3. Second Chern number and its physical consequencesB. TRI topological insulators based on lattice Dirac models

• The continuum Dirac model in (4+1)-d dimensions is expressed as

• Note: the above model bears superficial resemblance to (3+1)-d relativistic Dirac model. Here we don’t need a “5-vector” since we don’t require Lorentz invariance. The gamma matrices satisfy the Clifford algebra

• The lattice (tight-binding) version of this model is written as

• To get the k-space we do the same old Wannier to Bloch transformation defined by

3. Second Chern number and its physical consequencesB. TRI topological insulators based on lattice Dirac models

• To get the k-space we do the same old Wannier to Bloch transformation defined by

• The sum over i can be evaluated as

• Diagonalized Hamiltonian in k-space

3. Second Chern number and its physical consequencesB. TRI topological insulators based on lattice Dirac models

• Diagonalized Hamiltonian in k-space

• This Hamiltonian can be written in the compact form

• The second Chern number can be written as

• In terms of the compact Hamiltonian we get

• The critical values of m can be found as solutions to

• The function P[k] is a set of all the wavevectors obtained from index permutations of wavevector k. For example:

• The second Chern number C2 for the different regions in parameter space, separated by critical values of m, can be evaluated approximately near the critical points to give:

• On a hyper-cylinder we can define:

4. Dimensional reduction to (3+1)-d TRI insulatorsA. Effective action of (3+1)-d insulators

• Hamiltonian of Dirac model coupled to an external U(1) gauge field

• Consider a special “Landau”-gauge configuration satisfying:

• We have translational invariance in the w-direction; kw is a good quantum number. Hamiltonian can be rewritten as:

• Since different kw are decoupled we obtain the (3+1)-d model:

• To study the response properties of the (3+1)-d system, the effective action can be defined as:

• We can Taylor expand around the field configuration:

• The coefficient G3(θ0) is determined by the below Feynman diagram

E

m

Generic Insulator Flat Band Model

eE

e

4. Dimensional reduction to (3+1)-d TRI insulatorsA. Effective action of (3+1)-d insulators

• Copying coefficient G3(θ0) from last slide

• If we define the 4-D Berry connection can be defined as:

• Then we can show that the above expression for G3(θ0), in terms of Green functions, reduces to:

• The equivalence of these two forms can be proved in three steps:• Derive an expression h(k, t) showing adiabatic connection between the

expression for arbitrary h(k) and maximally degenerate h0(k)

• Explicit evaluation of G3(θ0) using Green functions for h0(k)

• Topological invariance of G3(θ0) under adiabatic deformations of h(k)• Step I: Adiabatic connection of h(k) to h0(k)

• Any single particle Hamiltonian h(k) can be diagonalized aswhere and

• With zero chemical potential, for an insulator with M full bands, the eigenvalues satisfy

• For t [0,1] we can define the interpolation

and• Then we get

• We obtain an adiabatic interpolation between

4. Dimensional reduction to (3+1)-d TRI insulatorsA. Effective action of (3+1)-d insulators

• Step II: Explicit evaluation of G3(θ0) using Green’s function for h0(k)

• The expression for h0(k) from step I:

• Properties of “projection” operators PG(k) and PE(k):

• The Green’s function for h0(k):

• Consider the maximally degenerate or flat-band Hamiltonian

E

m

Generic Insulator Flat Band Model

eE

e

• Using the Green’s function we can write

4. Dimensional reduction to (3+1)-d TRI insulatorsA. Effective action of (3+1)-d insulators

• Step II: Explicit evaluation of G3(θ0) using Green’s function for h0(k)

• Properties of “projection” operators PG(k) and PE(k):

• The Green’s function for h0(k):

• Recall the expression for G3(θ0)

• Recall C2 as well

• Plugging in the above expressions for the Green’s functions we get

where P1/2(k) = PG/E(k)

• Now, introduce the non-Abelian Chern-Simons term

• Analogy to the charge polarization defined as the integral of the adiabatic connection 1-form over a path in momentum space

4. Dimensional reduction to (3+1)-d TRI insulatorsB. Physical Consequences of the Effective Action S3D

• Extremizing the action S3D we get response equation

• Two cases: spatial and temporal variation of P3

• Case 1: Spatial variation, i.e. P3 = P3(z)

• For a uniform electric field Ex

• The integration over z in a finite range gives the 2D current density in the xy plane

• The net Hall conductance of the region z1 ≤ z ≤ z2 is:

action is well defined for an arbitrary3+1 -dimensional in-sulator Hamiltonianh k,x,t in which thedependenceonx,tis adiabatic. We obtained this effective theory by the dimen-sional reduction from a 4+1 -dimensional system, and wepresented it this way since we believe that this derivation isboth elegant and unifying. However, for readers who are notinterested in therelationship to higher dimensional physics,aself-contained derivation can also be carried out directly in3+1 dimensions,asweexplained earlier,by integrating out

the fermions in the presence of theA x,t and the x,texternal fields.

Thiseffectiveaction isknown in thefield theory literatureas axion electrodynamics,51–53 where the adiabatic fieldP3plays the role of the axion field.54,55 When theP3 field be-comes a constant parameter independent of space and time,this effective action is referred to as the topological term forthe vacuum.56,57 The axion field has not yet been experi-mentally identified, and it remains as a deep mystery in par-ticle physics. Our work shows that the same physics canoccur in acondensed matter system,where theadiabatic “ax-ion”field P3 x,t hasa direct physical interpretation and canbe accessed and controlled experimentally.

From the discussion above it is clear that 3D TRI topo-logical insulators realize a nontrivial solitonic backgroundfield. In Ref. 28 Fu and Kane suggested several candidatematerials which could be 3D topological insulators. These3D materials are topologically nontrivial because of bandinversion mechanism similar to that of the HgTe quantumwells. 20 Dai et al. 58 closely studied the strained bulk HgTe.Wewill keep thissystemin mind since it hasasimplephysi-cal interpretation, and its essential physics can be describedby the Dirac model presented earlier. We can consider thetrivial vacuum outside the material to have a constant axionfield with the value =0 and the interior of a 3D topologicalinsulator to have a = background field. The value =does not violate time reversalor the combined charge con-jugation and paritysymmetryCP in high-energy language.HgTe is a zero-gap semiconductor and has no topologicallyprotected features. However, when strained, the system de-velops a bulk insulating gap between thep-wave light-hole“conduction band” and thep-wave heavy-hole “valenceband” around the point. To study the topological featureswe must also include thes-wave band which in a conven-tional material such as GaAs would be a conduction band.Because of the strong spin-orbit coupling in HgTe, the bandstructure isactually inverted,i.e., thes-waveband becomesavalence band. For a moment we will ignore the heavy-holeband and only consider the light-hole ands-wave band.58

The effective Hamiltonian of these two bands is a massiveDirac Hamiltonian but with a negative mass. The negativemass indicatesaphaseshift of in thevacuumangle fromits original unshifted value in the trivial vacuum. The axiondomain wall structure at the surface of the topological insu-lator traps fermion zero modes which are simply the topo-logically protected surface states. If we include the effectsofthe heavy-hole band the dispersions of the bulk bands andsurface states are quantitatively modified. However, as longas the crystal is strained enough to maintain the bulk gap thetopological phenomena will be unaffected and the boundaryof the 3D topological insulator can still be described as an

axion domain wall. Thus, this material in condensed matterphysics provides a direct realization of axion electrodynam-ics.

B. Physical consequences of the effective action S3D

In this section we present the general physical conse-quences of the effective topological action 80 for3+1 -dimensional insulators coupled to aP3 polarization,

and in Sec. IVC we focus on its consequences for TRI in-sulators. Since the effective action is quadratic inA , it de-scribes a linear response to the external electromagneticfieldswhich dependson thespatial and temporal gradientsofP3. Taking a variation in S3D A, we obtain the responseequation,

j =1

2P3 A . 81

The physical consequencesEq. 81 can be understood bystudying the following two cases.

1. Hall effect induced by spatial gradient of P3

Consider a system in whichP3=P3 z only depends onz.For example, this can be realized by the lattice Dirac model73 with = z . This type of domain wall has also been

considered in Ref.59. In this case Eq. 81 becomes

j = zP3

2A , , , =t,x,y,

which describes a QH effect in thexy plane with the Hallconductivity xy= zP3/2 , as shown in Fig. 11 a . For auniform electric fieldEx in thex direction, the Hall currentdensity is jy= zP3/2 Ex. Thus the integration overz in afinite range gives the 2D current density in thexy plane,

FIG.11. Color online a Illustration of theHall effect inducedby a spatial gradient ofP3. The colors represent different values ofP3, which decrease from 0 at the bottom to 1/2 on top. The bluedashed arrow shows the direction of a uniform electric fieldEx andthered solid arrowsshow theHall current density induced,given bythe formula jy= zP3/2 Ex. b Energy spectrum of the3+1 -dimensional lattice Dirac model 73 in a magnetic fieldBz

toward thez direction. The boundary conditions are periodic in thex andy directions and open on thez direction. The red dashed andblack solid curves show the surface states on the top and bottomsurfaces, respectively, each of which is aBzLxLy/2 -fold degener-ate Landau Level. The parameters of model73 are chosen to bem=−3, c=1.

BWEIVERLACISYHPGNAHZDNA,SEHG

• Case 2: Temporal variation, i.e. P3 = P3(t)

• Using continuity j = ∂tP and the fact that B is constant and uniform we get:

• Such an equation describes the charge polarization induced by a magnetic field, which is a magneto-electric effect

• This can also be viewed as the higher dimensional charge pumping analog of the 1D case where the charge is pumped from one boundary to the other (see figure)

where

• Plugging this in the above equation we get

• Since g1 and g2 are Hermitian we can write

• Properties of the Hamiltonian in k-space:

• Define a time-reversal matrix:

4. Dimensional reduction to (3+1)-d TRI insulatorsC. Z2 topological classification of time-reversal invariant insulators

• Consider the second quantized Hamiltonian:

• We can define an interpolation for any two TRI band insulators as:

where• We can define an two different interpolations shown in the figure• We want to show Chern number difference of these two paths is an even number• To show this define two new interpolations defined as

• By their definition we can write

• To show C2[g1] = C2[g2] consider the eigenvalue equation

• Consider the action of g2 on the eigenstate of g1

• Taking the complex conjugate of both sides we get

• Using this we can write

4. Dimensional reduction to (3+1)-d TRI insulatorsC. Z2 topological classification of time-reversal invariant insulators

• Expression from last slide:

• Now, we can write the Berry phase gauge vector of the g1 and g2 systems as

• In other words, they are equivalent up to a gauge transformation, i.e.

• Therefore, we can say that

• Then we have effectively shown that

• For 4+1 D Dirac model, there are |C2[h]| nodal points (of the spectrum) in the 3D Brillouin zone

• If there exist a nodal point (k, θ) which is not time-reversal symmetric then it has a nodal partner (-k, -θ)

• For topologically nontrivial insulator |C2[h]| is odd

• Then number of nodes at the 8 time-reversal symmetric points (in 3D) must have odd number of spectral nodes

G

MA

A

B

q

- p

0

+pB

kx

ky

X

(3+1)d surface of(4+1)d system

X

M

GB

B

(2+1)-d surface of(3+1)d system

kx

ky

(a)Consequences of this topological quantum number

• The Hall conductance carried by a Dirac fermion is well-defined only when the fermion mass is non-vanishing, so that a gap is opened

• If the surface (initially) has 2n+1 Dirac cones. Breaking time-reversal symmetry on the surface gives the following Hall conductance from each Dirac cone

• We can explicitly see this in the model:• Then the Dirac Hamiltonian looks like:

4. Dimensional reduction to (3+1)-d TRI insulatorsD. Physical properties of Z2-nontrivial insulators

• The non-trivial insulator has a magnetoelectric polarization P3 = 1/2 mod 1; the effective action of the bulk system should be

where n = P3 – 1/2 ϵ ℤ. S3D is quantized (and independent of P3) for compact space-time; hence P3 is unphysical• For open boundaries consider semi-infinite nontrivial insulator occupying the space z ≤ 0; the rest (z > 0) is vacuum• Effective action for all of ℝ3 as

• Since P3 = 1/2 mod 1 for z < 0 and P3 = 0 mod 1 for z > 0, we have

• The domain wall of P3 carries a quantum Hall effect

4. Dimensional reduction to (3+1)-d TRI insulatorsD. Physical properties of Z2-nontrivial insulators

• Recall Dirac Hamiltonian from last slide

• Under a time-reversal transformation, Γ0 is even and Γ1,2,3,4 are odd; in other words only H0 respects time-reversal symmetry• Recall second Chern number for this model

• For θ = π and -4c < m < -2c we have C2 = 1. It can be shown that for θ = 0, the region -4c < m < -2c is adiabatically connected to m → -∞ such that C2 = 0.

• It can be noted that {Γ4,H0} = 0 in the bulk and on the surface

• The surface Hamiltonian of H0 can be written as

• Since the only thing that anticommutes with this above surface Hamiltonian, the surface Hamiltonian for H = H0 + H1 is

• In summary, the effect of a time-reversal symmetry breaking term on the surface is to assign a mass to the Dirac fermions which determines the winding direction of P3 through the domain wall

• An analogous Chern number, that can evaluate the winding, can be defined as g[M(x)], where M(x) is the T-breaking field. The surface action, in terms of g[M(x)], can be written as

4. Dimensional reduction to (3+1)-d TRI insulatorsD. Physical properties of Z2-nontrivial insulators

• Consider a ferromagnet-topological insulator heterostructure• The pointing of the magnetization is given by• For the first (parallel) case• For the second (antiparallel) case• Then for• Using this expression the currents on the top and bottom surfaces can

be determined and are indicated for the two cases by (horizontal) blue arrows

• The antiparallel magnetization leads to a vanishing net Hall conductance, while the parallel magnetization leads to σH = e2/h

• A top-down view of the device, to measure this quantum Hall effect, can be seen in the bottom left figure

• For the case of parallel magnetization, the trapping of the chiral edge states on the side surfaces of the topological insulator can be seen in the 3D view in the bottom right figure; these carry the quantized Hall current.

1. Magnetization-induced QH effect

TI

FM

FM

Ex

sH=1/2p

TI

FM

FM

Ex

sH=0(a) (b)

(c)

FM-TI-FM

(d)

TI

FM

FM

Ex

sH=1/2p

TI

FM

FM

Ex

sH=0(a) (b)

(c)

FM-TI-FM

(d)

TI

FM

FM

Ex

sH=1/2p

TI

FM

FM

Ex

sH=0(a) (b)

(c)

FM-TI-FM

(d)

TI

FM

FM

Ex

sH=1/2p

TI

FM

FM

Ex

sH=0(a) (b)

(c)

FM-TI-FM

(d)

• Recall, in a T-breaking field M(x), the surface Chern-Simons action:• Note: Although the Hall conductance of the two surfaces are the same in

the global x, y, z basis, they are opposite in the local basis defined with respect to the normal vector

• This corresponds to g[M(x)] + 1/2 = ±1/2 for the top and bottom surfaces respectively.• Drawing closer analogy to the integer quantum Hall effect (IQHE), we could alternatively say g[M(x)] = 0 (-1) for the top

(bottom) surface; i.e. there exists a domain wall between two adjacent plateaux of IQHE• Such a domain wall will trap a chiral Fermi liquid which is responsible for the net Hall effect• Note: in general there will also be other non-chiral states on this side surface; they are irrelevant since only one chiral edge

state is protected• Conditions for observing this type of quantum Hall: (i) kBT ≪ EM (magnetization-induced bulk gap), (ii) chemical potential

lies in the bulk gap• Question: is this special type of quantum Hall a topological phase or symmetry-protected topological phase?

FM

TI

qtopo

(a) (b)q

qtopo

B

PM

PM

TI

B

4. Dimensional reduction to (3+1)-d TRI insulatorsD. Physical properties of Z2-nontrivial insulators

• It is the rotation of polarization of light by a certain angle β = B d𝒱• We can tune polarization by tuning the applied magnetic field• Perhaps the same can be accomplished with external electric fields by

exploiting TME in a medium with nonzero θ• Electric fields are, in general, easier to generate than magnetic fields• Recall that the total action of the electromagnetic field including the

topological term is given by

3. Low-frequency Faraday rotation

• However, the effective theory only applies in the low-energy limit E ≪ Eg, where Eg is the gap of the surface state

• Assuming a FM-TI interface at z = 0, normally incident, linearly-polarized light in either region can be written as (primed variables belong to the TI):

• The ΔP3 terms in the action contribute unconventional boundary conditions at z = 0 given by (a = ax + iay, etc.)

• Solving the above equations simultaneously we get:

• By simple algebra the polarization plane rotated by an angle is given by

• For typical values of Eg = 10 meV (and ϵ, μ ~ 1 and Δ = 1/2) we get EM frequency ≪ 2.4 THz, which is in the far infrared or microwave region

• In principle, it is possible to find a topological insulator with a larger gap which can support an accurate measurement of Faraday rotation