Crystalline topological semi-metals

Andreas P. Schnyder

IFIMAC Conference Series, Universidad Autonoma de Madrid, Spain

Max Planck Institute for Solid State Research, Stuttgart

May 31st, 2017

Crystalline topological semi-metals

Figure 3: current generated in gapped nodal loop system by electrical field in ˆey (left panel) and ˆez

(right panel) direction.

which is depicted in Fig. 3.

Electrical field can also be applied in ˆex or ˆey direction, the corresponding currents are dj

d = e²

~ ER

8⇡²B_k cos ˆe_z, E =Eeˆ_x; dj

d = e²

~ ER

8⇡²B_k sin ˆez, E = Eeˆy.

(9)

1.2.2 Gapped case

Next we calculate the current in the gapped case, j = e²

~

Z d³k

(2⇡)³⌦k⇥E

= e²

~

Z d³k (2⇡)³

B_k²Cz

3 k_kEeˆ ⇥eˆE

(10)

We consider angular di↵erential current, dj

d = e²

~

EB_k²Cz

(2⇡)³ ZZ

S

k_k²

3dk_kdkzeˆ ⇥eˆE (11)

From above discussion, the integrand approximates delta function, which suggests contribution around nodal loop dominates. By using k_k = _B^R

k + _2RB^⇢

k cos✓ and kz = _C^⇢

z sin✓, the integration becomes,

dj d = e²

~

EB_k²Cz

(2⇡)³ ZZ

S

(_B^R

k + _2RB^⇢

k cos✓)² (p

⇢²+ ²)³

⇢ 2RB_kCz

d⇢d✓eˆ ⇥eˆE

= ⌧ e²

~ ER 8⇡²B_k

✓

1 µ

◆ ˆ

e ⇥eˆE.

(12)

where the integration range is ⇢ 2 (0,p

µ^{2 2}), which comes from E 2 ( , µ). With this expres- sion, we can calculate the currents and conductance with E in di↵erent direction.

Below, we calculate conductance when electrical field are in ˆex, ˆey and ˆex directions.

3

Review article: Rev. Mod. Phys. 88, 035005 (2016)

Outline

1. Introduction: Topological band theory

- Classification of crystalline topological materials


2

4. Conclusions & Outlook

2. Topological nodal-line semimetals

- Ca3P2, Zr5Si3

Review article: Rev. Mod. Phys. 88, 035005 (2016)

4. Dirac line nodes w/ non-symmorphic symmetries

- CuBi2O4

Figure 3: current generated in gapped nodal loop system by electrical field in ˆey (left panel) and ˆez

(right panel) direction.

which is depicted in Fig. 3.

Electrical field can also be applied in ˆexor ˆeydirection, the corresponding currents are dj

d =e²

~ ER

8⇡²B_kcos ˆez, E=Eeˆx; dj

d = e²

~ ER

8⇡²B_ksin ˆez, E=Eeˆy.

(9)

1.2.2 Gapped case

Next we calculate the current in the gapped case, j=e²

~ Z d³k

(2⇡)³⌦k⇥E

=e²

~ Z d³k

(2⇡)³ B_k²Cz

3 k_kEeˆ ⇥eˆE

(10)

We consider angular di↵erential current, dj d =e²

~ EB_k²Cz

(2⇡)³ ZZ

S

k_k²

3dk_kdkzeˆ ⇥eˆE (11)

From above discussion, the integrand approximates delta function, which suggests contribution around nodal loop dominates. By using k_k = _B^R

k +_2RB^⇢

kcos✓ and kz = _C^⇢

zsin✓, the integration becomes,

dj d =e²

~ EB_k²Cz

(2⇡)³ ZZ

S

(_B^R

k+_2RB^⇢

kcos✓)² (p

⇢²+ ²)³

⇢ 2RB_kCz

d⇢d✓eˆ ⇥eˆ_E

=⌧ e²

~ ER 8⇡²B_k

✓

1 µ

◆ ˆ e ⇥eˆE.

(12)

where the integration range is⇢2(0,p

µ^{2 2}), which comes fromE2( , µ). With this expres- sion, we can calculate the currents and conductance withEin di↵erent direction.

Below, we calculate conductance when electrical field are in ˆex, ˆeyand ˆexdirections.

3

3 momentum space the (anti)-commutation relations can be al-

tered as the centers of the two crystalline symmetry operations are di ↵ erent. We first consider the operations M _x and PT in two di ↵ erent orders

( x , y , z) ^M !

^x

( x , y , z + 1 / 2) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z 1 / 2) (11) ( x , y , z) ^PT ! ( x + 1 / 2 , y + 1 / 2 , z)

M

_x

! ( x 1 / 2 , y + 1 / 2 , z + 1 / 2) (12)

Since in the x direction M _x and PT reflecting at di ↵ erent centers and M _x has an additional half lattice shifting, the commutation relation between these two symmetry opera- tions posses an extra momentum-dependent phase pt ^s m ^s _x = e ^{i( k}

^x

^{+ k}

^z

⁾ m ^s _x pt ^s . The relation for the entire symmetry oper- ation is given by PT M _x = e ^{i( k}

^x

^{+ k}

^z

⁾ M _x PT [11, 12]. In the reflection planes k _x = 0 , ⇡ , due to the glide along the z direction the eigenvalues M _x = ± e ^ik

^z

^{/ 2} . Since PT ² = 1 and PT H (k ) PT ¹ = H (k), the kramers the- orem leads to 2-fold degenerate states ( | (k) i , PT | (k) i ) at any momentum in Brillouin zone. As M _x | _± (k) i =

± e ^ik

^z

^{/ 2} | _± (k) i at the M _x -reflection plane, M _x PT | _± (k) i = e ^i(k

^x

^k

^z

⁾ PT M _x | _± (k) i = ⌥ e ^i(k

^x

^k

^z

^{/ 2)} PT | _± (k ) i . As k _x =

⇡ , M _x PT | _± (k) i = ± e ^ik

^z

^{/ 2} PT | _± (k) i . In this regard,

| _± (k) i , PT | _± (k) i in the same eigenspace of M _x lead to the presence of 4-fold degenerate nodal lines protected by M _x reflection-glide symmetry only in the k _x = ⇡ plane. As shown in fig. 3(a,b), the two two-fold degenerate bands cor- respond to two di ↵ erent M _x eigenvalues. The 4-fold degen- erate band crossing (nodal line) is robust under the M _x pro- tection since the band hybridization mixes the M _x eigenval- ues and breaks M _x symmetry. On the other hand, as k _x = 0, M _x PT | _± (k) i = ⌥ e ^ik

^z

^{/ 2} PT | _± (k) i the two degenerate states correspond to di ↵ erent M _x eigenvalues. If a 4-band cross- ing occurs in k _x = 0 plane, two pairs of the crossings can be gapped without breaking any symmetries since each pair is in the same eigenspace of M _x . Therefore, robust nodal lines are absent in k _x = 0 plane as a trivial case. Since in most of the physical systems, time-reversal inversion operator and reflection operator usually anticommutes with each other (if ‘i’ is recovered in the reflection operator for spin-1 / 2 sys- tems, the anti-commutation relation is changed to commuta- tion relation); hence, symmetry-protected nodal lines is com- monly absent. As the reflection has a di ↵ erent reflection cen- ter than inversion symmetry, the emerge of the commutation relation between M _x and PT protects nodal lines in URSX plane (k _x = 0). Bi ₂ CuO ₄ is the first concrete material realiza- tion (c.f. the early theoretical proposals [11,12]).

To show additional 4-fold degeneracy in UR, we further include M _z symmetry by considering the relation between M _x and M _z

( x , y , z) ^M !

^x

( x , y , z + 1 / 2) ^M !

^z

( x + 1 / 2 , y + 1 / 2 , z 1 / 2) (13) ( x , y , z) ^M !

^z

( x + 1 / 2 , y + 1 / 2 , z) ^M !

^x

( x 1 / 2 , y + 1 / 2 , z + 1 / 2)

(14)

Similarly, m _z ^s m ^s _x = e ^{i( k}

^x

^{+ k}

^z

⁾ m ^s _x m _z ^s leads to M _z M _x = e ^{i( k}

^x

^{+ k}

^z

⁾ M _x M _z . This additional phase stems from the

R S X

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

Energy (eV)

(a) (b)

(c) (d)

R S X

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

Energy (eV)

0 0

k

_y

(1/a) 2 k

_z

(1/c)

FIG. 3. (a) bulk energy spectrum on the reflection plane k _x = ⇡ . en- ergy scale is weird Blue(red) indicates 2-fold degenerate energy band corresponding reflection eigenvalue M _x = e ^ik

^z

^{/ 2} ( e ^ik

^z

^{/ 2} ). Yang- hao check. maybe it’s other way around. (b) two double nodal rings appear at di ↵ erent energy levels. Green(purple) represents M _x = e ^ik

^z

^{/ 2} ( e ^ik

^z

^{/ 2} ) Check M _x eigenvalue. R point in (b) is ? (c) (100) surface spectrum shows surface states connecting the pro- jected double nodal ring. zoom in for the surface states (d) In S R and UX , any two 2-fold degenerate bands merge in UR as a 4-fold degenerate band.

the glide property of these two nonsymmorphic symme- try operators. In UR of the BZ ( ⇡, k _y , ⇡ ), the additional phase vanish. As M _x | _± (k) i = ⌥ i | _± (k) i , { M _z , M _x } = 0 lead to the states M _z | _± (k) i and M _z PT | _± (k) i in dif- ferent M _x eigenspaces ( ± i) from the M _x eigenspaces ( ⌥ i) of | _± (k) i and PT | _± (k) i . Hence, these four orthogonal states | _± (k) i , PT | _± (k) i , M _z | _± (k) i , M _z PT | _± (k) i share the same energy due to the symmetries. Back to Bi ₂ CuO ₄ , first due to PT symmetry, any band in BZ is two-fold degeneracy and each band in UR is four-fold degeneracy, as shown in fig.

3(d), due to the interplay of PT , M _x , and M _z symmetries. This analysis leads to that for the (magnetic) space group having a subgroup of this magnetic space group #56 . 367, UR is always 4-degenerate. For example, BaP ₂ (HO ₂ ) ₄ preserves time rever- sal symmetry and belongs SG #56 and WO ₃ TaTe ₄ SG 130.

As this double nodal line is protected by reflection glide symmetry, it is natural to ask if the surface states should be robust under symmetry protection. Although as shown in fig.3(c) (100) surface spectrum the boundary of the sur- face states connect the projected double nodal ring, the sur- face state cannot be protected by magnetic space group sym- metries in the absence of spin S U (2) symmetry[13]. The main reason is M _x operation, which is o ↵ -centered of the unit cell, is always k-dependent in momentum space. Momentum- dependence reflection unable to quantize Berry phase; hence, the bulk cannot be described by well-defined topological in- variant so the surface states are unstable. In the following, we use 1D reflection-symmetric toy model (7) to show unquan-

3. Quantum anomalies in nodal-line semimetals

- Parity anomaly & anomalous transport

TI/TSC d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8

Reflection FS1 p=8 p=1 p=2 p=3 p=4 p=5 p=6 p=7

FS2 p=2 p=3 p=4 p=5 p=6 p=7 p=8 p=1

R A MZ 0 MZ 0 MZ 0 MZ 0

R+ AIII 0 MZ 0 MZ 0 MZ 0 MZ

R AIII MZ Z 0 MZ Z 0 MZ Z 0 MZ Z 0

R+,R++

AI MZ 0 0 0 2MZ 0 MZ2 MZ2

BDI MZ2 MZ 0 0 0 2MZ 0 MZ2

D MZ2 MZ2 MZ 0 0 0 2MZ 0

DIII 0 MZ2 MZ2 MZ 0 0 0 2MZ

AII 2MZ 0 MZ2 MZ2 MZ 0 0 0

CII 0 2MZ 0 MZ2 MZ2 MZ 0 0

C 0 0 2MZ 0 MZ2 MZ2 MZ 0

CI 0 0 0 2MZ 0 MZ2 MZ2 MZ

R,R

AI 0 0 2MZ 0 TZ2 Z2 MZ 0

BDI 0 0 0 2MZ 0 TZ2 Z2 MZ

D MZ 0 0 0 2MZ 0 TZ2 Z2

DIII Z2 MZ 0 0 0 2MZ 0 TZ2

AII TZ2 Z2 MZ 0 0 0 2MZ 0

CII 0 TZ2 Z2 MZ 0 0 0 2MZ

C 2MZ 0 TZ2 Z2 MZ 0 0 0

CI 0 2MZ 0 TZ2 Z2 MZ 0 0

R+ BDI, CII 2Z 0 2MZ 0 2Z 0 2MZ 0

R+ DIII, CI 2MZ 0 2Z 0 2MZ 0 2Z 0

R+ BDI MZ Z 0 0 0 2MZ 2Z 0 MZ2 Z2MZ2 Z2 R+ DIII MZ2 Z2MZ2 Z2MZ Z 0 0 0 2MZ 2Z 0 R+ CII 2MZ 2Z 0 MZ2 Z2MZ2 Z2MZ Z 0 0 0 R+ CI 0 0 2MZ 2Z 0 MZ2 Z2MZ2 Z2MZ Z 0 Tabelle I Classification of topological insulators and superconductors (“TI/TSC”) as well as of stable Fermi surfaces (“FS1” and ”FS2”) and nodal points/lines in 27 symmetry classes with reflection symmetry, in terms of the spatial dimensiondof topological insulators and superconductors, and the codimensionpof Fermi surfaces (nodal lines). “FS1” denote Fermi surfaces (nodal lines) which are within mirror planes and at high-symmetry points, whereas “FS2” denote those that are away from high-symmetry points.Z2,MZ2andTZ2invariants only protect Fermi surfaces of dimension zero (dFS= 0) at high-symmetry points of the Brillouin zone. For entries labeled withZ2,MZ2,TZ2, Fermi surfaces located within the mirror plane but away from high symmetry points cannot be protected by aZ2orMZ2topological invariant. Nevertheless, the system can exhibit gapless surface states that are protected by aZ2orMZ2topological invariant. For gapless topological materials the presence of translation symmetry is always assumed. Hence, there is no distinction betweenTZ2andZ2for gapless topological materials.

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ₂ = −i!

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i, γ_j} = 2δ_ij (7)

mean field

γ_E=0^† = γ_E=0 (8)

⇒ γ_k^†_,E = γ_−k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| mˆ (k) : mˆ (k) ∈ S² π₂(S²) = (16) H_BdG = (2t[cosk_x + cosk_y] − µ)τ_z + ∆₀ (τ_x sin k_x + τ_y sink_y) = m(k) · τ (17)

m_x m_y m_z (18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ₂ = −i!

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i,γ_j} = 2δ_ij (7)

mean field

γ_E^† ₌₀ = γ_E=0 (8)

⇒ γ_k^†_,E = γ_−k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| m(k) :ˆ m(k)ˆ ∈ S² π₂(S²) = (16) H_BdG = (2t [cosk_x + cosk_y] − µ)τ_z + ∆₀ (τ_x sin k_x + τ_y sin k_y) = m(k) · τ (17)

m_x m_y m_z (18)

Energy

gap

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ₂ = −i!

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i,γ_j} = 2δ_ij (7)

mean field

γ_E^† ₌₀ = γ_E=0 (8)

⇒ γ_k^†_,E = γ_−k,−E (9) Ξψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| mˆ (k) : mˆ (k) ∈ S² π₂(S²) = (16) H_BdG = (2t [cosk_x + cosk_y] − µ)τ_z + ∆₀ (τ_xsin k_x + τ_y sink_y) = m(k) · τ (17)

m_x m_y m_z (18)

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

we have

k_x k_y π/a − π/a (1)

majoranas

γ₁ = ψ + ψ^† (2)

γ₂ = −i !

ψ − ψ^†"

(3) and

ψ = γ₁ + iγ₂ (4)

ψ^† = γ₁ − iγ₂ (5)

and

γ_i² = 1 (6)

{γ_i, γ_j} = 2δ_ij (7)

mean field

γ_E=0^† = γ_E=0 (8)

⇒ γ_k^†_,E = γ_−k,−E (9) Ξ ψ₊k,+E = τ_xψ₋^∗k,−E (10) Ξ² = +1 Ξ = τ_xK (11)

τ_x =

#0 1

1 0

$

(12) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (13) weak vs strong

|µ| < 4t (14)

n = 1 (15)

Lattice BdG Hamiltonian ˆ

m(k) = m(k)

|m(k)| m(k) :ˆ m(k)ˆ ∈ S² π₂(S²) = (16) H_BdG = (2t[cos k_x + cosk_y] − µ)τ_z + ∆₀ (τ_x sin k_x + τ_y sin k_y) = m(k) · τ (17)

m_x m_y m_z (18)

Energy gap

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

homotopy

ν = # k_x (1)

∆^±_k = ∆_s ± ∆_t |dk| (2)

∆_s > ∆_t ∆_s ∼ ∆_t ν = ±1 for ∆_t > ∆_s (3) and

π₃[U(2)] = q(k) : ∈ U(2) (4)

Lattice BdG H_BdG

h(k) = ε^kσ₀ + αg^k · σ (5)

∆(k) = (∆_sσ₀ + ∆_tdk · σ) iσ_y (6)

h_ex I_y ≃ e

!

! k_F,−

k_F,+

dk_y

2π sgn

"

#

µ

H_ex^µ ρ^µ₁(0, k_y)

$%

− t sin k_y + λ

Lx/2

#

n=1

ρ^x_n(0, k_y) cos k_y

&

. (7) and

j_n,ky = −t sin k_y '

c^†_nky↑c_nky↑ + c^†_nky↓c_nky↓(

(8) + λ cos k_y '

c^†_nky↓c_nky↑ + c^†_nky↑c_nky↓(

(9) The contribution j_n,k⁽¹⁾_y corresponds to nearest-neighbor hopping, whereas j_n,k⁽²⁾_y is due to SOC. We calculate the expectation value of the edge current at zero temperature from the spectrum E_l,ky and the wavefunctions )

)ψ_l,ky*

of H_k⁽¹⁰⁾_y , I_y = −e

! 1 N_y

#

ky

Lx/2

#

n=1

#

l,E_l<0

⟨ψ_l,ky|j_n,ky|ψ_l,ky⟩ (10) We observe that the current operators presence of the superconducting gaps or the edge;

these only enter through the eigenstates |ψ_l,ky⟩.

Momentum dependent topological number:

∝

3

#

µ=1

H_ex^µ ρ^µ₁(E, k_y) ρ^x₁ (11) N_QPI(ω, q) = − 1

πIm +

#

k

G₀(k, ω)T (ω)G₀(k + q, ω) ,

∝ -

S⃗_f) )

) T(ω) ) )

)S⃗_i.

(12)

a (13)

ξ_k^± = ε^k ± α |(14)gk|

crystal momentum

Festk¨ orperphysik II, Musterl¨ osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

homotopy

ν = # k_x (1)

∆^±k = ∆_s ± ∆_t |d^k| (2)

∆_s > ∆_t ∆_s ∼ ∆_t ν = ±1 for ∆_t > ∆_s (3) and

π₃[U(2)] = q(k) : ∈ U(2) (4)

Lattice BdG H_BdG

h(k) = ε^kσ₀ + αgk · σ (5)

∆(k) = (∆_sσ₀ + ∆_td^k · σ) iσ_y (6)

h_ex I_y ≃ e

!

! k_F,−

k_F,+

dk_y

2π sgn

"

#

µ

H_ex^µ ρ^µ₁(0, k_y)

$%

− t sin k_y + λ

Lx/2

#

n=1

ρ^x_n(0, k_y) cos k_y

&

. (7) and

j_n,ky = −t sin k_y '

c^†_nky↑c_nky↑ + c^†_nky↓c_nky↓(

(8) + λ cos k_y '

c^†_nky↓c_nky↑ + c^†_nky↑c_nky↓(

(9) The contribution j_n,k⁽¹⁾_y corresponds to nearest-neighbor hopping, whereas j_n,k⁽²⁾_y is due to SOC. We calculate the expectation value of the edge current at zero temperature from the spectrum E_l,ky and the wavefunctions )

)ψ_l,ky*

of H_k⁽¹⁰⁾_y , I_y = −e

! 1 N_y

#

ky

Lx/2

#

n=1

#

l,E_l<0

⟨ψ_l,ky|j_n,ky|ψ_l,ky⟩ (10) We observe that the current operators presence of the superconducting gaps or the edge;

these only enter through the eigenstates |ψ_l,ky⟩.

Momentum dependent topological number:

∝

3

#

µ=1

H_ex^µ ρ^µ₁(E, k_y) ρ^x₁ (11) N_QPI(ω, q) = −1

π Im +

#

k

G₀(k, ω)T(ω)G₀(k + q, ω) ,

∝ -

S⃗_f) )

) T(ω) ) )

)S⃗_i.

(12)

a (13)

ξk^± = ε^k ± α |(14)g^k|

crystal momentum

Topological band theory

Festk¨orperphysik II, Musterl¨osung 11.

Prof. M. Sigrist, WS05/06 ETH Z¨urich

Bloch theorem

[T (R), H] = 0 |ψ_n⟩ = e^ikr |u_n(k)⟩ (1) (2) H(k) = e^−ikrHe^+ikr (3) (4) H(k) |u_n(k)⟩ = E_n(k) |u_n(k)⟩ (5) we have

H(k) k_x k_y π/a − π/a k ∈ Brillouin Zone (6) majoranas

γ₁ = ψ + ψ^† (7)

γ₂ = −i !

ψ − ψ^†"

(8) and

ψ = γ₁ + iγ₂ (9)

ψ^† = γ₁ − iγ₂ (10)

and

γ_i² = 1 (11)

{γ_i, γ_j} = 2δ_ij (12)

mean field

γ_E^† ₌₀ = γ_E=0 (13)

⇒ γk^†,E = γ₋k,−E (14) Ξ ψ₊k,+E = τ_xψ₋^∗ k,−E (15) Ξ² = +1 Ξ = τ_xK (16)

τ_x =

#0 1

1 0

$

(17) c^†c c^†c ⇒ ⟨c^†c^†⟩c c = ∆^∗c c (18) weak vs strong

|µ| < 4t (19)

n = 1 (20)

•

Consider band structure:

time-reversal symmetry, particle-hole, 
 reflection, inversion (parity)

. symmetries to consider:

. top. equivalence classes distinguished by: ⁿ_Z ^{= i} 2⇡

Z

F dk 2 Z

filled


states topological invariant

•

(i) Topological equivalence for insulators: