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Hall quantum ional unu hs rniino fractional of transition phase quantum as rcinlqatmHl tt ihfiln frac- filling with state Hall quantum fractional iin nteohrhn,Laughlin hand, other the On sition. ,Lna ee,hv hw that shown have level, Landau 1, = hsqatmpaetasto,and transition, phase quantum this t enmtcodrprmtradthe and parameter order nematic he V / / s.B netgtn h spectrum the investigating By ase. aeapoone temperature-dependent pronounced a have 3 stebs ouetdcs fa nematic an of case documented best the is 2 z uvsa o otgs n h cln be- scaling the and voltages, low at curves yaia cln exponent, scaling dynamical 2 = t niorpcbcgon the background isotropic an ith e otenndsiaieHall non-dissipative the to ted iciaino h nematic the of disclination A . 2 4.I hs xeiet the experiments these In 24]. [2, N adulvlna ligfrac- filling near level Landau 2 = ,2 1, eory, incom- ne- 2 the symmetry is broken explicitly, these experiments pro- observation that, due to the breaking of time-reversal in- vide evidence for a large temperature-dependent nematic variance in the FQH fluid, the dynamics of the nematic susceptibility in these fluid states [25]. These experiments fluctuations is governed by a Berry phase term, whose strongly suggest that, at least in the N = 1 Landau level, coefficient they conjectured to be essentially the same the FQH phases the 2DEG may be close to a phase tran- as the (non-dissipative) Hall viscosity of the FQH fluid sition to an incompressible nematic state inside the topo- [37–39]. Maciejko and coworkers also further an interpre- logical fluid phase, i.e. a nematic FQH state. The notion tation of nematic fluctuations as a fluctuating geometry of a nematic FQH state was actually suggested early on (making contact with ideas put forward by Haldane on by Balents [26]. However, this concept did not attract the existence of geometric degrees of freedom in the FQH significant attention until the recent experiments of Xia liquid [33, 40, 41]). Similar ideas were discussed by two and coworkers which suggested the existence of strong of us in the context of a nematic transition in a spon- nematic correlations became available [25]. taneous anomalous quantum Hall state [42], and earlier The experiments of Xia and coworkers motivated Mul- on by one of us in a theory of thermal melting of the ligan and coworkers [27, 28] to formulate a theory of a pair-density-wave superconducting state [43]. The con- quantum phase transition inside the ν =7/3 FQH phase, jectured connection between the nematic fluctuations in from an isotropic fluid to a nematic FQH state inter- the FQH fluid and the Hall viscosity strongly suggest a preted as a quantum Lifshitz transition. The theory of relation with theories of the geometric response of these Mulligan et al. uses as a starting point the effective field topological fluids [44–48], which we will further elaborate theory of a Laughlin isotropic FQH fluid with filling frac- below. tion ν = 1/m (with m an odd integer) whose effective In this paper we address several open aspects of this Lagrangian is that of a (hydrodynamic) gauge field, aµ, problem that have remained unexplained. One of the with a Maxwell and a Chern-Simons term [29, 30]. In issues is the origin of the nematic quantum phase transi- this picture, the FQH quantum Lifshitz transition oc- tion which Maciejko et al. argued could be due to a soft- curs when the coefficient of the electric field term of the ening of the GMP collective mode. Here we will show Maxwell-like term of the effective action of the hydro- that the GMP mode can become gapless at wave vec- dynamic gauge field vanishes, and can be regarded as tor q = 0 if the effective interactions among the elec- a Chern-Simons version of the quantum Lifshitz model trons are sufficiently attractive in the quadrupolar chan- [31]. nel. It is known that in a Fermi liquid, a sufficiently While this theory successfully predicts many aspects attractive effective interaction in the quadrupolar chan- of the experiment (in particular the anisotropy) it has nel (i.e. a sufficiently negative charge-channel Landau several difficulties, the most serious of which is that in a parameter F2) can trigger a nematic instability through Galilean invariant system the coefficient of the term for a Pomeranchuk instability which results in a spontaneous hydrodynamic electric field is fixed by Kohn’s theorem quadrupolar distortion of the Fermi surface [49]. Here we [32]. Although this restriction can be violated by a rela- will postulate that at long wavelengths, in addition to the tively small amount by Landau level mixing effects [28], long-range Coulomb interaction, there is an attractive it is unlikely to become large enough to trigger a Lifshitz short-range quadrupolar interaction. Such an effective transition to a nematic state. Another puzzling aspect is interaction can arise due to the softening of the short- that the Chern-Simons Lifshitz theory of Mulligan et al. distance Coulomb interaction in Landau levels N 1. ≥ also applies to the integer Hall states. However, barring In fact, an early numerical study by Scarola and cowork- large enough Landau level mixing effects, it is hard to see ers [50] of the effective interactions of composite fermions how a system in the integer quantum Hall regime may [51] showed that in Landau levels with N 1 there is a ≥ break spontaneously rotational invariance. The experi- strong tendency for the FQH liquid to become unstable ment of Xia et al have also prompted several studies of (and was interpreted as an exciton instability.) From integer and fractional quantum Hall states in systems in the point of view of symmetry breaking, a q = 0 (‘exci- which the anisotropy is built-in explicitly in the geome- ton’) quadrupolar condensate is equivalent to an insta- try of the two dimensional surface in which the electrons bility to nematic state since they break the same spatial reside [33], including wave functions for states with fixed symmetries. The other focus of this work is to clarify the anisotropy [34]. relation between the nematic fluctuations (and possible Maciejko and coworkers [35] recently proposed an effec- order) in the FQH fluid to the response of this fluid to tive field theory of the spontaneous breaking of rotational changes on the actual background geometry of the surface invariance in a nematic state in the FQH regime with the on which the 2DEG resides. This is an important ques- form of a non-linear sigma model on the non-compact tar- tion since quantities such as the Hall viscosity measures get space SO(2, 1)+ manifold of the rotational degrees the response to shear deformations of the geometry and of freedom and the amplitude of the local nematic or- this is not quite the same as the the nematic response, der parameter. They proposed that the nematic transi- although, as we will see below, they are related. tion is triggered by a softening of the intra-Landau level In order to study the quantum nematic phase transi- Girvin-MacDonald-Platzman (GMP) collective mode of tion in a FQH fluid we first generalize the fermion Chern- the FQH fluid [36]. A key result from this work is the Simons theory of the FQH states [52] to include the ef- 3 fects of the attractive quadrupolar interaction, and show the effective Landau-Ginzburg theory from the fermion that indeed there can be a quantum phase transition in- Chern-Simons theory and in Section IV we show that side all Jain states of the FQH provided the quadrupolar there is a quantum phase transition to a nematic phase interaction is sufficiently attractive. In our treatment we for sufficiently strong attractive quadrupolar coupling. also include the coupling to the background geometry of In Section V we discuss the behavior of the Goldstone the 2D surface on which the 2DEG resides. We then use mode of the broken orientational symmetry in the ne- our recent results presented in Ref. [47] to show that the matic phase and the nature of the disclinations. The quadrupolar interaction couples to both the so-called sta- coupling to the background geometry is developed in Sec- tistical gauge field (of the fermion Chern-Simons theory) tion VI. Our conclusions are presented in Section VII. In and to the spin connection of the geometry. Our first several appendices we present details of the calculation of main result is the derivation of the effective action for the effective field theory. In Appendix A we present the the nematic degrees of freedom which, as expected, has calculation of the nematic correlators and in Appendix B the form proposed by Maciejko et al.. The fluctuations the calculation of mixed correlators of nematic and gauge of the nematic order parameter are strongly coupled to fields. A proof of gauge invariance is given in Appendix the GMP mode of the FQH fluid (which has quadrupo- C, and the nematic collective excitations are derived in lar character), and the nematic quantum phase transition Appendix D. is triggered when the q = 0 component of this mode be- comes gapless. Furthermore, the dynamics of the nematic degrees of freedom is controlled by a Berry phase term II. SPONTANEOUS BREAKING OF and, hence, has dynamics critical exponent z = 2. How- ROTATIONAL SYMMETRY IN FQH STATES ever its coefficient is not the Hall viscosity of the FQH fluid (as conjectured in Ref. [35]) but is given, instead, A. Composite Fermion Theory of FQH states by the Hall viscosity of the effective integer Hall effect of the composite fermions. Nevertheless, the Hall viscosity Here we begin with a short review of the composite of the system (both in the isotropic and in the nematic fermion theory of a FQH state [51, 52], specializing in phase), defined as the response to the shear deformation 1 the simpler case of the Laughlin state at filling ν = 3 , of the underlying geometry, is the same as the Hall vis- which can be easily generalized to the other states in Jain cosity of the FQH fluid obtained in Ref.[38] (and recently sequence ν = p/(2sp+1), where s,p Z. Let us consider rederived by us [47]). These results are reminiscent of the a theory of electron field Ψ in two∈ space dimensions in previous study by two of us [42] where we have studied an uniform magnetic field. The action for this system is the effective theory of the phase transition between an isotropic Chern insulator and a nematic Chern insulator. 1 = d2xdt Ψ†(x)D Ψ(x) (DΨ(x))† (DΨ(x)) We also demonstrate that in this theory the nematic tran- S 0 − 2m · Z  e  sition is reached while the Kohn mode remains unaffected 1 in both phases and at the phase transition. In addition d2x′d2xdt V ( x − x′ )Ψ†(x)Ψ(x)Ψ†(x′)Ψ(x′) −2 | | we also show that the components of the nematic order Z (2.1) parameter can be used to define an effective spin connec- tion (which is effectively the same as the “nematic gauge in which Dµ = ∂µ + iAµ is the covariant derivative of the field” phenomenologically introduced in Ref. [35]) and electron, me is the mass of the electron, and we have set that it couples to an external electromagnetic probe field the Planck constant ~, the speed of light c, and the elec- through a term with the form of the Wen-Zee term [53], a tric charge e to unity. The four-fermion term encodes the result also anticipated by Maciejko et al. We also derive two-body interaction between the electrons. The electro- the effective action for the spin connection of the back- magnetic gauge field Aµ can be written as Aµ = A¯µ +δAµ ij ground geometry and show that has the same form (with where A¯µ is for the uniform magnetic field B¯ = ǫ ∂iA¯j the same universal coefficients) in both phases. Finally perpendicular to the plane and δAµ is the probe field to we use our effective field theory to investigate the prop- measure the response of the FQH state. erties of a disclination of the nematic order parameter in The average electron densityρ ¯ and the uniform exter- the nematic phase, and show that it carries a fractional nal magnetic field B¯ are related to each other through (but non-universal) electric charge and that the Hall vis- the filling fraction ν cosity is modified by the disclination, which agrees with ν 1 the a symmetry-based argument of Ref.[35]. ρ¯ = B¯ = B¯ (2.2) 2π 6π This paper is organized as follows. The theory of spon- taneous rotational symmetry breaking is developed in where we have set ν =1/3 for the leaden Laughlin state. Section II. After summarizing the fermion Chern-Simons For a general Jain state the filling fraction is ν = p/(2sp+ gauge theory in Subsection IIA, in Subsection IIB we 1), where s and p are two integers. The Laughlin FQH introduce the quadrupolar interaction and its coupling state with ν = 1/3 can be pictorially understood as the with the statistical gauge field and with the spin connec- liquid state of the electrons in which, on average, each tion of the background metric. In Section III we derive electron is bound with the two flux quanta. For a general 4

Jain state, each electron is bound to 2s flux quanta and average values. Furthermore, the density fluctuation of becomes a composite fermion [51]. the electron δρ =Ψ†Ψ ρ¯ is bound with the flux of δa − µ This is a problem of strongly coupled electrons and 1 1 ij cannot tackled directly using weak coupling methods. To δρ(x)= δb(x)= ε ∂iδaj (2.7) make progress, we follow Ref.[52] and consider the equiv- 4π 4π alent system obtained by coupling the system of inter- This makes the density-density interaction between the acting electrons to the (dynamical) Chern-Simons term electrons to be quadratic in the gauge field δaµ. As the of the statistical gauge field aµ, using minimal coupling. action is quadratic in the composite fermion field Ψ, we (For a detailed discussion see Ref.[30].) The action action can integrate out the fermion and the fluctuating part of the equivalent problem is δaµ of the statistical gauge field to obtain the effective theory for the gauge field δAµ. From the effective the- 1 ory of δA , one can calculate the electromagnetic Hall = d2xdt Ψ†(x)D Ψ(x) (DΨ(x))† (DΨ(x)) µ S 0 − 2m · response and find the collective excitations of the FQH Z  e  1 state [55] which, with some caveats, agree qualitatively d2x′d2xdt V ( x x′ )Ψ†(x)Ψ(x)Ψ†(x′)Ψ(x′) long wavelengths with the experiments and numerical −2 | − | Z calculations. 1   + d2xdt ǫµνλa ∂ a (2.3) 8π µ ν λ Z B. Quadrupolar Interaction where Dµ = ∂µ + iAµ + iaµ is a new covariant derivative which includes the minimal coupling to both the electro- The composite fermion theory we just summarized is magnetic field A and to the statistical field a . This is µ µ so far is rotationally invariant and cannot describe a ne- the exact mapping of the original problem defined by the matic FQH state. Thus we should look for a new in- action of Eq. (2.1). The Chern-Simons term binds the gredient to the composite fermion theory to describe the two flux quanta to the electron and turns the electron nematic state and the transition toward the nematic state into the composite fermion [51, 54]. from the isotropic state. Since the density-density inter- We next consider uniform states which can be de- action of electrons and Chern-Simons term (at the level of scribed using the average field approximation in which the bare action of the composite fermion theory) cannot we smear out the two flux quanta bound to the electron induce the spontaneous breaking of the rotational sym- over the two-dimensional plane. This translates as choos- metry, we should look for an interaction which can favor ing the average part ofa ¯µ to partially cancel the external ¯ the anisotropic state rather than the isotropic state. To magnetic field Aµ. For the ν = 1/3 Laughlin state the this effect we follow the approach of the nematic Fermi effective field is fluid of Ref. [49] and add a quadrupolar interaction term 1 q to the action of the form A¯µ +¯aµ = A¯µ (2.4) S 3 1 2 2 ′ ′ ′ q = dt d xd x F2( x x )Tr[Q(x)Q(x )] Thus the composite fermion Ψ is subject to the mag- S −2 | − | Z Z (2.8) netic field which is 1 of the magnetic field experienced 3 where F ( x x′ ) is the Landau interaction in the by the electron. The composite fermion is in the integer 2 quadrupolar| channel− | whose spatial Fourier transform is quantum Hall effect at the filling ν = 1 and in effect is weakly coupled. We can write out the Lagrangian of the F F (q)= 2 (2.9) composite fermion. 2 1+ κq2 and κ > 0 parametrizes the interaction range. The cou- 2 † 1 † = d xdt Ψ (x)D0Ψ(x) (DΨ(x)) (DΨ(x)) pling constant F (i.e. the Landau parameter) has units S − 2m · 2 Z  e  of energy (length)6. Here we introduced the 2 2 1 × × d2x′d2x V ( x x′ )Ψ†(x)Ψ(x)Ψ†(x′)Ψ(x′) traceless symmetric tensor Q(x) −2 | − | Z 2 2 † Dx Dy DxDy + DyDx 1 2 µνλ  Q(x)=Ψ (x) − Ψ(x), + d xdt ǫ δaµ∂ν δaλ (2.5) D D + D D D2 D2 8π x y y x y − x Z   (2.10)

Here and below we denote by Dµ Here Dx and Dy are the spatial covariant derivatives de- fined in Eq.(2.6). 1 The full action (including the quadrupolar interaction Dµ = ∂µ + i A¯µ + iδaµ + iδAµ (2.6) 3 q) is manifestly rotationally invariant. In the case of Sa Fermi liquid, for large enough attractive quadrupolar the covariant derivative of the composite fermion (again, interactions, F2 < 0, there is a Pomeranchuk instabil- for the ν = 1/3 Laughlin state). The fields δaµ and ity which results in the spontaneous breaking of rota- δAµ are the fluctuation of the gauge fields about their tional invariance and the development of a nematic phase 5

[49]. Here too, if F2 < 0 and large enough in magni- fields couple to the the stress tensor tensor of the com- tude, the quadrupolar coupling can induce a transition posite fermions and thus play a role analogous to a back- to an anisotropic phase by developing the finite expec- ground metric. In this sense, we can regard the nematic tation value of Q(x). When Q = 0, the continuous fluctuation as providing a “dynamical metric” which rotational symmetry O(2) of theh i two-dimensional 6 space modifies the local geometry of the composite fermions is broken down to C2 generated by the discrete π rota- [42, 43]. tion of the plane. However, in the case of a Fermi fluid Thus we end up with the following action for the com- at zero external magnetic field the nematic phase leads posite fermions coupled to the Chern-Simons gauge field, to the spontaneous distortion of the Fermi surface and with a density-density interaction and a quadrupolar in- the development of an anisotropic effective mass for the teraction, quasiparticles in the anisotropic state. Furthermore, in 1 the absence of a coupling to the underlying lattice the = d2xdt Ψ†(x)D Ψ(x) (DΨ(x))† (DΨ(x)) S 0 − 2m · resulting nematic phase is a non-Fermi liquid. In the Z  e  case at hand, although there is no Fermi surface to begin 1 d2x′d2xdt V ( x x′ )δb(x)δb(x′) with, at the level of mean field theory, nematicity is also −32π2 | − | manifest as an effective anisotropy of the effective mass Z 1 2 µνλ of the composite fermions. + d xdt ǫ δaµ∂ν δaλ 8π Next, we include the quadrupolar interaction in the Z 1 κ the (fermionic) Chern-Simons theory of the FQH states 2 M 2 ∇ 2 + d xdt 2 + 2 Mi 4F2m 4F2m | | [52] of Eq.(2.3) e e i=1,2 Z h X M M 1 + 1 Ψ†(D2 D2)Ψ + 2 Ψ†(D D + D D )Ψ = d2xdt Ψ†(x)D Ψ(x) (DΨ(x))† (DΨ(x)) m x − y m x y y x S 0 − 2m · e e Z e (2.13)i 1 h i d2x′d2xdt V ( x x′ )δb(x)δb(x′) −32π2 | − | where, again, the covariant derivatives are given in Z 1 Eq.(2.6). So far we have not made any approximations. + d2xdt ǫµνλδa ∂ δa In the next section we will discuss the uniform states that 8π µ ν λ Z result by treating this theory in the average field approx- 1 dt d2xd2x′F ( x x′ )Tr[Q(x)Q(x′)] imation and by considering the effects of fluctuations at −2 2 | − | Z Z the one loop (“RPA”) level. (2.11) It turns out that from the theory we have defined the resulting quantum phase transition is strongly first order Here we used the Chern-Simons constraint (i.e. the and to a state with maximal nematicity (and without “Gauss law”) to represent the fluctuating density δρ of Landau quantization!). To avoid this pathological limit, the composite fermion in terms of the fluctuating statis- and to make the nematic phase stable (and accessible by a tical field δb which results in density-density interaction continuous quantum phase transition), we will introduce quadratic in the statistical gauge field. an extra term in the kinetic energy part of the action of However, the quadrupolar interaction cannot be writ- the form ten as a quadratic form in the statistical gauge field δa . µ 3 Instead, we perform a Hubbard-Stratonovich decoupling D2 ρπ¯ S = α d2xdtΨ† − Ψ (2.14) transformation to rewrite the quadrupolar interaction 6 − 2m − m Z  e e  term q in terms of two fields M1 and M2 (which can be regardedS as the two real components of a 2 2 real where, once again, D stands for the space components × of the (full) covariant derivative and D2 is the covariant symmetric matrix field). After decoupling the action q of Eq.(2.8) takes the form S Laplacian. A term of a similar type was introduced by Oganesyan et al. [49] in their theory of the nematic Fermi 1 κ fluid formed by a Pomeranchuk instability. Here too this 2 M 2 ∇ 2 q = d xdt 2 2 Mi (technically irrelevant) term will insure that the nematic S 4F2m − 4F2m | | e e i=1,2 Z h X state is stable, provided the coupling constant α is large M1 † 2 2 M2 † enough (as we will see below). For other ranges of α Ψ (Dx Dy)Ψ + Ψ (DxDy + DyDx)Ψ , me − me the quantum phase transition becomes first order, as it (2.12)i happens in theories of the electronic nematic transition in lattice systems [56]. Although the addition of this term Here we introduced suitable factors of the electron mass complicates the calculation somewhat, it does not change me to make the Hubbard-Stratonovich fields M1 and the physics in any essential way. We should note that this M2 dimensionless. F2 is the coupling constant of the term commutes with the gauge-invariant kinetic energy quadrupolar interaction of Eq.(2.10). and, consequently, it has the same eigenstates. Thus, this It is apparent that in Eq.(2.12) M1 and M2 play the term changes only the eigenvalues but it does not induce role of the order parameters for the nematic phase. These Landau level mixing. 6

C. Symmetries for the nematic order parameters. It has the conven- tional Landau-Ginzburg form supplemented by a topo- The action of Eq.(2.13) has two important symmetries. logical Berry phase term. We demonstrate that there is One is local gauge invariance, under which the Fermi field a continuous phase transition if the quadrupolar interac- tion F is bigger than a critical value. Furthermore, we Ψ(x) and the gauge field aµ(x) transform as 2 show that there is a Berry phase term for the nematic or- ′ −iΛ(x) ′ der parameter, which is similar to the Hall viscosity term, Ψ (x)= e Ψ(x), aµ(x)= aµ(x)+ ∂µΛ(x) (2.15) and the Berry phase term makes the quantum critical where Λ(x) is a (smooth) gauge transformation. point have the dynamical exponent z = 2. From a,M , we will see that there is a topological term, similar toL the The second symmetry is invariance under the coordi- Wen-Zee term, [53] which describes the response to the nate transformation of global rotations in real space, curvature induced by disclination (not the deformation x′ = R (ϕ)x (2.16) from the background geometry). In addition, a,M also i ij j contains an anisotropic Maxwell term that representsL the coupling of the Kohn collective mode to the nematic or- where Rij (ϕ) is the 2 2 rotation matrix by an angle of ϕ. The Fermi field is× invariant (a scalar) under rota- der parameter fields [42]. tions, Ψ(Rx′)=Ψ′(x). However the invariance of the action of Eq.(2.13) under global rotations requires that A. Gauge Field Lagrangian: L the Hubbard-Stratonovich fields M, which are conjugate a to the nematic order parameter field Qij of Eq.(2.10), transform not as a vector under rotations but as a di- Here we consider the term of the effective Lagrangian rector, i.e. a vector in without a direction. This means of Eq.(3.1) that includes only the gauge fields aµ and that it transforms under a rotation by twice the rotation δAµ. This part of the effective action does not know the angle in real space, nematic order parameter, and so it should be the same effective action of the gauge fields as in the isotropic FQH ′ states [52] Mi = Rij (2ϕ)Mj (2.17) 1 εµνλ Under this transformation, the Hubbard-Stratonovich = (δa + δA )Π0 (δa + δA )+ δa ∂ δa , La −2 µ µ µν ν ν 8π µ ν λ field is invariant under a rotation by π. Similarly, the (3.2) nematic order parameter, i.e. the traceless symmetric 0 2 2 matrix field of Eq.(2.10), transforms as a tensor Here Πµν (x y), given by under× rotations by an angle of 2ϕ. − 2 0 1 δ ZF Πµν (x y)= i = jµ(x)jν (y) , (3.3) − − ZF δaµ(x)aν (y) h i III. EFFECTIVE FIELD THEORY OF NEMATIC is the bare polarization tensor of the integer quantum ORDER PARAMETER Hall state of the composite fermion, and it is given in Ref.[52] whose results we use. The current-current time- The full action of Eq.(2.13) is a quadratic form in the ordered correlators shown in Eq.(3.3) are computed in composite fermions. These fermionic fields can be inte- the free composite fermion theory and ZF is the partition grated out allowing us to obtain an effective field theory function of composite fermions with an integer number for the nematic order parameter M1 and M2 coupled to of filled effective Landau levels. In the low energy and 0 the gauge fields. This procedure is safe provided on is ex- long wavelength limit Πµν (q,ω) is given by pending about a saddle point state with a finite energy 1 m gap. The resulting effective Lagrangian can be decom- Π0 (q,ω)= q2 e , 00 ¯ 2 posed as the three parts − π b(1 + αω¯c ) 0 1 me i jk Π (q,ω)= qj ω + ǫ qk, = a + M + a,M (3.1) 0j ¯ 2 L L L L − π b(1 + αω¯c ) π 1 m i where and include only the fluctuating gauge Π0 (q,ω)= q ω e ǫjkq , a M j0 j ¯ 2 k fields δaL + δA Land only the nematic order parameter − π b(1 + αω¯c ) − π 1 m i Mi,i = 1, 2, and a,M represents the coupling between Π0 (q,ω)= δ ω2 e ǫij ω L ij ij ¯ 2 the gauge fields and the nematic order parameter. − π b(1 + αω¯c ) − π For clarity, we discuss the three parts, a, M , and (q2δ q q ) L L ij − i j (3.4) a,M , of the full effective theory separately. Here, we − m (1 + αω¯2) Lbriefly show what we can learn from the three parts be- e c fore describing the details of each term. a is the ef- where ¯b = B/3 for the ν = 1/3 Laughlin state. Here fective Lagrangian for the statistical gaugeL fields of the we have included in the results of Ref.[52] the corrections isotropic FQH states [52]. is the effective Lagrangian due to the extra term in the kinetic energy of Eq.(2.14). LM 7

B. Order Parameter Lagrangian: LM potential and u the contact interaction). As in the Refs. [42] and [35], the effective theory of the nematic order pa- We next obtain the Lagrangian for the nematic fields, rameter field contains a Berry phase term associated with M , of Eq.(3.1) to the quartic order in the nematic order the non-dissipative response of the quantum Hall effect, Lparameter by calculating one-loop Feynman diagrams. which related to the Hall viscosity. This term makes time We will see that M exhibits the isotropic-anisotropic and space scale differently, and the associated quantum phase transition andL the quantum phase transition. To critical point has the dynamical exponent z = 2. calculate M , we need to compute the two-point and In our discussion we have neglected the role of the sym- four-pointL correlators of N = i δZ and the calculation metries of the underlying lattice. While lattice effects are i δMi is done in Appendix A. In the− main text, for simplicity irrelevant (and unimportant) for the topological proper- 1 ties of the FQH fluids they do matter for the nematic we discuss only the case of the FQH state at the filling 3 . However, as discussed in the Appendix A, it is straight- fluctuations and ordering. In the case of GaAs-AlAs het- forward to generalize the calculations to the other states erostructures, the 2DEG resides on surfaces which have in Jain sequence ν = p ,p,s Z. a tetragonal C4 symmetry. The extra terms of the La- 2sp+1 ∈ Integrating out the composite fermion and expanding grangian that break the symmetry from the full contin- uous rotations down to C are proportional to M 2 M 2 about the low-energy limit, i.e., taking the lowest terms 4 1 − 2 in frequency ω and momentum q, we obtain the effective and 2M1M2. We will discuss in another section that theory of the nematic fluctuations these terms gap out the Goldstone modes of the nematic phase. ǫij ρ¯ The parameters entering into the the effective La- = M ∂ M rM 2 LM 2(1+4αω¯2)2 i 0 j − grangian of Eq.(3.5) are obtained by a direct calculation c of the correlators, and are found to be κ¯ u (∇M )2 (M 2)2. (3.5) − 2 i − 4 1 ω¯ r = c , − 4F m2 − 2π¯l2(1+4αω¯2) where M 2 = M 2 + M 2. 2 e b c 1 2 κ In the effective Lagrangian of Eq.(3.5) we have ignored κ¯ = 2 two physically significant corrections terms. The La- − 2F2me grangian of Eq.(3.5) is invariant under the O(2) symme- 1 1 1 2 2 + 2 + 2 , try of arbitrary global rotations in the order parameter − π (1 + αω¯c ) 2(1+4αω¯c ) (1+9αω¯c ) h i space, i.e. ¯bω¯ 1 1 3 u = c 4π (1+4αω¯2)2 4(1+4αω¯2) − 4(1 + 16αω¯2) Mi Rij (φ)Mj , (3.6) c c c → h (3.8)i where R (φ) is the 2 2 rotation matrix by an arbitrary ij ¯ ¯ angle φ. However, as× we saw in Section II C, the only Hereω ¯c = b/me and lb = √3ℓ0 (for the Laughlin state symmetry (aside from gauge invariance) is a combina- at ν =1/3) are the effective cyclotron frequency and the tion of a rotation in space by an angle ϕ and a rotation in effective magnetic length of the composite fermion, where −1/2 the order parameter space by 2ϕ, which leave M M ℓ0 = B is the magnetic length. invariant. This means that the larger symmetry≡− of the From these results we can also see that the nematic Lagrangian of Eq.(3.5) is only approximate and that the order parameter will condense only when the quadrupo- Lagrangian must contain terms which reduce the symme- lar interaction is attractive and larger in magnitude than try accordingly. In fact, the effective Lagrangian allows the critical value for an extra (formally irrelevant) operator of the form 2 πl¯ F c = b (1+4αω¯2) (3.9) 2 | 2 | 2¯ω m2 c = λ M ∇ M (3.7) c e LSO − · Furthermore, since u > 0 the quantum phase transition 
 which is invariant under joint rotations in real space and is continuous and the nematic state is stable. in the order parameter space (and is formally a “spin- From Eq.(2.10) it is clear that the nematic order pa- orbit” type coupling). Such terms are well known to arise rameters formally couple to the quadrupole density in in the free energy of classical liquid crystals [57, 58]. the same way as the background metric couples to the The resulting effective Lagrangian of Eq.(3.5) has the energy-momentum tensor (although the extra term in same form as the effective theory of the nematic order the kinetic energy of Eq.(2.14) does not couple to the parameter in a Chern insulator[42], and of the effective nematic fields). We can regard the nematic order pa- field theory of the nematic FQH state of Maciejko and rameters as a “dynamical spatial metric” which modifies collaborators [35] Moreover, upon defining the complex spatial components of the metric tensor. From this obser- field Φ = M1 + iM2, it is easy to see that the lagrangian vation one may naively expect that the prefactor Berry of Eq.(3.5) is equivalent to the Lagrangian of a 2D di- phase term in Eq.(3.5) may be the Hall viscosity of the ρ¯ lute Bose gas (with r playing the role of the chemical FQHE ηH = 2ν when α = 0. 8

However, for α = 0, the prefactor of the Berry phase of the gauge fields contract with the (inverse of) met- term of Eq.(3.5) is the Hall viscosity term of the integer ric spatial tensorgij . The resulting terms in the effective quantum Hall state at ν = 1, and not of the actual Hall Lagrangian are viscosity of the fractional quantum Hall state. See the m 2M discussion of Sec. VI. This difference originates in the = e 1 (∂ δA˜ ∂ δA˜ )2 a,m ¯ 2 x 0 0 x fact that the “dynamical metric” associated to the ne- L 4πb(1+4αω¯c ) − maticity and the background metric are not equivalent. m 2M e 1 (∂ δA˜ ∂ δA˜ )2 For FQH states, the nematic order parameters only cou- ¯ 2 y 0 0 y −4πb(1+4αω¯c ) − ple with the stress energy tensor, while the background meM2 metric, not only couples with the stress energy tensor, + (∂ δA˜ ∂ δA˜ )(∂ δA˜ ∂ δA˜ ) ¯ 2 x 0 − 0 x y 0 − 0 y but also appears in the form of a spin connection, as dis- πb(1+4αω¯c ) cussed in detail in Ref.[47]. In the composite fermion (3.10) or composite boson theories, when we attach flux to the where δA˜ = δA + δa. These terms are second order in electron to form a composite particle, each flux quantum derivatives and are time-reversal and parity invariant. attached to the particle induces the additional angular However, there are contributions to which are momentum 1/2. This makes the composite particle cou- a,M first order in derivatives and hence breakL time-reversal ple to the spin connection though the particle is a scalar and parity. These contributions have the form of a Wen- and not a spinor. The orbital spin then couples to the Zee term [53, 61]. The Wen-Zee term can be understood local geometry to the spin connection much in the same as the response of the FQH states to a change of the way as relativistic fermions do. Thus, after we perform geometric curvature: the curvature will trap the gauge flux attachment to describe the FQH fluids, the com- charge. While this term can be ignored if the nematic posite fermion resulting from the flux attachment will order is uniform in space, it has interesting consequences minimally couple with the spin connection ω , as shown µ for the charge and the statistics of the disclination in the explicitly in Ref.[47]. The coupling through the spin con- nematic phase. nection with the background geometry is the origin of To obtain the Wen-Zee term for the nematic order the difference between the nematic order parameter and parameter we perform calculation of one-loop diagrams the (deformed) background metric. The derivation of with one current and one and two nematic fields, the correct Hall viscosity from the background metric deformation through the composite fermion theory was 1 = T M (δa + δA +2Z ) reported elsewhere [47]. Lwz − 2 iµ i µ µ µ The results of this section can be easily generalized to 1 p + RijµMiMj(δaµ + δAµ), (3.11) all the states in the Jain sequence ν = 2p+1 , with the 3 effective Lagrangian density. However, when p goes to infinity, the theory approaches to the half-filled Landau where we Tiµ and Rijµ denote the following three-point level and the gap vanishes. In this regime the system (time-ordered) correlators of the composite fermions becomes a non-Fermi liquid and the effective Lagrangian 1 δZF for the gauge field given by Eq.(3.2) now has a Landau Tiµ(r,t)=i2 = i Ni(r,t)jµ(0, 0) , Z δM δa − h i damping term [59, 60]. In this limit, at least formally, F i µ this theory is a generalization of the theory of the ne- 1 δZF Rijµ[ri,ti]= 3i matic quantum phase transition in Fermi fluids [49] to − ZF δMiδMjδaµ describe the compressible nematic quantum fluid at half- = Ni(r1,t1)Nj (r,t2)jµ(r3,t3) , (3.12) filled Landau levels (see Ref.[20] and references therein.) − h i where the correlators are time-ordered functions of the free composite fermion theory, and C. Order Parameter and Gauge Field Lagrangian: Z0 =0 La,M Zx =(δax + δAx)M1 + (δay + δAy)M2 Here we derive the third term of the effective La- Zy =(δax + δAx)M2 (δay + δAy)M1, (3.13) − grangian of Eq.(3.1), a,M , that describes the coupling between the gauge fieldsL and the nematic order param- Diagrammatically the correlators of Eq.(3.12) are rep- eters. This part of the effective Lagrangian will be im- resented by the Feynman diagrams of Fig.2. They are portant later for investigating the quantum numbers and computed explicitly in Appendix B. statistics of the disclinations of the nematic phase. After calculating the above correlators, we obtain the In the presence of nematic order, the natural coupling coupling between the geometric curvature induced by the between the nematic order parameter and the gauge field nematic fields and the statistical gauge field, which ex- is as a local anisotropy of the Maxwell term. Since the plicitly has the form of a Wen-Zee term order parameter acts as the spatial components of a met- 1 = ǫµνρωQ∂ (δa + δA ) (3.14) ric tensor [42], the indices of the field strength tensor fij Lwz 4π µ ν ρ ρ 9

Q where ωµ (with µ =0,x,y) is the the effective spin con- matic order parameters. It is given by nection induced by the local nematic order parameters, ρǫ¯ ij i.e. M 2 = 2 2 Mi∂0Mj r L 2(1+4αω¯c ) − ij Q ǫ κ¯ 2 u 2 2 (∇Mi) (M ) ω0 = 2 2 Mi∂0Mj, (1+4αω¯c ) − 2 − 4 ij 1 µνρ Q Q ǫ + ǫ ωµ ∂ν (δaρ + δAρ) ωx = 2 2 Mi∂xMj t(∂xM2 ∂yM1), 4π (1+4αω¯c ) − − 1 0 ij Π (δaµ + δAµ)(δaν + δAν ) Q ǫ 2 µν ω = Mi∂yMj + t(∂xM1 + ∂yM2), (3.15) − y (1+4αω¯2)2 1 1 c + εµνλδa ∂ δa + εµνλωQ∂ ωQ (3.18) 8π µ ν λ 24π µ ν ρ where In the last line we have added the gravitational Chern- 2 2 Simons term of the induced spin connection of the ne- t = 2 2 (3.16) matic fields, where we used the results of Ref. [45]. 1+ αω¯c − 2+8αω¯c

Q The spin connection ω of the nematic order parameter IV. CONDENSATION OF THE GMP MODE AT is different than the spin connection of the background THE NEMATIC PHASE TRANSITION geometry. The meaning of the spin connection can be clarified by looking at its curl, The FQH fluids have several types of collective exci- tations [36, 55] The Kohn mode is a cyclotron collec- Q Q 1 ∂xωy ∂yωx √gR (3.17) tive mode related with the inter-Landau level excitation. − ∝ 2 If the system has Galilean invariance, the energy of the where R is the geometric curvature of the dynamical met- Kohn mode at zero momentum only depends on the bare mass of the electron and is insensitive to any other micro- ric induced by the nematic order parameters Mi. Here g, the determinant of the metric, is given by g =1 4M 2. scopic detail [32] In a FQH fluid the Kohn mode is not the − Here the coupling term between the “spin connection” lowest energy collective excitation and at finite wave vec- q and gauge fields in Eq.(3.14) has a similar form of the tor can (and does) decay to lower energy modes. On Wen-Zee term of Ref.[53]. However, the coefficient in the other hand, the lowest energy collective mode, the Eq.(3.14) is not the orbital spin of the FQH state. Instead GMP mode, is stable. This mode is a quadrupolar intra- this coefficient is equal to the orbital spin of the integer Landau level fluctuation, and at long wavelengths it can quantum Hall state at ν = 1(when α = 0). This can be regard as a fluctuating quadrupole with structure fac- tor q4 (instead of q2 as in the case of the Kohn mode) be easily understood from the composite fermion theory ∼ because the composite fermions effectively are an the in- (see Refs.[55, 62]). Therefore, for a FQH state with the teger quantum Hall state and any response at the mean- quadrupolar interaction, we can expect that the interac- field approximation of the composite fermion will be the tion can change substantially the behavior of the GMP same as that of the integer quantum Hall phase. This mode by mixing with the nematic fluctuations (which fact still remains true even after integrating out the sta- are also quadrupolar). In this section, we consider the tistical gauge field. The difference again comes from the behavior of the GMP collective excitation of the FQH nonequivalence between the nematic order parameter and state at and near the quantum phase transition between the background metric. When we attach Chern-Simons the isotropic state and the nematic state. flux to the fermion in a background metric, the orbital To get the spectrum of the collective excitations, we spin induced by the flux attachment also gives rise to need the full polarization tensor of the electromagnetic additional geometry-gauge coupling term which has the response [52]. To this end we first calculate the polariza- form of a Wen-Zee term [47]. For the nematic order pa- tion tensor of the composite fermions 2 rameter, the coupling between the gauge field and the 0 1 δ ZF nematic order parameters only comes from the compos- Πµν = i , (4.1) − ZF δaµδaν ite fermion which forms an IQHE. The derivation of the correct Wen-Zee terms through the composite fermion where ZF is the partition function of the composite theory is reported in Ref.[47]. fermions. Because the composite fermion system is in an integer quantum Hall ground state, the poles of Πµν cor- respond to the Landau levels spaced byω ¯c, the effective cyclotron frequency of the composite fermions (modified D. Full Effective Action by the contributions of the extra terms of Eq.(2.14)). Next we compute the change in Πµν due to the effects Here we now ready to present the full effective La- of both the quadrupolar interaction and of the density- grangian of Eq.(3.1) in terms of the gauge fields and ne- density interaction and determine the full polarization 10 tensor for the external electromagnetic field Kµν . The Integrating out the statistical gauge field δaµ, we finally current and the nematic fields are defined in terms of the obtain the full response function Kµν for the external composite fermion Ψ by electromagnetic fields δS δS † 2 ji = ,Ni = =Ψ TiΨ. (4.2) K00 =q K0, δaµ δMi K0i =ωqiK0 + iǫikqkK1, where S is the full action of Eq.(2.13) supplemented by K =ωq K iǫ q K , the additional term of Eq.(2.14). i0 i 0 − ik k 1 We next compute the current-current correlators in- K =ω2δ K iǫ ωK + (q2δ q q )K , ij ij 0 − ij 1 ij − i j 2 cluding the mixing with the nematic fields to lowest or- A =Kµν δAµδAν , (4.6) ders in the quadrupolar coupling F2. To this end we first L calculate the polarization tensor Πµν to include de ef- where Kµ is given by fects of the nematic fluctuations to lowest order in the quadrupolar interaction F2. This calculation involves Π0 summing over all one-particle-reducible diagrams, i.e. an K = 0 − 16π2D infinite series of bubble diagrams with two external gauge 1 2 fields and arbitrary number of quadrupolar insertions 1 Π1 + 4π V (q)Π0q K1 = + + connecting the bubbles pairwise. The result of this RPA- 4π 16π2D 64π3D 2 2 2 2 type computation is Π2 V (q)(ω Π Π ) V (q)Π0Π2q K = + 0 − 1 + 2 16π2D D D Π (q,ω) =Π0 +2F m2 j N N j ij ij 2 e h i aih b j i (4.7) Xa,b +(2F m2)2 j N N N N j + and D is 2 e h i aih a bih b j i ··· a,b X q 2 2 2 1 2 V ( ) 2 2F m j N N j D = Π ω (Π1 + ) + Π0(Π2 )q (4.8) 0 2 e a,bh i aih b j i 0 − 4π − 16π2 =Πij + 2 (4.3) 1 (2FP2me) a,b NaNb − h i In the above expressions Π are frequency and 0 i (where we have set κ = 0). HereP Πij is the polariza- momentum-dependent functions whose explicit form can tion tensor for the statistical gauge field of the composite be found in Ref.[52. fermions with ν = 1, NaNb is the correlator matrix of The poles in K give the spectrum of the collective the nematic order parameters,h i and j N is the mixed µν h µ ai excitations of the FQHE of Ref.[52], generalized to in- correlator of a current and a nematic field, both of which clude both the quadrupolar and the density-density in- were calculated in the previous section. To simplify the teractions. At long wavelengths, this correlator has a notation, in Eq.(4.3) we dropped the explicit momentum pole at 3¯ω (with residue q2), the cyclotron frequency and frequency dependence of the correlators. c ∼ of the electron (recall thatω ¯c is the effective cyclotron Since we are interested in the low energy and long frequency of the composite fermion). This pole is identi- wave-length limit, we expand NaNb in the leading or- fied as the (cyclotron resonance) Kohn mode [32], slightly qh i der for both the momentum and the frequency ω and shifted here by the extra term we added to the kinetic obtain energy (Eq.(2.14)). 1 δZF On the other hand, we find that the attractive N1N1 = N2N2 =2i h i h i ZF δM1δM1 quadrupolar interaction pushes down to lower energies 3 4¯ωc the lowest collective excitation, the Girvin-MacDonald- = , 4 ¯2 2 2 2 2 Platzman (GMP) mode [36] (which has residue q ). l π(ω 4¯ωc )(1 + 4αω¯c ) b − This mode has the dispersion ∼ 1 δZF N1N2 = N2N1 =2i h i − h i ZF δM1δM2 2 3 F2m ω¯ κ 2 ω2 = ω2 + (α ω¯3 e c )(q¯l )2 (4.9) 2ωω¯c 1 1,2 c ¯4 b = , (4.4) − lb iπ¯l2(ω2 4¯ω2)(1 + 4αω¯2)2 b − c c Using the results for the polarization tensor Πij of where we have set Eq.(4.3) we find the following effective Lagrangian for ˜ the gauge fields 4F2 2 ω1 = +2¯ωc(1+4αω¯c ), 1 π 2 2 a = Πµν (δaµ + δAµ)(δaν + δAν ) ω ω¯′ 1 L − 2 1 c α1 = ′2 ′ − 2 ′2 2 , ′ ω¯ ω¯ ω1 + 2(ω ω¯ ) (c1ω1 c2)t 1 µνρ 2 ′ V (x x ) ′ c − c 1 − c − + ǫ δaµ∂ν δaρ d x − δb(x)δb(x ) 2 ′2 8π − 32π2 ω1 ω¯c 1 Z α2 = − (4.10) (4.5) − ω¯′2 +¯ω′ ω + 2(ω2 ω¯′2) (c ω + c )t2 c c 1 1 − c 1 1 2 11 and where we used the notation A. Goldstone Modes

The nematic order parameter breaks the (continu- F˜2ω¯d 8F˜2ω¯d c1 = − 1 , ous) rotational symmetry of two dimensional plane and 2(4¯ω2 ω2) − π(ω2 ω¯2) d − 1 1 − d ! thus there is an associated Goldstone mode and an am- ˜ 2 2 plitude mode (which is strongly mixed with the GMP F2 ω¯dω1 c2 = 2 2 2 , mode). The spectrum of the nematic Goldstone mode (4¯ωd ω1) − can be obtained straightforwardly from M . In the low 2 2 L F2meω¯c energy regime and deep enough in the nematic phase, F˜2 = , ¯l2 r = r < 0, we can consider the effective Lagrangian b −| | ω¯′ =¯ω (1 + αω¯2), of the phase fluctuations (the Goldstone mode). Simi- c c c larly to the effective Lagrangian in the Bogoliubov the- 2 ω¯d =¯ωc(1+4αω¯c ) ory of superfluidity (or in the composite boson theory 2 2 of the FQHE) in the nematic phase the amplitude fluc- t = . (4.11) 2 2 tuations yield an effective Lagrangian for the Goldstone 1+ αω¯c − 2+8αω¯c boson (the phase field θ) of the form

It is easy to check that at the nematic transition of 1 ρn 2 1 2 M = (∂0θ) ρnvn ∇θ + .... (5.1) Eq.(3.5), where the “nematic mass” r 0 at the crit- L 2 vn − 2 | | → c ical value of the quadrupolar interaction F2 (given in where the nematic stiffness ρn and the velocity of the Eq.(3.9)), the gap of the GMP mode vanishes, ω1 0. → Goldstone modes vn are given by It is also easy to see that near the phase transition α1 < 0, 2 F2meκ α2 < 0 , and F2κ < 0. Now, provided α1,2 l¯4 > 0 r κ¯ − b ρ = , v =4 r κ¯(1+4αω¯2) (5.2) near and at the transition, then the GMP mode will con- n | | 2 n c 2u(1+4p αω¯c ) | | dense at zero momentum. This will result a nematic p phase and the FQH fluid will spontaneously break the where the parameters r,κ ¯ and u are given in Eq.(3.8). rotational symmetry. This condition can be achieved In the nematic phase the Goldstone bosons are massless provided the range of the quadrupolar interaction, con- and have a linear dispersion trolled by κ, is large enough. On the other hand, if 2 F2meκ ω(q)= vn q (5.3) α1,2 l¯4 < 0, the GMP mode will condense at a | | − b finite momentum. This would result a crystalline phase On the other hand, the nematic Goldstone mode will in which electrons break spontaneously the translational be gapped if there is a explicit weak symmetry breaking and rotational symmetries of the two-dimensional plane. term. For instance, if the underlying lattice has tetrago- In both cases, the Kohn mode remains gapped at and nal symmetry, the point group symmetry of the 2DEG is near the transition, and thus the liquid crystalline phases be C4. For the nematic order parameter, which is invari- are incompressible electronic liquid states and have the ant under rotations by π, this term reduces the symme- quantized Hall response. try to a Z2 (Ising) symmetry. The appropriate symmetry Early numerical results by Scarola, Park and Jain [50] breaking term in the nematic Lagrangian has the form predicted that for certain type of interactions the FQH = γ (M 2 M 2) γ 2M M fluid would become unstable to an uniform exciton con- LSB − 1 1 − 2 − 2 1 2 2 2 densate associated with the GMP mode. Our results = γ1M cos4θ γ2M sin 4θ (5.4) show that their exciton condensate is equivalent to a − − quantum phase transition to a nematic state. where γ1 and γ2 are two coupling constants. In this case, the mass gap of the Goldstone mode is linearly propor- tional to the strength of these weak symmetry breaking terms.

V. GOLDSTONE MODE AND DISCLINATIONS IN NEMATIC PHASE B. Disclinations

We now discuss the properties of the nematic phase. In D = 2 space dimensions nematic order parameters There are several particle-like excitations in the phase. have topological singularities (or defects ) called discli- First of all, the Kohn mode and the Laughlin quasi- nations [58]. Due to the presence of the goldstone mode, particles remain massive. The only change is that their the disclinations experience logarithmic interaction be- propagation is anisotropic. In addition to of these exci- tween them. In 2D disclinations are half-vortices of the tations, there are two more excitations which are absent order parameter director field M. When two disclina- in the isotropic phase. tions are separated by a distance R, the energy cost for 12 the configuration is VI. RESPONSE OF THE NEMATIC FQH FLUID TO CHANGES IN THE GEOMETRY 1 2 M 2 M 2 E = d xκ 2 = κ ln(R/l0), (5.5) R>|x|>l0 x In this section, we would explore the response of ne- Z matic FQH fluid to a long wavelength change in the ge- ometry of the underlying surface (i.e. the crystal) on where l is a ultra-violet cut-off for the integral which can 0 which the 2DEG is defined such as a shear distortion. be taken to be the correlation length of the nematic order Changes in the geometry can be described in terms of a parameter in the nematic phase. Hence, at zero temper- background spatial metric g ature disclinations and anti-disclinations are bound in ij (neutral) pairs but, above a critical temperature Tc, they proliferate 1 2e1 2e2 gij = − − (6.1) 2e2 1+2e1 We will now show that in the nematic FQH state, the  −  disclination carries electric charge due to the Wen-Zee coupling between the spin connection defined by the ne- which modifies the form of the action of Eq.(2.13) to the Q matic fields ωµ and the gauge fields in the effective ac- following expression tion Eq. (3.18). To investigate the charge accumulated L at the disclination, we first integrate out the statistical 2 † 1 † gauge field δa in the effective action to find a Wen-Zee = d xdt Ψ (x)D0Ψ(x) (DΨ(x)) (DΨ(x)) µ S − 2me · term in the effective action Z   δb(x)2 d2xdt V − 32π2 1 µνρ Q Z ωQ,δA = ǫ ωµ ∂ν δAρ. (5.6) εµνλ L 12π + d2xdt δa ∂ δa 8π µ ν λ Z 1 where Aµ just an external weak electromagnetic probe. 2 M 2 + d xdt 2 This term, will allow us to compute the electric charge 4F2me Z h of the disclination. Maciejko etal. [35] use the term “ne- κ ∇ ∇ + 2 Mi Mi matic gauge field” to refer to what here we call the ne- 4F2m · Q e i=1,2 matic spin connection, ωµ . X κ 2 2 2 Let us consider the case in which there exists a π ne- + 2 2e1(∂x ∂y )+2e22∂x∂y Mi 4F2me − matic vortex centered at x = 0. (Recall that since the   nematic fields are directors their orientation is defined M1 + e1 † 2 2 + Ψ Dx Dy Ψ mod π.) We can calculate the electric charge accumu- me −   lated at the disclination. We find M2 + e2 † + Ψ DxDy + DyDx Ψ me 1 M 2δ(x) M cos2θ D2 ρπ¯ 3  x † δρ( )= ( 2 2 t| | 2 ) (5.7) αΨ Ψ 12π −(1+4αω¯c ) − x − − 2me − me 2e M  + · Ψ†D2Ψ (6.2) The first term indicates the spin connection of the ne- me matic disclination act as the gauge field of a single flux i at the disclination core. Thus, a disclination of the where the covariant derivative now is nematic field serves as a particle source which changes the local charge density. The second term indicates the b Dµ = ∂µ + i(Aµ + aµ + ω ), (6.3) charge density gets redistributed as a result of non-zero µ quadrupole moment. In classical electrodynamics, the b non-uniform charge density could give rise to an electron where ω is the spin connection of the background met- 2 2 quadrupole moment Qij = d xρ(r)(2xixj δij x ) and ric. Here we have used the result from our recent work vice versa. Since our nematic field couples− to the| | stress [47] that upon attaching two Chern-Simons flux to the tensor, a nematic order withR a disclination configura- fermion, the composite particle effectively carries spin 1 tion leads to a new charge density distribution, shown and couples to the spin connection of the background ge- in the second term in Eq.(5.7). The charge of the discli- ometry. Notice also that the quadrupolar interaction is nation depends on the strength of the order parameter modified by a change of the geometry. To simplify mat- M and is not quantized. This implies that the disclina- ters we considered only a contact density-density cou- |tions| will generally have irrational mutual statistics with pling (parametrized by the interaction strength V ). quasiparticles and irrational self statistics. Most of these We can now make use of the results of the preceding results were anticipated on phenomenological and sym- sections (and of the results of Ref.[47]) to derive the effec- metry grounds in the work of Maciejko et al. [35]. tive Lagrangian for the nematic fields M at low energies 13 and long distances. It is given by To better illustrate the effect of nematic fluctuations, we rewrite the action in terms of a separate dependence on ij ρ¯ Mi Mj the background metric and on the metric defined by the M =ǫ 2 + ei ∂0 2 + ej L 2 1+4αω¯c 1+4αω¯c nematic order parameter, and obtain ω¯     + c e 2 + ǫij ρ¯ e ∂ e ¯2 i 0 j 2¯πlb | | 1 1 1 + ǫµνλ A + ωb + ωm ∂ A + ωb + ωm 12π µ µ 2 µ ν λ λ 2 λ  1  u  r M 2 ǫµνλωm∂ ωm (M 2)2 (6.4) − | | − 48π µ ν λ − 4 Hereρ ¯ is the average electron density, and we denoted by ωm the spin connection for the sum of the background and nematic metrics,

m ij Mi Mj ω0 =ǫ 2 + ei ∂0 2 + ej , 1+4αω¯c 1+4αω¯c     m ij Mi Mj ωx =ǫ 2 + ei ∂x 2 + ej 1+4αω¯c 1+4αω¯c     ∂ (tM + e ) ∂ (tM + e ) , − x 2 2 − y 1 1 m ij  Mi Mj  ωy =ǫ 2 + ei ∂y 2 + ej 1+4αω¯c 1+4αω¯c     + ∂x(tM1 + e1)+ ∂y(tM2 + e2) , (6.5)  

ρ¯ M M ω¯ u =ǫij i + e ∂ j + e r M 2 + c e 2 + ǫij ρ¯ e ∂ e (M 2)2 M 2 i 0 2 j ¯2 i 0 j L 2 1+4αω¯c 1+4αω¯c − | | 2πlb | | − 4 1 3  3  1 1 + ǫµνλ A + ωb ∂ A + ωb ǫµνλωQ∂ A ǫµνλωb ∂ ωb 12π µ 2 µ ν λ 2 λ − 12π µ ν λ − 48π µ ν λ 1  µνρ    + 2 ǫ M1∂µe2 M2∂µe1 + e1∂µM2 e2∂µM1 ∂ν Aρ 12π(1+4αω¯c ) − −   1 µνρ 2 Q b + 2 ǫ M1∂µe2 M2∂µe1 + e1∂µM2 e2∂µM1 +(1+4αω¯c )ωµ ∂ν ωρ (6.6) 12π(1+4αω¯c ) − −  

In the isotropic phase, the nematic field is massive so it mab tensor. In the isotropic phase, the Hall viscosity H xx yy can be integrated out. This generates operators which [37, 39, 44, 47, 61, 63, 64], defined by η = ηxy = ηxy, are higher order in derivatives and are irrelevant in in is isotropic and for the ν = 1/3 Laughlin FQH state− it H 3 the low energy and long distance regime of our theory. is found to be given by η = 2 ρ¯, in agreement with Finally, we obtain the following simple expression for the earlier results. In the nematic phase, provided the ne- effective theory of background metric in the symmetric matic order is uniform in space, the Hall viscosity ηH = xx yy 3¯ρ phase, (ηxy ηxy)/2= 2 remains the same as in the isotropic FQH− fluid phase. However, since the system is spatially 3¯ρ =ǫij e ∂ e anisotropic, we can define the combination of the com- M i 0 j D xx yy ¯ H L 2 ponents of the viscosity η = (ηxy + ηxy)/2 Mη , 1 3 3 ∝ + ǫµνλ(A + ωb )∂ (A + ωb ) which indicates that there is a viscosity response for a 12π µ 2 µ ν λ 2 λ mixed shear and a dilation deformation which, however, 1 is not universal. In particular, when the nematic order is ǫµνλωb ∂ ωb (6.7) −48π µ ν λ uniform in space, the geometry quantity such as the Hall viscosity, orbital spin, central charge remains unchanged which is consistent with our recent results [47]. at the universal value. In the nematic phase, the nematic order couples to the electrons (and the composite fermions) as an effect mass On the other hand, in the presence of a disclination 14 in the nematic phase, from Eq.(6.6) we see that the Hall of the mean field theory of the composite fermions in a viscosity is modified. If M(x) is a configuration of the background nematic field. The actually Hall viscosity of nematic order parameter with a disclination at x = xv the FQH fluid (both in the isotropic and in the nematic with winding number nv, the Hall viscosity of the fluid phase) is obtained as a response to a shear distortion now is of the geometry in which the electrons move. Further- more, we uncover the existence of a geometric Chern- 1 M 2 x H x x Simons term between the nematic order parameters and η( )=η0 + | | 2 2 nvδ( v) 12π (1+4αω¯c ) − the gauge fields. The term is of the same form as Wen-Zee 2 2 term and the “spin connection” is interpreted in terms +2t[(∂x ∂y )M1 +4∂x∂yM2]) (6.8) − of the order parameter instead of the background metric. The first term is equal to the Hall viscosity of the Then the flux of the “spin connection” is proportional to isotropic phase. The second term shows the change of the disclination density in the nematic phase, and that the Hall viscosity due to the nematic disclination. Here, as a consequence the disclination carries non-quantized nv is the winding number of the disclination, and xv is gauge charge and statistics. the coordinate of the disclination core. The third term in- dicates the charge density redistribution results from the After the identification of the criticality and the nematic order as a quadrupole moment which affects the phases, we investigated the excitations near the quan- value of the Hall viscosity. However, the orbital spin and tum phase transition. As the the nematic quantum the gravitational Chern-Simons term remain the same in phase transition is approached, the mass gap of GMP both phases. mode of the FQH fluid is shown to vanish continuously. On the other hand, the Laughlin quasiparticles and the In the recent work of Maciejko [35]et.al. these au- Kohn mode remain gapped at the transition and thus thors considered an effective description of the nematic the Kohn’s theorem is not violated at and near the tran- FQH state and the transition between an isotropic state sition. Depending on the microscopic details of the inter- and the anisotropic state. They used a composite bo- actions, we showed that the GMP mode can close its gap son theory and wrote down the symmetry-allowed terms either at finite or at zero momentum, either giving rise in the effective Lagrangian. Interestingly they identified to a nematic or to crystal (or stripe) phase. Both liquid the coupling between the nematic order parameter and crystalline phases obtained through the softening of the the statistical gauge field by making an analogy to the GMP mode are incompressible electronic liquid crystals case of the magnetization of the quantum Hall ferromag- and thus are expected to have a fractionally quantized net. Furthermore, they found that the critical theory has Hall response. It is notable that the mechanism of the the dynamical scaling exponent z = 2 due to the Berry isotropic-nematic transition described here is special for phase term for the nematic order parameter, which we FQH states. For the integer quantum Hall states, the also found here. However, in their description the coef- lowest excitation is the Kohn mode, inter-Landau level ficient of the Berry phase term is the full Hall viscosity excitation, and it cannot close its gap at zero momentum of the FQH fluid. Instead, here we showed that that the without a large amount of Landau level mixing and a Berry phase term is not exactly equal to the Hall viscos- strong violation of Galilean invariance. ity of the FQH fluid but it is equal to the Hall viscosity of integer quantum Hall mean-field state of the composite fermions.

VII. CONCLUSIONS ACKNOWLEDGMENTS In this work, we have studied the nematic quan- tum phase transition inside a FQH state in a 2DEG in with an attractive quadrupolar interaction between elec- We would like to thanks Tankut Can, Andrey Gromov, trons. We used the Chern-Simons theory of composite Taylor Hughes, Shamit Kachru, Steve Kivelson, Rob fermions. Since the FQH state is gapped a critical at- Leigh, Joseph Maciejko, Roger Mong, Chetan Nayak, tractive quadrupolar coupling is needed for the system Kwon Park, Onkar Parrikar, Dam Thanh Son, Shivaji to develop a finite quadrupole density and to break ro- Sondhi, Paul Wiegmann, and Michael Zaletel for helpful tational symmetry spontaneously. The quantum phase discussions. G. Y. C. thanks the financial support from transition has dynamical exponent z = 2 and it is in ICMT postdoctoral fellowship. EF thanks the KITP the universality class of the quantum phase transition (and the Simons Foundation) and its IRONIC14 pro- in the dilute Bose gas. The z = 2 quantum criticality gram for support and hospitality. This work is sup- is a consequence of a Berry phase term present in the ported in part by the by the National Science Foundation, effective action for the nematic fields is related to (but under grants DMR-1064319 (GYC) and DMR 1408713 not the same) as the Hall viscosity. We showed that the (YY,EF) at the University of Illinois, PHY11-25915 at coefficient of the Berry phase term is the Hall viscosity KITP (EF), and by ICMT. 15

Appendix A: Calculation of the Nematic Correlators FIG. 1. Correlator of the nematic order parameters

In this appendix, we summarize the calculation of the correlators of the nematic order parameters which we ex- tensively use in the main text. First, we focus on the symmetric part of the correla- tors, which is the mass term for the order parameters. Here for simplicity, we set the magnetic length l¯b and Mi Mj electron bare mass me to be 1. At the end, we restore the magnetic length and mass in the expression by di- p mensional analysis. For the filling fraction ν = 2p+1 , the correlator is

δZ † 2 2 † 2 2 N1(r1,t1)N1(r2,t2) =2i = i Ψ (r1,t1)(Dx Dy)Ψ(r1,t1)Ψ (r2,t2)(Dx Dy)Ψ(r2,t2) h i δM1δM1 − hT − − i ∞ p−1 = i ei(ωm−ωl)(t2−t1)Θ(t t )φ† (r )(D2 D2)φ (r )φ† (r )(D2 D2)φ (r ) − 1 − 2 l,k1 1 x − y m.k2 1 m,k2 2 x − y l,k1 2 m>p 1 2 X Xl=0 kX,k  + e−i(ωm−ωl)(t2−t1)Θ(t t )φ† (r )(D2 D2)φ (r )φ† (r )(D2 D2)φ (r ) , (A1) 2 − 1 m,k2 1 x − y l.k1 1 l,k1 2 x − y m,k2 2  2 in which ωm =ω ¯cm + α(mω¯c) ,m Z is the cyclotron energy of the composite fermion at the m-th Landau level. For ν =1/3 filling, p = 1. Thus the∈ sum over m simply becomes the sum over m> 0 and we can set l = 0 at the end of calculation. To proceed, we perform the Fourier transformation of the correlator. Here we have denoted by

x x 2 1 1 1 x ik1 x −(y +k1 ) /2 x φl,k1 (r1)= e e Hl(y1 + k1 ), (A2) s√π2ll! the Landau wavefunctions in the Ay = 0 gauge. In Fourier space we find

x x i(x2−x1)(k −k +q )+iq (y2−y1) e 1 2 x y x 2 x 2 N N (q,ω) = C dk d2x dy e(−1/2)((yi+k1 ) +(yi+k2 ) ) h 1 1 i lm i i i ω (ω ω )+ iǫ m Z  − m − l x 2 2X x x 2 2 x Hl(y1 + k1 )(Dx Dy)Hm(y1 + k2 )Hm(y2 + k2 )(Dx Dy)Hl(y2 + k1 ) x −x − i(x2−x1)(−k +k +q )+iq (y2−y1) e 1 2 x y x 2 x 2 e(−1/2)((yi+k1 ) +(yi+k2 ) ) − ω + (ω ω ) iǫ m − l − H (y + kx)(D2 D2)H (y + kx)H (y + kx)(D2 D2)H (y + kx) , (A3) m 1 2 x − y l 1 1 l 2 1 x − y m 2 2  where We now change the variables to

kx + kx u˜ =y + 1 2 , i i 2 kx kx v˜ = 1 − 2 , 2 kx + kx u =y + 1 2 + iq /2, 1 1 2 y kx + kx u =y + 1 2 iq /2, 2 2 2 − y v =˜v iqy/2, 1 ∗ − C = (A4) v =˜v + iqy/2, (A5) lm 2l+ml!m!2π2 16

and we integrate out xi’s to obtain

N N (q,ω) = C du dv h 1 1 i lm i m X Z δ(˜v + qx/2) 2 ∗ [ e−ui −2vv H (u + v)(D2 D2)H (u v∗)H (u v)(D2 D2)H (u + v∗) ω (ω ω )+ iǫ l 1 x − y m 1 − m 2 − x − y l 2 − m − l δ(˜v qx/2) 2 ∗ − e−ui −2vv H (u v∗)(D2 D2)H (u + v)H (u + v∗)(D2 D2)H (u v)]. (A6) − ω + (ω ω ) iǫ m 1 − x − y l 1 l 2 x − y m 2 − m − l −

Since iD H (u v∗) = (u v∗)H (u v∗) (we In this way, the correlator can be simplified to the fol- − x m 1 − 1 − m 1 − choose the Landau gauge Dx = ∂x + i¯by,Dy = ∂y), we lowing expression have the following.

iD H =1/2H + nH − x n n+1 N−1 iD H (u v∗)=i( 1/2H (u v∗) − y m 1 − − m+1 1 − +mH (u v∗)) (A7) m−1 1 −

N N (q,ω) h 1 1 i δ(˜v + qx/2) 2 ∗ = C du dv[ e−ui −2vv m2H (u + v)H (u v∗)H (u v)H (u + v∗) lm i ω (ω ω )+ iǫ l+1 1 m−1 1 − m−1 2 − l−1 2 m m l X Z − − δ(˜v qx/2) 2 ∗ − e−ui −2vv m2H (u + v)H (u v∗)H (u v)H (u + v∗)] (A8) − ω + (ω ω ) iǫ l+1 1 m−1 1 − m−1 2 − l−1 2 m − l −

2 ¯2 We can use the expression for the inner product of the By dimensional analysis, we need to multiplyω ¯c /lb to two Hermite polynomials which is written in terms of the the correlator to restore the coefficients by setting the Laguerre polynomials magnetic length to be unity in the calculation.

2 du e−ui H (u + v)H (u v∗)= 1 l 1 m 1 − Z 2m√πl!(v∗)m−lLm−l( 2vv∗), l − (A9) 4¯ω3 N N (q,ω) = c + O(q2) 1 1 ¯2 2 2 2 h i lb π(ω 4¯ωc )(1 + 4αω¯c ) if l is not larger than m. Here v is related with qx, qy af- − m−l ∗ ω¯c 2 2 ter we integrate overv ˜. Ll ( 2vv ) is the polynomial = + O(q )+ O(ω ) 2 − ¯2 2 of q whose leading order is always a constant piece. We − lb π(1+4αω¯c ) can always express the result by expanding it in terms (A11) of ω and q by order. The leading order in q and ω of N1N1(q,ω) (coming from l =0,m = 2) includes a con- stanth piece. i 1 N N (q,ω) = The first term will contribute to the mass term of the h 1 1 i π(ω 2¯ω )(1 + 4αω¯2) − c c nematic order parameters. 1 2 2 + O(q ) −π(ω +2¯ωc)(1 + 4αω¯ ) c The anti-symmetric part, proportional to M ∂ M , of 4¯ω 1 0 2 = c + O(q2) (A10) the correlator can be calculated in the same way. The π(ω2 4¯ω2)(1 + 4αω¯2) − c c full result for the correlator is 17

N N (q,ω) = C du dv h 1 2 i lm i m X Z δ(˜v + qx/2) 2 ∗ [ e−ui −2vv H (u + v)(D2 D2)H (u v∗)H (u v)(D2 D2)H (u + v∗) ω (ω ω )+ iǫ l 1 x − y m 1 − m 2 − x − y l 2 − m − l δ(˜v q /2) 2 ∗ x −ui −2vv ∗ 2 2 ∗ 2 2 − e Hm(u1 v )(Dx Dy)Hl(u1 + v)Hl(u2 + v )(Dx Dy)Hm(u2 v)] − ω + (ωm ωl) iǫ − − − − − − 2 δ(˜v + qx/2) 2 ∗ m = C du dv[ e−ui −2vv H (u + v)H (u v∗)H (u v)H (u + v∗) lm i ω (ω ω )+ iǫ i l+1 1 m−1 1 − m−1 2 − l−1 2 m Z − m − l X 2 δ(˜v qx/2) 2 ∗ m + − e−ui −2vv H (u + v)H (u v∗)H (u v)H (u + v∗)] (A12) ω + (ω ω ) iǫ i l+1 1 m−1 1 − m−1 2 − l−1 2 m − l −

The leading order behavior in q and ω comes from the The coefficient of the leading term is the Hall viscosity term with l = 0 and m = 2. Within this approximation, of the integer quantum Hall state. the leading low frequency and low momenta behavior of the correlator is

2ω 2 N1N2(q,ω) = 2 2 2 2 + O(q ) h i iπ(ω 4¯ωc )(1 + 4αω¯c ) − (A13) 2 ¯2 Appendix B: Calculation of the Mixed Correlators Again, we now multiply the factorω ¯c /lb to restore the units properly to find the result of Nematic and Gauge Fields ω N N (q,ω) = i + O(q2)+ O(ω2) 1 2 ¯2 2 2 h i 2lb π(1+4αω¯c ) The nematic-gauge coupling term could be obtained in (A14) a similar way

† 2 2 † N (r1,t )j (r2,t ) = i Ψ (r ,t )(D D )Ψ(r ,t )Ψ (r ,t )Ψ(r ,t ) h 1 1 0 2 i − hT 1 1 x − y 1 1 2 2 2 2 i

= Clm duidv m X Z δ(˜v + qx/2) 2 ∗ [ e−ui −2vv H (u + v)(D2 D2)H (u v∗)H (u v)H (u + v∗) ω (ω ω )+ iǫ l 1 x − y m 1 − m 2 − l 2 − m − l δ(˜v qx/2) 2 ∗ − e−ui −2vv H (u v∗)(D2 D2)H (u + v)H (u + v∗)H (u v)] − ω + (ω ω ) iǫ m 1 − x − y l 1 l 2 m 2 − m − l − δ(˜v + qx/2) 2 ∗ = C du dv[ e−ui −2vv mH (u + v)H (u v∗)H (u v)H (u + v∗) lm i ω (ω ω )+ iǫ l+1 1 m−1 1 − m 2 − l 2 m m l X Z − − δ(˜v qx/2) 2 ∗ − e−ui −2vv mH (u + v)H (u v∗)H (u v)H (u + v∗)] (B1) − ω + (ω ω ) iǫ l+1 1 m−1 1 − m 2 − l 2 m − l −

The leading order term is p2, which comes from the Following the above calculation, we can obtain other lin- contribution of (l =0,m =∝ 2), (l =0,m = 1), and yields ear coupling terms between the nematic field and gauge the expression,

1 2 2 2 2 N1(p)j0( p) = ( 2 2 )(px py) h − i 2π 1+ αω¯c − 2+8αω¯c − (B2) 18

field,

(a) 1 2 2 N2(p)j0( p) = ( 2 2 )2pxpy h − i 2π 1+ αω¯c − 2+8αω¯c 1 2 2 N1(p)jx( p) = ( 2 2 )ωpx h − i 2π 1+ αω¯c − 2+8αω¯c 1 2 2 N1(p)jy( p) = ( 2 2 )( ωpy) h − i 2π 1+ αω¯c − 2+8αω¯c − 1 2 2 N2(p)jx( p) = ( 2 2 )ωpy h − i 2π 1+ αω¯c − 2+8αω¯c (b) 1 2 2 N2(p)jy( p) = ( 2 2 )ωpx (B3) h − i 2π 1+ αω¯c − 2+8αω¯c

(c)

FIG. 2. Diagrams contributing to the Wen-Zee term. Here These terms contribute partly to the Wen-Zee cou- the wiggly line represents the gauge field aµ and the dotted pling. For a complete expression of the Wen-Zee term, line represents the nematic field Mi. The thick line represents we also needs to evaluate the correlator between two ne- the composite fermion propagator. matic field and one gauge field

N (r1,t )N (r2,t )j (r3,t ) = h 1 1 2 2 0 3 i Ψ†(r ,t )(D2 D2)Ψ(r ,t )Ψ†(r ,t )(D D + D D )Ψ(r ,t )Ψ†(r ,t )Ψ(r ,t ) (B4) − hT 1 1 x − y 1 1 2 2 x y y x 2 2 3 3 3 3 i N1N2j0 (q,p)= N1(r1,t1)N2(r2,t2)j0(r3,t3) exp( iω(t2 t1) iω0(t3 t2)) exp(iq(r2 r1)+ ip(r3 r2)) h i h i − − − − − − (B5)

By redefining the variables,

kx + kx kx kx kx kx u = y + 1 2 + iqy/2, v = 1 − 2 iqy/2, v˜ = 1 − 2 ; 1 1 2 2 − 2 kx + kx kx kx kx kx u = y + 2 3 i(qy py)/2, v = 2 − 3 + i(qy py)/2, v˜ = 2 − 3 ; 2 2 2 − − 0 2 − 0 2 kx + kx kx kx kx kx u = y + 1 3 ipy/2, v + v = 1 − 3 + ipy/2, v˜ +v ˜ = 1 − 3 ; (B6) 3 3 2 − 0 2 0 2 19 we can write the time ordered correlator (for t1 >t2 >t3) as

N N j (q,p)= exp( u2 vv∗ v v∗ (v + v)(v∗ + v∗)) exp( iq/2 p/2) h 1 2 0i − i − − 0 0 − 0 0 − ∧ δ(˜v + qx/2)δ(˜v + px/2) C du dvdv 0 lmn i 0 [ω (ω ω )+ iǫ][ω (ω ω )+ iǫ] m,n m l 0 n l X Z − − − − H (u + v)(D2 D2)H (u v∗)H (u + v )(D D + D D )H (u v∗)H (u v∗ v∗)H (u + v + v ) l 1 x − y m 1 − m 2 0 x y y x n 2 − 0 n 3 − − 0 l 3 0 = exp( u2 vv∗ v v∗ (v + v)(v∗ + v∗)) exp( iq/2 p/2) − i − − 0 0 − 0 0 − ∧ δ(˜v + qx/2)δ(˜v + px/2) C du dvdv 0 lmn i 0 [ω (ω ω )+ iǫ][ω (ω ω )+ iǫ] m,n m l 0 n l X Z − − − − [H (u + v)mH (u v∗)H (u + v )(in)H (u v∗)H (u v∗ v∗)H (u + v + v ) l+1 1 m−1 1 − m+1 2 0 n−1 2 − 0 n 3 − − 0 l 3 0 + H (u + v)mH (u v∗)H (u + v )( im)H (u v∗)H (u v∗ v∗)H (u + v + v )] (B7) l+1 1 m−1 1 − m−1 2 0 − n+1 2 − 0 n 3 − − 0 l 3 0

For this three-point correlator, there always exists an an- of [l =0,n =0,m = 2], thus tisymmetric phase factor exp( iq/2 p/2)(known as the − ∧ Moyal phase, see e.g. Ref. [65]) which is responsible for N1N2j0 (q,p) t1>t2>t3 = h i | the Wen-Zee response. q p + q p − x y y x + .... 4π(ω (ω2 ω0))(ω0) − − (B8) The leading order contribution for three point time ordered correlator (at t1 >t2 >t3) comes from the choice In a similar way, we also have

N N j (q,p) = exp( u2 vv∗ v v∗ (v + v)(v∗ + v∗)) exp( iq/2 p/2) h 2 1 0i |t1>t2>t3 − i − − 0 0 − 0 0 − ∧ δ(˜v + qx/2)δ(˜v + px/2) C du dvdv 0 lmn i 0 [ω (ω ω )+ iǫ][ω (ω ω )+ iǫ] m,n m l 0 n l X Z − − − − H (u + v)(D D + D D )H (u v∗)H (u + v )(D2 D2)H (u v∗)H (u v∗ v∗)H (u + v + v ) l 1 x y y x m 1 − m 2 0 x − y n 2 − 0 n 3 − − 0 l 3 0 = exp( u2 vv∗ v v∗ (v + v)(v∗ + v∗)) exp( iq/2 p/2) − i − − 0 0 − 0 0 − ∧ δ(˜v + qx/2)δ(˜v + px/2) C du dvdv 0 lmn i 0 [ω (ω ω )+ iǫ][ω (ω ω )+ iǫ] m,n m l 0 n l X Z − − − − [H (u + v)(im)H (u v∗)H (u + v )(n)H (u v∗)H (u v∗ v∗)H (u + v + v ) l+1 1 m−1 1 − m+1 2 0 n−1 2 − 0 n 3 − − 0 l 3 0 + H (u + v)(im)H (u v∗)H (u + v )(m)H (u v∗)H (u v∗ v∗)H (u + v + v )] (B9) l+1 1 m−1 1 − m−1 2 0 n+1 2 − 0 n 3 − − 0 l 3 0

In the leading order, Appendix C: Proof of gauge invariance at the RPA level

qxpy qypx N2N1j0 (q, q0) t1>t2>t3 = − + .... h i | 4π(ω (ω2 ω0))(ω0) − − (B10)

The other time ordered correlator can be obtained in To calculate the collective excitation of the Nematic the similar way which finally gives Wen-Zee coupling. FQH state, we first treat the Quadrupolar interaction Finally, we have, perturbatively in the RPA level and then integrate out the gauge fluctuation. During the RPA procedure, we ij λνµ only keep the reducible diagrams for the infinite geomet- ǫ ǫ pλqν ric series. Here, we would proof that the polarization NiNj jµ (q,p)= 2 2 (B11) h i (1+4αω¯c ) 4π tensor of such RPA level correction is gauge invariant. 20

The polarization at RPA level has the form To proof its gauge invariance, we only need to proof

p ΠRPA(ω,p)=0 (C2) ΠRPA =Π0 + (2F m2) j N N j µ µν ij ij 2 e h i aih b j i Xa,b Or if write it in the real space, 2 2 + (2F2me) jiNa NaNb Nbjj + RPA h ih ih i ··· ∂µΠµν (x, y)=0 (C3) Xa,b 2 0 (2F2me) jiNa Nbjj 0 It is obvious that pµΠµν = 0. Thus we only need to prove =Πij + h 2 ih i (C1) 1 (2F2me) NaNb the gauge invariance of p j N = 0. − h i µh µ ai

† † † 2 2 † † 2 2 2∂r1 ji(r1)N1(r2) = i∂r1 D Ψ (r1)Ψ(r1)Ψ (r2)(D D )Ψ(r2) i∂r1 Ψ (r1)DiΨ(r1)Ψ (r2)(D D )Ψ(r2) i h i i h i x − y i− i h x − y i † † † 2 2 † † † 2 2 = i ∂r1 D Ψ (r1)Ψ(r1)Ψ (r2)(D D )Ψ(r2) + i D Ψ (r1)∂r1 Ψ(r1)Ψ (r2)(D D )Ψ(r2) h i i x − y i h i i x − y i † † 2 2 † † 2 2 i ∂r1 Ψ (r1)DiΨ(r1)Ψ (r2)(D D )Ψ(r2) i Ψ (r1)∂r1 DiΨ(r1)Ψ (r2)(D D )Ψ(r2) − h i x − y i− h i x − y i = i D†D†Ψ†(r )Ψ(r )Ψ†(r )(D2 D2)Ψ(r ) + i D†Ψ†(r )D Ψ(r )Ψ†(r )(D2 D2)Ψ(r ) h i i 1 1 2 x − y 2 i h i 1 i 1 2 x − y 2 i i D†Ψ†(r )D Ψ(r )Ψ†(r )(D2 D2)Ψ(r ) i Ψ†(r )D D Ψ(r )Ψ†(r )(D2 D2)Ψ(r ) − h i 1 i 1 2 x − y 2 i− h 1 i i 1 2 x − y 2 i = i D†D†Ψ†(r )Ψ(r )Ψ†(r )(D2 D2)Ψ(r ) i Ψ†(r )D D Ψ(r )Ψ†(r )(D2 D2)Ψ(r ) h i i 1 1 2 x − y 2 i− h 1 i i 1 2 x − y 2 i † † 2 2 † † 2 2 ∂ 1 j (r )N (r ) = ∂ 1 Ψ (r )Ψ(r )Ψ (r )(D D )Ψ(r ) Ψ (r )∂ 1 Ψ(r )Ψ (r )(D D )Ψ(r ) (C4) r0 h 0 1 1 2 i −h r0 1 1 2 x − y 2 i − h 1 r0 1 2 x − y 2 i In all we have,

D†2 1 † 2 2 † i∂ jµ(r1)N1(r2) = Ψ(r1)Ψ (r2)(D D )r2 Ψ(r2)Ψ (r1)( i∂0 + + µ)r1 − rµ h i h x − y − 2 i D2 (i∂ + + µ) Ψ(r )Ψ†(r )(D2 D2) Ψ(r )Ψ†(r ) − h 0 2 r1 1 2 x − y r2 2 1 i D†2 D2 = G(r , r )(D2 D2) G(r , r )( i∂ + + µ) (i∂ + + µ) G(r , r )(D2 D2) G(r , r ) (C5) h 1 2 x − y r2 2 1 − 0 2 r1 i − h 0 2 r1 1 2 x − y r2 2 1 i

Recall that the Green function has the property, gives the spectrum of the excitations.

2 2 D K00 = q K0, i∂0 + + µ G(r1, r2)= 2 r1 K0i = ωqiK0 + iǫikqkK1,   2 D † Ki0 = ωqiK0 iǫikqkK1, i∂0 + + µ Ψ(r1)Ψ (r2) − 2 r1 h i K = ω2δ K iǫ ωK + (q2δ q q )K ,   ij ij 0 − ij 1 ij − i j 2 = δ(r1 r2) (C6) Π − K = 0 , 0 −16π2D and similarly for the adjoint. Thus we have, 1 2 1 Π1 + V (q)Π q K = + 4π + 0 , 1 4π 16π2D 64π3D i∂ 1 jµ(r1)N1(r2) = 0 (C7) rµ 2 2 2 2 − h i Π2 V (q)(ω Π0 Π1) V (q)Π0Π2q K2 = + − + , Thus, we had shown that the polarization tensor at the 16π2D D D V (q) RPA level is gauge invariant. D = Π2ω2 (Π + θ)2 + Π (Π )q2, 0 − 1 0 2 − 16π2 1 θ = . (D1) Appendix D: Nematic collective excitations 4π

We solve D(q,ω) = 0 to find the poles of Kµν . To obtain the collective excitations of the nematic FQH state, we need to calculate the polarization tensorKij for 2 2 2 V (q) 2 Π0ω (Π1 + θ) + Π0(Π2 2 )q = 0 (D2) external electromagnetic gauge field. The poles in Kij − − 16π 21

As the left part of the equation D2 involves a sum of order O(q2). To solve this equation, we have to subtract infinite numbers of polynomials, it is impossible to solve the constant piece from the first two terms and set them it exactly. What we can try to do instead is to assume as zero. 2n dispersion ω = ω1 + αnq and find the solution asymp- ′ 2 ′2 2 totically near zero momentum. We only keep the lowest ω¯cω1 c1t ω1 ω¯c t c2 2 ′2 + = 2 ′2 + +2πθ (D3) terms in q and ω. As we can see, the first two terms have |ω ω¯c α | |ω ω¯c α | 1 − 1 − leading order O(1), while the last term have the leading This gives us the solution for the lowest excitation in the nematic FQH state, which is the GMP mode.

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