edge states of 3+1 bf model with boundarytheory.fi.infn.it/cappelli/sft2013/magnoli.pdf ·...

Topological insulatorsBF model in 3+1 D

The boundary of 3+1 D Bf modelConclusions

Edge states of 3+1 BF model with boundary

A. Amoretti, A. Blasi, A. Braggio, M. Carrega, G. Caruso, D.Ferraro, N. Maggiore, N.M.

May, 2013

A. Amoretti, A. Blasi, A. Braggio, M. Carrega, G. Caruso, D. Ferraro, N. Maggiore, N.M.Edge states of 3+1 BF model with boundary



Outline

I Genoa group research activity

I ADS-CFT

I Topological states of matter

I 3+1 BF model

I 3+1 BF model with boundary




Genoa group research activity

I ADS-CFT applications to condensed matter physics

I Topological field theories with boundary

I Fermionization of BF models with boundary




Applied AdS/CFT

Investigate strongly coupled quantum field theories viaclassical gravity

I growing list of applications:I hydrodynamics of quark gluon plasmaI holographic QCDI quantum critical systems

I strongly correlated electron systemsI cold atomic gases

I out of equilibrium dynamics




Bottom-up approach: Look for interesting behavior in simplemodels

I Assume that classical gravity in (asymptotically) AdSspacetime is dual to some strongly coupled QFT.

I Compute QFT correlators via AdS/CFT techniques.

I Add gauge and matter fields to gravity theory to describeinteresting physical models.




Holographic superconductors

Some simple gravity models describe quite well the behavior ofhigh Tc superconductors

I S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Phys. Rev.Lett. 101, (2008)

I S. Gubser and S. Pufu, JHEP 0811 (2008)




Coexistence of ferromagnetic and superconducting phases: canwe learn something from holography?




Quantum Spin Hall Effect

I no magneticfield

I strong spin-orbitcoupling

I two edges withoppositechirality and spin

I time reversalinvariance




3+1 D topological insulator

J. Moore, Nature 464,2010, experiment byHasan group

I observation of3D topologicalinsulator inBi2Se3

I surface helicalstates

I domain wallfermions inlattice gaugetheory




BF model in 3+1 D

I The Lagrangian is

S(3+1)BF =

k

2π

∫dxεµνρηFµνBρη

with Bρη a two-form gauge field.

The action is invariant under the symmetries:

δ(1)Aµ = −∂µθδ(2)Bµν = −(∂µφν − ∂νφµ).




BF model in 3+1 D

I Lorentz invariance is broken by the introduction of a planarboundary. Gauge conditions:

A3 = 0

Bi3 = 0,

where latin letters run over 0,1,2. Gauge fixing action:

Sgf =

∫d4x{bA3 + d iBi3},

where b and d i are respectively the Lagrange multipliers forthe fields A3 and Bi3.




BF model in 3+1 D

I Adding the sources the Action becomes:

S =

∫d4x{κεijk [2∂iAjBk3 + (∂iA3 − ∂3Ai )Bjk ]+

bA3 + d iBi3+ + J ijBijBij + 2J i3Bi3

Bi3 + J iAiAi+

JA3A3 + Jbb + J ididi},




The boundary Action

I The most general (quadratic) boundary Lagrangian is:

Lbd = δ(x3)[a1Ai B̃

i+a2m

2AiA

i+a3b+a4

2εijk∂iAjAk+a5diA

i],

where B̃ i ≡ εijkBjk




The equations of motion

I The equations of motion in the presence of the boundary:

J iAi+ 2κεijk∂jBk3 + α∂3B̃

i = −δ(x3)[a1B̃i + a2mAi + a3b+

+ a4εijk(∂jAk) + a5d

i ]

εijkJB̃k + κεijk(∂kA3 − ∂3Ak) = −a1δ(x3)εijkAk

J3 + b − κ∂i B̃ i = 0

2J i3Bi3+ d i + 2κεijk∂jAk = 0

A3 + Jb = −δ(x3)a3

B i3 + J id = −δ(x3)a5Ai ,

(1)




The Ward identities

I The Ward identities in the presence of the boundary: Theboundary term Lbd breaks the local Ward identities:

∂iJiAi

+ ∂3J3A3

+ ∂3b = −δ(x3)[a1∂i B̃i + a2m∂iA

i + a5∂idi ],

εijk∂jJB̃k + ∂3Ji3Bi3

+1

2∂3d

i = −δ(x3)a1εijk∂jAk .




The boundary conditions

I The boundary conditions can be found integrating the EOMin a small interval near the boundary:

(κ+ a1)B̃ i = −a2mAi − a3b − a4εijk∂jAk − a5d

i

(κ− a1)Ai = 0

a3 = 0

a5Ai = 0.




The Ward identities

I Using the boundary conditions the Ward identities can bewritten: ∫ ∞

−∞dx3∂iJ

iAi = κ∂i B̃

i∫ ∞−∞

dx3εijk∂jJB̃k = −κεijk∂jAk .




Algebra of observables

I Using the Ward identities we get the following algebra:

[B̃0(X ),Aα(X ′)]t=t′ = ∂αδ(2)(X ′ − X )

[B̃0(X ), B̃0(X ′)]t=t′ = 0

[Aα(X ),Aβ(X ′)]t=t′ = 0,




Physics on the boundary-1

I Boundary degrees of freedom:

∂i B̃i = 0 ⇒ B̃ i = εijk∂jζk

εijk∂jAk = 0 ⇒ Ak = ∂kΛ,

I Dualityεijk∂jζk = ∂ iΛ.

I Fermionic degrees of freedom?




Physics on the boundary 2

I Canonical commutation relations[Λ(X ),Π(Λ)(X ′)

]t=t′

= δ(2)(X − X ′)[ζ̃α(X ),Π(ζ)β(X ′)

]t=t′

= δαβ δ(2)(X − X ′),

Π(Λ) ≡ εαβ∂αζβ and Π(ζ)α ≡ ∂αΛ are the conjugate momenta

of the fields Λ and ζ̃α ≡ εαβζβ.

I The boundary algebra can be interpreted as a set of canonicalcommutation relations.





I 3D Lagrangian

L =∑

ΠΦ̇− H

= εαβ∂αζβ∂tΛ + ∂αΛεαβ∂tζβ − (εαβ∂αζβ)2 − (∂αΛ)2

I Fixed by commutation relations and duality.





I two conserved currents in the bulk:

Jµ = δLδAµ

= ∂νδL

δ∂νAµ= εµνρσ∂νBρσ

Sµν = δLδBµν

= εµνρσ∂ρAσ.

we interpret the edge excitations as a deformation of theboundary caused by the bulk currents flowing towards theedge. We parametrize the deformation by h(X ) and we

represent the edge currents Ji

and Sij

as:

Ji

=∫ h−x3

0dx3J i (2)

Sij

=∫ h−x3

0dx3S ij (3)

i = 0, 1, 2 (4)

where x30 is an auxiliary boundary, where the bulk and edge

currents match.A. Amoretti, A. Blasi, A. Braggio, M. Carrega, G. Caruso, D. Ferraro, N. Maggiore, N.M.Edge states of 3+1 BF model with boundary




I Currents and fields on the boundary:

Ji

= −(h + x30 )εijkBjk(X , 0)

Sij

= (h + x30 )εijlAl(X , 0).

I Al(X , 0) = ∂lΛ(X ) and B̃ i (X , 0) = εijk∂jζk(X ).

I From duality:

Ji

= −1

2εijkSjk

Ji

charge density current, Sij

spin density current.Relation between charge density current and spin densitycurrent which occurs on the surface of a (3+1)D TopologicalInsulator (Zhang).




Conclusions

I The abelian BF model appears as a possible candidate as aneffective theory for topological insulators.

I At the boundary we find helical surface states

I BF models in any dimensions give duality relations on theboundary.

I Fermions from bosons (tomographic representation).


edge states of 3+1 bf model with boundarytheory.fi.infn.it/cappelli/sft2013/magnoli.pdf ·...

Documents