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The Geometry of theQuantum Hall EffectDam Thanh Son (University of Chicago)
7th Crete Regional Meeting in String Theory
Refs:
DTS arXiv:1306.0638ongoing work with Siavash Golkar and Dung NguyenCarlos Hoyos, DTS arXiv:1109.2651DTS, M.Wingate cond-mat/0509786
Plan
• Review of quantum Hall physics
• Summary of results
• Nonrelativistic diffeomorphism
• Construction of the action
Fractional quantum Hall state
n=1
n=2
n=3
Huge ground state degeneracy without interactions
FQHE: interactions lift degeneracy
exp: the system is gapped at some values of filling fraction
⌫ =n
B/2⇡⌫ =
1
3,1
5, · · ·
Laughlin’s filling factors
Laughlin’s wave function
• In the symmetric gauge LLL states are
(z) = f(z)e�|z2|/4`2
(z) =Y
hiji
(zi � zj)3Y
i
e�|zi|2/4`2
Laughlin’s guess for the ground state wave fn ⌫ = 1/3
Not exact, although seems to be very good approximationimplies equal-time correlators, not at unequal times
Effective field theory
• Effective field theory: captures low-energy dynamics
• What are the low-energy degrees of freedom of a quantum Hall state?
• there are none (in the bulk): energy gap
• Thus the effective Lagrangian is polynomial over external fields and derivatives (generating functional)
Chern-Simons action
• To lowest order in derivatives:
S =�
4�
�d3x �µ��Aµ��A�
encodes Hall conductivity
�xy =�
2�
e2
�
Jµ =�S
�Aµ
Jy
= �xy
Ex
⇢ =⌫
2⇡B
Another formulation of CS theory
is the current
at the same time is the Lagrange multiplier enforcing
aµ = Aµ + @µ'
L =⌫
4⇡✏µ⌫�aµ@⌫a� � jµ(@µ'�Aµ + aµ)
jµ =�S
�Aµ
Note: is the phase of the condensate of composite bosons (ZHK)'
Universality beyond CS
Higher-derivatives corrections: of dynamical, not topological nature, hence non universal?
But there is universality beyond the CS action
Hall viscosityFollowing the evolution of the QH state with changing metric
hij = hij(t), deth = 1
2-dim space
nonzero Berry curvature
hT11 � T22i ⇠ ⌘Ah12
h11 � h22
h12
However in CS theory Tµ⌫ = 0
universal (not renormalizedby interactions)
Symmetries of NR theory
Microscopic theory
Dµ� � (�µ � iAµ)�
Invariance under time-independent diff ξ=ξ(x):
DTS, M.Wingate 2006
�Ai = ��k�kAi �Ak�i�k
�� = ��k�k�
�A0 = �⇠k@kA0
�hij = �ri⇠j �rj⇠i
S =
Zd
3x
ph
i
2
†$Dt � h
ij
2mDi
†Dj +
g
4m
F12ph
†
�
NR diffeomorphism
• These transformations can be generalized to be time-dependent: ξ=ξ(t,x)
Galilean transformations: special case ξi=vit
�� = ��k�k�
�A0 = �⇠k@kA0 �Ak ⇠k +
g
4"ij@i(hjk ⇠
k)
�Ai = ��k�kAi �Ak�i�k �mhik ⇠
k
�hij = �ri⇠j �rj⇠i
g = 0 version can be understood as NR reduction of relativistic diffeomorphism invariance
NR reductionStart with complex scalar field
Take nonrelativistic limit:
� = e�imcx0 ��2mc
S = �Zdx
p�g(gµ⌫Dµ�
⇤D⌫�+m
2�
⇤�)
gµ⌫ =
0
BB@
�1 +2↵0
mc2↵j
mc
↵i
mchij
1
CCA
Dµ� = (@µ � iAµ)�
S =
Zd
3x
ph
i
2
†$Dt � h
ij
2mDi
†Dj
�
Dµ = @µ � i(Aµ + ↵µ)
Relativistic diffeomorphism
: gauge transform
: general coordinate transformations
� = e�imcx0 ��2mc⇠0
⇠i
gµ⌫ =
0
BB@
�1 +2↵0
mc2↵j
mc
↵i
mchij
1
CCA
under diff Aµ = Aµ + ↵µ
�A0 = �⇠k@kA0
�Ai = ��k�kAi �Ak�i�k �mhik ⇠
k
�Ak ⇠k
Interactions
• Interactions can be introduced that preserve nonrelativistic diffeomorphism
• interactions mediated by fields
• For example, Yukawa interactions
�� = �⇠k@k�
S = S0 +
Zd
3x
ph�
† +
Zd
3x
ph(hij
@i�@j�+M
2�)
Is CS action invariant?
• CS action is gauge invariant, Galilei invariant
• but not diffeomorphism invariant
does not transform like a one-form Aµ
�SCS =⌫m
2⇡
Zd
3x ✏
ijEihjk ⇠
k g = 0
�A0 = �⇠k@kA0 �Ak ⇠k +
g
4"ij@i(hjk ⇠
k)
�Ai = ��k�kAi �Ak�i�k �mhik ⇠
k
�Aµ = �⇠k@kAµ �Ak@µ⇠k
Aµ = Aµ + ↵µ
Is CS action invariant?
• CS action is gauge invariant, Galilei invariant
• but not diffeomorphism invariant
does not transform like a one-form Aµ
�SCS =⌫m
2⇡
Zd
3x ✏
ijEihjk ⇠
k g = 0
�A0 = �⇠k@kA0 �Ak ⇠k +
g
4"ij@i(hjk ⇠
k)
�Ai = ��k�kAi �Ak�i�k �mhik ⇠
k
�Aµ = �⇠k@kAµ �Ak@µ⇠k
Aµ = Aµ + ↵µ
Requirements for EFT
• Respect general coordinate invariance
• Reproduce all topological properties of the quantum Hall state
• Have regular limit of
LLL degenerate with zero energy for any metric and B(Aharonov-Casher)
m ! 0, g = 2
Sg[A0, Ai, hij ]
What kind of geometry
• System does not live in a 3D Riemann space
• 2D Riemann manifold at any time slice
• can parallel transport along equal-time slices, but not between different times
Newton-Cartan geometry
dn = 0 ) n = dt
nµvµ = 1
dn = 0
choose t to be time coordinate
(hµ⌫ , nµ, vµ) hµ⌫n⌫ = 0
hµ⌫h⌫� = �µ� � vµn� hµ⌫v⌫ = 0
hµ⌫ =
✓0 00 hij
◆hµ⌫ =
✓v2 �vj�vi hij
◆
Connection in Newton-Cartan geometry
rµn⌫ = 0r�hµ⌫ = 0 h↵[µr⌫]v
↵ = 0
��µ⌫ = v�@(µn⌫) +
1
2h�⇢(@µh⇢⌫ + @⌫h⇢µ � @⇢hµ⌫)
Higher-dimensional interpretation
x
M = (x�, x
µ)
gMN =
✓0 n⌫
nµ hµ⌫
◆gMN =
✓0 v⌫
vµ hµ⌫
◆
��µ⌫ = v�@(µn⌫) +
1
2h�⇢(@µh⇢⌫ + @⌫h⇢µ � @⇢hµ⌫)
�LMN =
1
2gLR(@MgRN + @NgRM � @RgMN )
Improved gauge potentials
• With v one can construct a gauge potential that transforms as a one-form
Ai = Ai +mvi
�Aµ = �⇠k@kAµ � Ak@µ⇠k
What is v? Should be dynamically determined
A0 = A0 �mv2
2� g
4"ij@ivj
Effective field theory
related to “shift”
Dµ' = @µ'� Aµ + aµ�s!µ
Integrating out ⇢, vi, aµ
effective action for ) Aµ, hij
S =⌫
4⇡
Zd
3x ✏
µ⌫�aµ@⌫a� �
Zd
3x
ph ⇢v
µDµ'
+S0[⇢, vi, hij ]+
Zd
3x
g � 2
8m✏
µ⌫�nµF⌫�
Spin connection
e1
e2
!µ =1
2✏abea⌫rµe
b⌫
=1
2✏abeai@te
bi +
1
2✏ij@ivj
!0 =1
2✏abeair0e
bi
nonzeroin flat space�1�2 � �2�1 =
12�
g R
Wen-Zee shift�
2��µ��Aµ���� =
�
4�
�g A0R + · · ·
Q =�
d2x�
g j0 =�
d2x�
g� �
2�B +
�
4�R
�= �N� + ��
Total particle number on a manifold:
IQH states:
Laughlin’s states:
On a sphere: Q = �(N� + S), S =2�
�
‘shift’S = ⌫
S = 1/⌫
mixed CS
Physical consequences
• Kohn’s theorem
• Hall viscosity
• Hall conductivity: universal to order q^2
• Structure factor
Kohn’s theorem
Constant B, Response to homogeneous, time dependent E(t) independent of interactions: motion of center of mass
mvi = Ei + ✏ijvjB
Hall viscosity from WZ term
SWZ = � �B
16��ijhik�thjk + · · ·
derived by N.Read previously
hTxy
(Txx
� Tyy
)i ⇠ i⌘a!
⌘a =1
4S⇢
Avron, Seiler, Zograf
σxy(q)y
v
E
E
vx
Ex = E eiqx
jy = �xy(q)Ex
�xy(q)�xy(0)
= 1 + C2(q�)2 +O(q4�4)
C2 =S4� 1
correct for ν=1 state
Structure factor and shift
• Non-universal part of the action: leading contributions are
�zzF (i@t)�zz
positivity of spectral densities of stress-stress correlators:equal time density-density corr
Inequality saturated by Laughlin’s wave function
�µ⌫ = £vhµ⌫
(Haldane 2009)limk!0
s(k)
(k`)4� |S � 1|
8
Conclusion and outlook
• Quantum Hall states naturally live in Newton-Cartan geometry
• Symmetry determines the q^2 correction to Hall conductivity, other physical quantities
• Further questions:
• edge states?
• Relationship with conformal field theories?
• Meaning of Laughlin’s wave function?
• Implications for holographic realizations?
What is Hall viscosity?
ji = �vi
T ij = �vivj + P �ij � �Vij Vij =12(�iv
j + �jvi)
In 2 spatial dimensions, it is possible to write
T ij = · · ·� �H(�ikV kj + �jkV ki)
Hall viscosity (Avron Seiler Zograf)
breaks parity
Standard fluid dynamics: �t� + �iji = 0
�tji + �jT
ij = 0continuity eq.
Navier-Stokes eq.
dissipationless
Physical interpretation
• First term: Hall viscosity
yv
E
E
vx
�xvy + �yvx �= 0
Txx = Txx(x) �= 0
additional force Fx~∂x Txx
Hall effect: additional contribution to vy
Physical interpretation (II)
• 2nd term: more complicated interpretation
Fluid has nonzero angular velocity
�(x) =12�xvy = �cE�
x(x)2B
�B = 2mc�/e
Coriolis=Lorentz
Hall fluid is diamagnetic: d� = �MdB
M is spatially dependent M=M(x)
Extra contribution to current j = c z��M