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 = � � d 3 x � µ νλ A µ � ν A λ 4 � e 2 σ xy = ν encodes Hall conductivity � 2 π

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

Hall viscosity Avron et al 1995 • Turn on h xy (t) metric perturbations • observe T xx = - T yy ~ h’ xy (t) • there must be a term proportional first derivative of metric in the effective Lagrangian • How? curvature ~ 2nd derivative

Wen-Zee term • Hall viscosity: described by Wen-Zee term (W.Goldberger & N.Read unpublished; N.Read 2009 KITP talk) • Introduce spatial vielbein (viel=2) g ij =e ai e aj • We can now define the spin connection � i = 1 � 0 = 1 2 � ab e aj � i e bj 2 � ab e aj � 0 e bj Vielbein defined up to a local O(2) rotation e a i → e a i + �� ab e b ω µ → ω µ − ∂ µ λ i like an abelian gauge field

Vielbein and curvature ∂ 1 ω 2 − ∂ 2 ω 1 = 1 √ g R 2

Wen-Zee terms in addition to the Chern-Simons term 1 2 � � µ νλ ( � � µ � ν A λ + � � � µ � ν � λ ) will not be important for futher discussions The first term gives rise to • Wen-Zee shift • Hall viscosity

Wen-Zee shift • Rewrite S WZ as 2 � � µ νλ A µ � ν � λ = � � √ g A 0 R + · · · 4 � # of magnetic Total particle number: fluxes Euler � ν � � � d 2 x √ g j 0 = 2 π B + κ d 2 x √ g Q = 4 π R = ν N φ + κχ S = 2 κ On a sphere: Q = ν ( N φ + S ) , ν ‘shift’ IQH states: ν =n, κ =n 2 /2 Laughlin’s states: ν = 1 /n, κ = 1 /2

Hall viscosity from WZ term S WZ = − � B 16 � � ij h ik � t h jk + · · · stress ~ time derivative of metric η a = κ B 4 π = 1 4 S n derived by N.Read previously

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 DTS, M.Wingate 2006 Microscopic theory � i D t ψ − g ij � 2 ψ † ↔ � d t d 2 x √ g D µ ψ ≡ ( ∂ µ − iA µ ) ψ 2 mD i ψ † D j ψ S 0 = Gauge invariance: ψ → e i α ψ A µ → A µ + ∂ µ α General coordinate invariance: δψ = − ξ k ∂ k ψ ≡ L ξ ψ δ A 0 = ξ k ∂ k A 0 ≡ L ξ A 0 δ A i = − ξ k ∂ k A i − A k ∂ i ξ k ≡ L ξ A i δ g ij = − ξ k ∂ k g ij − g kj ∂ i ξ k − g ik ∂ j ξ k ≡ L ξ g ij Here ξ is time independent: ξ = ξ ( x )

NR diffeomorphism • These transformations can be generalized to be time-dependent: ξ = ξ (t, x ) δψ = − L ξ ψ δ A 0 = − L ξ A 0 − A k ˙ ξ k δ A i = − L ξ A i − mg ik ˙ ξ k δ g ij = − L ξ g ij Time dependent diffeomorphisms mix metric and gauge field Galilean transformations: special case ξ i =v i t

Where does it come from Start with complex scalar field � d x √− g ( g µ ν ∂ µ φ ∗ ∂ ν φ + φ ∗ φ ) S = − Take nonrelativistic limit: − 1 + 2 A 0 A i   ψ mc 2 mc φ = e − imcx 0   g µ ν = √   2 mc A i   g ij mc � i ∂ t ψ + A 0 ψ † ψ − g ij � � 2 ψ † ↔ 2 m ( ∂ i ψ † + iA i ψ † )( ∂ j ψ − iA j ψ ) d t d x √ g S = .

Relativistic diffeomorphism x µ → x µ + ξ µ ψ μ =0: gauge transform φ = e − imcx 0 √ 2 mc μ =i: general coordinate transformations

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 d t d 2 x √ g a 0 ( ψ † ψ − n 0 ) + 2 πε � � d t d 2 x d z √ g g ij ∂ i a 0 ∂ j a 0 + ( ∂ z a 0 ) 2 � � S = S 0 + e 2 δ a 0 = − ξ k ∂ k a 0

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 � S CS = � m � d t d 2 x � ij E i g jk ˙ � k 2 �

Is CS action invariant? • CS action is gauge invariant • CS action is Galilean invariant • CS action is not diffeomorphism invariant � S CS = � m � d t d 2 x � ij E i g jk ˙ � k 2 � Higher order terms in the action should changed by - δ S CS

Is CS action invariant? • CS action is gauge invariant • CS action is Galilean invariant • CS action is not diffeomorphism invariant � S CS = � m � d t d 2 x � ij E i g jk ˙ � k 2 � Higher order terms in the action should changed by - δ S CS But this cannot be achieved by local terms

Resolution • Higher order terms contain inverse powers of B ε µ νλ A µ ∂ ν A λ + m B g ij E i E j + · · · • Quantum Hall state with diff. invariance does not exist at zero magnetic field!

Diff invariant terms L 1 = ν ε µ νλ A µ ∂ ν A λ + m � � B g ij E i E j 4 π Kohn’s theorem ~ 1960 L 2 = κ ε µ νλ ω µ ∂ ν A λ + 1 � � 2 B g ij ∂ i B E j 2 π L 3 = − � ( B ) − m B � �� ( B ) g ij � i B E j σ xy (q) ground state energy density black: leading blue: subleading

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

σ xy (q): new prediction E x = E e iqx y E v j y = σ xy ( q ) E x x E v From effective field theory � xy ( q ) � xy (0) = 1 + C 2 ( q � ) 2 + O ( q 4 � 4 ) � 2 � n − 2 � C 2 = � a B 2 � �� ( B ) � � � c S / 4

Physical interpretation • First term: Hall viscosity y E v ∂ x v y + ∂ y v x � = 0 x E v T xx = T xx ( x ) � = 0 additional force F x ~ ∂ x T xx Hall effect: additional contribution to v y

Physical interpretation (II) • 2nd term: more complicated interpretation Fluid has nonzero angular velocity Ω ( x ) = 1 2 ∂ x v y = − cE � x ( x ) δ B = 2 mc Ω /e 2 B 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 (0) = 1 − 3 n � xy ( q ) 4 ( q � ) 2 + O ( q 4 � 4 ) ν =n � xy (0) = 1 + 2 n − 3 � xy ( q ) 1 ( q � ) 2 + O ( q 4 � 4 ) , � = 4 2 n +1 exact nonperturbative results!

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