introduction to band topology and dirac cone
TRANSCRIPT
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Introduction toBand Topology and Dirac Cone
Leandro Seixas Rocha
Instituto de Fısica, Universidade de Sao Paulo
March 16, 2012
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Contents
1 Bloch’s Band theory
2 Berry theory
3 Dirac cone
4 Z2 Band Topology
5 Topological Invariants in 3D
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Motivation
Recently, the topology of electronic states (or Bloch Hamiltonians) becamesimportant for description of Electronic Structure of materials.
Topology (Homotopy theory) were already being used to study topological defectsin media such as liquid crystals, Quantum Hall Effect (QHE), and High-energyphysics (Gauge field theory and Gravitation).
Topological states can be important for spintronics and fault-toleranttopological quantum computation.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Motivation
Recently, the topology of electronic states (or Bloch Hamiltonians) becamesimportant for description of Electronic Structure of materials.
Topology (Homotopy theory) were already being used to study topological defectsin media such as liquid crystals, Quantum Hall Effect (QHE), and High-energyphysics (Gauge field theory and Gravitation).
Topological states can be important for spintronics and fault-toleranttopological quantum computation.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Motivation
Recently, the topology of electronic states (or Bloch Hamiltonians) becamesimportant for description of Electronic Structure of materials.
Topology (Homotopy theory) were already being used to study topological defectsin media such as liquid crystals, Quantum Hall Effect (QHE), and High-energyphysics (Gauge field theory and Gravitation).
Topological states can be important for spintronics and fault-toleranttopological quantum computation.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch’s Band theory
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Solid-state review
Crystal
The physical systems studied here are crystals(2D or 3D), periodic by translation vectors ofBravais lattice
R =3∑
i=1
ni ai , ∀ni ∈ Z, (1)
where {ai} are the basis vectors, given by eachmaterial.
For each unit cell, there is a basis with thepositions of atoms.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Solid-state review
Crystal
The physical systems studied here are crystals(2D or 3D), periodic by translation vectors ofBravais lattice
R =3∑
i=1
ni ai , ∀ni ∈ Z, (1)
where {ai} are the basis vectors, given by eachmaterial.
For each unit cell, there is a basis with thepositions of atoms.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Solid-state review
Reciprocal lattice and Brillouin zone
The dual space of Bravais lattice is theReciprocal lattice, formed by the vectors
G =3∑
i=1
mi bi , ∀mi ∈ Z, (2)
where {bi} are vectors satisfying the conditionai · bj = 2πδij .
The electronic states are labeled by the vectors kin Brillouin zone (BZ), obeying the condition
|k| < |k− G|, ∀G. (3)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Solid-state review
Reciprocal lattice and Brillouin zone
The dual space of Bravais lattice is theReciprocal lattice, formed by the vectors
G =3∑
i=1
mi bi , ∀mi ∈ Z, (2)
where {bi} are vectors satisfying the conditionai · bj = 2πδij .
The electronic states are labeled by the vectors kin Brillouin zone (BZ), obeying the condition
|k| < |k− G|, ∀G. (3)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian
Bloch’s theorem
The eigenstates of a periodic Hamiltonian T−1R HTR = H, are
〈r|ψnk〉 = e ik·r 〈r|unk〉 , (4)
with 〈r + R|unk〉 = 〈r|unk〉.
The Schrodinger equation for these states are
H(k)|unk〉 = En(k)|unk〉, (5)
where
H(k) = e ik·rHe−ik·r (6)
is the Bloch Hamiltonian.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian
Bloch’s theorem
The eigenstates of a periodic Hamiltonian T−1R HTR = H, are
〈r|ψnk〉 = e ik·r 〈r|unk〉 , (4)
with 〈r + R|unk〉 = 〈r|unk〉.
The Schrodinger equation for these states are
H(k)|unk〉 = En(k)|unk〉, (5)
where
H(k) = e ik·rHe−ik·r (6)
is the Bloch Hamiltonian.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Fundamental domain and Brillouin Zone Topology
The Bloch Hamiltonian is periodic in k-space
H(k + bi ) = H(k), ∀k ∈ BZ , and i = 1, 2, 3. (7)
The Fundamental domain of Bloch Hamiltonian is the Brillouin Zone (BZ).
If H(k1) = H(k2), then k1 ∼ k2, is called identification of points k1 and k2.
Brillouin Zone as Torus (T 3BZ ).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Fundamental domain and Brillouin Zone Topology
The Bloch Hamiltonian is periodic in k-space
H(k + bi ) = H(k), ∀k ∈ BZ , and i = 1, 2, 3. (7)
The Fundamental domain of Bloch Hamiltonian is the Brillouin Zone (BZ).
If H(k1) = H(k2), then k1 ∼ k2, is called identification of points k1 and k2.
Brillouin Zone as Torus (T 3BZ ).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Fundamental domain and Brillouin Zone Topology
The Bloch Hamiltonian is periodic in k-space
H(k + bi ) = H(k), ∀k ∈ BZ , and i = 1, 2, 3. (7)
The Fundamental domain of Bloch Hamiltonian is the Brillouin Zone (BZ).
If H(k1) = H(k2), then k1 ∼ k2, is called identification of points k1 and k2.
Brillouin Zone as Torus (T 3BZ ).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Fundamental domain and Brillouin Zone Topology
The Bloch Hamiltonian is periodic in k-space
H(k + bi ) = H(k), ∀k ∈ BZ , and i = 1, 2, 3. (7)
The Fundamental domain of Bloch Hamiltonian is the Brillouin Zone (BZ).
If H(k1) = H(k2), then k1 ∼ k2, is called identification of points k1 and k2.
Brillouin Zone as Torus (T 3BZ ).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian space
Set of all Bloch Hamiltonians (for occupied state) is called M.
Band structure is defined by
H : T 3BZ →M, [k 7→ H(k)] . (8)
M` is the space of all Bloch Hamiltonians with `-fold degenerate eigenvalues.
Mj ,` is the space of Bloch Hamiltonians with j-fold and `-fold degenerateeigenvalues.
Topology of Bloch Hamiltonian space is called Band topology.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian space
Set of all Bloch Hamiltonians (for occupied state) is called M.
Band structure is defined by
H : T 3BZ →M, [k 7→ H(k)] . (8)
M` is the space of all Bloch Hamiltonians with `-fold degenerate eigenvalues.
Mj ,` is the space of Bloch Hamiltonians with j-fold and `-fold degenerateeigenvalues.
Topology of Bloch Hamiltonian space is called Band topology.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian space
Set of all Bloch Hamiltonians (for occupied state) is called M.
Band structure is defined by
H : T 3BZ →M, [k 7→ H(k)] . (8)
M` is the space of all Bloch Hamiltonians with `-fold degenerate eigenvalues.
Mj ,` is the space of Bloch Hamiltonians with j-fold and `-fold degenerateeigenvalues.
Topology of Bloch Hamiltonian space is called Band topology.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian space
Set of all Bloch Hamiltonians (for occupied state) is called M.
Band structure is defined by
H : T 3BZ →M, [k 7→ H(k)] . (8)
M` is the space of all Bloch Hamiltonians with `-fold degenerate eigenvalues.
Mj ,` is the space of Bloch Hamiltonians with j-fold and `-fold degenerateeigenvalues.
Topology of Bloch Hamiltonian space is called Band topology.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonian
Bloch Hamiltonian space
Set of all Bloch Hamiltonians (for occupied state) is called M.
Band structure is defined by
H : T 3BZ →M, [k 7→ H(k)] . (8)
M` is the space of all Bloch Hamiltonians with `-fold degenerate eigenvalues.
Mj ,` is the space of Bloch Hamiltonians with j-fold and `-fold degenerateeigenvalues.
Topology of Bloch Hamiltonian space is called Band topology.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Band Topology
Topology
Topology is the mathematical study of the properties that are preserved throughdeformations, twistings, and stretchings of objects. Tearing, however, is not allowed.a
aWeisstein, Eric W. “Topology.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Topology.html
The space of Bloch Hamiltonians (M) is a topological space.
Two Bloch Hamiltonians H(k) and H′(k) are topologically equivalent if there isa smooth function (Homotopy)
F (λ), λ ∈ [0, 1] (9)
with H = F (0) and H′ = F (1).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Band Topology
Topology
Topology is the mathematical study of the properties that are preserved throughdeformations, twistings, and stretchings of objects. Tearing, however, is not allowed.a
aWeisstein, Eric W. “Topology.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Topology.html
The space of Bloch Hamiltonians (M) is a topological space.
Two Bloch Hamiltonians H(k) and H′(k) are topologically equivalent if there isa smooth function (Homotopy)
F (λ), λ ∈ [0, 1] (9)
with H = F (0) and H′ = F (1).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Band Topology
Topology
Topology is the mathematical study of the properties that are preserved throughdeformations, twistings, and stretchings of objects. Tearing, however, is not allowed.a
aWeisstein, Eric W. “Topology.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Topology.html
The space of Bloch Hamiltonians (M) is a topological space.
Two Bloch Hamiltonians H(k) and H′(k) are topologically equivalent if there isa smooth function (Homotopy)
F (λ), λ ∈ [0, 1] (9)
with H = F (0) and H′ = F (1).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Visualization of Homotopy
The smooth function F (λ), also known as homotopy, can be seen asdeformation of Bloch Hamiltonians.
Figure: Homotopy between mug and torus.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Topological invariant
Generally we cannot determine the homotopy between topological spaces.
Since the deformation is continous, integer number associated to topologicalspaces should be preserved under deformations. (Idea of Algebraic topology)
Topological invariant
Topological invariant is a properties (integer) of a topological space (M) which isinvariant under deformations.
Two topologically equivalent spaces must share all the topological invariants.
Topological invariant is a properties of entire space.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Topological invariant
Generally we cannot determine the homotopy between topological spaces.
Since the deformation is continous, integer number associated to topologicalspaces should be preserved under deformations. (Idea of Algebraic topology)
Topological invariant
Topological invariant is a properties (integer) of a topological space (M) which isinvariant under deformations.
Two topologically equivalent spaces must share all the topological invariants.
Topological invariant is a properties of entire space.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Topological invariant
Generally we cannot determine the homotopy between topological spaces.
Since the deformation is continous, integer number associated to topologicalspaces should be preserved under deformations. (Idea of Algebraic topology)
Topological invariant
Topological invariant is a properties (integer) of a topological space (M) which isinvariant under deformations.
Two topologically equivalent spaces must share all the topological invariants.
Topological invariant is a properties of entire space.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Topological invariant
Generally we cannot determine the homotopy between topological spaces.
Since the deformation is continous, integer number associated to topologicalspaces should be preserved under deformations. (Idea of Algebraic topology)
Topological invariant
Topological invariant is a properties (integer) of a topological space (M) which isinvariant under deformations.
Two topologically equivalent spaces must share all the topological invariants.
Topological invariant is a properties of entire space.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Band topology
Topological invariant
Generally we cannot determine the homotopy between topological spaces.
Since the deformation is continous, integer number associated to topologicalspaces should be preserved under deformations. (Idea of Algebraic topology)
Topological invariant
Topological invariant is a properties (integer) of a topological space (M) which isinvariant under deformations.
Two topologically equivalent spaces must share all the topological invariants.
Topological invariant is a properties of entire space.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry theory
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Parallel transport
Parallel transport (Tulio Levi-Civita, 1917)
A differential displacement in k vector gives
|unk〉 −→ |unk+dk〉 = |unk〉+ ∇k|unk〉 · dk. (10)
Parallel component is calculated applying the projector P‖ = |unk〉〈unk|
|unk〉P‖−→ |unk〉 〈unk|unk+dk〉 = |unk〉+ |unk〉〈unk|∇k|unk〉 · dk. (11)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Parallel transport
Parallel transport (Tulio Levi-Civita, 1917)
A differential displacement in k vector gives
|unk〉 −→ |unk+dk〉 = |unk〉+ ∇k|unk〉 · dk. (10)
Parallel component is calculated applying the projector P‖ = |unk〉〈unk|
|unk〉P‖−→ |unk〉 〈unk|unk+dk〉 = |unk〉+ |unk〉〈unk|∇k|unk〉 · dk. (11)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry connection
Berry connection
The Bloch states |unk〉 and |unk+dk〉 are “connected” by Berry connection
An(k) = −i〈unk|∇k|unk〉. (12)
The parallel differential displacement of Bloch state gives
〈unk|unk+dk〉 = 1 + iAn(k) · dk, (13)
For finite parallel transport, from ki to kf , we have
〈unki|unkf〉 = e
i∫ kf
kidk′·An (14)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry connection
Berry connection
The Bloch states |unk〉 and |unk+dk〉 are “connected” by Berry connection
An(k) = −i〈unk|∇k|unk〉. (12)
The parallel differential displacement of Bloch state gives
〈unk|unk+dk〉 = 1 + iAn(k) · dk, (13)
For finite parallel transport, from ki to kf , we have
〈unki|unkf〉 = e
i∫ kf
kidk′·An (14)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry connection
Berry connection
The Bloch states |unk〉 and |unk+dk〉 are “connected” by Berry connection
An(k) = −i〈unk|∇k|unk〉. (12)
The parallel differential displacement of Bloch state gives
〈unk|unk+dk〉 = 1 + iAn(k) · dk, (13)
For finite parallel transport, from ki to kf , we have
〈unki|unkf〉 = e
i∫ kf
kidk′·An (14)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry phase and holonomy
Berry phase
Integrating in a loop C (= ∂S), we found⟨uinitial
nk |ufinalnk
⟩= e i
∮C dk′·An(k′). (15)
Berry phase
A Bloch state |unk〉 acquires the Berry phasea
γn(S) =
∮∂S
dk · An(k), (16)
when this state is parallel transported through a loop ∂S in k-space.
aM.V. Berry, Proc. Roy. Soc. London Ser. A, 392, 45 (1984).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry phase and holonomy
Berry phase
Integrating in a loop C (= ∂S), we found⟨uinitial
nk |ufinalnk
⟩= e i
∮C dk′·An(k′). (15)
Berry phase
A Bloch state |unk〉 acquires the Berry phasea
γn(S) =
∮∂S
dk · An(k), (16)
when this state is parallel transported through a loop ∂S in k-space.
aM.V. Berry, Proc. Roy. Soc. London Ser. A, 392, 45 (1984).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry phase and holonomy
Berry phase as holonomy
The Berry phase in a closed path onto a curved surface is seen as holonomy ofBerry connection1.
Figure: Holonomy (Hannay’s angle) of parallel transport of a vector onto a sphere.
1B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry curvature
Berry curvature
Using the Stokes’ theorem for the Berry phase, we have
γn(S) =
∫Sd2k · Bn(k), (17)
where
Bn(k) = ∇k × An(k) (18)
is the Berry curvature.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry curvature
Berry curvature (cont’d)
From Berry curvature expression, we have
Bn(k) = −i∇k × 〈unk|∇kunk〉 = −i〈∇kunk| × |∇kunk〉. (19)
Using the completeness relation and Schrodinger equation derivative
Bn(k) = −i∑n 6=n′
〈unk|∇kH|un′k〉 × 〈un′k|∇kH|unk〉(En(k)− En′(k))2
. (20)
At degeneracy En(k0) = En′(k0), the Berry curvature is singular,
limk→k0
Bn(k) =∞. (21)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry curvature
Berry curvature (cont’d)
From Berry curvature expression, we have
Bn(k) = −i∇k × 〈unk|∇kunk〉 = −i〈∇kunk| × |∇kunk〉. (19)
Using the completeness relation and Schrodinger equation derivative
Bn(k) = −i∑n 6=n′
〈unk|∇kH|un′k〉 × 〈un′k|∇kH|unk〉(En(k)− En′(k))2
. (20)
At degeneracy En(k0) = En′(k0), the Berry curvature is singular,
limk→k0
Bn(k) =∞. (21)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Berry curvature
Berry curvature (cont’d)
From Berry curvature expression, we have
Bn(k) = −i∇k × 〈unk|∇kunk〉 = −i〈∇kunk| × |∇kunk〉. (19)
Using the completeness relation and Schrodinger equation derivative
Bn(k) = −i∑n 6=n′
〈unk|∇kH|un′k〉 × 〈un′k|∇kH|unk〉(En(k)− En′(k))2
. (20)
At degeneracy En(k0) = En′(k0), the Berry curvature is singular,
limk→k0
Bn(k) =∞. (21)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Chern number
Chern number
For 2D systems, the Brillouin Zone is the surfaceT 2
BZ .
Taking the parallel transport along a loop(C = ∂S) and the inverse loop (−C = ∂Sc), theBerry phase acquires is
eγn(Sc )eγn(S) = 1⇒ eγn(Sc⋃S) = 1, (22)
and using S ∪ Sc = T 2BZ , we have
γn(T 2BZ ) = 2π × integer. (23)
Figure: Loops onto Brillouinzone (T 2
BZ ).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Chern number
Chern number
For 2D systems, the Brillouin Zone is the surfaceT 2
BZ .
Taking the parallel transport along a loop(C = ∂S) and the inverse loop (−C = ∂Sc), theBerry phase acquires is
eγn(Sc )eγn(S) = 1⇒ eγn(Sc⋃S) = 1, (22)
and using S ∪ Sc = T 2BZ , we have
γn(T 2BZ ) = 2π × integer. (23)
Figure: Loops onto Brillouinzone (T 2
BZ ).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Chern number
Chern number (cont’d)
Chern number
From each Bloch state |unk〉, we can associate the integer number
C1n =
1
2π
∫T 2
BZ
d2k · Bn(k), (24)
known as First Chern number (or just Chern number).
We can also define a Chern number for all occupied states (En(k) < EF )
C1 =occ∑
n
C1n ,
∞∑n=1
C1n = 0. (25)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Chern number
Chern number vs. Euler characteristic
Chern number is a topological invariant of Bloch Hamiltonian.
The Chern number is similar to Euler characteristic.
Gauss–Bonnet theorem
The Gauss–Bonnet theorem is
1
2π
∫MdS ·K = χ(M), (26)
where χ(M) = 2(1− g) is the Euler characteristic and g is the genus of surface M.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac cone
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Symmetry and degeneracy
Symmetry and degeneracy
Symmetries can leads point-like degeneracies.
A general symmetry operator is represented by an unitary operator S, leading
H(k)S−→ H′(k) = SH(k)S−1, and |unk〉
S−→ |u′nk〉 = S|unk〉. (27)
If at k = k0, S is a symmetry operator, then
SHS−1 = H ⇒ [H,S] = 0. (28)
Consequently
H (S|unk0〉) = En (S|unk0〉) , and H|unk0〉 = En|unk0〉, (29)
i.e., the states |unk0〉 and S|unk0〉 are degenerate.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Symmetry and degeneracy
Symmetry and degeneracy
Symmetries can leads point-like degeneracies.
A general symmetry operator is represented by an unitary operator S, leading
H(k)S−→ H′(k) = SH(k)S−1, and |unk〉
S−→ |u′nk〉 = S|unk〉. (27)
If at k = k0, S is a symmetry operator, then
SHS−1 = H ⇒ [H,S] = 0. (28)
Consequently
H (S|unk0〉) = En (S|unk0〉) , and H|unk0〉 = En|unk0〉, (29)
i.e., the states |unk0〉 and S|unk0〉 are degenerate.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Symmetry and degeneracy
Symmetry and degeneracy
Symmetries can leads point-like degeneracies.
A general symmetry operator is represented by an unitary operator S, leading
H(k)S−→ H′(k) = SH(k)S−1, and |unk〉
S−→ |u′nk〉 = S|unk〉. (27)
If at k = k0, S is a symmetry operator, then
SHS−1 = H ⇒ [H,S] = 0. (28)
Consequently
H (S|unk0〉) = En (S|unk0〉) , and H|unk0〉 = En|unk0〉, (29)
i.e., the states |unk0〉 and S|unk0〉 are degenerate.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Symmetry and degeneracy
Symmetry and degeneracy
Symmetries can leads point-like degeneracies.
A general symmetry operator is represented by an unitary operator S, leading
H(k)S−→ H′(k) = SH(k)S−1, and |unk〉
S−→ |u′nk〉 = S|unk〉. (27)
If at k = k0, S is a symmetry operator, then
SHS−1 = H ⇒ [H,S] = 0. (28)
Consequently
H (S|unk0〉) = En (S|unk0〉) , and H|unk0〉 = En|unk0〉, (29)
i.e., the states |unk0〉 and S|unk0〉 are degenerate.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem
von Neumann–Wigner theorem
The codimensiona of a double degeneracy is three.
aCodimension of degeneracy is the number of null parameters that are fixed to hold thedegeneracy.
Consider two degenerate quantum states |u1nk0〉 and |u2
nk0〉. A symmetry operator
S acting on these states gives
S|uαnk0〉 =
2∑β=1
D(n)αβ (S)|uβnk0
〉, (30)
i.e., the states {|uαnk0〉} span the basis for Γnk0 representation (Partner states).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since [H,S] = 0, so
H|uαnk0〉 =
2∑β=1
Hαβ|uβnk0〉, (31)
i.e., the eigenstates of Hamiltonian are the same of S.
In matrix form (H11 H12
H21 H22
)(|u1
nk0〉
|u2nk0〉
)= En
(|u1
nk0〉
|u2nk0〉
). (32)
In the basis {1l, σ1, σ2, σ3}, we have
H = h01l + h1σ1 + h2σ2 + h3σ3 =
(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
). (33)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since [H,S] = 0, so
H|uαnk0〉 =
2∑β=1
Hαβ|uβnk0〉, (31)
i.e., the eigenstates of Hamiltonian are the same of S.
In matrix form (H11 H12
H21 H22
)(|u1
nk0〉
|u2nk0〉
)= En
(|u1
nk0〉
|u2nk0〉
). (32)
In the basis {1l, σ1, σ2, σ3}, we have
H = h01l + h1σ1 + h2σ2 + h3σ3 =
(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
). (33)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since [H,S] = 0, so
H|uαnk0〉 =
2∑β=1
Hαβ|uβnk0〉, (31)
i.e., the eigenstates of Hamiltonian are the same of S.
In matrix form (H11 H12
H21 H22
)(|u1
nk0〉
|u2nk0〉
)= En
(|u1
nk0〉
|u2nk0〉
). (32)
In the basis {1l, σ1, σ2, σ3}, we have
H = h01l + h1σ1 + h2σ2 + h3σ3 =
(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
). (33)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since |uαnk〉 are degenerate eigenstates, we have(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
)=
(En(k0) 0
0 En(k0)
), (34)
and, consequently
h0(k0) = En(k0), hi (k0) = 0, i = 1, 2, 3. � (35)
The conditions hi (k0) = 0 for i = 1, 2, 3 is the proof of von Neumann–Wignertheorem.
Note that, if any parameter hi is non-zero, the degeneracy is broken.
The h0 can be eliminated without loss of generality.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since |uαnk〉 are degenerate eigenstates, we have(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
)=
(En(k0) 0
0 En(k0)
), (34)
and, consequently
h0(k0) = En(k0), hi (k0) = 0, i = 1, 2, 3. � (35)
The conditions hi (k0) = 0 for i = 1, 2, 3 is the proof of von Neumann–Wignertheorem.
Note that, if any parameter hi is non-zero, the degeneracy is broken.
The h0 can be eliminated without loss of generality.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since |uαnk〉 are degenerate eigenstates, we have(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
)=
(En(k0) 0
0 En(k0)
), (34)
and, consequently
h0(k0) = En(k0), hi (k0) = 0, i = 1, 2, 3. � (35)
The conditions hi (k0) = 0 for i = 1, 2, 3 is the proof of von Neumann–Wignertheorem.
Note that, if any parameter hi is non-zero, the degeneracy is broken.
The h0 can be eliminated without loss of generality.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
von Neumann–Wigner theorem
von Neumann–Wigner theorem (cont’d)
Since |uαnk〉 are degenerate eigenstates, we have(h0 + h3 h1 − ih2
h1 + ih2 h0 − h3
)=
(En(k0) 0
0 En(k0)
), (34)
and, consequently
h0(k0) = En(k0), hi (k0) = 0, i = 1, 2, 3. � (35)
The conditions hi (k0) = 0 for i = 1, 2, 3 is the proof of von Neumann–Wignertheorem.
Note that, if any parameter hi is non-zero, the degeneracy is broken.
The h0 can be eliminated without loss of generality.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac cone
Dirac cone
In the vicinity of k0, the condition hi = 0(H(k0) = 0) can be relaxed.
From continuity of the Hamiltonian
H(k) = h(k) · σ =
(h3 h1 − ih2
h1 + ih2 −h3
). (36)
Dirac cone
The band structure E±(k) = ±√
h21 + h2
2 + h23 of the
Hamiltonian H(k) are called Dirac cone.
If h(k) is a linear function of k, the cone in h-spaceform also a cone in k-space.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac monopole
Dirac monopole
The influence of a Dirac cone on the band topology may be studied from thesimple Hamiltonian
H0(k) = Cσ · k, (37)
where C is a constant.
The Berry curvature for this Hamiltonian is
Bn(k) = −i∑n′ 6=n
〈unk|σ|un′k〉 × 〈un′k|σ|unk〉2 |k|2
, (38)
and, consequently
B±(k) = ±1
2
k
|k|3. (39)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac monopole
Dirac monopole
The influence of a Dirac cone on the band topology may be studied from thesimple Hamiltonian
H0(k) = Cσ · k, (37)
where C is a constant.
The Berry curvature for this Hamiltonian is
Bn(k) = −i∑n′ 6=n
〈unk|σ|un′k〉 × 〈un′k|σ|unk〉2 |k|2
, (38)
and, consequently
B±(k) = ±1
2
k
|k|3. (39)
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac monopole
Dirac monopole (cont’d)
The Berry curvature B±(k) is similar to magnetic field of a (Dirac) magneticmonopole2.
The singularity in the Berry curvature is also called Dirac monopole.
Back to h-space, k 7→ h(k), the Berry curvature for the Dirac monopole is
B±(k) = ±1
2
h(k)
|h(k)|3J, (40)
where J is determinat of Jacobian matrix.
2P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac monopole
Dirac monopole (cont’d)
The Berry curvature B±(k) is similar to magnetic field of a (Dirac) magneticmonopole2.
The singularity in the Berry curvature is also called Dirac monopole.
Back to h-space, k 7→ h(k), the Berry curvature for the Dirac monopole is
B±(k) = ±1
2
h(k)
|h(k)|3J, (40)
where J is determinat of Jacobian matrix.
2P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac monopole
Dirac monopole (cont’d)
The Berry curvature B±(k) is similar to magnetic field of a (Dirac) magneticmonopole2.
The singularity in the Berry curvature is also called Dirac monopole.
Back to h-space, k 7→ h(k), the Berry curvature for the Dirac monopole is
B±(k) = ±1
2
h(k)
|h(k)|3J, (40)
where J is determinat of Jacobian matrix.
2P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Dirac monopole
Review
Two Bloch Hamiltonian can be topologically equivalents (homotopic) if they sharethe same topological invariants.
For 2D system, the Chern number (a topological invariant) is calculated as
C1n =
1
2π
∫T 2
BZ
d2k · Bn(k), (41)
where Bn = −i∇k × 〈unk|∇k|unk〉.Dirac cone can lead to a non-trivial Chern number.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 Band Topology
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Time Reversal Symmetry (TRS)
Time Reversal Operator
T : r→ r, T : k→ −k (42)
Time Reversal Operator (acting on Hilbert space)
ΘH(k)Θ−1 = H(−k) (43)
Θ is an anti-linear operator.
The operator can be written as
Θ = UK , (44)
where U is a linear operator, and K is complex conjugate operator.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Time Reversal Symmetry (TRS)
Time Reversal Operator
T : r→ r, T : k→ −k (42)
Time Reversal Operator (acting on Hilbert space)
ΘH(k)Θ−1 = H(−k) (43)
Θ is an anti-linear operator.
The operator can be written as
Θ = UK , (44)
where U is a linear operator, and K is complex conjugate operator.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Time Reversal Symmetry (TRS)
Time Reversal Operator
T : r→ r, T : k→ −k (42)
Time Reversal Operator (acting on Hilbert space)
ΘH(k)Θ−1 = H(−k) (43)
Θ is an anti-linear operator.
The operator can be written as
Θ = UK , (44)
where U is a linear operator, and K is complex conjugate operator.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Time Reversal Symmetry (TRS)
Time Reversal Operator
T : r→ r, T : k→ −k (42)
Time Reversal Operator (acting on Hilbert space)
ΘH(k)Θ−1 = H(−k) (43)
Θ is an anti-linear operator.
The operator can be written as
Θ = UK , (44)
where U is a linear operator, and K is complex conjugate operator.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Kramers’ degeneracy theorem
Kramers’ degeneracy theorem
For a time reversal system with n odd fermions, the energy levels are at least doublydegenerate.
Proof. Consider that the |unk〉 and Θ|unk〉 are the same states. Thus
Θ|unk〉 = e iδ|unk〉 ⇒ Θ2|unk〉 = +|unk〉 (45)
But, for odd number of fermions,
Θ2 = −1 (for odd n). (46)
Consequetly, |unk〉 and Θ|unk〉 are degenerate states.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Kramers’ points
For some points {Γα}, α = 1, . . . ,N , called Kramers’ points, the BlochHamiltonian obey
H(Γα) = H(−Γα), ⇒ [Θ,H(Γα)] = 0. (47)
The coordinates of Kramers’ points are
Γα =1
2
(D∑
i=1
ni bi
), ni = 0, 1. (48)
For D-dimensional k-space, there are N = 2D Kramers’ points.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Kramers’ points
For some points {Γα}, α = 1, . . . ,N , called Kramers’ points, the BlochHamiltonian obey
H(Γα) = H(−Γα), ⇒ [Θ,H(Γα)] = 0. (47)
The coordinates of Kramers’ points are
Γα =1
2
(D∑
i=1
ni bi
), ni = 0, 1. (48)
For D-dimensional k-space, there are N = 2D Kramers’ points.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Kramers’ points
For some points {Γα}, α = 1, . . . ,N , called Kramers’ points, the BlochHamiltonian obey
H(Γα) = H(−Γα), ⇒ [Θ,H(Γα)] = 0. (47)
The coordinates of Kramers’ points are
Γα =1
2
(D∑
i=1
ni bi
), ni = 0, 1. (48)
For D-dimensional k-space, there are N = 2D Kramers’ points.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Time Reversal Symmetry and Kramers’ degeneracy theorem
Kramers’ points (cont’d)
Square BZHexagonal BZ
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Effective Brillouin Zone (EBZ)
From TRS, the Brillouin Zone is “reduced by half”, by an Effective BrillouinZone (EBZ).
H(k), k ∈ EBZ and ΘH(k)Θ−1, −k ∈ EBZ (49)
EBZ is topologically equivalent to cylinder (C 1 = S1 × [0, 1]).
EBZ is non-closed surface. It is need be closed for topological invariantscalculations. Problem for closing EBZ.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Effective Brillouin Zone (EBZ)
From TRS, the Brillouin Zone is “reduced by half”, by an Effective BrillouinZone (EBZ).
H(k), k ∈ EBZ and ΘH(k)Θ−1, −k ∈ EBZ (49)
EBZ is topologically equivalent to cylinder (C 1 = S1 × [0, 1]).
EBZ is non-closed surface. It is need be closed for topological invariantscalculations. Problem for closing EBZ.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Effective Brillouin Zone (EBZ)
From TRS, the Brillouin Zone is “reduced by half”, by an Effective BrillouinZone (EBZ).
H(k), k ∈ EBZ and ΘH(k)Θ−1, −k ∈ EBZ (49)
EBZ is topologically equivalent to cylinder (C 1 = S1 × [0, 1]).
EBZ is non-closed surface. It is need be closed for topological invariantscalculations. Problem for closing EBZ.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Contraction
Contraction
A contractiona is a function f (θ, λ), with θ ∈ [0, 2π) and λ ∈ [0, 1], such as f (θ, 0) isidentified with the EBZ boundary, and f (θ, 1) = Q (constant).
aJ.E. Moore and L. Balents, Phys. Rev. B 75, 121306 (R) (2007).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Contraction (cont’d)
Two contraction f1(θ, λ) and f2(θ, λ) is homotopic (topologically equivalent) tosphere.
The sphere has “hemispheres correspondence”.
The Chern number onto caps is an even number (2n).
The total Chern number on EBZ is definied up an even number.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Contraction (cont’d)
Two contraction f1(θ, λ) and f2(θ, λ) is homotopic (topologically equivalent) tosphere.
The sphere has “hemispheres correspondence”.
The Chern number onto caps is an even number (2n).
The total Chern number on EBZ is definied up an even number.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Contraction (cont’d)
Two contraction f1(θ, λ) and f2(θ, λ) is homotopic (topologically equivalent) tosphere.
The sphere has “hemispheres correspondence”.
The Chern number onto caps is an even number (2n).
The total Chern number on EBZ is definied up an even number.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Effective Brillouin Zone (EBZ)
Contraction (cont’d)
Two contraction f1(θ, λ) and f2(θ, λ) is homotopic (topologically equivalent) tosphere.
The sphere has “hemispheres correspondence”.
The Chern number onto caps is an even number (2n).
The total Chern number on EBZ is definied up an even number.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 topological invariant
Z2 topological invariant
Contracting the EBZ, we obtain S2EBZ ,
The Chern number integrated over entire S2EBZ (for all occupied states) is
C1 =1
2π
∫S2
EBZ
d2k · B (50)
The Z2 topological invariant is
ν =1
2π
[∫EBZ
d2k · B−∫∂(EBZ)
dk · A
]mod 2, (51)
where A =occ∑n
An and B =occ∑n
Bn.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 topological invariant
Z2 topological invariant
Contracting the EBZ, we obtain S2EBZ ,
The Chern number integrated over entire S2EBZ (for all occupied states) is
C1 =1
2π
∫S2
EBZ
d2k · B (50)
The Z2 topological invariant is
ν =1
2π
[∫EBZ
d2k · B−∫∂(EBZ)
dk · A
]mod 2, (51)
where A =occ∑n
An and B =occ∑n
Bn.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 topological invariant
Z2 topological invariant
Contracting the EBZ, we obtain S2EBZ ,
The Chern number integrated over entire S2EBZ (for all occupied states) is
C1 =1
2π
∫S2
EBZ
d2k · B (50)
The Z2 topological invariant is
ν =1
2π
[∫EBZ
d2k · B−∫∂(EBZ)
dk · A
]mod 2, (51)
where A =occ∑n
An and B =occ∑n
Bn.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Topological Invariants in 3D
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonians in 3D
Brillouin Zone in 3D
In 3D, the BZ is topologically equivalent to T 3 = S1 × S1 × S1 (hyper-torus).
The BZ is
T 3BZ =
k = x1
2b1 + y
1
2b2 + z
1
2b3
def= (x , y , z), where −1 ≤ x , y , z ≤ 1︸ ︷︷ ︸
entire BZ
(52)
With the Time-Reversal Symmetry (TRS), the BZ is reduced to EBZ.
EBZ =
(x , y , z);−1 ≤ x , y ≤ 1 and 0 ≤ z ≤ 1︸ ︷︷ ︸“Half” BZ
. (53)
There are 8 Kramers’ points in 3D: Γα, α = 1, . . . , 8.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonians in 3D
Brillouin Zone in 3D
In 3D, the BZ is topologically equivalent to T 3 = S1 × S1 × S1 (hyper-torus).
The BZ is
T 3BZ =
k = x1
2b1 + y
1
2b2 + z
1
2b3
def= (x , y , z), where −1 ≤ x , y , z ≤ 1︸ ︷︷ ︸
entire BZ
(52)
With the Time-Reversal Symmetry (TRS), the BZ is reduced to EBZ.
EBZ =
(x , y , z);−1 ≤ x , y ≤ 1 and 0 ≤ z ≤ 1︸ ︷︷ ︸“Half” BZ
. (53)
There are 8 Kramers’ points in 3D: Γα, α = 1, . . . , 8.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonians in 3D
Brillouin Zone in 3D
In 3D, the BZ is topologically equivalent to T 3 = S1 × S1 × S1 (hyper-torus).
The BZ is
T 3BZ =
k = x1
2b1 + y
1
2b2 + z
1
2b3
def= (x , y , z), where −1 ≤ x , y , z ≤ 1︸ ︷︷ ︸
entire BZ
(52)
With the Time-Reversal Symmetry (TRS), the BZ is reduced to EBZ.
EBZ =
(x , y , z);−1 ≤ x , y ≤ 1 and 0 ≤ z ≤ 1︸ ︷︷ ︸“Half” BZ
. (53)
There are 8 Kramers’ points in 3D: Γα, α = 1, . . . , 8.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonians in 3D
Brillouin Zone in 3D
In 3D, the BZ is topologically equivalent to T 3 = S1 × S1 × S1 (hyper-torus).
The BZ is
T 3BZ =
k = x1
2b1 + y
1
2b2 + z
1
2b3
def= (x , y , z), where −1 ≤ x , y , z ≤ 1︸ ︷︷ ︸
entire BZ
(52)
With the Time-Reversal Symmetry (TRS), the BZ is reduced to EBZ.
EBZ =
(x , y , z);−1 ≤ x , y ≤ 1 and 0 ≤ z ≤ 1︸ ︷︷ ︸“Half” BZ
. (53)
There are 8 Kramers’ points in 3D: Γα, α = 1, . . . , 8.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Bloch Hamiltonians in 3D
Brillouin Zone in 3D (cont’d)
Figure: EBZ in grey and Kramers’ points in red.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 ⊕ 3Z2 theory
6 planes T 2
The EBZ can be sliced in planes with T 2
topology.
There are 6 planes: x = 0, x = 1, y = 0,y = 1, z = 0 and z = 1.
For each plane, there are a topologicalinvariant Z2: x0, x1, y0, y1, z0 and z1.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 ⊕ 3Z2 theory
6 planes T 2
The EBZ can be sliced in planes with T 2
topology.
There are 6 planes: x = 0, x = 1, y = 0,y = 1, z = 0 and z = 1.
For each plane, there are a topologicalinvariant Z2: x0, x1, y0, y1, z0 and z1.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 ⊕ 3Z2 theory
6 planes T 2
The EBZ can be sliced in planes with T 2
topology.
There are 6 planes: x = 0, x = 1, y = 0,y = 1, z = 0 and z = 1.
For each plane, there are a topologicalinvariant Z2: x0, x1, y0, y1, z0 and z1.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 ⊕ 3Z2 theory
Constraints from TRS
For each Dirac monopole (in blue), there areanother Dirac monopole in the same plane byTRS. Thus,
z0 = z1 = 0. (54)
If I choose the EBZ as:{(x , y , z);−1 ≤ y , z ≤ 1 and 0 ≤ x ≤ 1}, then thetopological invariants are: y0, y1, z0 and z1.
There are always 4 independent topologicalinvariants Z2 in 3D. Denoted by ν0, ν1, ν2 and ν3.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 ⊕ 3Z2 theory
Constraints from TRS
For each Dirac monopole (in blue), there areanother Dirac monopole in the same plane byTRS. Thus,
z0 = z1 = 0. (54)
If I choose the EBZ as:{(x , y , z);−1 ≤ y , z ≤ 1 and 0 ≤ x ≤ 1}, then thetopological invariants are: y0, y1, z0 and z1.
There are always 4 independent topologicalinvariants Z2 in 3D. Denoted by ν0, ν1, ν2 and ν3.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Z2 ⊕ 3Z2 theory
Constraints from TRS
For each Dirac monopole (in blue), there areanother Dirac monopole in the same plane byTRS. Thus,
z0 = z1 = 0. (54)
If I choose the EBZ as:{(x , y , z);−1 ≤ y , z ≤ 1 and 0 ≤ x ≤ 1}, then thetopological invariants are: y0, y1, z0 and z1.
There are always 4 independent topologicalinvariants Z2 in 3D. Denoted by ν0, ν1, ν2 and ν3.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Strong and Weak TI
The topological invariant ν0 is defined for classify TI in strong and weak.
The topological phase is denoted by
(ν0; ν1ν2ν3) (55)
Stacking of several 2D TI is a weak TI.
Only genuine 3D TI are strong (ν0 = 1).
Strong TI are robust by disorder.
By the Bulk-edge correspondence, the topological invariant of bulk is equal tonumber of Dirac cone (mod 2) on surface.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Strong and Weak TI
The topological invariant ν0 is defined for classify TI in strong and weak.
The topological phase is denoted by
(ν0; ν1ν2ν3) (55)
Stacking of several 2D TI is a weak TI.
Only genuine 3D TI are strong (ν0 = 1).
Strong TI are robust by disorder.
By the Bulk-edge correspondence, the topological invariant of bulk is equal tonumber of Dirac cone (mod 2) on surface.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Strong and Weak TI
The topological invariant ν0 is defined for classify TI in strong and weak.
The topological phase is denoted by
(ν0; ν1ν2ν3) (55)
Stacking of several 2D TI is a weak TI.
Only genuine 3D TI are strong (ν0 = 1).
Strong TI are robust by disorder.
By the Bulk-edge correspondence, the topological invariant of bulk is equal tonumber of Dirac cone (mod 2) on surface.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Strong and Weak TI
The topological invariant ν0 is defined for classify TI in strong and weak.
The topological phase is denoted by
(ν0; ν1ν2ν3) (55)
Stacking of several 2D TI is a weak TI.
Only genuine 3D TI are strong (ν0 = 1).
Strong TI are robust by disorder.
By the Bulk-edge correspondence, the topological invariant of bulk is equal tonumber of Dirac cone (mod 2) on surface.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Strong and Weak TI
The topological invariant ν0 is defined for classify TI in strong and weak.
The topological phase is denoted by
(ν0; ν1ν2ν3) (55)
Stacking of several 2D TI is a weak TI.
Only genuine 3D TI are strong (ν0 = 1).
Strong TI are robust by disorder.
By the Bulk-edge correspondence, the topological invariant of bulk is equal tonumber of Dirac cone (mod 2) on surface.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Strong and Weak TI
The topological invariant ν0 is defined for classify TI in strong and weak.
The topological phase is denoted by
(ν0; ν1ν2ν3) (55)
Stacking of several 2D TI is a weak TI.
Only genuine 3D TI are strong (ν0 = 1).
Strong TI are robust by disorder.
By the Bulk-edge correspondence, the topological invariant of bulk is equal tonumber of Dirac cone (mod 2) on surface.
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
Examples
Theory
Ref: H. Zhang et al., Nature Phys. 5,438-442 (2009).
Experiment (ARPES)
Ref: Y. Xia et al., Nature Phys. 5,398-402 (2009).
Bloch’s Band theory Berry theory Dirac cone Z2 Band Topology Topological Invariants in 3D
Strong and Weak TI
The End