Hall Viscosity and Electromagnetic Response
IINT-PUB-11-037
Hall Viscosity and Electromagnetic Response
Carlos Hoyos and Dam Thanh Son Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA
We show that, for Galilean invariant quantum Hall states, the Hall viscosity appears in theelectromagnetic response at finite wave numbers q . In particular, the leading q dependence of theHall conductivity at small q receives a contribution from the Hall viscosity. The coefficient of the q term in the Hall conductivity is universal in the limit of strong magnetic field. PACS numbers: 73.43.Cd
Introduction. —Quantum Hall states have been shownto possess, in addition to the Hall conductivity, a newproperty called the Hall viscosity [1, 2]. The Hall viscos-ity breaks parity, is dissipationless and can be defined atzero temperature. It has been shown recently [3, 4] thatthe Hall viscosity is related to a topological property ofthe quantum Hall state—the Wen-Zee shift [5].One may ask how the Hall viscosity can be measured.As originally defined, the Hall viscosity is related to thestress response of the system to metric perturbations.Such perturbations can be, in principle, mimicked bylattice vibrations (sound waves). It has also been sug-gested that the Hall viscosity determines the stress cre-ated by an inhomogeneous electric field [7]. In this paperwe show that, for quantum Hall states of systems withGalilean invariance and made up of particles of the samecharge/mass ratio, the Hall viscosity can be, in princi-ple, determined from electromagnetic response alone. Weshall show this result first using intuitive physical argu-ments, and then by employing the formalism of nonrel-ativistic diffeomorphism invariance, applied to the low-energy effective action of the Hall liquid.
Main result. —Consider a quantum Hall state in finitemagnetic field B . First we concentrate on the case whenthe interaction between particles is short-ranged. (Thecase of Coulomb interaction will be treated later in thepaper.) Let us turn on a static longitudinal electric field E = − ∇ φ where φ is the scalar potential. We take φ tovary in space with some wave vector q pointing along the x direction and measure the Hall current j y (see Fig. 1).The proportionality between j y and E x is the wave-vectordependent Hall conductivity, j y ( q ) = σ xy ( q ) E x ( q ) . (1)In the limit q → σ xy ( q ) approaches the universal value,determined by the rational filling factor ν : σ xy (0) = νe / (2 π (cid:126) ). In general, σ xy has a nontrivial dependenceon the wave number q .We will show that, for a Galilean invariant system ofelectrons, the coefficient C of the first correction in thelow- q expansion of the Hall conductivity σ xy ( q ) σ xy (0) = 1 + C ( q(cid:96) ) + O ( q (cid:96) ) , (2) can be related to the Hall viscosity η a and the function (cid:15) ( B ) which is is the energy density (energy per unit area)as function of the external magnetic field (cid:15) ( B ) at fixedfilling factor, C = η a (cid:126) n − πν (cid:96) (cid:126) ω c B (cid:15) (cid:48)(cid:48) ( B ) . (3)Here (cid:96) = (cid:112) (cid:126) c/ | e | B is the magnetic length, ω c = | e | B/mc is the cyclotron frequency, and n is the density of elec-trons.Using the relationship between η a and the shift S : η a = (cid:126) n S / S /
4, which makes clear thatthe magnitude of this contribution is robust (i.e., doesnot depend on interactions). The second contributioninvolves the function (cid:15) ( B ) and is not universal. How-ever, its magnitude can be extracted independently bymeasuring currents created by weak inhomogeneous per-turbations of the magnetic field δB , j = − c(cid:15) (cid:48)(cid:48) ( B )ˆ z × ∇ δB. (4)Hence, by measuring the electromagnetic response of thesystem to inhomogenous electric and magnetic fields, onecan determine the Hall viscosity.The situation becomes simpler in the limit of highmagnetic fields (i.e., that of no mixing between Landaulevels) in which the energy (cid:15) ( B ) becomes that of non-interacting electrons in a magnetic field. For the inte-ger quantum Hall state with ν = N , the energy density (cid:15) ( B ) = ( N / π ) (cid:126) ω c /(cid:96) , and the shift S = N , so we have σ xy ( q ) σ xy (0) = 1 − N q(cid:96) ) + O ( q (cid:96) ) for ν = N . (5)The result coincides with what has been computed in theliterature ( σ xy is proportional to Σ in the notations ofRef. [8]). For fractional quantum Hall states with ν < (cid:15) ( B ) = ( ν/ π ) (cid:126) ω c /(cid:96) , therefore C = S −
1. Inparticular, for Laughlin’s states with ν = 1 / (2 k +1), theshift S = 2 k +1 [5], so σ xy ( q ) σ xy (0) = 1 + 2 k −
34 ( q(cid:96) ) + O ( q (cid:96) ) , ν = 12 k +1 . (6) a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p In general, for any quantum Hall state, we can find the q correction to σ xy ( q ) from the value of the shift S andthe total energy, as a function of the magnetic field. Physical argument. —Before presenting the mathemat-ical proof of the statement made above, we will give aphysical derivation. We will show that the two contribu-tions to C come from two different physical effects.First let us note that to first approximation, the Hallfluid moves along the y direction with a velocity thatdepends on x (see Fig. 1), v y ( x ) = − cE x ( x ) B . (7)This velocity is determined by balancing electric andmagnetic forces acting on a fluid volume. However, theflow (7) is a shear flow with a nonzero strain rate. TheHall viscosity leads to an additional stress in the system,which in turn induces a correction to the current. y vEEv x FIG. 1: Pattern of flow in an inhomogeneous electric field.
Let us compute the magnitude of the this effect. Thestrain rate V xy = ∂ x v y induces, through the Hall vis-cosity, an additional contribution to the stress, σ xx = − σ yy = 2 η a V xy The x -dependence of σ xx leads to an ad-ditional force acting on each volume element of the fluidalong the x axis: f x = − ∂ x σ xx . This force induces acorrection to the Hall current equal to δj y = − cB f x = − η a c B E (cid:48)(cid:48) x ( x ) . (8)We thus find the first correction to σ xy , σ (1) xy ( q ) = η a c B q . (9)The second effect is related to the fact that the fluidflow, in addition to having a shear rate, also has a nonzerolocal angular velocity:Ω( x ) = 12 ∂ x v y = − cE (cid:48) x ( x )2 B . (10)This local rotation acts as an effective magnetic field,equal to δB = 2 mc Ω /e (found by equating the Coriolisforce with the Lorentz force from δB .) On the otherhand, the quantum Hall fluid is a diamagnetic material.with magnetic moment density M = − ∂(cid:15)/∂B . For aconstant magnetic field, M is constant. But due to thefluctuations δB there is an inhomogeneous contributionto the magnetic moment density, δM = − ∂ (cid:15)∂B δB = (cid:15) (cid:48)(cid:48) ( B ) mc E (cid:48) x ( x ) eB . (11) This fluctuating magnetic moment density leads to anadditional electromagnetic current, j = c ˆ z × ∇ M : j y = (cid:15) (cid:48)(cid:48) ( B ) mc E (cid:48)(cid:48) x ( x ) | e | B . (12)We find the second contribution to the Hall conductivity, σ (2) xy ( q ) = − mc (cid:15) (cid:48)(cid:48) ( B ) | e | B q . (13)The finite-wave-number correction to the Hall conduc-tivity is σ (1) xy + σ (2) xy . Elementary algebraic manipulationsbring it to the form of Eqs. (2) and (3). Diffeomorphism invariance. —We now formally provethe result derived above by constructing a low-energy ef-fective theory of the quantum Hall state. As the quantumHall state is gapped, the effective action is as a local func-tional of the external fields. Expanding in momentum tolowest order, it is simply the Chern-Simons action. Inorder to reproduce the q correction to σ xy we need to gobeyond leading order.We shall make use of the nonrelativistic diffeomor-phism invariance, introduced in Ref. [9]. Our strategyis to couple our system to gravity and find out the sym-metries of the action. These symmetries are inheritedby the low-energy effective theory, and impose nontrivialconstraints to the effective Lagrangian.We consider a quantum Hall state in the presence of anexternal gauge field A µ ( t, x ) and a spatial metric g ij ( t, x ).For example, for the case of free fermions we assume theaction to be S = (cid:90) d t d x √ g (cid:104) i ψ † ∂ t ψ − ∂ t ψ † ψ ) + A ψ † ψ − g ij m ( ∂ i ψ † + iA i ψ † )( ∂ j ψ − iA j ψ ) (cid:105) . (14)We will set (cid:126) = 1 and absorb an e/c factor into thenormalization of the gauge potential A i . Most of thetime we will set the spatial metric to be flat ( g ij = δ ij )at the end of calculations, but it will be useful to considera general metric in the intermediate steps.The action (14) is invariant under reparametrizationof spatial coordinates x k → x k + ξ k , where ξ k dependsboth on space and time, ξ k = ξ k ( t, x ). The passive formof the transformations is δA = − ξ k ∂ k A − A k ˙ ξ k , (15) δA i = − ξ k ∂ k A i − A k ∂ i ξ k − mg ik ˙ ξ k , (16) δg ij = − ξ k ∂ k g ij − g kj ∂ i ξ k − g ik ∂ j ξ k , (17) δψ = − ξ k ∂ k ψ. (18)The Galilean transformation is a special case with ξ k = v k t . As explained in Ref. [9], the transformations abovecan be motivated by taking a nonrelativistic limit of rel-ativistic diffeomorphisms.Interactions can be introduced in a way which pre-serves the diffeomorphism invariance. For example, byadding to (14) S = S + (cid:90) d t d x √ g (cid:16) λψ † ψφ − g ij ∂ i φ∂ j φ − m φ φ (cid:17) (19)one introduces an attractive potential of range m − φ be-tween the particles. The new action is diffeomorphisminvariant if φ transforms as a scalar δφ = − ξ k ∂ k φ . Ageneric potential decaying faster than an exponential canbe represented by an infinite number of mediating fields,and so coupling to the external metric can be made com-patible with diffeomorphism invariance.Coulomb interactions can also be introduced, but nowthe field mediating the interaction propagates in threespatial dimensions. We can assume that the spatial met-ric does not depend on the third direction S = S + (cid:90) d t d x √ g a ( ψ † ψ − n )+ 2 πεe (cid:90) d t d x d z √ g (cid:2) g ij ∂ i a ∂ j a + ( ∂ z a ) (cid:3) . (20)( ε is the dielectric constant). We have included a neu-tralizing background with density n . The full actionis diffeomorphism invariant if a transforms as a scalar: δa = − ξ k ∂ k a . Power counting. —We now start constructing the low-energy effective field theory of the quantum Hall states.For incompressible states, there is no low-energy excita-tions, and we can integrate out ψ . If interactions areshort-ranged, the fields φ mediating interactions can alsobe integrated out. Thus the effective Lagrangian is alocal function of the external fields A µ , g ij and theirderivatives. The effective action must be invariant un-der (15)—(17).To organize a derivative expansion, one needs a power-counting scheme with a small parameter. There is an am-biguity in choosing the scheme, as the time derivative ∂ t and spatial derivatives can be chosen to be independentexpansion parameters. For definiteness, in this paper weuse the following scheme. All quantities will be regardedas proportional to some powers of a small parameter (cid:15) ,times some powers of ω c and (cid:96) . The external fields areassumed to vary slowly in space and time, ∂ i ∼ (cid:15)(cid:96) − , ∂ t ∼ (cid:15) ω c . (21)As for the magnitude of external perturbations, we as-sume δA ∼ (cid:15) ω c , δA i ∼ (cid:15) − (cid:96) − , δg ij ∼ . (22)In this scheme, we allow for order one variations of themetric, the magnetic field ( δB ∼ (cid:15) (cid:96) − ) and the chemi-cal potential ( A ). In further formulas, the electric and magnetic fields are defined as E i = ∂ i A − ∂ A i , B = F √ g = (cid:15) ij ∂ i A j √ g ≡ ε ij ∂ i A j , (23)so E i = O ( (cid:15) ) and B = O (1). Chern-Simons and Wen-Zee terms. —Two importantingredients in our construction of the effective field theoryare the Chern-Simons action and the Wen-Zee action.The Chern-Simons action is S CS = ν π (cid:90) d t d x (cid:15) µνλ A µ ∂ ν A λ , (24)and is of order (cid:15) in our power counting scheme. This willbe the leading term in the effective action. To constructthe Wen-Zee action, we first define the spin connection.We introduce a spatial vielbein e ai , a = 1 , g ij = e ai e aj and (cid:15) ab e ai e bj = ε ij . The vielbein is defined up tolocal O(2) rotations in a space. If we now define theconnection ω µ , ω = 12 (cid:15) ab e aj ∂ e bj , (25) ω i = 12 (cid:15) ab e aj ∇ i e bj = 12 ( (cid:15) ab e aj ∂ i e bj − ε jk ∂ j g ik ) , (26)then under local O(2) rotations ω µ transforms like anAbelian gauge potential ω µ → ω µ − ∂ µ λ . By using ω µ wecan construct the following gauge invariant action S WZ = κ π (cid:90) d t d x (cid:15) µνλ ω µ ∂ ν A λ . (27)This action is of order (cid:15) in our power counting schemeand has to be included if we work to that order. The ωdω Chern-Simons term, on the other hand, is of order (cid:15) and will not be considered.The coefficient κ is related to the shift S . Indeed, the“torsion magnetic” field ∂ ω − ∂ ω = √ gR where R isthe scalar curvature. Integrating by parts, the Wen-Zeeaction contains a term κ π (cid:15) µνλ ω µ ∂ ν A λ (cid:39) κ π √ g A R + · · · (28)which gives a contribution to the particle number densitythat is proportional to the scalar curvature. If the quan-tum Hall state lives on a closed two dimensional surface,then the total number of particles is Q = (cid:90) d x √ g j = (cid:90) d x √ g (cid:16) ν π B + κ π R (cid:17) = νN φ + κχ (29)where N φ is the total number of magnetic fluxes and χ = 2(1 − g ) is the Euler character. Comparing to thedefinition of S in Ref. [5], we find κ = ν S . For theinteger Quantum Hall state with ν = N , κ = N /
2. ForLaughlin’s states κ = 1 / S WZ = − κB π (cid:15) ij δg ik ∂ t δg jk + · · · (30)which implies the presence of an odd term in the stresstensor two point function, or Hall viscosity. The value ofthe Hall viscosity is η a = κB/ π = S n . This relation-ship between the Hall viscosity and the shift has beenderived previously in Ref. [4]. Most general effective action. —It is straightforward toverify that both S CS and S WZ are not diffeomorphisminvariant, and need to be corrected. In fact, to order O ( (cid:15) ), the most general effective action can be written as S = (cid:82) d t d x √ g (cid:80) i =1 L i , where L i ( i = 1 , . . . ,
5) are fiveindependent general diffeomorphism invariant (to order (cid:15) ) terms L = ν π (cid:16) ε µνλ A µ ∂ ν A λ + mB g ij E i E j (cid:17) , (31) L = κ π (cid:16) ε µνλ ω µ ∂ ν A λ + 12 B g ij ∂ i B E j (cid:17) , (32) L = − (cid:15) ( B ) − mB (cid:15) (cid:48)(cid:48) ( B ) g ij ∂ i B E j , (33) L = − K ( B ) g ij ∂ i B ∂ j B, (34) L = R h ( B ) , (35)where (cid:15) ( B ), K ( B ), and h ( B ) are functions of B . Thefunction (cid:15) ( B ) has the physical meaning of the energydensity of the quantum Hall state as a function of themagnetic field B , L and L do not enter the quantitiesof of our interest. The next to leading order term in L enforces compliance with Kohn’s theorem. The two-pointfunction of the electromagnetic current j µ is obtainedby taking the second derivative of the effective actionwith respect to A µ , then setting perturbations to zero.Equivalently we can differentiate the effective action oncewith respect to the external fields to get the current. Wefind, in flat space j i = ν π (cid:15) ij E j − B (cid:104) κ π − m(cid:15) (cid:48)(cid:48) ( B ) (cid:105) (cid:15) ij ∂ j ( ∇ · E )+ · · · (36)where · · · refers to term that vanish when the magneticfield is not perturbed. Equations (2) and (3) are repro-duced from this formula. Inclusion of Coulomb interactions .—In the case withCoulomb interactions, one needs to take into account thescreening of the electric field. The expansion (2,3) there-fore applies not to σ xy ( q ) but to˜ σ xy ( q ) = (cid:20) e χ ( q )2 π(cid:15) q (cid:21) σ xy ( q ) (cid:39) (cid:104) ν κ π ( q(cid:96) ) (cid:105) σ xy ( q )(37) where κ = e / (4 π(cid:15)(cid:96)ω c ) and χ ( q ) is the static suscepti-bility, the small- q behavior of which is is determined byKohn’s theorem: χ ( q ) = νmq / (2 πB ). In the limit ofhigh magnetic fields where κ (cid:28)
1, the distinction be-tween σ xy and ˜ σ xy disappears. Conclusions. —We have shown that the Hall viscositydoes not only appear in the response to gravitationalfluctuations, but also, under certain circumstances, ina purely electromagnetic response function. For this oneneeds Galilean invariance and that all particles have thesame charge/mass ratio, a condition satisfied in the mostinteresting physical systems.One notes that topological arguments alone are insuf-ficient to determine the coefficient of the q term in thefinite wave number Hall conductivity. But topology, cou-pled with nonrelativistic diffeomorphism invariance, ispowerful enough to find this coefficient [e.g., Eq. (6)].It would be interesting to explore consequences of diffeo-morphism invariance for other systems with topologicalorder, e.g., the p x + ip y paired state or the superfluid Bphase of He or the compressible ν = 1 / [1] J. E. Avron, R. Seiler, and P. G. Zograf, Phys. Rev. Lett. , 697 (1995).[2] J. E. Avron, J. Stat. Phys. , 543 (1998).[3] N. Read, Phys. Rev. B , 045308 (2009).[4] N. Read and E. H. Rezayi, Phys. Rev. B , 085316(2011).[5] X. G. Wen and A. Zee, Phys. Rev. Lett. , 953 (1992).[6] W. Goldberger and N. Read, unpublished.[7] F. D. M. Haldane, arXiv:0906.1854.[8] Y. H. Chen, F. Wilczek, E. Witten and B. I. Halperin,Int. J. Mod. Phys. B , 1001 (1989).[9] D. T. Son and M. Wingate, Ann. Phys. (NY)321