Effective Field Theory of Fractional Quantized Hall Nematics
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Effective Field Theory of Fractional Quantized Hall Nematics
Michael Mulligan, Chetan Nayak, and Shamit Kachru Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA Microsoft Station Q, Santa Barbara, CA 93106, USA Department of Physics, Stanford University and SLAC, Stanford, CA 94305, USA
We present a Landau-Ginzburg theory for a fractional quantized Hall nematic state and the transition to itfrom an isotropic fractional quantum Hall state. This justifies Lifshitz-Chern-Simons theory – which is shownto be its dual – on a more microscopic basis and enables us to compute a ground state wave function in thesymmetry-broken phase. In such a state of matter, the Hall resistance remains quantized while the longitudinalDC resistivity due to thermally-excited quasiparticles is anisotropic. We interpret recent experiments at Landaulevel filling factor ν = 7 / in terms of our theory. Introduction. A fractional quantized Hall nematic (FQHN) is a phase in which a fractional quantized Hallconductance coexists with the broken rotational symmetrycharacteristic of a nematic, as in the model introduced in Ref.1. The idea that a phase of matter could have both topologicalorder and conventional broken symmetry is not new; forinstance, quantum Hall ferromagnets are another example[2, 3]. See [4] for a more recent discussion in a relatedsystem. However, the FQHN has the unusual feature thatthe broken symmetry and the topological order are equallyimportant for determining the system’s transport properties.Furthermore, the model also predicts an unusual quantumcritical point separating the FQHN from an ordinary isotropicfractional quantum Hall state.Remarkably, a recent experiment may have observed aFQHN [5]. An in-plane magnetic field B k is applied to the ν = 7 / fractional quantum Hall plateau. When the angle θ between the total magnetic field and the normal is zero, thesystem is essentially isotropic: for T < mK, R xx ≈ R yy .At T = 15 mK, there is a well-developed Hall plateau with R xy = R yx = he . At T > mK, there is a small ( ≈ ) difference between R xx and R yy , which may be due todevice geometry, alignment of the contacts, or a small intrinsicanisotropy acquired by the samples during the growth process.For tilt angles θ > ◦ and T < mK, R xy = R yx = he while R xx − R yy increases with decreasing temperature. Infact, dR xx /dT < while dR yy /dT > at the lowest ob-served temperatures. Thus, this experiment finds transportwhich is reminiscent of the nematic phases found at half-filling of higher Landau levels, such as ν = 9 / , / , . . . without an in-plane field [6, 7] and also at ν = 5 / and / in the presence of an in-plane field [8, 9], except for one verystriking difference: the Hall resistance remains quantized inthe anisotropic phase.We interpret these observations as a slightly rounded tran-sition between an isotropic fractional quantum Hall phase at θ < θ c < ∼ ◦ and an FQHN at θ > θ c . The rounding ofthe transition is caused by the in-plane field. We believe itto be a weak rotational symmetry-breaking field because thesystem is in an isotropic metallic phase for even larger tilts atthe nearby fraction ν = 5 / [10] and because the anisotropyat 300 mK actually decreases as the tilt is increased from ◦ to ◦ . We conjecture that the most important effect of thein-plane field is to vary the effective interaction between theelectrons, thereby driving the (almost) spontaneous breakingof rotational symmetry. We are thus led to apply our model[1] to this experiment.To this end, we give a more microscopic derivation of ourmodel as a Landau-Ginzburg theory. We thereby recover atheory which is equivalent, through particle-vortex duality, tothe effective field theory introduced in Ref. 1. In order tocompare theory and experiment more closely, we extend ourprevious analysis of zero-temperature, finite-frequency trans-port to finite-temperature DC transport; in order to do this,we must enlarge our model to include the effects of gappedcharged quasiparticles. The development of nematic order in-duces strongly temperature-dependent anisotropy in the quasi-particle effective masses. We predict that both longitudinalconductances will eventually vanish at the lowest tempera-tures, although one of them will have non-monotonic temper-ature dependence at slightly higher temperatures. We finallymake predictions for transport at and near the transition point. Landau-Ginzburg Theory.
One can map the problem ofspinless planar electrons in a transverse magnetic field B withCoulomb repulsion, to an equivalent system of a bosonic orderparameter φ of unit charge coupled to a Chern-Simons gaugefield a µ [11]. The action takes the form: S LG = Z d xdt (cid:16) φ † i ( ∂ t − i ( A t + a t )) φ − m e | ( ∂ i − i ( A i + a i )) φ | + ν π ǫ αβγ a α ∂ β a γ − Z d y ( φ † φ ( x ) − ¯ ρ ) V ( x − y )( φ † φ − ¯ ρ ) (cid:17) . (1) A µ is the background electromagnetic field satisfying ǫ ij ∂ i A j = B ; ¯ ρ is the mean charge density of bosons (orequivalently, electrons); m e is the electron band mass; V(x)is a general two-body potential; and the Chern-Simons gaugefield a µ attaches πν − units of statistical flux to each parti-cle [12]. In particular, for ν − an odd integer, the resultingAharonov-Bohm phases transmute the bosons into fermions.We assume that the low-energy effective theory for dis-tances longer than the magnetic length, obtained by integrat-ing out short-distance fluctuations of φ , a µ , has the same formas the microscopic action (1), but with the bare microscopicparameters /m e and V ( x − y ) replaced by renormalizedones, ¯ r and V eff ( x − y ) . Such an ansatz allows one to de-rive many of the properties of the standard fractional quantumHall states [11, 13]. Here, we will make the same ansatz, butwithout assuming that ¯ r remains positive. We note that eventhe ‘microscopic’ action (1) must be viewed as an effectivelow-energy action that describes the partially filled N = 1 Landau level with ν = 2 + 1 / . The electrons are confinedto a quantum well of finite-width; a strictly two-dimensionaltheory is an effective theory at energy scales far below thesplitting between energy sub-bands for motion perpendicu-lar to the plane. Thus, the application of the in-plane field B || , through its modification of the motion perpendicular tothe plane, will modify the parameters in S LG . Consequently,the effective parameters at distances longer than the magneticlength will also be modified, but not in a simple or, at present,transparent way. It is easy to check that reasonable local vari-ations of V eff do not cause qualitative changes to the physicsof (1) [13]. We leave to a future study the question of higher-body potential terms resulting from a projection of the degreesof freedom into a specific Landau level.Therefore, we conjecture that as the in-plane field B || isvaried, the most significant variation is of the parameter ¯ r . Inother words, we study the instabilities of (1) as the kineticstructure of the theory is modified.Since we will be considering ¯ r < , we add the followingterm with c > to the action in order to maintain stability ofthe vacuum: δS = − c Z d xdt | ( ∂ i − i ( A i + a i )) φ | . (2)This theory exhibits a transition between an isotropic frac-tional quantum Hall phase, when ¯ r > , and an anisotropicphase with well-quantized Hall conductance (after inclusionof disorder or a lattice) when ¯ r < , just as in [1]. The twophases are separated by a quantum critical point with z = 2 dynamical scaling, arising at ¯ r = 0 . Kohn’s Theorem.
On might object to any variation of ¯ r from its bare value on the basis of Kohn’s theorem [14]. (SeeSection 5 of [13] for a discussion.) In a Galilean-invariantsystem of N identical mutually interacting particles of unitcharge and mass m e subject to a constant external magneticfield B , Kohn’s theorem states that the density-density corre-lation function has the low momentum limit, lim q → h ρ ( ω, q ) ρ ( − ω, − q ) i q = m e ω − ω c , (3)The locations of the two poles are determined by the cyclotronfrequency ω c = B/m e . For fixed B , the cyclotron frequencyis determined by the bare mass of the particles, independentof the their relative interactions. The residues of the poles areequal to ± / B . The form of this correlator is ensured bya Ward identity and satisfies an f-sum rule. Kohn’s theoremroughly states that the center-of-mass of the system alwaysdecouples from the relative coordinate motion of the parti-cles; it effectively behaves as a single charge N particle of mass N m e , exhibiting circular motion at a frequency ω c in abackground magnetic field B . In quantum Hall systems, thequantum well explicitly breaks translational symmetry in the z -direction (i.e. perpendicular to the plane). However, thein-plane center-of-mass motion still decouples from the otherdegrees of freedom, so long as the magnetic field is strictlyperpendicular to the plane. Thus, Kohn’s theorem holds evenin this case.If we now compute the density-density correlator using theaction (1), we find precisely the form dictated by Kohn’s theo-rem (3). However, the modification /m e → ¯ r would changethe location of the pole. This manifestly constitutes a violationof Kohn’s theorem.However, the experiment of Ref. 5 does not satisfy the as-sumptions of Kohn’s theorem. The large in-plane field, com-bined with the confining well potential (perpendicular to theplane), manifestly breaks Galilean invariance and does not al-low a decoupling of the center-of-mass mode. The in-planefield couples motion along the z -direction to motion in theplane, while the confining potential in the z -direction cou-ples the z -component of the center-of-mass position to the z -component of the relative coordinates. The N = 1 Landaulevel in the devices considered in Ref. 5 is particularly sus-ceptible to perturbations mixing planar and z -direction mo-tion because the gap to the N = 0 Landau level of the nextquantum well sub-band is small [10].In summary, our theory, in which ¯ r is not fixed, appliesto situations, such as those in the experiment of Ref. 5, inwhich Kohn’s theorem does not hold. Our theory cannot de-scribe a fictional system in which the two-dimensional layeris infinitely-thin and the transition is driven purely by tuningthe inter-electron interaction (without any in-plane field) sincesuch a system would necessarily satisfy Kohn’s theorem. Tomake our point more concrete, we show in the Appendix that,as a result of the violation of the conditions of Kohn’s the-orem, the location of the cyclotron pole can vary as B || isincreased. Duality.
We have computed the long wavelength trans-port properties of the various phases of (1) directly from theLandau-Ginzburg theory and found them to exactly match theresponse determined from the Lifshitz-Chern-Simons (LCS)theory of [1]. This is expected because there is a low-energy equivalence between the (more) microscopic theory(1), (2) and the LCS theory which we demonstrate by ex-panding about the relevant ground state in the three cases ¯ r > , ¯ r = 0 , ¯ r < , and mapping the low-energy theory tothe action governing the similar phase of the LCS theory, us-ing particle-vortex duality [15]. For convenience, we assumea short-ranged repulsive interaction, V eff ( x ) = V δ ( x ) with V > , throughout. This choice is motivated by expectedscreening effects of the microscopic electrons. Nevertheless,the precise form of V eff plays very little role in the considera-tions below as long as it is local.For ¯ r ≥ , there is a saddle point configuration given by h φ † φ i = ¯ ρ , h a µ i = − A µ , with filling fraction ¯ ρ/B = ν/ π .The low-energy action for fluctuations about this ground statewhen ¯ r > is S eff (¯ r >
0) = Z d xdt (cid:16) − δρ ( ∂ t θ − δa t ) − ¯ r ρ ( ∂ i θ − δa i ) + ν π ǫ αβγ δa α ∂ β δa γ − V ( δρ ) (cid:17) . (4) δρ and θ govern the fluctuations of the norm and phase of thebosonic order parameter φ , δa µ represents the fluctuation ofthe Chern-Simons gauge field, and we have taken the back-ground field fluctuations to vanish. S eff (¯ r > can be rewrit-ten by introducing the field J i (the spatial components of the U (1) current associated with the background gauge field): S eff (¯ r >
0) = Z d xdt (cid:16) − δρ ( ∂ t θ − δa t ) − J i ( ∂ i θ − δa i )+ 12¯ r ¯ ρ J i + ν π ǫ αβγ δa α ∂ β δa γ − V δρ (cid:17) . (5)Now, integrating out J i trivially reproduces the previous La-grangian; but we can instead find a dual description of thetheory by keeping J i in the Lagrangian and integrating outthe other degrees of freedom. θ appears linearly and functionsas a Lagrange multiplier ensuring conservation of J µ . We canguarantee this by rewriting J µ = π ǫ µντ ∂ ν n τ . Writing thetheory in terms of n , and integrating out δa µ , we find S LCS (¯ r >
0) = Z d xdt (cid:16) g e ( ∂ i n t − ∂ t n i ) − g m ( ∂ i n j − ∂ j n i ) + 14 πν ǫ αβγ n α ∂ β n γ (cid:17) . (6)This is Maxwell-Chern-Simons theory at level ν − with g e =4 π ¯ r ¯ ρ and g m = π V . This matches the behavior of the LCStheory of [1] in the fractional quantum Hall phase (¯ r > .When ¯ r = 0 (the z = 2 critical point), it is necessary tokeep the δS term. Nevertheless, the dualization proceeds al-most identically. The leading terms in the expansion of theaction in small fluctuations about the saddle point are S eff (¯ r = 0) = Z d xdt (cid:16) − δρ ( ∂ t θ − δa t ) − J i ∂ (cid:0) ∂ i ∂ j ( ∂ j θ − δa j ) − c ¯ ρ J i (cid:1) + ν π ǫ αβγ δa α ∂ β δa γ − V δρ (cid:17) . (7)This is a formal expression because of the inverse Laplacian inthe second term. Current conservation, which is imposed bythe θ equation of motion, allows us to replace J with the emer-gent gauge field n . Imposing the gauge conditions δn = 0 and ∂ i n i = 0 , and integrating out a µ , we obtain a gauge-fixedversion of the LCS Lagrangian. Covariantizing the gauge-fixed action yields S LCS (¯ r = 0) = 1 g Z d xdt (cid:16) κ ∂ ( ∂ i n t − ∂ t n i ) −
12 ( ∂ i n j − ∂ j n i ) + g πν ǫ αβγ n α ∂ β n γ (cid:17) , (8)where κ = 2 c ¯ ρV , and g = 4 π /V . This is precisely thetheory governing the critical point in [1], with the e i field inte-grated out. (The z = 2 nature of the e i field action ∼ ( ∂ i e j ) in that theory, gives rise to the peculiar inverse Laplacian inthe action above).Lastly, we discuss the anisotropic ¯ r < phase. The groundstate is still homogeneous, h φ † φ i = ρ ′ ρ ′ = ¯ ρ + | ¯ r | / cV ,but anisotropic, since h a µ i = − A µ − v µ , with, v = 0 and v i = | ¯ r | / c . At this saddle point, the chemical potential isshifted upwards. The leading terms in the low-energy action,expanding around the symmetry-breaking vacuum with thecondensate lying along the x-axis, take the form (where againwe have introduced a current J i ) S eff (¯ r <
0) = Z d xdt (cid:16) − δρ ( ∂ t θ − δa t ) − J x [( ∂ x θ − δa x ) − | ¯ r | ρ ′ J x )] − J y ∂ y [ ∂ y ( ∂ y θ − δa y ) − cρ ′ J y ]+ ν π ǫ αβγ δa α ∂ β δa γ − V δρ (cid:17) . (9)The θ equation of motion imposes current conservation for thedensity δρ and current J i . Integrating out δa µ once more, weobtain S LCS (¯ r <
0) = 1 g Z d xdt (cid:16) κ ∂ ( ∂ x n t − ∂ t n x ) + g | r | ( ∂ y n t − ∂ t n y ) −
12 ( ∂ i n j − ∂ j n i ) + g πν ǫ αβγ n α ∂ β n γ (cid:17) , (10)where κ = 2 cρ ′ V , | r | = 4 | ¯ r | ρ ′ V , and g is as above. Thisagrees with the LCS theory in the anisotropic phase in [1]. It isgapless, as may be seen from the n i propagators, which evincea contribution from the Goldstone mode for spontaneously-broken SO(2) rotational symmetry. Note that a symmetrybreaking vacuum along the x-direction of the LG theory corre-sponds to a symmetry breaking vacuum along the y-directionin the LCS theory.The effects of disorder are implemented by allowing spa-tially varying ¯ r ( x ) in the Landau-Ginzburg description. Thelow-energy equivalence implies that introducing such disorderin the LG theory will lift the Goldstone mode of the sponta-neously broken SO(2) symmetry and will lead to a quantizedHall conductance, as it did in in the anisotropic phase of theLCS theory [1]. The pseudo-Goldstone mode should be vis-ible in low-energy Raman scattering experiments. Alterna-tively, we could introduce a lattice by including terms in theaction which explicitly lower the rotational symmetry fromSO(2) to D . In this case, the third term in (9) takes, instead,the form J y [( ∂ y θ − δa y ) − | ¯ r ′ | ρ ′ J y )] , where ¯ r ′ is propor-tional to the effective lattice potential; consequently, there isno Goldstone mode for rotational symmetry-breaking. Ground State Wave Function in the ¯ r < Phase.
We nowcompute the ground state wave function in the ¯ r < phasefollowing the method described in [13]. For D symmetry,which is more experimentally-relevant, it takes the form: Ψ( z i ) = Y i We now compute the con-tribution to the finite temperature DC conductivity tensor fromthermally-excited charged quasiparticles. The LCS theoryis more convenient than the equivalent Landau-Ginzburg de-scription because (massive) charged quasiparticles are vor-tices of the Landau-Ginzburg theory and fundamental parti-cles of the LCS theory. This computation demonstrates thathighly-anisotropic finite-temperature transport can result fromour model but is not an attempt to give a precise fit to exper-imental data, which would require a more careful analysis ofthe effects of disorder, the lattice, and subleading interactions.We include the effects of the massive quasiparticles byadding to the ‘first-order’ form of the LCS action, S LCS = 1 g Z d xdt (cid:16) e i ∂ t n i + n t ∂ i e i − r e i − κ ∂ i e j ) − 12 ( ǫ ij ∂ i n j ) + g πν ǫ µνλ n µ ∂ ν n λ − λ e i ) + α e x + e y ) + 12 π ǫ µνλ A µ ∂ ν n λ (cid:17) , (12)the matter action, S matter = Z d xdt Φ ∗ (cid:16) i∂ t + n t − ∆ + ( i∂ i + n i ) + u e x ( i∂ x + n x ) + u e y ( i∂ y + n y ) (cid:17) Φ . (13)Thus, we study the total action S = S LCS + S matter . In S LCS , we have not integrated out the e i field. At tree-level,the quartic e terms in S LCS are marginal; the operator withcoefficient λ preserves the full spatial SO (2) symmetry, whilethe operator with coefficient α explicitly breaks it down to D .We assume α is small and positive, reflecting a small explicitbreaking of SO (2) inherent in the real material. The last termin S LCS is the coupling to the external electromagnetic field A µ . The statistical gauge field endows the massive quasipar-ticles represented by Φ with their fractional statistics. Theirrelevant energy-energy coupling parameterized by u is theleading term that directly communicates the D spatial rota-tional symmetry breaking of the r < ground state to thematter field. By ignoring a possible e i | Φ | coupling, we areassuming that the magnitude of the symmetry-breaking orderparameter h e i i in the r < regime is much less than the quasi-particle gap ∆ .We concentrate on the finite temperature DC conductivitywhen r < , however, the actual expressions obtained arevalid for all r , if interpreted appropriately. (The functionalform of the optical conductivity was already determined in[1]; it differs in the two phases, and shows striking features atthe critical point.) Let us assume that h e x i is non-zero in the r < regime at zero temperature. To quadratic order, S matter becomes S matter = Z d xdt (cid:16) Φ ∗ ( i∂ t + n − ∆)Φ+ Φ ∗ ((1 + u h e x i )( i∂ x + n x ) + ( i∂ y + n y ) )Φ (cid:17) . (14)At temperatures less than ∆ , we can integrate out the quasi-particles and write an effective action solely in terms of thefields appearing in S LCS . It is convenient to express the re-sulting effective action in Fourier space, obtaining S = S LCS + 12 Z d qdω n µ ( − ω, − q )Π µν ( ω, q ) n ν ( ω, q ) . (15)The kernel Π µν appearing in the second term contains thequasiparticle contribution to the conductivity, σ qpij , σ qpij = lim ω → iω h j i ( − ω, j j ( ω, i = lim ω → iω Π ij ( ω, q = 0) , where j i ( ω, q ) = δS matter δn i ( − ω, − q ) is the quasiparticle current op-erator. Computing the DC conductivity from (15), we find: σ ij = 12 π lim ω → ǫ ik ǫ jl ( kǫ kl + 2 πσ qp kl ) − . (16)This implies that ρ xy = − ρ yx = k while ρ xx = 2 πσ qp yy and ρ yy = 2 πσ qp xx . Thus, we see that one of the most remarkablefeatures of the experimental results in Ref. 5 has a natural ex-planation in our model: ρ xy remains quantized while ρ xx , ρ yy can be temperature-dependent if σ qp is diagonal. Secondly,we note that the anisotropy in the DC resistivity comes en-tirely from the induced anisotropy in the quasiparticle kineticenergy. By contrast, the transport due to the fluctuations in S LCS showed frequency-dependent anisotropy that resultedfrom subleading terms in the gauge field action [1]. So thereis additional anisotropy in the AC transport that is not presentin the DC transport. In particular, AC transport shows low fre-quency conductivity that vanishes linearly and cubically alongthe two orthogonal directions. The two types of anisotropycome from different physical mechanisms – anisotropy in thegauge field kinetic energy versus anisotropy in the quasiparti-cle kinetic energy – although the ultimate cause is the same.It remains to calculate Π µν . We summarize the calculationof Π ii for spatial i below. We introduce dissipation by assum-ing the quasiparticles have an elastic scattering lifetime equalto τ and single-particle gap ∆ / . Due to the anisotropy in-troduced by h e x i in S matter , the longitudinal current-currentcorrelation functions along the two spatial directions are re-lated, h j x ( ω n , j x ( − ω n , i = (1 + u h e x i ) f ( iω n , T ) h j y ( ω n , j y ( − ω n , i = (1 + u h e x i ) − f ( iω n , T ) , where f ( iω n , T ) = Tπ X m Z dqq G ( iω n + m , q ) G ( iω m , q ) (17)and after rescaling q x to obtain the rotationally invariant form, G − ( iω m , q ) = iω m − ∆ / − q + iπτ Arg (∆ / − iω n ) . (18)Here, we use the fact that the imaginary part of the correla-tion function (which gives the real part of the conductivity) iscutoff independent so that the rescaling of the cutoffs can beneglected. The diamagnetic contribution to the sum vanishes.Replacing the sum over Matsubara frequencies, ω m =2 πT , by a contour integral, we have Im f ( ω + iδ, T ) = π ωT τ e − ∆ / T , where we have made use of the large τ limit.Therefore, the longitudinal quasiparticle DC conductivities, σ qp xx,yy = π (1 + u h e x i ) ± / T τ e − ∆ / T , (19)where the + ( − ) refers to σ qp xx ( σ qp yy ). Inserting these expres-sions into (16), we find that ρ xx − ρ yy ≈ π u h e x i T τ e − ∆ / T + O ( e − ∆ /T ) . (20)(19) and (20) are assumed to be valid at temperatures T < ∆ / , but high enough such that variable-range hopping canbe ignored. Thus, we have demonstrated theoretically theexistence of a FQHE that has both anisotropic zero temper-ature AC transport as well as anisotropic finite temperatureDC transport. H m K L R e s i s ti v it y H W L FIG. 1: Longitudinal resistivities ρ xx,yy along the easy x-axis (red)and hard y-axis (blue) obtained from the conductivities in (19) areplotted versus temperature. The microscopic parameters entering theexpressions in (19) are found phenomenologically from the resis-tances measured in [5] as explained in the main text. The precise temperature dependence of the DC resistiv-ity is determined by the behavior of h e x i and the quasipar-ticle scattering time τ . At temperatures near the roundedfinite-temperature phase transition, h e x i can be identified withthe order parameter of the particular finite-temperature phasetransition. We expect this classical phase transition to be de-scribed by a theory lying on the Ashkin-Teller half-line orequivalently, the moduli space of the c = 1 Z orbifold theory.All theories along the line possess a global Z symmetry andan order parameter for the Z -broken phase with critical expo-nent / < β < ∞ [17] [18]. Since the Kosterlitz-Thouless critical point lies at the boundary point of this half-line, weexpect the particular critical theory governing the transitionto be largely determined by the degree of SO (2) rotationsymmetry-breaking in the experimental system. Since the in-plane field appears to be a weak symmetry-breaking field, asdiscussed in the Introduction, we expect the transition to befairly sharp.For definiteness below, we take the transition to be in theuniversality class of the critical four-state Potts model whichlies at the boson radius r = 1 / √ on the orbifold line.In Fig.1 are plotted the resistivities, ρ xx,yy . The order pa-rameter for this transition hOi ∼ h e x i ∼ ( − t ) / , where t = ( T − T c ) /T c [19]. (There is not a significant qualitativedifference in the plots if the system has the full SO (2) rota-tion symmetry so we have suppressed a separate discussion ofthe KT transition.)The microscopic parameters in (19) are determined us-ing the resistance measurements performed in [5] as follows.For simplicity, we assume rotationally invariant relations be-tween resistivities and resistances, ρ xx = f ( L x , L y ) R xx and ρ yy = f ( L x , L y ) R yy , where f ( L x , L y ) is some function de-pending on the sample lengths L x,y . In other words, we ig-nore possible geometrical enhancements that may be presentin translating between these two sets of quantities. (See [20]for a discussion of the importance of this distinction in thecontext of anisotropic transport in ν = 9 / , / , ... half-filled Landau levels [6].) The measured temperature depen-dences of ρ xx and ρ yy at zero in-plane field are fit well by Ar-rhenius plots, ρ xx,yy = A exp( − ∆ / T ) , with ∆ = 225mK and A = 10 − h/e . We continue to use these values inthe anisotropic regime, which we identify with the region ¯ r < , when the in-plane field is of sufficient magnitude.The temperature-independent value of A over the temperaturerange < T < implies a quasiparticle scat-tering lifetime τ ∼ T . An estimate of . × − h/e forthe maximum value of R xx observed at a tilt angle of ◦ and achieved as T → from above implies u h e x i ∼ − t ) / with T c = 50mK . We stress that the fitting of pa- H mK L R e s i s ti v it y H W L FIG. 2: Longitudinal resistivity ρ yy along the hard y-axis for threeseparate values of the parameter m = u h e x i / ( − t ) / . From top tobottom (blue, red, yellow), m = 50 , , . rameters used to obtain Fig.1 is meant to be as optimistic aspossible so as to determine the microscopic parameters of ourtheory if it is to apply to the experiment.The height of the peak observed in Fig.1 dependssensitively on the temperature-independent value of ( u h e x i ) / ( − t ) / = ( ur ) / (( λ − α ) ( − t ) / ) . In Fig.2, weplot ρ yy for three different values of this parameter startingwith the value used in Fig.1. As the figure indicates, the peakdecreases as this parameter is lowered. H m K L R e s i s ti v it y H W L FIG. 3: In-plane field rounded longitudinal resistivities ρ xx,yy alongthe easy x-axis (red) and hard y-axis (blue) obtained from the con-ductivities in (19) are plotted versus temperature. A mean fieldcrossover function has been used in the expression for the resistiv-ities. We expect the in-plane magnetic field B || to act as a smallsymmetry-breaking field on this finite temperature transition.This will lead to a rounding of the resistivity curves in Fig.1.The order parameter now behaves as h e x i ∼ B /δ || g ± (cid:16) ( ± t ) β B /δ || (cid:17) , (21)where the ± is determined by the sign of t , and the criticalexponents β = 1 / at the four-state Potts point and δ = 15 along the orbifold line. Integral expressions for the scalingfunctions g ± are known [21]; a precise functional form, how-ever, is not. Scaling dictates that g − ( x = 0) = g + ( x = 0) arefinite and non-zero, g − ( x ) ∼ x as x → ∞ , and g + ( x ) = 0 for x > x crit ∼ .Since we do not have an explicit functional form for g ± ,let us simply model the transition using mean field theory inorder to obtain a picture for the rounding of the transition.(We do not mean to imply that the crossover function for the Z transition is in any way similar φ mean field crossoverfunction. Rather, we only want a picture for how the tran-sition might be rounded.) The φ mean field critical expo-nents β = 1 / and δ = 3 . Specification of the free energy, F = T c tφ + φ − φh , allows the calculation of g ± viaminimization of F with respect to φ . We select the root, h φ i = h / (cid:16) − / x + 2 / (9 + p 81 + 12sgn( t ) x ) / / (9 + p 81 + 12sgn( t ) x ) / ) (cid:17) , (22) where x = | T − T c | / /h / . It satisfies the scaling re-quirements detailed in the previous paragraph. Substituting u h e x i = 7 h φ i at h = 1 into our expressions for the re-sistivities using the same values for the overall scale of theresistivity and behavior of the scattering time τ as above, wefind Fig. 3.When r ≥ , the form (16) and (19) of the finite tempera-ture DC conductivity matrix still holds. However, h e x i is zeroand the longitudinal conductivity along the two directions co-incides. Note that non-zero AC conductivity at the r = 0 critical point requires disorder exactly like the r < regime[1]. Discussion. In this paper, we have given an explanationof one of the most striking aspects of the data of Ref. 5:the anisotropy of the longitudinal resistances coexisting withquantized Hall resistance. Our theory further predicts that,while one of the resistances will increase with decreasingtemperature at temperatures just below the (rounded) finite-temperature phase transition at which nematic order develops,as observed [5], both longitudinal resistances will, eventually,go to zero at the lowest temperatures, which is yet to be ob-served. Transport beyond the linear regime, the nature of themassive quasiparticles in the anisotropic phase, and a morecomplete determination of the values of the parameters in theeffective Lagrangian in terms of microscopic variables are in-teresting open problems.We thank J. Chalker, J. P. Eisenstein, E. Fradkin, S. Kivel-son, H. Liu, J. McGreevy, S. Shenker, S. Simon and J. Xia forhelpful discussions, and thank the Aspen Center for Physicsfor hospitality. M. M. acknowledges the hospitality of theStanford ITP, the Galileo Galilei ITP and INFN, and OxfordUniversity while this work was in progress. M. M. was sup-ported in part by funds provided by the U. S. Department ofEnergy (D. O. E.) under cooperative research agreement DE-FG0205ER41360. C.N. was supported in part by the DARPA-QuEST program. [1] M. Mulligan, C. Nayak, and S. Kachru, Phys. Rev. B ,085102 (2010).[2] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi,Phys. Rev. B , 16419 (1993).[3] S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Ty-cko, Phys. Rev. Lett. , 5112 (1995).[4] D. A. Abanin, S. A. Parameswaran, S. A. Kivelson, and S. L.Sondhi, Phys. Rev. B , 035428 (2010).[5] J. Xia, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West,arXiv:1109.3219 (2011).[6] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, andK. W. West, Phys. Rev. Lett. , 394 (1999).[7] E. Fradkin, S. Kivelson, M. Lawler, J. Eisenstein, andA. Mackenzie, Annu. Rev. Condens. Matter Phys. , 153(2010).[8] W. Pan, R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeif-fer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. , 820(1999). [9] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, andK. W. West, Phys. Rev. Lett. , 824 (1999).[10] J. Xia, V. Cvicek, J. P. Eisenstein, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. , 176807 (2010).[11] S. C. Zhang, T. H. Hansson, and S. Kivelson, Phys. Rev. Lett. , 82 (1989).[12] One can rewrite this action in terms of ˜ a µ = ν − a µ so that theChern-Simons term will have an integer coefficient ν − and thetheory can be defined on arbitrary manifolds.[13] S. C. Zhang, Int. J. Mod. Phys. B6 , 25 (1992).[14] W. Kohn, Phys. Rev. , 1242 (1961).[15] M. P. A. Fisher and D. H. Lee, Phys. Rev. B , 2756 (1989).[16] K. Musaelian and R. Joynt, J. Phys. , 105 (1996).[17] The critical exponent β of the order parameter field alongthe c = 1 orbifold line can be determined using the relation β = ν ( η/ between critical exponents. ν = 2 − x where x is the scaling dimension of the ‘energy’ operator of the CFT. / < ν < ∞ as the boson radius ∞ > r > / √ .The twist operator of the CFT has scaling dimension fixed at η/ / along the critical line (since it simply acts to changeboundary conditions). The critical four-state Potts model and Z parafermion CFT have β = 1 / and / , respectively,while β formally diverges at the KT point where the correla-tion length diverges as exp(( T /T c − − / ) as the criticaltemperature is approached. The exponent characterizing the de-cay of the order parameter at the critical point with respect toa symmetry-breaking field, δ = (4 − η ) /η = 15 , along thecritical line.[18] Note that a conventional Landau-Ginzburg description of theclassical phase transition between a nematically ordered andisotropic state is described by a free energy functional of a sym-metric, traceless n × n matrix N αβ . Landau-Ginzburg theorypredicts a continuous transition when n = 2 while a first-ordertransition is expected when n ≥ because of an allowed cubicterm in the free energy. The Z -invariant critical point we studydescribes a continuous symmetry-breaking transition with vec-tor order parameter h e i i . It is the Z -invariant director h e i i thatenters physical quantities like the resistivity and allows the de-scription of a continuous nematic transition in terms of certainresponse functions.[19] F. Y. Wu, Rev. Mod. Phys. , 235 (1982).[20] S. H. Simon, Phys. Rev. Lett. , 4223 (1999).[21] S. L. Lukyanov and A. B. Zamolodchikov, Nucl.Phys. B493 ,571 (1997). Appendix. In this Appendix, we show that, as a result ofthe violation of the conditions of Kohn’s theorem, the locationof the cyclotron pole can vary as B || is increased. (It is alsopossible that additional spectral weight shows up at O ( q ) ,but we shall not study this possibility in any detail.) We dothis by identifying the leading pole in the density-density re-sponse with the gap between the lowest and first-excited statesin the center-of-mass part of the quantum mechanical many-body wave function. This identification is correct for vanish-ing in-plane field B || and we believe it holds for perturbativelysmall values of B || as well, where separation of variables intocenter-of-mass and relative coordinates is well-defined. Ofcourse, this simple example can, at best, give us a few cluesabout the real system, which is far more complicated. One ofthese is that the pole moves towards the origin (at least ini-tially) as B || is increased from zero. This justifies our studyof the model (1) with varying ¯ r . We begin with the quantum mechanics problem of two mu-tually interacting three-dimensional electrons in a backgroundmagnetic field. This can be easily generalized to an arbitrarynumber of particles. We take their motion along the x − y plane to be unconstrained but subject to a confining potentialalong the z -direction. The Hamiltonian is H = X i =1 , h m e (cid:16) ∂ x i + ( i∂ y i − ( Bx i + B || z i )) − ∂ z i (cid:17) + A ( z i ) i + V ( | x − x | ) , (23)where x i = ( x i , y i , z i ) labels the position of the two parti-cles. (The gauge chosen for the vector potential is consis-tent with a spatial geometry that is of a finite length alongthe x, z -directions, and infinite along the y -direction. Note,however, we are essentially ignoring the finite length alongthe x -direction in the discussion below so we can think of itas being large compared to the length scale provided by theconfining potential along the z -direction.)We consider the component of the magnetic field lyingalong the x -direction to be a perturbation to the system. Itis convenient to switch coordinates to the center-of-mass andrelative coordinate frame. Choosing X = 12 ( x + x ) , ρ = x − x , (24)the Hamiltonian becomes H = 12(2 m e ) (cid:16) − ∂ X x + ( i∂ X y − BX x + B || X z )) − ∂ X z (cid:17) + 12( m e / (cid:16) − ∂ ρ x + ( i∂ ρ y − 12 ( Bρ x + B || ρ z )) − ∂ ρ z (cid:17) + X ± A ( X z ± ρ z ) + V ( | ρ | ) . (25)Aside from the confining potentials A ( z , ) = A ( X z ± ρ z ) , the center-of-mass and relative coordinates are decou-pled.At B || = 0 , motion in the z -direction decouples from themotion in the plane and we are left with a collection of two-dimensional electrons indexed by their band or energy alongthe z -direction. (Here, we are assuming the pair potential onlydepends on the separation of the electrons in the x − y plane;the well width is assumed small compared to the magneticlength B − / . This is not the case in the experiments of Refs.5, 10, so the violations of Kohn’s theorem will be larger thanin our simple model.) Given a z -eigenfunction, the center-of-mass part of the wave function executes oscillatory motion atthe cyclotron frequency B/ (2 m e ) . This is the generalizationof Kohn’s theorem to the situation where electrons are con-fined along the direction parallel to the magnetic field. Whenthe spacing between the energy levels of the z i -eigenfunctionsgreatly exceeds the cyclotron frequency, it is possible to ig-nore higher sub-bands when considering low-energy proper-ties of the system. However, this is not the case in the experi-ments of Refs. 5, 10.Now consider B || = 0 . There is now a direct mixing be-tween motion in the z -direction and motion in the plane. Thismixing mediates a coupling at higher orders in B || betweenthe planar center-of-mass and relative degrees of freedom.Thus, there there is no requirement of a pole at the cyclotronfrequency in the density-density correlator. This follows fromthe fact that the full three-dimensional Galilean symmetry (ex-cept for X y , ρ y translations) is broken when there is both anin-plane field and a non-zero confining potential along the di-rection normal to the plane. If either the confining well orin-plane field are removed, there will be a Kohn pole at ω c .We would like to better understand departures of the polefrom the cyclotron frequency in this more general situationwith non-zero in-plane field. Namely, we would like to knowhow the location of the pole varies with B || . We can ob-tain some intuition by studying a special case for the formof the confining potential. Take the confining potential to bequadratic, A ( z ) = λ z . Then, because A ( X z + 12 ρ z ) + A ( X z − ρ z ) = λ ( X z + 14 ρ z ) , (26)the center-of-mass and relative coordinate motion are still de-coupled. This decoupling is not generic; a quartic potential,for example, couples X z and ρ z together. However, we willargue that some conclusions drawn from the quadratic caseare general.We know that at B || = 0 , the Kohn pole corresponds tothe splitting between the ground and first excited states of thecenter-of-mass motion. The relative coordinate is irrelevantboth when B || = 0 and for a quadratic electric potential, andso we drop it from our discussion. Thus, the Hamiltonian westudy perturbatively in B || is H = H + H , (27)where H = 12(2 m e ) (cid:16) − ∂ X x + ( i∂ X y − BX x ) − ∂ X z (cid:17) + λ X z ,H = B || Bm e ( i∂ X y − BX x ) X z + O ( B || ) . (28)First, we note that translation invariance along the X y -direction allows us to replace derivatives with respect to X y with the momentum k y along this direction. Next, we shift the X x coordinate by defining ˜ X x = k y c eB − X x . The Hamiltonianhas the form, H = 14 m e (cid:16) − ∂ X x + 4 B ˜ X x − ∂ X z + 4 m e λ X z (cid:17) + B || Bm e ˜ X x X z + O ( B || ) , (29)where terms proportional to B || are taken to be a perturbation.Our goal is to determine the spectral flow as a function of B || of the ground and first excited energy levels. Dividing outby the irrelevant X y factor (which determines the degeneracy of the Landau levels in a rectangular sample, but is inconse-quential here), the eigenfunctions of the above coupled har-monic oscillator Hamiltonian at B || = 0 take the form: Ψ( ˜ X x , X z ) m,n = c m,n exp( − M ω c X x ) exp( − M ω z X z ) × H m ( p M ω c ˜ X x ) H n ( p M ω z X z ) , (30)where ω c = B/m e , ω z = λ/ √ m e , M = 2 m e , H n ( X ) denote Hermite polynomials, and the c m,n are normalizationconstants. We assume that ω c < ω z < ∞ .(For electrons moving in a Ga-As quantum well at ν = 7 / ,we can estimate ω c and ω z . Given a band mass m e ∼ . m f ,where m f is the free electron mass and transverse magneticmagnetic field of . T, we estimate an ω c ∼ × − eV ∼ . λ has engineering dimension equal to [Mass] / so wetake it to be proportional to /w / , where w is the well widthwhich is 40 nm for the experiment in [5]. We fix the orderof the proportionality constant via the estimate of the Landaulevel sub-band gap given in Fig. 2 of [10]. We find ω z ∼ x/w / , where x = 10 − . Thus, ω z /ω c ∼ . As the fillingfraction is lowered, the ratio ω z /ω c is increased and so thediscussion below becomes less relevant as the two scales aretoo far apart. Note also that this ratio approaches unity as theproportionality constant between the band and free electronmasses is lowered.)The perturbative shift in the energy of a state to second or-der in H is given by the formula, E m,n = E (0) n,m + h m, n | H | m, n i + X | k i6 = | m,n i |h k | H | m, n i| E (0) m,n − E (0) k , (31)where E (0) m,n is the unperturbed energy of the state Ψ( ˜ X x , X z ) m,n := | m, n i . We are interested in the difference E , − E , . At zeroth-order, this difference is equal to the cy-clotron frequency, ω c . The first-order term on the RHS of (31)vanishes because the perturbation is linear in both ˜ X x and X z (the ground state is nodeless). Now consider the second-orderterm. The state (or collection of states since we are ignoringthe k y dependence) mixed with the ground state | , i by theperturbation is | , i , while | , i is the state mixed with | , i .Because the wave functions factorize, |h , | H | , i| = |h| , | H | , i| , (32)however, | E (0)0 , − E (0)1 , | = ω c + ω z > ω z − ω c = | E (0)1 , − E (0)0 , | , (33)with both energy denominator differences being negative. Sowhile both E , and E , are shifted downwards, the groundstate is shifted less than the first excited state because of thedifference in magnitude of the energy denominators. Thisimplies that the location of the would-be Kohn pole is de-creased from the cyclotron frequency. (There is no contradic-tion with general level repulsion expectations as we are study-ing a systems with more than two states.) Notice that thisresult requires mixing between the different X z -bands and isnot present if we take the gap between these energy levelsto infinity. Contributions from other excited states only oc-cur at higher orders in perturbation theory. Note also that wedropped a term in the perturbing Hamiltonian quadratic in B || and so it could, in principle, compete at the same order as thesecond-order result above. However, this term has no conse-quence on the energy difference as it shifts both energies bythe same amount.The above analysis implies that the location of the leadingpole in a small momentum expansion of the density-densitycorrelator moves towards the origin as an in-plane field is ap-plied. This conclusion was drawn using a certain form of con-fining potential in the direction transverse to the x − y plane.How general are these results? If we consider a more gen-eral form of the confining potential, there will be a couplingbetween the center-of-mass and relative degrees of freedom.Nevertheless, for small B || , our results hold generally. Inthis limit, we can ignore the coupling between the center-of-mass and relative degrees of freedom. The perturbation cou-ples the planar and z motion of the center-of-mass at leading order while the coupling between the planar center-of-massand relative degrees of freedom only occurs at higher order inperturbation theory in B || . Although the X z eigenfunctionswill take a different functional form and the spacing betweenthese eigenfunctions will no longer be in regular multiples of ω z , the above argument goes through unchanged as long asthe gap between the lowest and first-excited X z eigenfunc-tion is greater than ω c . If B || is not small, then we cannotignore the coupling between the center-of-mass and relativedegrees of freedom. This can further modify the distributionof spectral weight, but we do not have any simple argumentfor whether this coupling will move the pole, broaden the poleinto a Lorentzian, or change its spectral weight. At any rate,we can say that the coupling between the center-of-mass andrelative degrees of freedom will almost certainly cause furtherdeviations from expectations based on Kohn’s theorem. Hap-pily, the effective field theory (1) describes a system wherethe would-be Kohn pole is different from the bare cyclotronfrequency through the variation of ¯ rr