Effective Field Theory of Relativistic Quantum Hall Systems
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Effective Field Theory of Relativistic Quantum Hall Systems
Siavash Golkar, Matthew M. Roberts, and Dam Thanh Son
Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Ilinois 60637, USA
Motivated by the observation of the fractional quantum Hall effect in graphene, we consider theeffective field theory of relativistic quantum Hall states. We find that, beside the Chern-Simons term,the effective action also contains a term of topological nature, which couples the electromagneticfield with a topologically conserved current of 2 + 1 dimensional relativistic fluid. In contrast to theChern-Simons term, the new term involves the spacetime metric in a nontrivial way. We extractthe predictions of the effective theory for linear electromagnetic and gravitational responses. Forfractional quantum Hall states at the zeroth Landau level, additional holomorphic constraints allowone to express the results in terms of two dimensionless constants of topological nature.
PACS numbers: 73.43.-f,81.05.ue
Introduction.—
Recently both integer [1, 2] and frac-tional [3, 4] quantum Hall effects have been observed ingraphene. In many respects, graphene behaves like a rel-ativistic system; in particular the integer quantum Hall(IQH) plateaux corresponds to the Hall conductivity (inunit of the quantum ( e /h ) ν = 4( n + ), with the fac-tor 4 due to the spin and valley degeneracies and theoffset of 1 / < ν < ν , and a relativistic version of the shift, denotedhere as κ . The latter is defined as the offset between thetotal charge Q and the magnetic flux in units of the fluxquantum N φ when the system is put on a sphere withno quasiparticles or quasiholes present: Q = νN φ + κ .Together with a single, nonuniversal function of the mag-netic field, denoted as f ( b ), all parity-odd transports ofthe quantum Hall system are completely determined upto second order in the expansion over momentum. Inparticular, the system possesses a Hall viscosity [5] pro-portional to κ . This relationship is the relativistic analogof the exact relationship between the Hall viscosity andthe shift in nonrelativistic systems [6].There is a further simplification for quantum Hallstates on the zeroth Landau level (which correspondsto − ≤ ν ≤ f ( b ) which is otherwise non-universal. In this case, allparity-odd transport is universal to second order in theexpansion over momentum. For example, the frequencyand momentum dependence of the Hall conductivity isfound to be σ xy ( ω, q ) = ν π (cid:16) ω ω c (cid:17) + κ − ν π ( qℓ B ) , (1)where ℓ B = p ~ c/eB is the magnetic length and ω c = v F /ℓ B , with v F being the Fermi velocity. Equation (1)parallels a similar formula in the nonrelativistic case [7,8]. The dependence on the shift is exactly the same,after the identification κ ∼ ν S , but there is a differencein a constant in front of ( qℓ B ) . This is the reflection ofthe fact that the electromagnetic current is not a simplelowest-Landau level operator. Power-counting. —From now on we set ~ = v F = 1and absorb the electron charge e into the magnetic field.The low-energy dynamics in the bulk of a gapped quan-tum Hall system can be described by a local effectiveaction which is a functional of the external probes. Wewill turn on both the electromagnetic and gravitationalperturbations. We regard the external magnetic field B to be O (1) and consider perturbations of the externalgauge field that are of the same order as the background: F µν = O (1). Denoting the momentum scale by p , onethen has A µ = O ( p − ). The perturbations of the metricis assume to be of order one: g µν = O (1), so, for example,the Riemann tensor is O ( p ).The term in the effective Lagrangian with the lowestpower of p is the Chern-Simons term, S CS = ν π Z d x √− g ǫ µνλ A µ ∂ ν A λ , (2)and is O ( p − ). At the next, O (1), order, there is onlyone gauge invariant scalar F µν F µν , so the most generalcontribution to the action at this order is S ǫ = − Z d x √− g ǫ ( b ) , (3)where b = ( F µν F µν ) / and ǫ can be any function of b . We also introduce a unit timelike vector u µ , u = − u µ = 12 b ǫ µνλ F νλ , (4)which corresponds to the local frame in which the electricfield vanishes. The Bianchi identity dF = 0 becomes ∇ µ ( bu µ ) = 0. The action S ǫ depends on the metric,and varying the action with respect to the metric onefinds that the stress-energy tensor is, to this order T µν =( ǫ + P ) u µ u ν + P g µν , where P = bǫ ′ ( b ) − ǫ ( b ). The stress-energy tensor has the form of that of an ideal fluid.To construct higher-order terms, it is convenient to use b and u µ instead of F µν . Both b and u µ are O (1) in ourpower-counting scheme. The definition of u µ involves b inthe denominator, but this does not create any problemsince the effective theory is supposed to work only atfinite magnetic field. The presence of the unit vector u µ reminds us of the Einstein-aether theory [9], but thereare importance difference due to the fact that we arein (2+1) dimensions. One can write down two obviousterms at order O ( p ), f ( b ) ǫ µνλ u µ ∂ ν u λ , f ( b ) u µ ∂ µ b. (5)However, by introducing a new function f ( b ) so that f ( b ) = bf ′ ( b ), the second term can be shown to van-ish by integration by parts. There exists, however, onemore term at order O ( p ). The construction of this terminvolves a topological current which we now describe. Topological current.—
One notices that the followingcurrent is identically conserved, J µ = 18 π ε µνλ ε αβγ u α (cid:16) ∇ ν u β ∇ λ u γ − R νλβγ (cid:17) . (6)The current has a topological interpretation. The totalcharge calculated on any space-like surface is simply theEuler character of this surface. The only exception isthe special case of a Euclidean space time containing an S , where the total charge becomes the Euler charactermultiplied by the winding number of the S → S mapfrom the 2D spatial slice to the space of unit vector u µ (this winding number in Lorentz signature is equal to 1).We can motivate this in the simple case when u µ ∼ (1 ,~ J = 18 π (2) R, (7)so the total charge is proportional to the Euler character-istic χ of the hypersurface, Q = χ/ − g , where g isthe genus of the surface. A more thorough investigationof this new topological current will be presented in [10]. The second topological term. —We can now add to theeffective action a term κ R d x √− g A µ J µ . Since J µ is identically conserved, this term is gauge invariant up toa boundary term, similar to the Chern-Simons term. Butin contrast to the Chern-Simons term, the new term ex-ists only for background where the magnetic field doesnot vanish anywhere. This restriction is natural for theeffective field theory of a quantum Hall state. Moreover,the term is O ( p ) in our power counting scheme, thus wehave to take that into account when working to this or-der.The term under consideration is the relativistic coun-terpart of the mixed Chern-Simons term A ∧ dω whichappears in the nonrelativistic case [11, 12]. In the (2+1)Drelativistic theory where the spin connection is non-Abelian such a term cannot be directly written down.Thus, the final action, including all terms to order O ( p )is L = ν π ε µνλ A µ ∂ ν A λ − ǫ ( b ) + f ( b ) ε µνλ u µ ∂ ν u λ + κ π ε µνλ ε αβγ A µ u α (cid:16) ∇ ν u β ∇ λ u γ − R νλβγ (cid:17) . (8)It is interesting to note that when ǫ ( b ) ∼ b / and f ( b ) ∼ b , the action is fully Weyl invariant. This wouldbe the case if the microscopic theory underlying the quan-tum Hall state is a conformal field theory. Relativistic shift. —The coefficient κ is related to a rela-tivistic version of the shift [11]. The charge density is thevariation of the action with respect to A . One can seethat the total charge on a closed surface comes only fromthe Chern-Simons and the κ term in the Lagrangian (8), Q = Z d x (cid:16) ν π F + κ π J (cid:17) . (9)By using (7) this can be written as Q = νN φ + κχ/ N φ is the total number of magnetic flux quantathreading the manifold and χ is the Euler character ofthe manifold. This relationship is a relativistic versionof the the shift, which is normally defined as S in theequation Q = ν ( N φ + S ) on a sphere [11]. We havedefined κ to remain finite at ν = 0.For an integer quantum Hall states with ν = N f ( n + ),where N f is the total number of “flavor” degeneracy ofthe Landau levels (in graphene N f = 4) the total chargecan be found by summing up all charges of the filledLandau levels on a sphere. We find κ = N f n ( n + 1).Note that κ = 0 for the ν = ± κ is related to the shift S of the correspond-ing state in the usual nonrelativistic theory. For illustra-tion let us consider a state with 0 < ν < N φ magnetic flux quantathreading it, is identical to the Landau levels of a nonrel-ativistic fermion on sphere with N ′ φ = N φ − N φ = N ′ φ +1).For the latter, the formula Q = ν ( N ′ φ + S NR ) is taken asthe definition of S NR , which implies κ = ν ( S NR − ν = state has κ = . Discrete symmetries. —Let us assume the the micro-scopic theory respects C , P , and T , and discuss if the ef-fective theory breaks these symmetries. Recall that [13]under C , A µ → − A µ ; under P , x → − x , A → A , A → − A , and A → A ; and under T t → − t , A → A and A i → − A i . All these symmetries arebroken by the background magnetic field, and C is fur-ther broken if there is a nonzero chemical potential. Butone combination, P T , leaves both the magnetic field andthe chemical potential invariant. It is easy to see that allterms we have considered are invariant with respect to
P T . When the chemical potential is zero, however, wecan classify our terms also with respect to CP and CT .The latter is the particle-hole symmetry of the lowestLandau level. Under these combinations ν , κ , and f ( b )all change signs. Therefore the presence of a nonzero ν , κ ,or f ( b ) at zero chemical potential signals a spontaneousbreaking of CP and CT symmetries. The ν = 0 IQHstate of graphene has κ = 0, consistent with unbroken CP and CT . On the other hand, it is easy to construct amultiflavor Moore-Read state [14] at ν = 0 which breaksthese symmetries. Momentum density. —As the first application of theeffective field theory, we compute the momentum density T i in the background of inhomogeneous magnetic field B = b ( x, y ). To that end, we turn on a perturbation inthe g i component of the metric tensor and read out themomentum density from the action: δS = R d xT i g i .We find T i = − ǫ ij ∂ j (cid:16) κ π b + f ( b ) (cid:17) . (10) FQH states on the zeroth Landau level. —We now showthat, for the FQH states on the zeroth Landau level withnegligible mixing with other Landau levels, the function f ( b ) is completely determined by the topological coeffi-cients ν and κ . This comes from a holomorphic constraintrelating the momentum density and the particle density.For concreteness, we choose the following representa-tion for the 2 × γ = σ , γ = iσ , γ = − iσ , (11)for which the free Hamiltonian has the form H = − iγ γ i D i − A = − i (cid:18) D ¯ D (cid:19) − A . (12)Here D ≡ D z , ¯ D ≡ D ¯ z , and we use complex coordinates: z = x + iy , ¯ z = x − iy . The n = 0 Landau level are ψ = ( ϕ, T , where ϕ satisfies the holomorphic constraint ¯ Dϕ = ¯ Dϕ ∗ = 0. Now let us look at the stress-energytensor, T µν = − i ψγ ( µ ↔ D ν ) ψ, (13)assuming a static, but spatially inhomogeneous, mag-netic field, and no electric field. For the 0 i componentswe can ignore time derivatives as A = 0 and the lowestLandau level has zero energy. We see that T i = − i ϕ ∗↔ D i ϕ, (14)which, by using the holomorphic constraints, can betransformed into T i = − ǫ ij ∂ j n, (15)where n = ϕ ∗ ϕ is the particle number density on thelowest Landau level. Comparing that to (10), we findthat, for FQH states in the zeroth Landau level, f ( b ) = 18 π ( ν − κ ) b. (16)The calculation above neglects possible mixing betweenLandau levels, as well as the possible corrections to thestress-energy tensor (13) due to interactions. Both ef-fects are small when the interaction energy scale is muchsmaller than the distance between Landau levels √ B . Response functions. —We now compute different re-sponse functions of the relativistic quantum Hall statesto external fields.First we compute the Hall viscosity. The Hall viscos-ity is defined through response to uniform shear metricperturbations. For simplicity we turn on only spatiallyhomogeneous perturbations of the spatial components ofthe metric, g ij = δ ij + h ij ( t ). The relevant term in theaction is κ Z d x √− gA µ J µ = − κB π Z d x ǫ jk h ij ∂ t h ik , (17)where we have performed integration by parts. We find η H = κB π . (18)The relationship between the Hall viscosity and κ is iden-tical to the nonrelativistic result η H = n S / S → S NR − < ν <
1. Note that the Hall viscosity depends only onthe topological number κ . It is natural since η H can bedetermined by adiabatic transport and hence should notdepend on non-universal functions like f ( b ).Next we look at the components of the stress tensorwhen one turns on a static, spatially inhomogeneous,electric field. The result is T ij = P δ ij + κ π ( ∂ i E j + ∂ j E i ) − (cid:16) κ π + f ′ ( B ) (cid:17) δ ij ∇ · E , (19)which can be written in terms of the drift velocity v i = ǫ ij E j /B and the shear rate V ij = ( ∂ i v j + ∂ j v i − δ ij ∂ · v ), T ij = P δ ij − η H ( ǫ ik V kj + ǫ jk V ki )+ δ ij ( η H + Bf ′ ( B )) ∇ × v . (20)The traceless part of the stress tensor reflects the nonzeroHall viscosity of the quantum Hall fluid [5, 15, 16].The two point functions of currents give us the re-sponse to external electromagnetic field, derived from thequadratic part of the effective action in flat space, L = − (cid:18) κ πB + f ( B ) B (cid:19) ǫ ij E i ∂ t E j − f ′ ( B ) B E i ∂ i B. (21)In particular, we find the correction to Hall conductivity σ xy (for longitudinal electric fields) at nonzero frequen-cies and wavenumbers, σ xy ( ω, q ) = ν π + (cid:18) κ π + 2 f ( B ) B (cid:19) ω B − f ′ ( B ) B q B . (22)At lowest Landau level, the formula becomes σ xy ( ω, q ) = ν π + νπ ω B + κ − ν π q B . (23)In Galilean invariant systems, the frequency depen-dence of the conductivity matrix is completely deter-mined, at q = 0, by Kohn’s theorem [17]. In relativisticsystems Kohn’s theorem no longer applies. Nevertheless,Eq. (23) implies that, in Lorentz invariant systems, the ω correction is completely fixed by the filling fraction.We now discuss the relevance of our formulas forgraphene, where the Lorentz invariance of the low-energyfree theory is broken by the Coulomb interaction [18].In the limit of weak Coulomb interaction our formulashould work, to leading order of the interaction strength,for integer quantum Hall states. For fractional quantumHall states in the zeroth Landau level, if one is inter-ested in the response for ω ≪ q , the effect of retarda-tion of the interaction should be small. In this case, onecan replace the instantaneous Coulomb interaction bya Lorentz invariant interaction without changing the re-sponse functions. Thus the q correction to σ xy is reliablefor graphene FQH states at the zeroth Landau level. Summary. —We constructed an effective field theorydescription for relativistic quantum Hall liquids. The the-ory contains one additional topological coefficient besidesthe Hall conductivity. For states at the zeroth Landaulevel, there is an additional holomorphic constraint whichcompletely determines the effective Lagrangian in termsof the two topological numbers. Our formalism elucidatesthe intricate relationship between topology and geome-try in the problem. In particular, there is an importantterm in the Lagrangian with a topologically determinedcoefficient ( κ ), but the term itself depends nontriviallyon the metric of space. It would be interesting to extend the theory to higherorders in momentum expansion and compare the predic-tions for electromagnetic and gravitational responses tothe nonrelativistic case [19, 20].We note that on a manifold with a boundary, conser-vation of the topological current requires the addition ofa boundary action. However, this is only possible if thefield u µ is parallel to the boundary. If this is not thecase, the gauge non-invariance of the action should beabsorbed by the anomaly of the boundary theory. Theimplication of this should be further investigated.Finally, our effective description may serve as a bench-mark for holographic models of quantum Hall effect,many of which have underlying Lorentz invariance (see,e.g., Ref. [21]).The authors thank Nicholas Read and Paul Wiegmannfor discussions. This work is supported, in part, by DOEgrant DE-FG02-13ER41958. S.G. is supported in part byNSF MRSEC grant DMR-0820054. D.T.S. is supportedin part by a Simons Investigator grant from the SimonsFoundation. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, andA. A. Firsov, Nature , 197 (2005).[2] Y. Zhang, Y. -W. Tan, H. L. Stormer, and P. Kim, Nature , 201 (2005).[3] X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y. Andrei,Nature , 192 (2009).[4] K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer,and P. 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