Topological insulators. Pavel Buividovich (Regensburg). Hall effect. Classical treatment. Dissipative motion for point-like particles ( Drude theory). Steady motion. Cyclotron frequency. Drude conductivity. Current. Resistivity tensor. Classical Hall effect.
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Topological insulators
PavelBuividovich
(Regensburg)
Hall effect
Classical treatment
Dissipative motion for point-like particles (Drude theory)
Steady motion
Cyclotron frequency
Drude conductivity
Current
Resistivity tensor
Classical Hall effect
Clean system limit:
INSULATOR!!!
Importance of
matrix structure Naïve look at longitudinal components:
INSULATOR AND CONDUCTOR SIMULTANEOUSLY!!!
Classical Hall effect: boundaries
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
Quantum Hall Effect
Quantum Hall Effect
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
Chern insulator [Haldane’88]
Open B.C. in y direction, numerical diagonalization
Chern insulator [Haldane’88]
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
Quantum Hall effect: general formula
Quantum Hall effect and Berry fluxTKNN invariant
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
Digression: Berry connection
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
Example: two-band model
CS number change =
Massless fermions =
Pinch at the surface
Along with current, also charge density is generated
Response in covariant form
Electromagnetic response and effective action
Effective action for this response
Electromagnetic Chern-Simons
= Magnetic Helicity
Winding of
magnetic flux
lines
Topological inequivalence of insulators
Consider the Quantum Hall state
in cylindrical geometry
ky is still a good quantum number
Collection of 1D Hamiltonians
QHE and adiabatic pumping
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
QHE and adiabatic pumping
Polarization
EM response
Classical dipole moment
But what is X for PBC???
Mathematically,
X is not a good operator
Resta formula:
Quantum theory of electric polarization[King-Smith,Vanderbilt’93 (!!!)]
Model: electrons in 1D periodic potentials
Bloch Hamiltonians
a
Discrete levels at finite interval!!
Many-body fermionic theorySlater determinant
Quantum theory of electric polarization
King-Smith and Vanderbilt formula
Polarization =
Berry phase of 1D theory
(despite no curvature)
Quantum theory of electric polarization
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
From (2+1)D Chern Insulators to (1+1)D Z2 TIs
From (2+1)D Chern Insulators to (1+1)D Z2 TIs
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.)
Relative Chern parity and level crossing
Once again, EM response for electrically polarized system
Corresponding effective action
For bulk Z2 TI with periodic BC P(x) = 1/2
Relative Chern parity and θ-term
Consider the 4D single-particle hamiltonianh(k)
Similarly to (2+1)D Chern insulator, electromagnetic response
C2 is the “Second Chern Number”
(4+1)D Chern insulators (aka domain wall fermions)
Effective EM action
Parallel E and B in 3D generate current along 5th dimension
In continuum space
Five (4 x 4) Dirac matrices:{Γµ , Γν} = 2 δµν
Lattice model = (4+1)D Wilson-Dirac fermions
In momentum space
(4+1)D Chern insulators: Dirac models
Critical values of mass CS numbers
(where massless modes exist)
(4+1)D Chern insulators: Dirac models
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”
Z2 classification of time-reversal invariant topological insulators in (3+1)D and in (2+1)Dfrom (4+1)D Chern insulators
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,
Effective EM action of 3D TRI topinsulators
Electrostatic potential A0
Band inversion at intermediate concentration
Real 3D topological insulator: Bi1-xSbx
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
(4+1)D CSI Z2TRI in (3+1)D Z2TRI in (2+1D)
This is also interpolation between h1 and h2
Berry curvature of φ vanishes on the boundary
Periodic table of Topological Insulators
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
Kramers theorem
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
Kramers theorem
States at TRIM are always doubly degenerate
Kramers degeneracy
Z2 classification of (2+1)D TI
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)
Kane-Mele model: role of SO coupling
Two edge states with opposite spins: left/up, right/down
Spin-momentum locking
Insensitive to disorder as long as
T is not violated
Magnetic disorder
is dangerous
Graphene tight-binding model with nearest- and
next-nearest-neighbour interactions
Topological Mott insulators
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. Altland 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
Some useful references (and sources of pictures/formulas for this lecture :-)