Quantum Hall effect: what can be learned from curved space? Dam - - PowerPoint PPT Presentation

Quantum Hall effect: what can be learned from curved space?

Dam Thanh Son (INT, University of Washington) Carlos Hoyos, DTS 2011

In memory of my father Dam Trung Bao (1929-2011)

Outline

• This talk is not be related to AdS/CFT, string

theory

• but we will see how thinking about curve space

helps us understand flat-space physics

Quantum Hall state

• simplest example: noninteracting electrons filling n Landau levels

(interger QH effect)

• Fractional QH effect: much more complicated theory (Laughlin)
• gapped, no low-energy degree of freedom
• The effective action can be expanded in polynomials of external

fields

• To lowest order: Chern-Simons action

S = 4

• d3x µνλAµνAλ

encodes Hall conductivity

σxy = ν 2π e2

What is missing

• CS action does not involve metric
• Stress-energy tensor = 0
• It is not how real quantum Hall system behaves

Hall viscosity

• Turn on hxy(t) metric perturbations
• observe Txx = - Tyy ~ h’xy(t)
• there must be a term proportional first derivative
• f metric in the effective Lagrangian
• How? curvature ~ 2nd derivative

Avron et al 1995

Wen-Zee term

• Hall viscosity: described by Wen-Zee term

(W.Goldberger & N.Read unpublished; N.Read 2009 KITP talk)

• Introduce spatial vielbein (viel=2) gij=eai eaj
• We can now define the spin connection

i = 1 2abeajiebj 0 = 1 2abeaj0ebj Vielbein defined up to a local O(2) rotation

ea

i → ea i + abeb i

ωµ → ωµ − ∂µλ

like an abelian gauge field

Vielbein and curvature

∂1ω2 − ∂2ω1 = 1 2 √g R

Wen-Zee terms

1 2 µνλ( µνAλ + µνλ) in addition to the Chern-Simons term

will not be important for futher discussions

The first term gives rise to

• Wen-Zee shift
• Hall viscosity

Wen-Zee shift

• Rewrite SWZ as
• 2 µνλAµνλ =

4 √g A0R + · · ·

Q =

• d2x √g j0 =
• d2x √g

ν 2π B + κ 4π R

• = νNφ + κχ

Total particle number: IQH states: ν=n, κ=n2/2 Laughlin’s states: ν=1/n, κ=1/2 On a sphere:

Q = ν(Nφ + S), S = 2κ ν

‘shift’

# of magnetic fluxes Euler

Hall viscosity from WZ term

SWZ = − B 16 ijhikthjk + · · · ηa = κB 4π = 1 4Sn

derived by N.Read previously

stress ~ time derivative of metric

Flat space physics

• But is this Wen-Zee term be important for physics in flat

space?

• In this talk we will argue that it is
• Reason: nonrelativistic diffeomorphism
• For a nonrelativistic system of particles with the same charge/

mass ratio, there is a nonrelativistic principle of equivalence

• accelerated frame ~ electric field
• rotating frame ~ magnetic field (Coriolis force ~ Lorentz

force)

• nonrelativistic diffeomorphism mixes metric and EM field

Symmetries of NR theory

Microscopic theory

Dµψ ≡ (∂µ − iAµ)ψ

Gauge invariance:

ψ → eiαψ Aµ → Aµ + ∂µα

General coordinate invariance:

δgij = −ξk∂kgij − gkj∂iξk − gik∂jξk δAi = −ξk∂kAi − Ak∂iξk δψ = −ξk∂kψ ≡ Lξψ ≡ LξAi ≡ Lξgij

Here ξ is time independent: ξ=ξ(x)

δA0 = ξk∂kA0≡ LξA0

DTS, M.Wingate 2006 S0 =

• dt d2x √g

i 2ψ†↔ Dtψ − gij 2mDiψ†Djψ

NR diffeomorphism

• These transformations can be generalized to be

time-dependent: ξ=ξ(t,x)

δψ = −Lξψ δA0 = −LξA0−Ak ˙ ξk δAi = −LξAi−mgik ˙ ξk δgij = −Lξgij

Galilean transformations: special case ξi=vit Time dependent diffeomorphisms mix metric and gauge field

Where does it come from

Start with complex scalar field

gµν =     −1 + 2A0 mc2 Ai mc Ai mc gij    

S = −

• dx √−g (gµν∂µφ∗∂νφ + φ∗φ)

Take nonrelativistic limit:

S =

• dt dx √g

i 2ψ†↔ ∂tψ + A0ψ†ψ − gij 2m(∂iψ† + iAiψ†)(∂jψ − iAjψ)

• .

φ = e−imcx0 ψ √ 2mc

Relativistic diffeomorphism

μ=0: gauge transform μ=i: general coordinate transformations

xµ → xµ + ξµ

φ = e−imcx0 ψ √ 2mc

Interactions

• Interactions can be introduced that preserve

nonrelativistic diffeomorphism

• interactions mediated by fields
• For example, Coulomb interactions: mediated by

photon propagating in 3+1 dimensions

S = S0 +

• dt d2x √g a0(ψ†ψ − n0) + 2πε

e2

• dt d2x dz √g
• gij∂ia0∂ja0 + (∂za0)2

δa0 = −ξk∂ka0

Is CS action invariant?

Is CS action invariant?

• CS action is gauge invariant

Is CS action invariant?

• CS action is gauge invariant
• CS action is Galilean invariant

Is CS action invariant?

• CS action is gauge invariant
• CS action is Galilean invariant
• CS action is not diffeomorphism invariant

Is CS action invariant?

• CS action is gauge invariant
• CS action is Galilean invariant
• CS action is not diffeomorphism invariant

SCS = m 2

• dt d2x ijEigjk ˙

k

Is CS action invariant?

• CS action is gauge invariant
• CS action is Galilean invariant
• CS action is not diffeomorphism invariant

SCS = m 2

• dt d2x ijEigjk ˙

k Higher order terms in the action should changed by -δSCS

Is CS action invariant?

• CS action is gauge invariant
• CS action is Galilean invariant
• CS action is not diffeomorphism invariant

SCS = m 2

• dt d2x ijEigjk ˙

k Higher order terms in the action should changed by -δSCS But this cannot be achieved by local terms

Resolution

• Higher order terms contain inverse powers of B

εµνλAµ∂νAλ + m B gijEiEj + · · ·

• Quantum Hall state with diff. invariance does not exist at

zero magnetic field!

Diff invariant terms

L1 = ν 4π

• εµνλAµ∂νAλ + m

B gijEiEj

• L2 = κ

2π

• εµνλωµ∂νAλ + 1

2B gij∂iB Ej

• L3 = −(B) − m

B (B)gijiB Ej

black: leading blue: subleading

ground state energy density

Kohn’s theorem ~ 1960 σxy(q)

Kohn’s theorem

• Response of the system on uniform electric field

does not depend on interactions

• Effective action captures first order in omega

corrections to conductivities at q=0

σxy(q): new prediction

y v E E v x

Ex = E eiqx jy = σxy(q)Ex From effective field theory

xy(q) xy(0) = 1 + C2(q)2 + O(q44)

C2 = a n − 2

• 2

c B2(B) S/4

Physical interpretation

• First term: Hall viscosity

y v E E v x ∂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

Ω(x) = 1 2∂xvy = −cE

x(x)

2B

δB = 2mcΩ/e

Coriolis=Lorentz

Hall fluid is diamagnetic: d = −MdB M is spatially dependent M=M(x) Extra contribution to current j = c ˆ z M

Current ~ gradient of magnetization

j = c ˆ z M

High B limit

• In the limit of high magnetic field: ϵ(B) known: free

fermions

• n Landau levels for IQH states
• first Landau level for FQH states with ν<1
• Wen-Zee shift is known

xy(q) xy(0) = 1 − 3n 4 (q)2 + O(q44)

xy(q) xy(0) = 1 + 2n − 3 4 (q)2 + O(q44), = 1 2n+1

exact nonperturbative results! ν=n

Conclusions

• Thinking about the curved space is productive in

nonrelativistic physics

• Reason: NR principle of equivalence
• NR diffeomorphism mixes metric and EM field
• Nontrivial consequences in quantum Hall physics
• Wen-Zee term in the action leads to one

contribution to the Hall conductivity at finite q