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Effective Field Theories of Topological Insulators
Eduardo FradkinDepartment of Physics and
Institute for Condensed Matter TheoryUniversity of Illinois
AtMa Chan, Shinsei Ryu, Taylor Hughes and EF, ArXiv:1210.4305
Talk at the Workshop“Entanglement Spectra in Complex Quantum Wave Functions”, Max
Planck Institute for the Physics of Complex Systems, Dresden, Germany, November 12-16, 2012
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Motivation
• To provide an effective field theory of topological phases
• We know how to do this for the fractional quantum Hall states (Wen)
• Hydrodynamic low energy theory for the charge degrees of freedom
• Distinction between topologically protected response and topological phase: when is a topological phase?
• Candidates: in 2D Chern-Simons; in 3D: “axion”; “BF” theory
• How are topological insulators related to topological phases?
• 2D vs 3D; U(1) vs Z2
• Fractionalization2
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• To provide an effective field theory of topological phases
• We know how to do this for the fractional quantum Hall states (Wen)
• Hydrodynamic low energy theory for the charge degrees of freedom
• Distinction between topologically protected response and topological phase: when is a topological phase?
• Candidates: in 2D Chern-Simons; in 3D: “axion”; “BF” theory
• How are topological insulators related to topological phases?
• 2D vs 3D; U(1) vs Z2
• Fractionalization2
Monday, November 12, 12
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Strategy
• We will use “functional bosonization” to derive the effective field theory
• It is equivalent to bosonization in D=1+1 as an operator identity for free gapless fermions
• For D>1+1 we will get an effective hydrodynamic field theory for gapped fermions
• Extension to interacting fermions and fractionalization
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Functional Bosonization
Bosonization of the fermion path integral
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Z[Aex
] =
RD
⇥¯ ,
⇤exp
�iSF [
¯ , , Aex
]
�
To compute current correlators
hjµ1(x1
)jµ2(x2
) · · · i =1i
�
�A
ex
µ1(x
1
)1i
�
�A
ex
µ2(x
2
)· · · lnZ[Aex]
Monday, November 12, 12
Use gauge invariance of the fermion path integral: shift Aex to Aex + a, where a is a gauge transformation:
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Z[Aex + a] = Z[Aex].
fµ⌫ [a] = 0
Z[Aex] =ZD[a]
pure
Z[Aex + a]
Z[A
ex
] =
ZD[a, b]Z[a]⇥ exp
⇣� i
2
Zd
Dx bµ⌫···✏
µ⌫···↵�(f↵� [a]� f↵� [A
ex
])
⌘
hjµ1(x1)jµ2(x2) · · · i = h✏µ1⌫1�1···@⌫1b�1···(x1)✏µ2⌫2�2···
@⌫2b�2···(x2) · · · i
j
µ(x) ⌘ ✏
µ⌫�⇢···@⌫b�⇢···(x) , @µj
µ = 0
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• This procedure is meaningful only if the effective action of the gauge field is local
• This works in 1+1 dimensions for gapless relativistic fermions
• For D>1+1 it works only as en effective action for low energy degrees of freedom if there is a finite energy gap
• This leads to a hydrodynamic description
• For systems with a Fermi surface one obtains the Landau theory of the Fermi liquid
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Example: Polyacetylene
• Fermions in d=1 with a spontaneously broken translation symmetry: broken chiral symmetry (Class AIII)
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✓ R
L
◆! ei�3✓
✓ R
L
◆⇢(x)! ⇢(x + a)) ✓ = kF a
Topological invariant ⌫ =✓(+1)� ✓(�1)
2⇡
Charge conjugation (particle-hole) ✓ = ⌫⇡ mod 2⇡
Z[A
ex
] =
ZD[a, b] exp
i
Zd
DxL!
L = �b✏µ⌫@µ(a⌫ �Aex
⌫ ) +✓
2⇡✏µ⌫@µa⌫ + · · ·
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D=2+1 Chern Insulator (Class A or D)
• Free fermions with broken time reversal invariance: integer quantum Hall states and the quantum anomalous Hall state
• The states are characterized by a topological invariant, the Chern number Ch
• In this case we obtain
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L = �bµ✏µ⌫�@⌫(a� �Aex
� ) +Ch
4⇡✏µ⌫�aµ@⌫a�.
•The first term is the BF Lagrangian•The hydrodynamic field b couples to flux tubes•The statistical gauge field a coupes to quasiparticle worldlines
Quantized Hall conductance �xy
= Che2
h
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3D Topological Insulator (Class AIII and DIII)
• Example: massive relativistic fermions with a conserved U(1) charge.
• This system has a topological invariant: the winding number
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L = �bµ⌫✏µ⌫�⇢@�(a⇢ �Aex
⇢ ) +✓
8⇡2
✏µ⌫�⇢@µa⌫@�a⇢ �1
4⇡2g2
@µa⌫@µa⌫ + · · ·
with ✓ = ⌫⇡ mod 2⇡
If time-reversal (particle-hole) is imposed, the topological class is Z2
The effective action for the external gauge field has an axion term (Qi, Hughes, Zhang, 2009)
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In D=4+1dimensions the effective field theory is a BF theory with a topological term for the field a whose coefficient is a topological invariant, the second Chern number Ch2
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L = �bµ⌫�✏µ⌫�⇢�@⇢(a� �Aex
� ) +Ch
2
24⇡2
✏µ⌫�⇢�aµ@⌫a�@⇢a� + · · ·
This theory can be mapped onto the result of Cho and Moore:
L =12⇡
✏µ⌫�⇢aµ@⌫b�⇢ +12⇡
✏µ⌫�⇢Aex
µ @⌫b�⇢ + C✏µ⌫�⇢@µa⌫@�Aex
⇢
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•Conclusion: all free fermion insulators (or adiabatically equivalent states) have a hydrodynamic effective field theory with the form of a BF action with or without a topological term. •In general the hydrodynamic field is a tensor (antisymmetric) field •The structure of the topological term depends on the dimension•For systems on closed manifolds this approach involves an average over boundary conditions
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Physical Picture
The effective field theory we found (without the topological term) has the same form as the topological field theory of a d=2 gapped superconductor (Hansson, Oganesyan and Sondhi)
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L = �2k
4⇡bµ✏µ⌫�@⌫(a� �Aex
� ) + · · ·
•Whereas in our case k=1, in the SC k=2. •For k=1 this is a trivial insulator• for k>1 this is a fractionalized state•In the case of the SC fractionalization follows from pairing: the Bogoliubov quasiparticles have spin but no charge
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Topological Insulator in 2+1 Dimensions(Class A or D)
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K =✓
0 �k�k Ch
◆
L = �2k
4⇡bµ✏µ⌫�@⌫(a� �Aex
� ) +Ch
4⇡✏µ⌫�aµ@⌫a�
=Kij
4⇡↵i
µ✏µ⌫�@⌫↵j� +
k
2⇡bµ✏µ⌫�@⌫Aex
�
(↵1µ, ↵2
µ) = (bµ, aµ)
• In general the effective field theory has the K-matrix form (as in the fractional quantum Hall fluids) (Wen and Zee) with a degeneracy k2 on the torus •For k=1 this is the theory of the QAH insulator (see Cho and Moore)• A theory with k>1 is a fractionalized topological Chern insulator•To derive this state one has to resort to flux attachment or parton constructions
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The effective action for the external gauge field in D=3+1 is
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S
e↵
[Aex] =1
8⇡
2
Zd
4
x ✓✏
µ⌫⇢�@µA
ex
⌫ @⇢Aex
�
If the bulk mass gap is allowed to have a chiral twist, one finds a current
jµ = ✏µ⌫�⇢@⌫b�⇢ =1
4⇡2
✏µ⌫�⇢@⌫
�✓@�Aex
⇢
�
If the chiral angle has a jump from 0 to π at the surface one obtains a surface Hall conductivity with 1/2 of the quantized value
�xy
=14⇡
=e2
2h
Electromagnetic Response
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Fractionalized Z2 Topological Insulator
• The fractionalized U(1) topological insulator in 2+1 is effectively equivalent of a FQH state
• Not much is known on a D=3+1 fractionalized Z2 insulator
• One can obtain an effective field theory using a parton construction (in 3+1 dimensions flux attachment does not work since braiding is not stable)
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L = �✏µ⌫�⇢bµ⌫@�
kX
I=1
aI⇢ +
✓
8⇡2
kX
I=1
✏µ⌫�⇢@µaI⌫@�aI
⇢ � e✏µ⌫�⇢@⌫b�⇢Aex
µ
Projection: aIµ ⌘ aµ ) L =
✓
8⇡2k✏µ⌫�⇢@µAex
⌫ @�Aex
⇢
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• This approach clearly works in all dimensions but the consequences depend on the dimensionality
• The edge excitations carry fractional charge and statistics
• In this case there is an extended duality (infinite modular group) (Fradkin and Kivelson, 1996) leading to a complex phase diagram (currently under study)
• The projection procedure can be regarded as a form of generalized “pairing” which by projecting our gauge degrees of freedom leads to fractionalization
• These ideas are known (from the case of the FQH states) to lead to non-abelian statistics in 2D
• In 3D this will also work for the edge states
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Conclusions and Outlook
• The “functional bosonization” approach leads to the determination of the structure of the effective field theory for topological phases
• It works in all dimensions (where these phases occur)
• It relies only on gauge invariance, the symmetries and locality (gap)
• It works for phases with both Z and Z2 topological invariants
• But, it does not tell us for which local Hamiltonians these phases occur
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