Graphene, Index Theorem and Graphene, Index Theorem and Topological - - PowerPoint PPT Presentation

Graphene, Index Theorem and Graphene, Index Theorem and Topological Degeneracy Topological Degeneracy

JKP Michael Stone

Dresden, 24 September 2006 cond-mat/0607394

Quantum Information, Physics and Topology

• Encoding and manipulating QI in small physical systems is pledged by

decoherence and control errors.

• Error correction can be employed to resolve this problem by using a (huge)
• verhead of qubits and quantum gates.
• An alternative method is to employ intrinsically error protected systems such

as topological ones => properties are described by integer numbers! protected by macroscopic properties: hard to destroy.

• E.g. you can employ system with degenerate ground states:

– Make sure degeneracy is protected by topological properties (V) – Make sure degenerate states are locally indistinguishable (X) – Encode information in these degenerate levels TOPOLOGICAL DEGENERACY

Overview

• Graphene: two dimensional layer of graphite –hexagonal lattice of C atoms

– Fullerene: C60, C70 – Nanotubes

• Conducting properties of these materials: zero energy modes.
• Can be used as miniaturized elements of circuits.

g=0 g=1 g=2

• Zero modes provide degeneracy of ground state: G zero modes deg.

Topological quantum computation

• Kitaev’s toric code
• Honeycomb lattice (same as graphene, but with “real” fermions)

G

2 ⇒

• Conductivity can depend on topology.

Euler characteristic

• Index theorem (Atiyah-Singer)

– Smooth, orientable, compact, Riemannian manifolds, M, with genus, g. – Define elliptic operator D on M. Includes curvature and gauge fields. – The index theorem relates the number of zero energy modes of D with g.

Different geometries of Graphene

Fullerene (C60): Nanotubes:

Graphene: structure

The Hamiltonian of graphene is given by

) ( cos 2 cos 2 cos 2 3 ) ( ) 1 ( ) 1 ( ) ( 2

, ,

v u k v k u k t k E e e t e e t H a b b a t a a t H

v k i u k i v k i u k i k j i j i j i j i j i

r r r r r r r r

r r r r r r r r r

− ⋅ + ⋅ + ⋅ + ± = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − + + − = + − = − =

⋅ ⋅ ⋅ − ⋅ − > < + + > < +

∑ ∑

i

a

t

u r

v r

fermionic modes E(kx,ky): Fourier transformation: Fermi points: E(k)=0 A B

Graphene: structure

p t ip p ip p t H v u k v k u k t k E

y x y x p

r r r r r r r r r r

r

⋅ ± = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ± ≈ − ⋅ + ⋅ + ⋅ + ± = σ 2 3 2 3 ) ( cos 2 cos 2 cos 2 3 ) (

Linearise energy around a conical point,

) (k E r

Relativistic Dirac equation at the tip of a pencil! Two types of spinors:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

− − + +

B K A K B K A K , , , , ,

p K k r r r + =

K+ K-

±

K

are the Fermi points and A and B are the two triangular sub-lattices

z

σ

Note: rotation maps to states with the same energy, but opposite momenta

Graphene: curvature

To introduce curvature: cut sector and reconnect sites. This creates a single pentagon with no other deformations present. Results in a conical configuration. To preserve continuity of the spinor field when circulating the pentagon one can introduce two additional fields:

• Spin connection Q:
• Non-abelian gauge field, A:

Resulting 4x4 Dirac equation can be decoupled by simple rotation to a pair of 2x2 Dirac equations (k=1,2):

3 / π

z

dx Q σ π

µ µ

6 − =

∫

y

dx A τ π

µ µ

2 − =

∫

Mixes + and – spinors

k k a k a a

E iA iQ p e t ψ ψ σ

µ µ µ µ µ

= − −

∑

,

) ( 2 3 2 π

µ µ

± =

∫

dx Ak

Mixes A and B components elliptic operator

Index Theorem

Consider finite matrices,

+ − − +

⎯→ ⎯ ⎯→ ⎯

+

V V V V

M M

,

+

M M,

Mu Mu MM u Mu M λ λ λ = ⇒ = ≠

+ +

) ( , For

Define operator: with eigenvalues +1, -1 for Consider the dimension of the null subspace of Then Non-zero eigenvalues cancel in pairs. Expression is t independent.

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 1 1

5

γ

− + ν

ν ,

− + V

V ,

− + V

V ,

− + ⊕

= V V V

Define

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

+ + +

MM M M Q M M Q ,

2

) ( index ) (

) ( ) ( 5

2 2 2

Q e e e Tr

MM Sp t M M Sp t tQ

≡ − = − =

− + − − −

∑ ∑

+ +

ν ν γ

λ λ

non-zero modes come in pairs Same number of zero modes as Q

Index Theorem

Consider 2-dimensional Dirac operator defined over a compact surface

D Q =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ∂ + − ∂ − ∂ − − ∂ = + = ) ( ) ( ) ( ) (

y y x x y y x x y y x x

iA i iA iA i iA p p D σ σ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ∇ − − ∇ − = − −∇ =

xy xy xy z

F F F D

2 2 2 2

σ

One can show that

...] ) 2 1 3 1 ( 1 [ 4 1

2 2 2 ) (

2

+ ∇ + + + =

′ = + −∇ −

t F F t F t e

xy z xy xy z r r F t

xy z

σ σ π

σ

That gives

∫

=

− −∇ −

x d F e Tr

xy F t z

xy z

2 ) (

2 1 ] [

2

π σ

σ

Higher order terms in t have to be checked...

Index Theorem

) ( index ) (

) ( ) ( 5

2 2 2

Q e e e Tr

MM Sp t M M Sp t tQ

≡ − = − =

− + − − −

∑ ∑

+ +

ν ν γ

λ λ

∫

=

− −∇ −

x d F e Tr

xy F t z

xy z

2 ) (

2 1 ] [

2

π σ

σ

Assume it holds for infinite dimensions (continuum). If D is defined on compact manifold then RHS is an integer (topological invariant). Open boundary conditions can give a discrepancy caused by boundary terms. Thus, the number of zero modes depends on the gauge field configuration. Continuous deformations of the gauge field will not change the number of zero modes. Including surface curvature does not change the above result (only in 2-dims).

Index Theorem

The Index theorem states:

∫∫

= − =

− +

F D π ν ν 2 1 ) ( index

integer! The integral is taken over the whole compact surface. F: field strength of gauge vector potential, A. For compact manifolds the term on the r. h. s. is an integer. It is a topological number: small deformations does not change its value. From this theorem you can obtain the least number of zero modes. The exact number is obtained if or is equal to zero.

+

ν

−

ν

[Atiyah and Singer, Ann. of Math. 87, 485 (1968);...]

Index Theorem: Euler characteristic

∫∫

= − = + − = R g F E V π χ 2 1 ) 1 ( 2

Euler characteristic for lattices on compact surfaces: Consider folding of graphene in a compact manifold. The minimal violation is

• btained by insertion of pentagons or heptagons that contribute positive or

negative curvature respectively. Consider – number of pentagons – number of hexagons – number of heptagons

5

n

6

n

7

n

7 6 5 7 6 5 7 6 5

2 / ) 7 6 5 ( 3 / ) 7 6 5 ( n n n F n n n E n n n V + + = + + = + + =

From the Euler characteristic formula:

) 1 ( 12

7 5

g n n − = −

Fullerenes: “Nanotubes”:

12

5 =

⇒ = n g 1

7 5

= − ⇒ = n n g

Index Theorem: Graphene application

∫∫

= − =

− +

F D π ν ν 2 1 ) ( index

∫ ∫∫

= A F

Stokes’s theorem

) 1 ( 3 ) ( 2 2 1

7 5

g n n − ± = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛± π π

Thus, one obtains:

⎩ ⎨ ⎧ = − − = − = −

− +

2 for ), 1 ( 3 1 for ), 1 ( 3 k g k g ν ν

Least number of zero modes:

| 1 | 6 g −

Index Theorem: Graphene application

| 1 | 6 ) ( index g D − = − =

− +

ν ν

C60: g=0 Nanotubes: g=1 Zero mode pairs No zero modes

[J. Gonzalez et al. Phys. Rev. Lett. 69, 172 (1992)]

Ultra-cold Fermi atoms and optical lattices

Single species ultra cold Fermi atoms superposed by

• ptical lattices that form a hexagonal lattice.
• Very low temperatures: T~0.1TF
• Arbitrary filling factors: e.g. 1/2

See dependence of conductivity on disorder, impurities and lattice defects: e.g. insert pentagons at the edge of the lattice of effect of empty sites. Similar index theorem can be devised for open boundary conditions. Measurement of conductivity in Fermi lattices has already been performed in the laboratory:

t

[Duan et al. Phys. Rev. Lett. 91, 090402 (2003)] [Ott et al. Phys. Rev. Lett. 92, 160601 (2004)]

Conclusions

• Index Theorem for compactified graphene sheets.
• Agrees well with known models of fullerenes and nanotubes.
• Gives conductivity properties for higher genus models:

sideways connected nanotubes.

• Predicts stability of spectrum under small deformations.
• Relate to topological models:

– obtain topologically related degeneracy: – encode and manipulate quantum information. – apply reverse engineering to find new models with specific degeneracy properties.

• Related experiments with ultra-cold Fermi atoms can give insight to the

properties of graphene. May be easier to implement than solid state setup.

| 1 | 6

2

g −

Thank you for your attention! [cond-mat/0607394]

Postdoc/PhD positions at University of Leeds

• n Topological QC

available now! Email: jp370@damtp.ac.uk