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
2π
2π
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].
