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## PowerPoint Slideshow about ' Topological insulators' - imelda-matthews

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### Hall effect

### Classical Hall effect

### Classical Hall effect: boundaries

### Quantum Hall Effect

### Quantum Hall Effect

### Chern insulator [Haldane’88]

### Chern insulator [Haldane’88]

### Quantum Hall effect: general formula

### Quantum Hall effect and Berry fluxTKNN invariant

### Digression: Berry connection

### Example: two-band model

### Electromagnetic response and effective action

### Topological inequivalence of insulators

### QHE and adiabatic pumping

### QHE and adiabatic pumping

### Quantum theory of electric polarization[King-Smith,Vanderbilt’93 (!!!)]

### Quantum theory of electric polarization

### Quantum theory of electric polarization

### From (2+1)D Chern Insulators to (1+1)D Z2 TIs

### From (2+1)D Chern Insulators to (1+1)D Z2 TIs

### Relative Chern parity and level crossing

### Relative Chern parity and θ-term

### (4+1)D Chern insulators (aka domain wall fermions)

### (4+1)D Chern insulators: Dirac models

### (4+1)D Chern insulators: Dirac models

### Z2 classification of time-reversal invariant topological insulators in (3+1)D and in (2+1)Dfrom (4+1)D Chern insulators

### Effective EM action of 3D TRI topinsulators

### Real 3D topological insulator: Bi1-xSbx

### (4+1)D CSI Z2TRI in (3+1)D Z2TRI in (2+1D)

### Periodic table of Topological Insulators

### Kramers theorem

### Kramers theorem

### Z2 classification of (2+1)D TI

### Kane-Mele model: role of SO coupling

### Spin-momentum locking

### Topological Mott insulators

### Some useful references (and sources of pictures/formulas for this lecture :-)

Classical treatment

Dissipative motion for point-like particles (Drude theory)

Steady motion

Drude conductivity

Current

Resistivity tensor

- Hall resistivity (off-diag component of resistivity tensor)
- - Does not depend on disorder
- Measures charge/density
- of electric current carriers
- - Valuable experimental tool

INSULATOR!!!

Importance of

matrix structure Naïve look at longitudinal components:

INSULATOR AND CONDUCTOR SIMULTANEOUSLY!!!

Conductance happens exclusively due to boundary states!

Otherwise an insulating state

Non-relativistic Landau levels

Model the boundary by a confining potential V(y) = mw2y2/2

- Number of conducting states =
- no of LLs below Fermi level
- Hall conductivity σ ~ n
- Pairs of right- and left- movers
- on the “Boundary”
- NOW THE QUESTION:
- Hall state without magnetic
- Field???

Originally, hexagonal lattice, but we consider square

Two-band model, similar to Wilson-Dirac [Qi, Wu, Zhang]

Phase diagram

m=2 Dirac point at kx,ky=±π

m=0 Dirac points at (0, ±π), (±π,0)

m=-2 Dirac point at kx,ky=0

Open B.C. in y direction, numerical diagonalization

Response to a weak electric field, V = -e E y

(Single-particle states)

Electric Current (system of multiple fermions)

Velocity operator

vx,yfrom

Heisenberg

equations

Berry connection

Berry curvature

Integral of Berry curvature = multiple of 2π

(wave function is single-valued on the BZ)

Berry curvature in terms of projectors

TKNN = Thouless, Kohmoto, Nightingale, den Nijs

Adiabatically time-dependent Hamiltonian H(t) = H[R(t)] with parameters R(t). For every t, define an eigenstate

However, does not solve the Schroedingerequation

Substitute

Adiabatic evolution along the loop yields a nontrivial phase

Bloch momentum: also adiabatic parameter

General two-band Hamiltonian Projectors

Berry curvature in terms of projectors

- Two-band Hamiltonian: mapping of sphere on the torus,
- VOLUME ELEMENT
- For the Haldane model
- m>2: n=0
- 2>m>0: n=-1
- 0>m>-2: n=1
- -2>m : n = 0

CS number change =

Massless fermions =

Pinch at the surface

Along with current, also charge density is generated

Response in covariant form

Effective action for this response

Electromagnetic Chern-Simons

= Magnetic Helicity

Winding of

magnetic flux

lines

Consider the Quantum Hall state

in cylindrical geometry

ky is still a good quantum number

Collection of 1D Hamiltonians

Switch on electric field Ey, Ay = - Ey t “Phase variable”

2 πrotation ofΦ , timeΔt = 2 π/ LyEy

Charge flow in this timeΔQ = σHΔt Ey Ly = CS/(2 π) 2 π = CS

Every cycle of Φ moves CS unit charges to the boundaries

More generally, consider a parameter-dependent Hamiltonian

Define the current response

Similarly to QHE derivation

Polarization

EM response

But what is X for PBC???

Mathematically,

X is not a good operator

Resta formula:

Model: electrons in 1D periodic potentials

Bloch Hamiltonians

a

Discrete levels at finite interval!!

Many-body fermionic theorySlater determinant

King-Smith and Vanderbilt formula

Polarization =

Berry phase of 1D theory

(despite no curvature)

- Formally, in tight-binding models X is always integer-valued
- BUT: band structure implicitly remembers about continuous
- space and microscopic dipole moment
- We can have e.g. Electric Dipole Moment
- for effective lattice Dirac fermions
- In QFT, intrinsic property
- In condmat, emergent phenomenon
- C.F. lattice studies of CME

1D Hamiltonian Particle-hole symmetry

Consider two PH-symmetric hamiltoniansh1(k) and h2(k)

Define continuous interpolation

For

Now h(k,θ) can be assigned

the CS number

= charge flow in a cycle of θ

- Particle-hole symmetry implies P(θ) = -P(2π - θ)
- On periodic 1D lattice of unit spacing,
- P(θ) is only defined modulo 1 P(θ) +P(2π - θ) = 0 mod 1
- P(0) or P(π) = 0 or ½ Z2 classification
- Relative parity of CS numbers
- Generally, different h(k,θ) = different CS numbers
- Consider two interpolations h(k,θ) and h’(k,θ)
- C[h(k, θ)]-C[h’(k,θ)] = 2 n

Now consider 1D Hamiltonians with open boundary conditions

CS = numer of left/right zero level crossings in [0, 2 π]

Particle-hole symmetry: zero level at θ also at 2 π – θ

Odd CS zero level at π(assume θ=0 is a trivial insul.)

Once again, EM response for electrically polarized system

Corresponding effective action

For bulk Z2 TI with periodic BC P(x) = 1/2

- TI = Topological field theory in the bulk:
- no local variation can changeΦ
- Current can only flow at the boundary where P changes
- Theta angle = π, Charge conjugation only allows
- theta = 0 (Z2 trivial) or theta = π(Z2 nontrivial)
- Odd number of localized statesat the left/right boundary

Consider the 4D single-particle hamiltonianh(k)

Similarly to (2+1)D Chern insulator, electromagnetic response

C2 is the “Second Chern Number”

Effective EM action

Parallel E and B in 3D generate current along 5th dimension

Five (4 x 4) Dirac matrices:{Γµ , Γν} = 2 δµν

Lattice model = (4+1)D Wilson-Dirac fermions

In momentum space

Critical values of mass CS numbers

(where massless modes exist)

Open boundary conditions in the 5th dimension

|C2| boundary modes on the left/on the right boundaries

Effective boundary Weyl Hamiltonians

2 Weyl fermions =

1 Domain-wall fermion (Dirac)

Charge flows into the bulk

= (3+1)D anomaly

Consider two 3D hamiltonians

h1(k) and h2(k), Define extrapolation

“Magnetoelectric polarization”

Time-reversal implies P(θ) = -P(2π - θ)

P(θ) is only defined modulo 1 => P(θ) +P(2π - θ) = 0 mod 1

P(0) or P(π) = 0 or ½ => C[h(k, θ)]-C[h’(k,θ)] = 2 n

Dimensional reduction from (4+1)D effective action

In the bulk, P3=1/2 theta-angle = π

Electric current responds to the gradient of P3

At the boundary,

- Spatial gradient of P3: Hall current
- Time variation of P3: current || B
- P3 is like “axion” (TME/CME)
- Response to electrostatic field near boundary

Electrostatic potential A0

Band inversion at intermediate concentration

Consider two 2D hamiltonians

h1(k) and h2(k), Define extrapolation

h(k,θ) is like 3D Z2 TI Z2 invariant

This invariant does not depend on parametrization?

Consider two parametrizationsh(k,θ) and h’(k,θ)

Interpolation

between them

This is also interpolation between h1 and h2

Berry curvature of φ vanishes on the boundary

Chern invariants are only defined in odd dimensions

Time-reversal operator for Pauli electrons

Anti-unitary symmetry

Single-particle Hamiltonian in momentum space

(Bloch Hamiltonian)

If [h,θ]=0

Consider some eigenstate

Every eigenstate has a partner at (-k)

With the same energy!!!

Since θ changes spins, it cannot be

Example: TRIM

(Time Reversal Invariant Momenta)

-k is equivalent to k

For 1D lattice, unit spacing

TRIM: k = {±π, 0}

Assume

States at TRIM are always doubly degenerate

Kramers degeneracy

- Contact || x between two (2+1)D Tis
- kxis still good quantum number
- There will be some midgap states crossing zero
- At kx= 0, π (TRIM) double degeneracy
- Even or odd number of crossings Z2 invariant

- Odd number of crossings = odd number of massless modes
- Topologically protected (no smooth deformations remove)

Simple theoretical model for (2+1)D TRI topological insulator

[Kane,Mele’05]: graphene with strong spin-orbital coupling

- Gap is opened

- Time reversal is not broken

- In graphene, SO coupling

is too small

Possible physical implementation

Heavy adatom in the

centre of hexagonal lattice

(SO is big for heavy atoms

with high orbitals occupied)

Two edge states with opposite spins: insulatorleft/up, right/down

Insensitive to disorder as long as

T is not violated

Magnetic disorder

is dangerous

Graphene insulator tight-binding model with nearest- and

next-nearest-neighbour interactions

By tuning U, V1 and V2 we

can generate an effective SO

coupling.

Not in real graphene,

But what about artificial?

Also, spin transport on the surface of 3D Mott TI

[Pesin,Balents’10]

- “Primer on topological insulators”, A. insulatorAltland and L. Fritz

- “Topological insulator materials”, Y. Ando, ArXiv:1304.5693

- “Topological field theory of time-reversal invariant insulators”, X.-L. Qi, T. L. Hughes, S.-C. Zhang, ArXiv:0802.3537

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