review of a short course on TI

Jingcheng Liang1, ∗
1
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China
(Dated: July 23, 2025)

CONTENTS

I. SSH model 1
A. Hamiltonian of SSH model 1
B. chiral symmetry 1
C. topological invariant 2
D. fractionalization of electron 3
E. exact calculation of zero energy states 3

II. Berry phase and Chern number 3

A. Discrete case 3
B. continuum case 4
C. Berry phase and adiabatic dynamics 5
D. Berry’s formula for Berry curvature 6

III. Polarization and Berry Phase 6

A. Bloch states 6
B. Wannier states 6

References 7
This note reviews the important points of the book Ref. [1].

I. SSH MODEL

A. Hamiltonian of SSH model

The SSH model is described by a two-band Hamiltonian:

ˆ · σ̂
H(k) = d(k) (1)

with

dx (k) = v + w cos k (2)

dy (k) = w sin k (3)
dz (k) = 0 (4)

B. chiral symmetry

Now we need a different symmetry other than the unitary symmetry(It can be made to disappear if one restricts
himself to one sector of the Hilbert space). One of such symmetries is the chiral symmetry:

Γ̂Ĥ Γ̂† = −H (5)

∗ liangjc@iphy.ac.cn
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FIG. 1. The endpoints of d(k) as k goes across the Brillouin zone

It is Hermitian and unitary and further Γ2 = 1. The chiral symmetry can be defined by the sublattice projectors:
1
P̂A = (1 + Γ̂) (6)
2
1
P̂B = (1 − Γ̂) (7)
2
In the ssh model, they have an explicit form:
N
X
PA = |m, A⟩ ⟨m, A| (8)
m=1
N
X
PB = |m, B⟩ ⟨m, B| (9)
m=1

The chiral symmetry is represented by the operator Σz ,

Σz = PA − PB (10)
It is easy to show that
Σz HΣz = −H (11)

when H only contains terms like |m, A⟩ m′ , B or |m, B⟩ m′ , A . This relation holds even when the hopping ampli-
tudes depend on the position. For the two band Hamiltonian Eq. (1), we have dz (k) = 0 under chiral symmetry. The
endpoint of d(k) is then a closed directed loop on the plane, allowing a well-defined winding number about the origin.

C. topological invariant

Since d(k) is restricted on the 2d plane, we may define the topological invariant according to how many times the
d(k) crosses a line from 0 to infinity. See Fig. 1. If we define
d
d˜ = (12)
|d|
Then the winding number should be
Z
1 ˜
ν= d(arg d) (13)
2π
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Therefore, we have,
Z π
1 ˜ × d ˜
ν= (d(k) d(k))dk (14)
2π −π dk
The Hamiltonian with chiral symmetry,
 
0 h(k)
H(k) = (15)
h∗ (k) 0

the topological invariant can also be defined by

Z π
1 d
ν= dk log h(k) (16)
2πi −π dk
To change the topological invariant, one needs to either break the chiral symmetry or close the gap of the Hamiltonian.
We may also use the net number of edge states at one end of the system to define the topological invariant. They
are the same as the ν defined through the bulk band properties. The bulk topological invariant can predict low energy
physics at the edge. This is an example of the Bulk-Boundary Correspondence(BBC).

D. fractionalization of electron

The explanation in this book is that the two edge states hybridize and form two states. At half-filling, only the
negative energy state is occupied. This state has equal weight at two ends of the chain, hence one end only carries
one-half of the electron charge.
However, the above explanation is not convincing. From the point of view of many-body physics, the density of the
chain is still uniform with no exceptional half charge.

E. exact calculation of zero energy states

When the hopping amplitudes are position-dependent, the Hamiltonian is

N
X N
X −1
H= (vm |m, B⟩ ⟨m, A| + h.c.) + (wm |m + 1, A⟩ ⟨m, B| + h.c.) (17)
m=1 m=1

For zero energy state, we have

N
X
H (am |m, A⟩ + bm |m, B⟩) = 0 (18)
1

boundaries are

vN aN = v1 b1 = 0 (19)

So in generic case, there is no exact zero energy case. But we can still have asymptotic zero-energy states.

II. BERRY PHASE AND CHERN NUMBER

A. Discrete case

1. The relative phase of two quantum states are

γ12 = − arg ⟨Ψ1 |Ψ2 ⟩ (20)

However, it is not invariant under a local gauge transformation,

Ψj → eiαj Ψj , γ12 → γ12 + α1 − α2 (21)

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2. For N states ordered in a loop, the Berry phase is defined as

γL = − arg(⟨Ψ1 |Ψ2 ⟩ ⟨Ψ2 |Ψ3 ⟩ . . . ⟨ΨN |Ψ1 ⟩) (22)

It is gauge invariant.
3. The Berry flux of a plaquette Fnm is the Berry phase of the four states that make up the plaquette. We have
the Berry phase of a loop can be expressed using the Berry flux:
 
N
X −1 M
X −1
e−iγL = exp−i Fnm  (23)
n=1 n=1

4. Chern number
On a torus, the product of the Berry flux of all plaquettes are 1,
M Y
Y N
e−iFnm = 1 (24)
m=1 n=1

The Chern number is defined as the sum of all the Berry fluxes:
1 X
Q= Fnm (25)
2π nm

A modified Berry flux F̃nm is

F̃nm = γ(nm),(n+1,m) + γ(n+1,m),(n+1,m+1) + γ(n+1,m+1),(n,m+1) + γ(n,m+1),(n,m) (26)

A plaquette nm contains Qnm vortices,

Fnm − F̃nm
Qnm = (27)
2π
and
X
Q= Qnm (28)
nm

B. continuum case

1. Berry connection
We have
Ψ(R) Ψ(R + dR)
e−i∆γ = ; ∆γ = i Ψ(R) ∇R Ψ(R) · dR (29)
| Ψ(R) Ψ(R + dR) |

The Berry connection is defined as

A(R) = i Ψ(R) ∇R Ψ(R) = −Im Ψ(R) ∇R Ψ(R) (30)

2. Berry Phase
For a curve L in parameter space, the Berry phase is defined as
 I 
γ(L) = − arg exp −i A · dR (31)
L

3. Berry Curvature
For the gauge R → Ψ(R) that is not smooth around a point R0 of the parameter space, we may always find a
gauge R → Ψ′ (R) that is locally smooth and generates the same map as R → Ψ(R) Ψ(R) . This can be proved
by considering the Hamiltonian H = − Ψ(R) Ψ(R) whose ground state is Ψ(R) , the Ψ′ (R) is constructed using
perturbation theory around point R0 .
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For a smooth gauge, we may convert the line integral into a surface integral:
I Z Z
A · dR = (∂x Ay − ∂y Ax )dxdy = Bdxdy (32)
∂V V V

For 2D surface embedded in the three dimensional parameter space, we can apply the three dimensional Stokes
theorem:
I Z
A · dR = B · dS (33)
∂V V

where B(R) = ∇R × A(R).

4. Chern number
The Chern number is an extension of the discrete case and is the surface integral of the Berry curvature over the
whole parameter space S:
Z
1
Q=− Bdxdy (34)
2π S

From the discrete case, we know that the Chern number is an integer so it can not change as we continuously change
the Hamiltonian and the Berry curvature.

C. Berry phase and adiabatic dynamics

We have a Hamiltonian that depends on the parameter R and we have the snapshot basis:

H(R) n(R) = En (R) n(R) (35)

If the nth band is separated from the other bands, and we start from

R(t = 0) = R0 ; ψ(t = 0) = n(R0 ) (36)

By virtue of the adiabatic approximation, we take the Ansatz,

En (R(t′ ))dt′ |n(R(t))⟩

Rt
Ψ(t) = eiγn (t) e−i 0 (37)

Substituting into the time dependent Schrödinger equation,

d
i ψ(t) = H(R(t)) ψ(t) (38)
dt
We have

dγn d
− n(R) + i n(R) = 0 (39)
dt dt

Eq. (39) is the consequence of the adiabatic approximation, since the time derivative of n(R) is generally not pro-
portional to it derivative. Something has been thrown away so that we can have the ansatz Eq. (37). For detail
calculation, see Griffiths P385.
From Eq. (39) we have

d dR
γn (t) = i n(R) ∇R n(R) (40)
dt dt
For cyclic change of Hamiltonian, the adiabatic phase is equivalent to the Berry phase:
I
γn (L) = i n(R) ∇R n(R) dR (41)
L

The Berry phase can be measured by the interferometric setup.

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D. Berry’s formula for Berry curvature

The first formula is derived using the definition B = ∇ × A:

Bj = −Imϵjkl ∂k ⟨n|∂l n⟩ = −Imϵjkl ⟨∂k n|∂l n⟩ (42)

The second one is:

X ⟨n| ∇H |m⟩ × ⟨m| ∇H |n⟩
B (n) = −Im (43)
(En − Em )2
m̸=n

Eq. (43) can be derived from the second equality of Eq. ( 42) after taking the derivative of H(R) |n⟩ = En |n⟩.

III. POLARIZATION AND BERRY PHASE

We consider the Rice-Mele model in this section. It is obtained from the SSH model by adding an extra staggered
onsite potential.
N
X N
X −1
H= (v |m, B⟩ ⟨m, A| + h.c.) + (w |m + 1, A⟩ ⟨m, B| + h.c.)
m=1 m=1
N
X
+u (|m, A⟩ ⟨m, A| − |m, B⟩ ⟨m, B|) (44)
1

A. Bloch states

Ψ(k) = |k⟩ ⊗ u(k) (45)

with
N
1 X imk
|k⟩ = √ e |m⟩ , f or k ∈ {δk , 2δk , . . . , N δk } with δk = 2π/N (46)
N m=1

and u(k) are eigenstates with energy E(k) of the bulk momentum space Hamiltonian,

v + we−ik
 
u
H(k) = (47)
v + wrik −u

The projector to the occupied subspace:

X
P = Ψ(k) Ψ(k) (48)
k∈BZ

B. Wannier states

The Wanier states are defined by the following properties:

w(j ′ ) w(j) = δj ′ j , Orthonomal set (49)

N
X
w(j) w(j) = P, Span the occupied subspace (50)
j=1

m + 1 w(j + 1) = m w(j) , Related by translation (51)

2
lim w(N/2) (x − N/2) w(N/2) < ∞, Localization (52)
N →∞
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[1] J. K. Asbóth, L. Oroszlány, and A. Pályi, A short course on topological insulators, Lecture notes in physics 919 (2016).