.Simple and intuitive tight binding model.Helped me understand what topological materials are all about.


The Model

Spinless model on the Honeycomb lattice (2 sites)

On-site energy

H = ∆∑i

(−)τic†ici + t1∑<ij>

(c†icj + h.c

)+ t2

∑<<ij>>

(ic†icj + h.c

)

Real first neighbor hoppingsImaginary second neighbor hoppings


The Model


On-site energy

H = ∆∑i


(c†icj + h.c

)+ t2

∑<<ij>>

(ic†icj + h.c

)

Real first neighbor hoppings

Imaginary second neighbor hoppings


The Model


On-site energy

H = ∆∑i


(c†icj + h.c

)+ t2

∑<<ij>>

(ic†icj + h.c

)

Real first neighbor hoppingsImaginary second neighbor hoppingsNicodemos Varnava The Haldane Model September 27, 2017 3 / 12

Defining non-trivial states

The atomic limit: Strength of hoppings and hybridization is zero i.eflat bands

Figure: ∆ = 0.7, t1 = t2 = 0

Definition: A state is topological when it is impossible to turn offthe interactions (atomic limit) without closing the gap.


Trivial vs Topological

∆ = 0.7, t1 = −1.0, t2 = 0 and t2 = −0.06

1

1Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 5 / 12


∆ = 0.7, t1 = −1.0, t2 = −0.1347 and t2 = −0.24

2


What is topological about the topological state?

Hybrid Wannier Representation:

|hnk1l2〉 =1

2π

∫ 2π

0e−ik2l2 |ψnk1k2〉 dk2

State |hnk1l2〉 is localized at the l2/a unit cell in the y direction andextended in x.Well defined expectation value of the y operator.




|hnk1l2〉 =1

2π

∫ 2π


State |hnk1l2〉 is localized at the l2/a unit cell in the y direction andextended in x.

Well defined expectation value of the y operator.




|hnk1l2〉 =1

2π

∫ 2π


State |hnk1l2〉 is localized at the l2/a unit cell in the y direction andextended in x.Well defined expectation value of the y operator.



1

Trivial

Topological



1

Trivial Topological


Quantized Hall Conductivity

The semiclassical Boltzmann theory tells as: k̇ = − e~E

An electric field in the x direction will move states in the y direction

σxy = Ce2

hC = Chern number = number of unit cells after adiabatic cycle





σxy = Ce2

h

C = Chern number = number of unit cells after adiabatic cycle





σxy = Ce2

hC = Chern number = number of unit cells after adiabatic cycle


Bulk-Boundary Correspondence

Charge moving from the bottom to the top surface.

The only way is if the boundary is conducting.

3


Bulk-Boundary Correspondence

Charge moving from the bottom to the top surface.The only way is if the boundary is conducting.

3


Summary

Topological states cannot be connected to the atomic limit withoutclosing the gap.

Global properties are quantized.Weak perturbation will not affect them.


Summary

Topological states cannot be connected to the atomic limit withoutclosing the gap.Global properties are quantized.

Weak perturbation will not affect them.


Summary

Topological states cannot be connected to the atomic limit withoutclosing the gap.Global properties are quantized.Weak perturbation will not affect them.


The Haldane Model - physics.rutgers.edu · Nicodemos Varnava The Haldane Model September 27, 2017 7...

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