Gauge Dynamics and Topological Insulators - damtp

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Gauge Dynamics and. Topological Insulators. David Tong. Back in Swansea 2013. Based on arXiv:1305.2414 with Benjamin Béri and Kenny Wong ...
Gauge  Dynamics  and     Topological  Insulators   David  Tong     Back  in  Swansea   2013  

Based  on  arXiv:1305.2414     with  Benjamin  Béri  and  Kenny  Wong  

Story  1:  Massless  Fermions  

=E

R∝T Non-Abelian Magnetic Fields dt R∝T 2/3 R ∝ f SU(2)  Yang-­‐Mills  in  d=2+1  

oughout this paper we discuss2/3 to Dirac fermion R ∝ f SU (2) Yang-Mills coupled ∝ action T R ∝ f 2/3 k in d = 2 + 1 dimensions and start with R the � 3 ψ 1 / 1 + iψ¯2 Dψ / 2 S = − d x 1 2 Tr Fµν F µν + iψ¯2/3 1 Dψ 2e R∝f ψ1 ψ1 h Dirac fermions ψ1 and ψ2 transform in the fundamental representation ψ2 2) gauge symmetry. The parity anomaly prohibits an odd number of funda ψ2 the pair of fermions above is the minimum ions [10, 11], which means that ψ1 ψ2 Global  Symmetries   U (1)F

Our conventions: We use signature (−, +, +). The non-Abelian field strength is Fµν = a (1)1 +   e the SU (2) generators are TU = 2Fσ a . We chose+  the representation of gamma matric ψ U (1) 1 F _   σ 1 , σ 2 }. +  

ement gement

U (1)A ⊂ SU (2)F

U (1)A ⊂ SU (2)F 3

U (1) U (1) F A ⊂ SU (2)F

er we are interested in the properties of the theory in the w 2 avoid the flow toBackground   strong coupling, wetheory on aaU background possible . The admits (1)F ×SU (2)F fl MagnePc   Fturn ield   as athe doublet. eld which we choose to lie in Cartan subalgebra of the ga

Left to its own devices, the theory becomes s energiesB below it is expected to confi σ 3 wethisarescale, Fin12this = paper interested in the propertie 2 regime. To avoid the flow to strong coupling, w magneticthe fieldSU which choose togroup lie in theto CarU breaks (2)we gauge

� e2 , this magnetic field B is3 he theory remains weakly coupled. The goal of this paper F12 = σ Note   •  Preserves  Pme  reversal  T   2 ons of the theory in the background of this magnetic field.

•  Semi-­‐classical   limit    B � e2 , this magnetic field breaks the When     scale and the theory remains weakly coupled. T

oceeding, we pause to make simple ofpoint. Abelian magne theaexcitations the theory in the background reversal symmetry and charge conjugation: under both they Before proceeding, we pause to make a simpl However, for our non-Abelian magnetic field, Band→charge −B conju is s both time reversal symmetry B → −B.ofHowever, for our non-Abelian magn tion. (It is in the Weyl group the Cartan subalgebra)

conjugation, a fact which will be important in Section 4.

In the background the(2) U (1) photon remains massless. U (1)A(2.2), ⊂ SU F Dynamics   in  a  M agnePc   Field:     charged and form Landau levels. ThereSpin   is a 0 simple, intuitive wa energies of these Landau levels. Consider first a massless scalar a magnetic background B. The familiar Landau level quantizati levels 2

Massless  relaPvisPc  scalar:  

E = B(2n + 1)

2 Escalar = qB (2n + 1)

n = 0, 1, 2, . . .

scalar   knowledgement E2   The solutions for the fermions and W-bosons take the same form,

ingredient: a Zeeman splitting which splits the degeneracy between

hanks to Nick Dorey for many useful discussions. I’m supported by 2   The action (2.1) can be obtained by dimensionally reducing 3 + 1 dimen ty. Landau   levels  

erences

coupled to a single fundamental Dirac fermion Ψ, � 1 ¯ DΨ / S = − d4 x Tr Fµν F µν + iΨ 2 2e

Upon dimensional reduction, the Dirac spinor Ψ becomes ψ and ψ of (2.1).

1 2 J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. as opposed to a Weyl, spinor ensures that the theory is free from the Witten 4724 (1995) [arXiv:hep-th/9510017].

A

F

A

F

Dynamics  in  a  MagnePc  Field:  Spin  1/2  E 2 = B(2n + 1) E 2 = B(2n + 1)

E 2 = B(2n + 1) + Massless  relaPvisPc  fermion:  

scalar   Acknowledgement E2  

E 2 = B(2n + 1) + 2Bs fermion  

s=±

1 2

Acknowledgement My thanks to Nick Dorey for many useful discussions. I’m supported My thanks to Nick Dorey for many useful discu Landau     ociety. Society. levels  

References

ReferencesZeeman  spliXng  

[1] J. Polchinski, “Dirichlet-Branes and Ramond-Ra 1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. R 4724 (1995) [arXiv:hep-th/9510017]. 4724 (1995) [arXiv:hep-th/9510017]. Massless  modes!  



+

A

F

E 2 = B(2n + 1)

E 2 = B(2n + 1) Degeneracy   of  Lowest  Landau  Level   E22 = B(2n + 1) + 2Bs E =of  B(2n + 1) + 2Bs •  8  species   fermion  

•  Dirac  spinor  is  2  component;  SU(2)  gauge;  SU(2)  flavour  

1 •  Lowest  Landau   as  ±4  e1xcitaPons   ss h= =± •  All  others  have  8  e2xcitaPons  

2

B Bxσ 3 3 •  Pick  gauge   A Ay = y = 2 xσ 2 −iky

�� � � # ξ(k) 2 −B/4(x+2y/B) −B/4(x+2k/B)2

ψky) (x)∼= ee−iky ee ψk (x,







# 0



B dk B ∼ k  labels  degeneracy   expof  L−iky − x+ andau   l evel   2π 2π 4 6 � � � � dk B 4 B = exp +iky − x+ 2π 2π 4 4

E2  

2k B 2k B

�2 � �

�2 � �

ξi− (k) 0 0 ξi+ (k)





(1.1)

(1.2)

A

F

E 2 = B(2n 2 + 1) + 2Bs

E = B(2n + 1)

1 s=± Dynamics  in  a  MagnePc  Field:  Spin   1   2 E = B(2n2 + 1) + 2 2

E = B(2n + 1)

Massless  relaPvisPc  W-­‐boson:  

scalar   Acknowledgement E2  

My thanks to Nick Dorey for Landau     Society. levels  

References

s = −1, 0, +1

1 sB= ± Ay = xσ 3 2 2 E = B(2n + 1) + 2Bs 2 s = −1, 0, +1� ψkW-­‐boson   (x, y) ∼ e−iky e−B/4(x+2k/B)

fermion  

2

ξ

B 3 Ay = xσ 2� �



� B dk B 2 4 manyψ−useful discussions. I’m supported exp −iky − x+ i ∼ 2π 2π 4 B −iky −B/4(x+2k/B e e � ψ�k (x, y) ∼ � � dk B 2 4 B ψi+ = exp +iky − x+ 4 B � �2π 2π





B dk B ∼ exp 2 −iky − E = −B Phys. [1] J. Polchinski, “Dirichlet-Branes and Ramond-Ramond 2π 2π Charges,” 4 � 4724 (1995) [arXiv:hep-th/9510017]. � � Tachyonic  modes!   � dk B 4 B + ψi = exp +iky − 2π 2π 4 ψi−

4

What  Becomes  of  the  W-­‐Bosons?  

•  W-­‐bosons  are  tachyonic   •  They  condense  and  form  a  laXce  

Nielsen  and  Olesen;  Ambjorn  and  Olesen  

 

W-­‐Boson  LaXce   Plan:  Work  to  linear  order  

•  This  is  not  valid   •  It’s  easy  to  make  it  valid   •  But  it  doesn’t  mader   •  We  will  compute  topological  quanPPes  





4

B

ξi (k)

E 2 = −B

E 22 = −B W-­‐Boson  LaXce   E = −B



2 /2 −2πiny/d −B/2(x+2πn/Bd)2 −πin � 2= 2 W = −iW e e −πin /2 W −2πiny/d e−B/2(x+2πn/Bd) x y � 2 2 −iWy = W e −πin /2 e−2πiny/d e−B/2(x+2πn/Bd) −iWy = W e e n∈Z e

n∈Z n∈Z

wledgement 2

Bd 2 = 4π Bd = 4π

Square  laXce  

4π 4π =√ Bd = √3 3

Triangular  laXce  

nks to Nick Dorey for many useful discussions. I’m supported b Bd22

ences ent

ent

olchinski, many “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Re Dorey I’m supported supported by by the Royal Royal Dorey for for many useful useful discussions. discussions. I’m the

E 2 = −B

W-­‐Boson  LaXce  

Wx = −iWy = W

wledgement



e

−πin2 /2 −2πiny/d −B/2(x+2πn/Bd)2

e

e

n∈Z

W-­‐boson  winds  around  vortex  cores  

nks to Nick Dorey for many useful discussions. I’m supported b

ences

olchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Re

ψ2

4

2

U (1)F

� 2 t2he  Fermions?   What  Becomes   B >oef   v U (1)A ⊂ SU (2)−B/4(x+2k/B) 2 F −iky

ψk (x, y) ∼ e

e

ξ(k) 0



2

B(2n + 1) SUE(2)= → U (1) Need  to  do  (very)  degenerate   p erturbaPon   t heory   � � � � � �2 � � − ξi (k) B dk 2k 4 B 2 − E = B(2n + 1) + 2Bs ψi ∼ exp −iky − x+ 2π 2π 4 B 0 � � 1� �� � � � � (x)γs µ=W±µ (x)ψ � 2 ∆H = d2 x � Bψ1,k (x) + (1 ↔ 2) 1,k 0 dk B 2k 4 2 + � ψi = k,k exp +iky − x+ 2π 2π 4 B ξi+ (k) B

� Ay = xσ 3 ζ(p1 , p2 ) = e−2πinp22/Bd ψk=p1 /2+2πn/d � � n E 2 = −B

where   ψk (x, y) ∼ e

ψi− ∼

� 4





−iky −B/4(x+2k/B)2

e



B x = dk W −iWy = W7 2π B



2π dk





0

��



� 2 /2 −2πiny/d �2 2 − −πin −B/2(x+2πn/Bd)

B e n∈Z 4

exp −iky −

ξ(k)

B

x+



e2k B

2k

eξi (k)

�2 � �

0

0



(1.1)

SU 4 4 (2) → U 2 2(1)

SU (2) → U (1)

2 2 BB>>e2ev v2

�Becomes   of  the  Fermions?   � What   � � d2 x µ ∆H ψ1,k 2 = µ (x)γ Wµ (x)ψ1,k� (x) + (1 ↔ 2) ∆H = SU d (2) x→→UU(1) ψ (x)γ Wµ (x)ψ1,k� (x) + (1 ↔ 2) SU(2) (1)1,kk,k� k,k�

� Diagonalise  with     �� −2πinp2 /Bd � � ζ = e ψk=p1 /2+2πn/d (p ,p ) 1 2 22 µµ � ∆H = ψψ1,k (x) ++ (1(1 ↔↔ 2)2) � (x) = dd xx (x)γ WW µ (x)ψ 1,k1,k 1,k(x)γ µ (x)ψ � k,k k,k�

ζ(p1 , p2 ) =



n

e−2πinp2 /Bd ψk=p1 /2+2πn/d

p1 ∈ [0, 4π/d) � � −2πinp2 /Bd n ζζ(p(p1 ,p,p2 )) == ee−2πinp2 /Bd ψψ k=p 1 /2+2πn/d k=p 1 2 1 /2+2πn/d n n

pp1 ∈∈[0, [0,4π/d) 4π/d) 1 p2 ∈ [0, Bd) p2 ∈ [0, Bd) p� ∈ T2 2 p� ∈ T 7 7

p2 ∈ [0, Bd)

7

p� ∈ T2

This  is  the  (magnePc)  Brilliouin  zone  (BZ)  

7

And  something   nice   p� ∈ T2 happens….  

∆H =



d2 p ζ1† (�p) W (p1 , p2 ) ζ1 (�p) + (1 ↔ 2)

owledgement

nks to Nick

Physical  W-­‐boson  laXce  becomes     energy   over   BZ!   Dorey forfuncPon   many useful discussions.

I’m supported by

ences

Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev

Dirac  Cones  in  the  Brillouin  zone   Energy  

E

p1

B

p2

B

The  E e= v = d3 x im( C Fψ¯Berry Smass 1 ψ1 − ψ2 ψ2 ) 2π

BZ

ignificance of this choice lies in the observation that it preserves time reve SU (2) → U (1) � hat will prove to be important To seeCthis, note that in d = 2+1 dime σxylater. = ¯ is odd underfilled bands rmionic bilinear ψψ time reversal. However, we can define • � Preserves   � both  charge  conjugaPon  C    and  Pme  reversal  T     mwhich ust   ccompanied   ∆Hsymmetry = d2• x both   ψ1,k (x)γbµe  Wawe (x)ψ +y   (1 ↔ 2) exchange ψ1 and ψ2 . The op µ 1,k (x) b n of this in simultaneously k,k is invariant•  under new  SU symmetry (2)F → Uand (1)A it also preserves charge conjug Flavour  this symmetry   etails of the action � of time reversal invariance as well as charge conjugati −2πinp /Bd ζ(p1 , p2 ) = e ψk=p /2+2πn/d ded in Appendix D. n �



2

1

knowledgement

ethanks addition of the mass term breaks the SU (2)F flavour symmetry down to a to Nick Dorey for many useful discussions. I’m supported by the Roy 7 oup. ety. Under the pair of Abelian symmetries U (1)F ×U (1)A , ψ1 has charge (+

Now  we  have  an  insulator   Energy  

E

p1

B

p2

B

Again,  the  E e v � S = SY M − d x Tr Dµ φD φ − λ Tr φ − v 2 2 S = SY M − d3 x Tr Dµ φDµ φ −4 Tr φ2 − 2 4 2

•  SU   (2) → U (1) nowledgement

2 2 •  W-­‐bosons  tachyonic  for   B > e v

hanks to Nick Dorey for many useful discussions. I’m supported by the ement

•  Numerical  soluPons  found  (triangular  laXce  preferred)   (But  under   assumpPon  I’m of  existence   of  a  laXce   structure)   Acknowledgement ck Dorey for many•  useful discussions. supported by the Royal

y.

erences My thanks

•  SoluPon   is  cfor ertainly   only  useful meta-­‐stable   in  quantum  I’m theory   to Nick Dorey many discussions. supported by the •  Monopoles  destroy  magnePc  field  

Society. Polchinski, “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Let •  Is  soluPon  stable  classically?     724 (1995) [arXiv:hep-th/9510017]. “Dirichlet-Branes and Ramond-Ramond Charges,” Phys. Rev. Lett. 75,

References

rXiv:hep-th/9510017].