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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.

Nicodemos Varnava The Haldane Model September 27, 2017 2 / 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.

Nicodemos Varnava The Haldane Model September 27, 2017 2 / 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.

Nicodemos Varnava The Haldane Model September 27, 2017 2 / 12

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

Nicodemos Varnava The Haldane Model September 27, 2017 3 / 12

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 hoppings

Imaginary second neighbor hoppings

Nicodemos Varnava The Haldane Model September 27, 2017 3 / 12

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 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.

Nicodemos Varnava The Haldane Model September 27, 2017 4 / 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.

Nicodemos Varnava The Haldane Model September 27, 2017 4 / 12

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

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

Trivial vs Topological

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

2

2Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 6 / 12

Trivial vs Topological

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

2

2Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 6 / 12

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.

Nicodemos Varnava The Haldane Model September 27, 2017 7 / 12

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.

Nicodemos Varnava The Haldane Model September 27, 2017 7 / 12

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.

Nicodemos Varnava The Haldane Model September 27, 2017 7 / 12

Trivial vs Topological

1

Trivial

Topological

1Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 8 / 12

Trivial vs Topological

1

Trivial Topological

1Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 8 / 12

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

Nicodemos Varnava The Haldane Model September 27, 2017 9 / 12

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

h

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

Nicodemos Varnava The Haldane Model September 27, 2017 9 / 12

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

Nicodemos Varnava The Haldane Model September 27, 2017 9 / 12

Bulk-Boundary Correspondence

Charge moving from the bottom to the top surface.

The only way is if the boundary is conducting.

3

3Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 10 / 12

Bulk-Boundary Correspondence

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

3

3Vanderbilt. Berry Phases in Electronic StructureNicodemos Varnava The Haldane Model September 27, 2017 10 / 12

Summary

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

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

Nicodemos Varnava The Haldane Model September 27, 2017 11 / 12

Summary

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

Weak perturbation will not affect them.

Nicodemos Varnava The Haldane Model September 27, 2017 11 / 12

Summary

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

Nicodemos Varnava The Haldane Model September 27, 2017 11 / 12

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