arXiv:1602.01482v3 [cond-mat.str-el] 3 May 2016 indrn h atfwyasi eea ra fquan- of areas several physics. in matter years condensed few past tum the during tion n nudroe Bi underdoped in and as others, such YBa among superconductors, in cuprate ranging, temperature systems high different from experimen- many discovered Electronic in been now tally system. by physical have a phases of of nematic breaking symmetries spontaneous spatial the the by characterized states, are crystal liquid which electronic as known matter, elec- tronic quantum-mechanical strongly-interacting of phases correlated from strongly a arises in material. phase micro- electrons nemato- nematic of the self-organization constituent electronic the to the an back molecules, of gen traced nature be cigar-shaped scopic can order classi- cousins, orientational their crystal in Unlike liquid system symmetry. cal electronic translation correlated breaking with- strongly out spontaneously broken a is invariance of rotational which state a is xeietleiec o neetoi eai tt was state nematic electronic an for evidence experimental ecnutr uha Ca(Fe as such perconductors h iae uhnt Sr ruthenate bilayer the lcrncnmtcpae aebe ou fatten- of focus a been have phases nematic Electronic h lcrncnmtcsaeblnst ls of class a to belongs state nematic electronic The oee,tefis n oti aetems spectacular most the date this to and first the However, eai unu hs rniino opst em liquid Fermi composite of transition phase quantum Nematic .ITOUTO N MOTIVATION AND INTRODUCTION I. 2 al nttt o hoeia hsc,Uiest fCal of University Physics, Theoretical for Institute Kavli 2 Cu 3 eateto hsc,KraAvne nttt fScience of Institute Advanced Korea Physics, of Department 3 O iudsae h ffciefil hoyfrteisotropic-ne the for theory ga field effective Chern-Simons z The quasiparticle the fermion state. of composite liquid fluctuations the destroy the can Both parameter Chern-S nemati underlying at the attachment. comp the flux of flux with the the fluctuations interplaying into local on metric levels the based dynamical that Landau is show half-filled We theory d in We This transition. liquid liquids. transition. electron Fermi attrac phase conventional an an the the of by that case of triggered the and show is in We phase as nematic electrons, levels. the and Landau isotropic half-filled the in arising liquid h iciain are neeti hre eso hta Goldstone that a show topologi as We interpreted the be that charge. can show electric experiments We conductivity an identity. carries Ward disclination, a the by Hal compressib the obtained of also this computation fluctu is of (one-loop) breaking perturbative theory time-reversal a present the field We from effective originate “H the terms non-universal that Both a show as also interpreted dy effective We be the governs can which which term type fluctuations, phase Berry a is there 6+ yaia xoetdet h adudmigo h es F dense the of damping Landau the to due exponent dynamical 3 = epeetater fteiorpcnmtcqatmpaet phase quantum isotropic-nematic the of theory a present We x 6–8 o ra ag fdpn levels, doping of range broad a for 2 Sr 2 3 nvriyo liosa raaCapin lios6180 Illinois Urbana-Champaign, at Illinois of University CaCu 1 Ru 2,3 eateto hsc n nttt o odne atrTh Matter Condensed for Institute and Physics of Department 2 1 ih You, Yizhi hs ednyt exhibit to tendency whose O − x 2 7 1 Co O . 11 neetoi nematic electronic An 8+ eesadtergoercresponse geometric their and levels x ) δ 2 , 9 As oio-ae su- iron-based to ,2 1, 2 , 10 i on Cho, Young Gil n loto also and Dtd a ,2016) 4, May (Dated: 4,5 fri at abr,SnaBraa aiona93106 California Barbara, Santa Barbara, Santa ifornia ,1 3, tie rudsae nitrrtto tl sdi the in used still interpretation a an literature. of state, evidence ground as striped interpreted originally were experiments lcrcfil sse,wt hr ne fnarrow-band of fields. onset electric sharp larger threshold a for with sharp noise extremely seen, Landau an is the field regime, of electric this center in the at and, from and level, away samples observed Hall same quantum is the integer plateau reentrant in a contrast, temperatures, for same In detected the characteristic ever regime. fields, were this break- electric in state, symmetry ground threshold charge-density-wave translation no hence, a of and, signs bias e.g. low any ing, at show linear not are do curves I-V the regime hsaiorp a aryrpdices tatem- a at increase rapid fairly a perature has anisotropy This rcino ecn at percent a of fraction ihmgei ed ntemdl ftesecond, the of middle the in in fields (2DEG) magnetic gases high electron two-dimensional in discovered acsa ag s350i h laetsmlsa the at samples cleanest to the down in originally resis- 3,500 temperatures, of as lowest ratio large a properties (with as anisotropy transport tances spatial longitudinal strong the a that exhibit found was It scmrsil n h rcinlqatmHl (FQH) observed. Hall not quantum is fractional 2DEG effect the the and which in compressible regimes is in higher), (and level Landau h etro h adulvlfrLna levels Landau for level in Landau made the were of measurements center transport the Hall and tudinal ec,i h opesbeaiorpcrgm,down regime, anisotropic compressible the in Hence, n dad Fradkin Eduardo and ai hs rniini hw ohave to shown is transition phase matic n rv h ytmit non-Fermi a into system the drive and s to fteCenSmn ag fields. gauge Chern-Simons the of ation eai oegpe yltieeffects. lattice by gapped mode nematic mn ag ed soitdwt the with associated fields gauge imons n ehooy ajo 0-0,Korea 305-701, Daejeon Technology, and icst n loso htti term this that show also and viscosity l eoac bevdi radio-frequency in observed resonance metric. dynamical the of viscosity” all aiso h eai re parameter order nematic the of namics st em iudcoet nematic a to close liquid Fermi osite re aaeesata neffective an as act parameters order c rv h hoyo h eai state nematic the of theory the erive ieqarplritrcinbetween interaction quadrupolar tive T unu hs rniinbetween transition phase quantum a xiaino h eai fluid, nematic the of excitation cal efli a e-e-yeterm. Wen-Zee-type a has fluid le ≃ asto ntecmoieFermi composite the in ransition ahetpoeuewihmaps which procedure tachment g edadtenmtcorder nematic the and field uge 5m rmanmnlaiorp fa of anisotropy nominal a from mK 65 risse.W hwthat show We system. ermi -00 USA 1-3080, 12,13 nhl-le Landau half-filled in s T 1 eory, nteeeprmns longi- experiments, these In ∼ 1,14,15 .Ipraty nthis in Importantly, K. 1 eetees these Nevertheless, T & 5mK). 25 N N ≥ 2, = 2. 2 to the lowest temperature accessible in the experiments compressible nature of nematic fractional quantum Hall (which currently go down to about 10 mK), the 2DEG states strongly constrains the behavior of the nematic behaves as a compressible charged fluid with a large fluctuations and largely determines the structure of the anisotropy which onsets below a well defined tempera- effective behavior at low energies. These studies have also ture. This behavior strongly suggested that there is a shown that the nematic transition inside the FQH state (thermal) phase transition of the 2DEG, rounded by a is triggered by a softening and condensation of the stable very weak native anisotropy (with a characteristic energy collective mode of the FQH fluid, the Girvin-MacDonald- scale estimated to be 3 5 mK, whose microscopic ori- Platzman (GMP) mode, at zero momentum. ∼ −16 gin has remained unclear ) to a low-temperature elec- The close vicinity of nematic order of a compressible 5 tronic nematic state. The nematic nature of the state state or a FQH state (which is hence incompressible) was verified by detailed fits of the transport anisotropy strongly suggests that the nature of the effective inter- data to classical Monte Carlo simulations of the thermal actions in the 2DEG in Landau levels N 1 favor both 17,18 fluctuations of nematic order. paired and nematic ordered states. In this≥ context, it Subsequent experiments in 2DEGs in quantum wells, is surprising that, in spite of all the work on nematic- earlier by tilting the magnetic field,19,20 and, more re- ity in the incompressible state, there has been almost no cently, by the application of hydrostatic pressure in the work on the compressible nematic state for more than a absence of an in-plane magnetic field,21 have revealed decade. Aside from the notable semi-phenomenological the existence of a complex phase diagram in which com- theory of the quantum Hall nematic of Radzihovsky and pressible nematic phases were found even in the first Dorsey,46 based both on a microscopic theory and on Landau level, N = 1, competing with the famous, pre- quantum hydrodynamics, and of the work of Wexler sumably non-Abelian, paired, FQH state at filling frac- and Dorsey,47 who made estimates of the dislocation- tion ν = 5/2. More recent tilted field experiments have unbinding transition of a quantum Hall stripe state to also revealed the existence of an incompressible nematic a quantum Hall nematic based on the Hartree-Fock the- FQH state in the N = 1 Landau level at filling frac- ory of the stripe state, except for a variational Monte tion ν =7/3, competing with the isotropic Laughlin-like Carlo wave function study by Doan and Manousakis,48 FQH state at that filling fraction.22,23 No nematic state the compressible nematic state has not been studied. has ever been reported in the lowest, N = 0, Landau In principle there are two logical pathways to reach level. a nematic phase by a quantum phase transition: a) by Early Hartree-Fock theories of the 2DEGs near the quantum melting of a stripe phase, or b) by a (Pomer- center of the Landau level, for Landau level index N anchuk) instability of an isotropic Fermi liquid type state. large enough, have predicted a stripe-like ground state, Although the close vicinity of the isotropic compressible i.e. a compressible state in which the electron density Fermi liquid state to the observed nematic state in Lan- is spontaneously modulated along one direction.24–26 For dau levels N 1 suggests that the latter may be a suit- this reason, the anisotropic states in the compressible able starting≥ point, the fact that exact diagonalization regimes in Landau levels N 2 were originally referred studies find local stripe order38 suggests that the actual ≥ to as striped states. Most microscopic theories for the physics is likely to lie somewhere in between these two anisotropic state at ν = 9/2 (and in higher Landau lev- regimes. Also, it is possible that the state is nematic els) were built on this proposal.27–33 The resulting pic- above some critical temperature while the ground state ture of the stripe state is an array of “sliding” Luttinger may be a stripe phase. However, the fact that there is liquids.34–37 strong evidence for rotation symmetry breaking but not On the other hand, an exact diagonalization study by of translation symmetry breaking, down to the lowest ex- Rezayi and Haldane38 for a system of up to 16 electrons perimentally accessible temperatures, suggests that the for half-filled Landau levels in a toroidal geometry gave ground state may be a nematic state (perhaps close to a strong evidence for both a paired FQH state and a stripe- quantum phase transition to a stripe phase.) like state as a function of the effective interactions in the The purpose of this paper is to develop a theory of the Landau level. We should note that in such small system compressible nematic state of the 2DEG in large mag- (and in a toroidal geometry) finite-size effects can blur netic fields. Throughout this work we will use the map- the distinction between a stripe state and a nematic state, ping of electrons in Landau levels to composite fermions but it is an evidence for at least short-range stripe order. in the same Landau levels but now coupled to a Chern- Interest in nematic quantum Hall states attracted re- Simons gauge field,49,50 i.e. a flux attachment transfor- newed attention after the experimental discovery of an mation. At a formal first-quantized level this mapping incompressible nematic phase inside the fractional quan- is an exact identity. However the resulting theory has tum Hall state in the N = 1 Landau level at filling frac- no small parameter and in practice a mean field the- tion ν = 7/3 by Xia and coworkers.22,23 This state has ory, the average field approximation, must be used. For been studied theoretically by several groups.39–45 These FQH states, which have a finite energy gap already at studies have revealed that nematic fluctuations are inti- the level of the mean field theory, this approach has been mately related to the geometric response of the quantum shown to yield exact predictions of the universal prop- Hall fluid and, in particular, to the Hall viscosity. The in- erties of the fractional quantum Hall fluid, including the 3

Hall conductance, the charge and statistics of the ex- cal soliton, the nematic disclination, which in this fluid citations, degeneracy on a torus, and the Hall viscosity carries an (unquantized) electrical charge. The resulting and related geometric responses.50–52 On the other hand, effective field theory of the nematic fractional quantum here we will be interested in compressible phases which Hall state obtained by this approach has the same struc- do not have a gap (by definition) and hence the theory ture and properties as the one proposed on symmetry is not as well controlled as in the FQH regime. As our grounds by Maciejko and coworkers.45 starting point we will consider the isotropic Fermi liquid The theory of the compressible nematic state that we state of Halperin, Lee and Read53,54 (HLR) of compos- will present here is naturally connected to our earlier ite fermions,55 which is based on the same mapping, and work on the incompressible nematic FQH state. Thus, look for a Pomeranchuk quantum phase transition to a we will represent the problem of the half-filled Landau nematic state. This is a natural point of view which has level and, in fact, for all the compressible limiting states been used extensively as a description of nematic Fermi of the Jain sequences, at filling fraction ν = 1/(2n), as fluids.56 However, the HLR Fermi liquid is a non-Fermi a system of composite fermions minimally coupled (i.e. liquid to begin with which makes the application of these in gauge-invariant way) to both the external electromag- ideas not straightforward. netic field Aµ and to the statistical gauge field aµ which At the level of mean field theory, the HLR (or Jain) implements the flux attachment. Hence, here too, the ac- Fermi liquid states are the limiting state of the FQH tion also includes a Chern-Simons term (with a suitable states in the Jain sequences55 with filling fraction ν = coefficient). p/(2np 1), where p and n are two non-negative inte- Also, in addition to the Coulomb interaction, which gers. In± Jain’s picture, a FQH state of electrons can only involves a coupling of the local densities, as in the be viewed (in mean field theory) as an integer quantum case of a Fermi liquid,56 we will also include an attractive Hall state with filling fraction p of composite fermions, interaction in the quadrupolar channel, i.e. an attractive which is made of gluing 2n fluxes± to every electron, in interaction between the nematic densities. The coupling a partially screened magnetic field B/(2np 1). In the constant of this quadrupolar interaction is nothing but ± compressible limit, p , the effective magnetic field the F2 Landau parameter of a Fermi liquid. By gauge felt by the composite| | fermions → ∞ vanishes (on average) or, invariance, the quadrupolar coupling of the fermions also equivalently, the effective charge of the fermions vanishes involves both the gauge fields, Aµ and aµ, since the ne- in the same limit. In this regime, the charge-neutral com- matic densities are bilinear of the Fermi fields which nec- posite fermions fill a Fermi sea and form a (composite) essarily involve spatial derivatives. Fermi liquid.57 This simple picture, and its subsequent We will not attempt here to provide a microscopic extensions, has given a successful description of numer- derivation of the value (and sign) of the effective ous experiments in the compressible regime.58–61 Also, quadrupolar interaction. Nevertheless, it is well known a current picture of the microscopic origin of the non- that the effective interactions of composite fermions are abelian paired state in the first, N = 1, Landau level, at quite different than those of electrons,61 and depend on filling fraction ν = 5/2, is a paired state in the px + ipy the Landau level index, as well as on properties of the het- channel of composite fermions.62,63 It is then natural to erostructure (or quantum well) which define the 2DEG. look for a similar quantum transition to a compressible In addition, the estimate of values of Landau parame- nematic state from the HLR state. ters, which is notoriously difficult even for conventional In Ref. [41] we worked out a theory of the incompress- Fermi liquids, is much harder in the case of composite ible nematic state in a fractional quantum Hall state us- fermions. The currently available numerical estimates64 ing the flux attachment via fermion Chern-Simons gauge for F2 obtain values that for N 1 are very close to 1. ≥ − theory49 as a quantum phase transition in a Laughlin However, the physics of the compressible state is ac- FQH state. Much as in the case of the nematic tran- tually quite different from the FQH states, and the ex- sition in a conventional Fermi liquid,56 we showed that tension of this theory to the compressible state involves the quantum phase transition (the Pomeranchuk insta- several problems. One is the lack of a small expansion bility) can be caused by an effective quadrupolar inter- parameter to control the theory. In the incompressible action of among the electrons if it becomes sufficiently FQH states this is not a serious problem provided that attractive. Furthermore, we showed that the electrons one focused only on the long distance and low energy feel fluctuations of the nematic order parameter as an regime where it behaves as a topological fluid. This sim- effective dynamical metric field. A direct consequence plification is absent in the compressible state since it is of this coupling is that the effective Lagrangian of the gapless. Even though at the level of mean field theory nematic fluctuations has a time-reversal breaking parity- the state is predicted to be a Fermi liquid, the coupling odd Berry phase term, which is closely related to (but of the fluctuations of the statistical gauge field turns the not equal to) the Hall viscosity. Due to this parity odd (mean-field) HLR state into a non-Fermi liquid. Thus, al- term, the quantum critical theory has dynamical expo- ready at the leading perturbative order, imaginary part of nent z = 2. There are also Wen-Zee-like term in terms the the composite fermion self-energy Σ′′(ω) overwhelms 53,65 2/3 of the “effective” dynamical metric, i.e., nematic order the real part, i.e. Σ′′(ω) ω for short-range parameters. In the nematic phase, there is a topologi- interactions although it is milder∝ | for| Coulomb interac- 4 tions, Σ′′(ω) ω ( a “marginal” Fermi liquid), and In what follows we will set aside these important the composite∝ fermion | | quasiparticles become ill-defined. caveats, and develop a theory of the compressible ne- In addition, a calculation of the Landau parameters Fl matic state as an instability of the HLR composite Fermi for the HLR theory66 revealed that, for all angular mo- liquid (CFL) state. In Section II we present the theory mentum channels l 1, all the Landau parameters are of the nematic composite Fermi liquid, following closely ≥ equal to the Pomeranchuk value, Fl = 1. Therefore, the the structure and results of the theory of the nematic quasiparticle picture breaks down and− even the relic of a Fermi fluid of Oganesyan, Kivelson and Fradkin,56 and of Fermi surface appears to be prone to instabilities (such the composite Fermi liquid of Halperin, Lee and Read53 as a nematic instability). Current numerical estimates Here we introduce the gauge-invariant quadrupolar in- in the half-filled Landau level yield negative values of teraction, and present the basic structure of the effective 64 F2 and close to the Pomeranchuk instability value. Al- action for the nematic order parameter fields. In Sec- ready the conventional theory of the nematic transition is tion III we derive the parity-even part of the nematic non-trivial since at the Pomeranchuk point the Fermi liq- fluctuations, and in Section IV we derive the parity-odd uid breaks down, and in the HLR state the Fermi liquid component which yields the Hall viscosity using a dia- picture has been already broken down due to the cou- grammatic approach. In Section V we derive a set of pling to the fluctuating gauge field. Nevertheless, prop- important Ward identities which we use throughout the erties of the fluid determined by gauge-invariant currents paper. In Section VI we discuss the effective quantum and densities are well behaved, in the sense that they are dynamics of the nematic fields and their electromagnetic free of infrared singularities, although the results are at response which is relevant to resonance experiments. In best qualitative since the theory does not have a small Section VII we derive the Wen-Zee term for the CFL. parameter.67,68 Here we find that, due to the non-local nature of this A further complication is the lack of particle-hole sym- term in the compressible state, not only its coefficient is 41 metry in the HLR theory.69 This problem is the focus not quantized (as it is in the incompressible FQH state ) of intense current work.70–78 It has also been a focus but its relation with the Berry phase term for the nematic of attention in the theory of the paired (Pfaffian) FQH fields (i.e. the effective Hall viscosity of the CFL) is not state.79,80 Particle-hole symmetry in the half-filled Lan- straightforward. In Section VIII we derive the geometri- dau level can exist only in the absence of Landau level cal response of the CFL and in particular the Hall viscos- mixing and if only quadratic interactions are allowed. On ity of this compressible fluid. Here too, contrary to what the other hand, although the flux attachment transfor- happens in the FQH states, this response is not quan- mation is an exact mapping, at the level of the average tized and it is not universal. In Section IX, we discuss field approximation there is a large reorganization of the the connection of our theoretical results on the nematic Hilbert space which involves a large mixing of Landau CFL states with various experiments showing transport levels. For this reason the Jain wave functions are pro- anisotropy in half-filled Landau levels. Section X is de- jected onto the Landau level. However, in the field theory voted to our conclusions and open questions. Details of approach there is no such projection. our calculations and of previous results are presented in the Appendices. In Appendix A we give a summary of In the incompressible FQH states the effects of Lan- the theory of the quantum phase transition to a nematic dau level mixing become negligible at long distances and state in a Fermi liquid of Oganesyan and coworkers,56 at low energies provided that the quantum fluctuations and in Appendix B we summarize the HLR theory of the (“one-loop” or “RPA”) are included, as a consequence half-filled Landau level.53 Details of the derivation of the of incompressibility, Galilean and gauge invariance.81,82 Berry-phase-type term for the nematic fields are given in The correct universal properties, encoded in the effec- Appendix D and for the Wen-Zee term for the nematic tive topological field theory, of the FQH states are re- fields in Appendix E. The derivation of the nematic cor- produced only after these leading quantum corrections relators is given in Appendix F, and the vertex correction are included. These quantum corrections at long dis- for the nematic polarization in Appendix G. tance and at low energies turn the composite fermions into anyons with fractional charge and fractional statis- tics and, in this sense, there are no composite fermions in the spectrum of states of the FQH fluids. On the II. THEORY OF NEMATIC PHASE other hand, because of the absence of a small param- TRANSITION OF THE COMPOSITE FERMI eter, the theory yields quantitatively incorrect values LIQUID for dimensionful (and non-universal) quantities, often by significant amounts although improvements have been In this Section we consider the nematic-isotropic quan- made.68 These problems become more complex in the tum phase transition inside the CFL. We will construct case of the HLR state and, for this reason, theories of the theory using the theory of the nematic transition the compressible state projected onto the Landau level in a Fermi liquid of Oganesyan, Kivelson and Fradkin56 have been introduced.83,84 These theories are technically (OFK), summarized in Appendix A, and the theory of more complex and non-local, and are only qualitatively the isotropic CFL of Halperin, Lee and Read,53,54,85,86 understood. summarized in Appendix B, as our starting points. 5

Our starting point is the action for electrons in a half- of the nematic order parameter, the traceless symmetric filled Landau level of HLR with a quadrupolar interaction tensor Qˆ(r),

2 2 D 1 D2 D2 2D D = dtd rΨ†(r,t) iD + + µ Ψ(r,t), ˆ x y x y t Q(r)= Ψ†(r) − 2 2 Ψ(r), (2.3) S 2m 2m 2DxDy D D Z h i  y − x 1 2 2 dtd rd r′ V (r r′)δρ(r,t)δρ(r′,t) − 2 − where Dx and Dy are the x and y components of the Z covariant derivative which is explicitly dependent on the 1 2 2 dtd rd r′ F (r r′)Tr Qˆ(r,t) Qˆ(r′,t) , 2 gauge field Aµ. Hence, the nematic order parameter cou- − 2 − · 41 Z  (2.1) ples to the electromagnetic gauge field as a quadrupole. where Dµ = ∂µ + iAµ is the covariant derivative and the index µ = t,x,y (not to be confused with the chemical A. Flux attachment and nematic order potential which is also denoted by µ!). Here V (r r′) is the pair interaction potential, and the quadrupolar− in- Now we proceed to attach the flux using the Chern- teraction F2 in momentum space is represented as Simons term and follow the same strategy as in the con- F ventional CFL theory of Appendix B to find the fol- F (q)= 2 (2.2) 2 1+ κq2 lowing effective theory. In addition to this, we perform a Hubbard-Stratonovich transformation to decouple the Here we will use the same prescription we used in Ref.[41] quadrupolar interaction in terms of a field M(r,t) = in the context of the nematic FQH state, and we will be (M1(r,t),M2(r,t)) (see also Appendix A), to find an ac- careful to include the gauge field Aµ in the definition tion of the form

2 2 D 4 1 2 2 1 = dtd rΨ†(r,t) iD + + µ + αD Ψ(r,t) d rd r′dt δb(r,t)V (r r′)δb(r′,t) S t 2m − 2 16π2 − Z µνλ h i Z 2 ε 2 M1 2 2 M2 + dtd r (δa δA )∂ (δa δA )+ dtd r Ψ†(r,t)(D D )Ψ(r,t)+ Ψ†(r,t)(2D D )Ψ(r,t) 8π µ − µ ν λ − λ 2m x − y 2m x y Z Z 1 h i dtd2r M 2 + κ (∇M)2 (2.4) − 2F2 Z h i

6 where the covariant derivative now is Dµ = ∂µ + iδaµ, clude a term of order D to stabilize the nematic state, where δaµ is the fluctuating component of the Chern- whereas here, in the compressible case, a term of order Simons gauge field (See Appendix B), and where we have D4 is sufficient. used the Chern-Simons constraint to replace the density fluctuation δρ(r,t) with the Chern-Simons flux fluctua- tion δb(r,t) (as shown in Appendix B.) Our goal in this paper is to derive the effective the- The action of Eq.(2.4) is the theory that we will an- ory for the external electromagnetic gauge field δAµ and alyze in this paper. This action now also contains a 4 for the nematic order parameters M. In this section we higher-order gradient term, αD , in the fermion disper- will sketch the calculation and highlight the important sion needed to stabilize the nematic phase, i.e., making features of the results. The derivations of the main re- the sign of the quartic term of the free energy of the ne- 56 sults are presented in the following Sections and in the matic order parameter positive. However, we will be Appendices. mainly interested in the leading scaling behaviors of the various correlators in q and ω for small q k where ≪ F we linearize the kinetic energy of the fermion near kF to calculate the correlators. Then the higher-order disper- We will proceed in two stages. First we will expand sion does not affect the leading scaling behaviors of the the effective action resulting from integrating out the dynamic properties of the correlators. Therefore, from fermions about the isotropic HLR state. The result is here and on, for the most part we we will drop the αD4 an effective action that depends also on the fluctuating term when calculating the dynamical properties of the piece of the Chern-Simons gauge field δaµ (as it is done in CFL. We note that in our earlier work on the nematic Appendix B for the HLR theory). After integrating out fractional quantum Hall states41 it was necessary to in- the composite fermions we obtain the following effective 6 action Fermi liquid. The effective action a,M in Eq.(2.6) has the form (see Eq.(C6) of AppendixS C for details) 2 D 4 eff [δaµ, M, δAµ]= itr ln iDt + + µ + αD S − 2m 1 [M,δa ]= M (q,ω)T (q,ω)δa (q,ω) M1 h M2 a,M µ i iν ν 2 2 S −2 q,ω + Dx Dy + 2DxDy Z 2m − 2m (2.7) i 2 1 µνλ
This term is a mixed bilinear form in the nematic field M + d rdt ε (δaµ δAµ) ∂ν (δaλ δAλ) 8π − − and in the Chern-Simons gauge field δa , and represents Z µ 1 2 2 1 the quadrupolar coupling. d rd r′dt δb(r,t)V (r r′)δb(r′,t) −2 16π2 − The other new term, represented by the effective ac- Z tion a,M,a of Eq.(2.6), has the form of (see Eq.(C8) of 2 1 2 2 d rdt M + κ (∇M) AppendixS C for details) − 2F2 Z h i(2.5) 1 [δa , M]= δa (q,ω)V [M]δa (−q, ω), Sa,M,a µ −2 µ µν ν − Since we are interested in deriving an effective field the- Zq,ω ory near the nematic (Pomeranchuk) quantum phase (2.8) transition, we will expand the fermion determinant (the which represents the parity-even coupling Maxwell-type first two lines on the right hand side of this effective ac- terms of the Chern-Simons gauge fields to the local fluc- tion) up to the quartic order in the nematic fields M tuations of the nematic order parameters. In this last (and quadratic orders in their spatial derivatives). We term, the nematic fields M couple to the gauge fields as will also expand the effective action up to the quadratic a fluctuating spatial metric. order in the fluctuations of the Chern-Simons gauge field The couplings between the Chern-Simons gauge fields δaµ. Notice that the only trace of broken time reversal and the nematic fields in Eq.(2.6) imply that these fields invariance in this effective action is in the Chern-Simons mix. This has important consequences for the effective action (the third line of this effective action), and that dynamics of the nematic fields. This is found in our third the fermion determinant represents a system of compos- and last step in which we now integrate out the fluctua- ite fermions at finite density (the HLR composite Fermi tions of the Chern-Simons gauge fields. Since the effec- liquid) coupled to the fluctuations of the Chern-Simons tive action of the Chern-Simons gauge fields has a Chern- gauge fields δaµ and to the nematic fluctuations M. Simons term which is odd under parity and time-reversal, The result of this expansion is an effective action for this step leads to parity-odd terms in the effective action the nematic fields M and an effective action for the fluc- of the nematic fields. Also, from the form of the coupling tuations of the Chern-Simons gauge fields δaµ. The ef- to the external electromagnetic fields, we will now obtain fective action has the general form an effective action for these probe fields (the same as in the HLR theory) plus their quadrupolar coupling to the [M,δaµ, δAµ]= n[M]+ CFL[δaµ, Aµ] nematic fields. This last effective coupling leads to the S S S + a,M [δaµ, M]+ a,M,a[δaµ, M] (2.6) signatures of the nematic fluctuations (and order) in the S S current correlation functions. The first two terms of the right hand side are the ex- In Ref.[41] we presented a theory of a nematic FQH pected terms for the effective action for nematic fields state based on a Chern-Simons gauge theory of flux at- alone, Sn[M], identical to the result of OKF for effec- tachment. An important feature of that theory is that tive nematic theory (shown in Eq.(A7) of Appendix A, already at the mean filed level (i.e. the average field and subsequent equations), and terms for the Chern- approximation) the effective action of the nematic fields Simons gauge fields alone, CFL identical to the HLR re- has a Berry phase term, originating from the broken time sult (given explicitly in Eq.(B6)S of Appendix B.). Thus, reversal invariance, which dictates the quantum dynam- the nematic order parameter fields M condense at the ics. The actual coefficient of this term can be exactly Pomeranchuk instability and become overdamped (with obtained at the level of mean field theory, and further dynamical critical exponent z = 3.) Likewise, the gauge fluctuation correction does not modify the result. Chern-Simons gauge fields are overdamped and also have This coefficient is part of the actual, universal, value of z = 3 dynamic critical exponent. In both theories, the Hall viscosity. However, in the case of a nematic com- the fermionic quasiparticles are destroyed by these over- posite Fermi liquid the situation is quite different since damped fluctuations. at the mean field level (where the gauge fluctuation is ig- The physics of the nematic composite Fermi fluid orig- nored) the gapless composite fermions do not see directly inates in the last two terms of the effective action of a broken time reversal invariance which is encoded in the Eq.(2.6). Although the nematic order parameters are Chern-Simons action of the gauge field fluctuations. We charge-neutral, they still couple to the gauge fields but will see below that the fluctuations of the Chern-Simons as a quadrupole. This coupling leads to two new terms gauge fields will induce a Berry phase type term for the in the effective action, not present either in the theory of nematic fields although this term will be non-local and the nematic Fermi fluid, or in the theory of the composite its coupling constant is unquantized (and non-universal). 7

The same holds for the Wen-Zee term and the Hall vis- we are working with a model with short range interac- cosity. tions and hence we will set the effective mass to be given by Eq.(2.10). Further, the high-frequency cutoff Λ¯ is nat- urally the Fermi energy, i.e., B. Effective field theory of nematic fluctuations ¯ CFL 2 Λ= EF = ℓ0− /m. (2.11) Before proceeding further, we first relate the various parameters appearing in Eq.(2.4), i.e., the Fermi momen- With these relations at hand, we can express all quanti- tum and the effective mass of composite fermions, to the ties in terms of the magnetic length l0 and of the inter- natural scales in a landau level: the magnetic length l0 action 1/V (0). and effective interaction strength between electrons. In We will work close to the Pomeranchuk quantum phase the Landau level, the density of electron is naturally re- transition to the nematic state, where the distance to lated with the magnetic length. On the other hand, the the nematic quantum critical point (the Pomeranchuk density is related with the Fermi momentum kF , result- instability) is parametrized by ing in the standard relations V (0) 1 1 δ = 4 + . (2.12) kF = 4πρ = , (2.9) 4πl0 2F2 l0 p where we use units in which ~/ec = 1. On the other We will keep terms in the effective action up to (and in- hand, the mass of the composite fermion is renormalized cluding) quartic order in the nematic fields. NF is the by the interactions between electrons. In the limit of density of states at the Fermi surface. Here, as in the a large magnetic field the effective mass is expected to HLR work on the composite Fermi liquid, we will keep be determined by the scale of electron-electron interac- only terms in the effective action which are quadratic in tions alone, as shown by the work of Halperin, Lee and the fluctuations of the gauge fields δaµ. This amounts Read.53 We can estimate the mass of composite fermions to working in the random phase approximation (RPA). in terms of the interactions via the dimensional analysis Higher order terms are (presumably) unimportant and (see Appendix B). Hence we obtain an estimate of the we will neglect them. The effective action derived be- effective mass m of the composite fermions in terms of low is effectively a loop expansion in the fluctuations of the density-density interaction V (q) as the gauge fields. Although in most cases we will need to consider diagrams with up to two internal gauge field C propagators, in the case of the Wen-Zee term the leading m = , (2.10) V (0) non-vanishing three has three internal gauge field propa- gators. where C is a numerical constant. In the case of Coulomb Before presenting the details of our theory, we first interactions, HLR showed that the energy scale is the summarize the main results. Here we show that the ef- Coulomb energy at the scale of the magnetic length. Here fective action for the nematic fields M is

[M, δA ]= M (p, Ω) (p,ω)M ( p, Ω) d2rdtλM 4 Seff µ i Lij j − − − Zp,Ω Z µνρ + χiΩ εij Mi(p, Ω)Mj ( p, Ω)+ βǫ ωµ(p, Ω) ∂ν δAρ ( p, Ω) p,Ω − − p,Ω − − Z Z   1 δA (p, Ω) (V (p, Ω)[M]+ K (p, Ω)) δA ( p, Ω), (2.13) − 2 µ µν µν ν − − Zp,Ω 56 2 3 where λ = (1/l0) (2α)/5, with α being the coefficient of the quartic term in the single-particle dispersion (see Appendix A), and the coefficient β is given by β = 2 . In Eq.(2.13) (p,ω) is the inverse propagator of the 3π√3 Lij nematic fields

κ 2 2 1 iΩ 1 iΩ p δ cos (2θp) 3 sin(2θp) cos(2θp) 3 2F2 2πl0 p 2πl0 p ij (p, Ω) = − − − . (2.14) 1 iΩ κ 2 2 1 iΩ L  sin(2θp) cos(2θp) 3   p δ sin (2θp)  3   2πl0 p − 2F2 − − 2πl0 p      where, as in Eq.(2.12), δ denotes the distance to the quantum critical point (i.e. the Pomeranchuk instability), and 56 θp is teh angle of the momentum p with the x axis. The result of Eq.(2.14) was first derived by Oganesyan et al.. 8

2 The second term in Eq.(2.13) is a Berry phase term and its (non-universal) coefficient χ = 2 is the Hall viscosity 3πl0 of the CFL (see Section IV). Finally, the tensor Vµν [M] represents the parity-even coupling between the Maxwell terms of the electromagnetic gauge fields and the nematic fields (and couple as a metric fluctuation) is given by

M1 (p2 p2) (M p + M p )ω ( M p + M p )ω 1 2 x y 1 x 2 y 2 y 2 x V (p,ω)[M]= K (p,ω) (M p + M− p )ω M ω2 − M ω2 . (2.15) µν 2π 0  1 x 2 y 1 2  ( M p + M p )ω M ω2 M ω2 − 2 y 2 x 2 − 1  

The function K0(p,ω) is given in Appendix B, and external electromagnetic gauge field δAµ, it is clear that Kµν (p,ω) is the polarization tensor of the electromag- the gauge invariance is respected, since ∂ν Vνλ = 0 and netic field in the HLR theory of the CFL. ∂ν Kνλ = 0, which implies gauge invariance. We first remark that, due to the Landau damping We present the detailed calculation for the main results terms in the inverse propagator ij (p, Ω), the nematic in the following sections. phase transition of the compressibleL half-filled Landau level has a dynamical critical exponent z = 3. However, the effective action of Eq.(2.13) has also a Berry-phase- III. PARITY-EVEN COMPONENTS OF THE type term for the nematic order parameters induced by NEMATIC FLUCTUATIONS Chern-Simons gauge fluctuation. Although this term is formally subleading to the Landau damping term, it is kept since it is the leading parity-odd contribution to the nematic fields. By symmetry, the nematic order parameter acts as a locally-fluctuating dynamical metric to electrons and to composite fermions and modifies the local frames. In the nematic phase, where the order parameters have a non-vanishing expectation value, this coupling leads to an anisotropic electromagnetic response. These effects are encoded through the term Vµν of the polarization tensor for the external probe electromagnetic gauge field FIG. 1. Nematic correlator: the full lines represent the com- δAµ. The isotropic part Kµν of the polarization tensor posite fermion propagator, and the broken lines represent the was calculated in Ref.[53] (see Eq.(B11)). Here ωµ plays nematic order parameters. the role the “spin connection” of the “dynamical metric” defined by the nematic order parameter fields, and can be explicitly written out in terms of the nematic order The propagator of the nematic order parameters in the parameter fields as Fermi liquid state is (here denotes time ordering and T ij i, j =1, 2) ω0 = ǫ Mi∂0Mj (2.16) ij 1 ω = ǫ M ∂ M (∂ M ∂ M ) i MiMj 0(p,ω) ij− (p,ω) (3.1) x i x j − x 2 − y 1 hT i ≡ L ij ωy = ǫ Mi∂yMj + (∂xM1 + ∂yM2) The diagonal component 11 (represented diagrammati- cally in Fig.1) has the explicitL form Due to the Wen-Zee-like term, i.e., the term µνλ ∼ 2 ǫ ωµ∂ν δAλ in the action, the disclination of the ne- k2 k2 matic order parameters inside the nematic phase mini- (p, Ω) = i g(k + p, Ω+ ω)g(k,ω) x − y L11 − 2m mally couples with the electromagnetic gauge field and Zk,ω ! carries the (non-quantized) electric charge, which, in this k4 k3 iΩ = F cos2(2θ ) F + o(Ω/p), compressible state, will be eventually screened by the 4πm − p 2π p gapless electrons.   V (0) 1 iΩ The effective theory is both gauge invariant and ro- = cos2(2θ ) + o(Ω/p) (3.2) 4πl4 − p 2πl3 p tationally invariant. First of all, the inverse propagator 0 0   in Eq.(2.13) and Eq.(2.15) of the nematic order pa- ij where Lrameter is constructed in the way that it is apparently rotationally symmetric. We will come back to this later. 1 g(k,ω)= 2 (3.3) k2 k Secondly, the action is also gauge invariant. The Wen- ω − F + iη sign(ω) Zee-like term is apparently gauge invariant because it − 2m involves the field strength F = ∂ δA ∂ δA explic- is the time-ordered free composite fermion propagator. νρ ν ρ − ρ ν itly. For the full polarization tensor Vµν + Kµν for the In the same way, we can calculate the other components 9 of ij (p,ω) and replace kF ,m in terms of l0, V (0), For the compressible half-filled Landau levels, the L mean field state of the composite fermion without the 1 iΩ gauge fluctuation is a CFL with well-defined Fermi sur- 12(p, Ω) =cos(2θp) sin(2θp) , face. Furthermore, the state is time-reversal even so we L 2πl3 p 0   do not expect any Berry phase term for the nematic or- V (0) 1 iΩ (p, Ω) = sin2(2θ ) , (3.4) der parameters to emerge. However, once we include the L22 4πl4 − p 2πl3 p 0 0   dynamics of the gauge fluctuation δaµ and go beyond the mean-field theory, the time-reversal symmetry is ex- 56 which agree with the results of OKF. plicitly broken due to the Chern-Simons term for δaµ Now we can include the gauge fluctuations. This is and thus the Hall viscosity is expected to arise from the included through the loop expansion in the gauge fields, fluctuations. This is to be expected since, for the same and we calculate only the one-loop corrections. In Ap- reasons, the Hall conductivity of the CFL in the HLR pendix F and Appendix G we show that these correc- theory also comes from gauge fluctuations. tions are subleading to the leading term, and thus do To study the contribution from the gauge fluctuation not change the dynamic scaling behavior of the critical to the Berry phase term, we formally integrate out the theory of the nematic-isotropic phase transition. fermions at one-loop order, and rewrite the theory in terms of the fluctuating gauge field δaµ and the nematic order parameter. Throughout this section we use the po- larization functions of the compressible fermions Πµν , Π0 IV. PARITY-ODD COMPONENTS OF THE and Π2, defined in Eqs.(C3), (C4) and (C5), respectively, NEMATIC FLUCTUATIONS: THE HALL given explicitly in Appendix C. We first consider the lin- VISCOSITY ear coupling between the nematic field and the gauge field at the mean field level, 40,41 In our earlier work, we investigated the isotropic- 1 nematic phase transition in FQH states and Chern in- a,M = Mi(q,ω)Tiν (q,ω)δaν (−q, ω), (4.1) S − 2 − sulators. We concluded that the theory describing the Zq,ω phase transition to the anisotropic state in such chiral where Tiν (q,ω) is the 2 3 matrix (with i = 1, 2 and topological phases always contains a Berry phase term for ν = t,x,y) × the nematic order parameters, which is odd under time reversal and parity. In the nematic FQH states the coef- 2 2 m qx qy qxω qyω ficient of the Berry phase term is related (but not equal Tiν (q,ω)= Π2(q,ω) − − . (4.2) 2π 2qxqy ωqy qxω to) with a dissipationless Hall viscosity.41,51 In Ref.[41] − − −  we concluded that the nematic fluctuation, regarded as a is the effective vertex. This coupling can be regarded as dynamical metric, only couples with the stress tensor of the symmetric part of the Wen-Zee-like term which we the composite fermion while the background metric also will discuss later. appears in the covariant derivative as the spin connection Beyond this, there is another term involving the fluctu- of the composite particle. Accordingly, the Berry phase ating nematic order parameter and Chern-Simons gauge of the nematic order parameter is the odd Hall viscosity fields. of the mean-field state of the composite fermion theory, not of the electron fluid. Hence, in the incompressible 1 a,M,a = δaµ(q,ω) µν (q,ω)δaν (−q, ω), FQH states, the Berry phase term is equivalent to the S −2 V − Zq,ω Hall viscosity of the composite fermion filling up integer (4.3) number of the effective Landau levels.41,51 where

M1 (q2 q2) (M q + M q )ω ( M q + M q )ω Π (q,ω) 2 x y 1 x 2 y 2 y 2 x (q,ω)= 0 (M q + M− q )ω M ω2 − M ω2 . (4.4) µν 2π 1 x 2 y 1 2 V ( M q + M q )ω M ω2 M ω2  − 2 y 2 x 2 − 1  

This term is the coupling to the dynamical metric or ne- by the nematic order parameter. matic order parameter of the Maxwell term for the fluc- Together with the original isotropic HLR result for the tuating gauge field δaµ. As the nematic order parameter action of the fluctuating Chern-Simons field δaµ can be considered as a dynamical metric that modifies the local metric of the composite fermion, the gauge bo- 1 = δa (q,ω)Π (q,ω)δa ( q, ω), (4.5) son coupled to the fermion is also modified accordingly Sa −2 µ µν ν − − Zq,ω 10 we can obtain the Berry phase term of the nematic order The effective action = + of the gauge field S Sa Sa,M,a parameter by integrating out the fluctuating gauge fields δaµ, defined by Eq.(2.7) and Eq.(2.8), coupled with the and performing the loop expansions. In the following, we nematic order parameter M, is fix the gauge a0 = 0 to facilitate the calculation.

1 1 2 2 2 − = δai(q,ω) Πij + (tij ω Π0)M1 + (tij ω Π0)M2 δaj ( q, ω) (4.6) S −2 q,ω,k,Ω − Z h i 1 2 where the 2 2 matrices tij and tij denote the Pauli matrices σ3 and σ1, respectively, and M are the nematic fields. By integrating× out the fluctuations of the gauge fields, and performing the loop expansion, we obtain leading corrections to the (time-ordered) correlators of the nematic order parameter M (shown in the Feynman diagram of Fig. 2)

1 2 i 1 2 j δM δM (Ω, p)= i tr Π− (ω, q) ω Π t Π− (ω +Ω, q + p) ω Π t h i j i ij 0 ij 0 Zq,ω Ω Π q2  √ρ¯  = i ǫij 2 + .... (4.7) 2 Π q2 + q2/4 Π q2 + q2/4 Zq,ω 2 2 where we have approximate the form of the polarization viscosity seemingly depends only on the particle density. tensor Πij (ω, q) valid for low frequency and momentum This value of the Berry phase χ is one of our main results with ω q vF (see Appendix B). in this paper. It plays a key role in the effective dynam- ≪ | | ics of the nematic order parameters. However, we should caution that, contrary to the case of the FQH states, this value of the Hall viscosity is not protected in the CFL, and should be regarded as an estimate.

Our result indicates that the gauge field fluctuations generate a Berry phase term which will in turn contribute to the Hall viscosity of the compressible half-filled Lan- dau levels. The way in which the Berry phase term is generated in this case is different from that of the incom- FIG. 2. Berry phase term. Here the bubble is the fermion pressible FQH states. In the incompressible FQH states, loop corrected by the fluctuating gauge field. the Berry phase of nematic order parameters is already present at the mean field level of the composite fermion which forms an integer quantum Hall state, due to the ex- Hence, the anti-symmetric (and hence off-diagonal) 1 plicitly broke the time-reversal symmetry of the compos- part of the nematic correlator − gives a Berry phase ij ite fermions in the effective Landau level.41,51 Further- type term in the effective actionL for the nematic fields more, the Chern-Simons gauge fields in the FQH state are gapped and do not affect the value of the Berry phase M = χǫij (iΩ)M (p, Ω)M ( p, Ω), (4.8) Sij i j − − term in the low energy regime. In contrast, for com- Zp,Ω pressible half-filled Landau levels, the mean field state where is a Fermi liquid alone which seemingly respects time- reversal symmetry, which is broken by the Chern-Simons 2 ¯ 2 term for the gauge fields. Their fluctuations can affect χ = Λm = 2 (4.9) 3π 3πl0 the low-energy dynamics of the nematic order parame- ters and induce the non-zero Berry phase term for the with Λ¯ the high-frequency cutoff of the CFL and m is nematic order parameters. In particular, while in the the effective mass of the composite fermions. Using the incompressible FQH states the coefficient of the Berry HLR results53 (see Appendix B), since the UV cutoff phase term has a universal relation with the composite ¯ CFL Λ= EF (the Fermi energy of the CFL), and the value fermion density, in the case of the compressible state this of the effective mass m (which in the HLR theory is ar- “Hall viscosity” is non-universal as it depends explicitly gued to depend only on the scale of the Coulomb interac- on the UV energy cutoff Λ¯ of the compressible composite ¯ 2 tion) one finds that mΛ = ℓ0− , and hence that the Hall Fermi fluid. 11

V. WARD IDENTITIES The density and current correlators, ρρ and JxJy , are given by the polarization tensor of theh i externalh elec-i We now present another way to compute and check the tromagnetic gauge field beyond the mean field level which Berry phase term, Eq.(4.8). Here we will use the Ward include the fluctuations of the Chern-Simons gauge field. identity between the current operators J and the stress tensor Tij (and the linear momentum density T0j ). We 2 start with the explicit expressions for these operators in 1 Π0q ρρ (ω, q)= the CFL, h i 4(2π) D(ω, q) iω 1 JxJy (ω, q)= , Ji = [Ψ†(r,t)(∂i + iδai)Ψ ((∂i + iδai)Ψ†)Ψ(r,t)], h i 2(2π) 2m − 1 1 Π0qyω iqx T = Ψ†(r,t)(∂ + iδa )(∂ + iδa )Ψ(r,t), ρJy (ω, q)= , ij 2m i i j j h i −4(2π) D(ω, q) − 2(2π) T = mJ . (5.1) 1 Π0qxω iqy 0j j J ρ (ω, q)= , h x i −4(2π) D(ω, q) − 2(2π) which are manifestly gauge invariant. 1 1 V (q) The conservation law of the energy-momentum tensor D(ω, q)= Π2ω2 Π q2(Π + ) . (5.6) 2π 0 − 4 − 0 2 16π2 (i.e. local conservation of energy and momentum), in the   presence of the Chern-Simons gauge field fluctuations, implies that Using these relations we obtain a result consistent with ij the previous approach, ∂µTµi = δb ǫ Jj , (5.2) where δb = εij ∂iδaj is the fluctuating flux of the gauge field δa . Upon expanding out in components the ex- T1T2 (Ω, p)= ∂p2 4 ρJxρJy (Ω, p) µ h i h i pression of Eq.(5.2), we have 2 ¯ 2 = iΩ mΛ= iΩ 2 (5.7) 3π 3πl0 ∂0mJx + ∂xTxx + ∂yTyx = δbJy, ∂ mJ + ∂ T + ∂ T = δbJ . (5.3) 0 y x xy y yy − x In what follows we will use the notation T = T T , 1 xx − yy and T2 = Txy +Tyx. Focusing only on the anti-symmetric response, we find the following relation between the cor- VI. EFFECTIVE DYNAMICS AND relation functions of current, stress tensor and density SUSCEPTIBILITY OF THE NEMATIC ORDER operators,

∂2 m2 J (r ,t )J (r ,t ) + ∂2 T (r ,t )T (r ,t ) t1 h x 1 1 y 2 2 i r1 h 1 1 1 2 2 2 i With the results of the calculation of the Berry phase =4 ρ(r1,t1)Jx(r1,t1)ρ(r2,t2)Jy(r2,t2) . (5.4) term and Landau damping terms for the nematic order h i parameters at hand, we can proceed to derive the ex- Hence the Berry phase term, i.e. the Hall viscosity de- pression for the nematic correlators including both con- termined by the parity-odd correlation function of the tributions. The effective theory for the nematic order stress tensor,41,51,52,87–90 is related with the correlation parameter field M close to the nematic transition is function of the composite operators of densities and cur- rents (shown on on the right side of Eq.(5.4)). In mo- mentum and frequency space (and in the low frequency [M]= M (p, Ω) M ( p, Ω) d3xλM 4 limit, Ω 0) this results takes the form S i Lij j − − − → Zp,Ω Z (6.1) p2 T T (p, Ω) = [4 ρρ (p − q,ω + Ω) J J (q,ω) h 1 2i h i h x yi Zq,ω +4 ρJ (p − q,ω + Ω) J ρ (q,ω)] h yi h x i in which the correlator ij (p,ω) is given by the sum of (5.5) the two contributions, whichL yields the result

κ 2 2 1 iΩ 2 1 iΩ p δ cos (2θp) 3 ( ) iΩ( 2 ) + sin(2θp) cos(2θp) 3 ( ) 2F2 2πl0 p 3πl0 2πl0 p ij = − 2 − − 1 iΩ κ 2 2 1 iΩ , (6.2) L iΩ( 2 ) + sin(2θp) cos(2θp) 3 ( ) p δ sin (2θp) 3 ( ) − 3πl0 2πl0 p − 2F2 − − 2πl0 p ! 12 where δ is the distance to the Pomeranchuk instability ω vF q, then the off-diagonal terms are sub-dominant of Eq.(2.12). As in the case of the nematic Fermi fluid, and≪ can be ignored. Hence the overdamped critical mode ij (p,ω) is nothing but the inverse susceptibility of the is not affected by the gauge fluctuations. This leads nematicL order parameters, and zeros of the determinant to the conclusion that the criticality of the isotropic- of ij (p,ω) yield the dispersion relation of the nematic anisotropic phase transition of half-filled LL still exhibits collectiveL modes of the CFL. Clearly the nematic sus- z = 3 critical dynamical exponent. ceptibility is finite for δ > 0 and diverges as δ 0, as expected at a continuous quantum phase transition.→ Except for the Hall viscosity term, which originates from the gauge fluctuations which explicitly break the A. Nematic Susceptibility time-reversal symmetry, Eq.(6.2) is almost the same as the result for the nematic correlator of OKF.56 The Berry phase term, which makes the effective theory From the effective theory Eq.(6.2), we can read-off time-reversal odd, mixes the transverse and longitudinal the dynamic nematic susceptibility χM , i.e. the nematic modes of the nematic order parameters. However, if we propagator. Inside the isotropic phase, δ > 0, the sus- focus only on the parameter regime where q k and ceptibility is given by ≪ F

κ 2 2 1 iΩ 2 1 iΩ p δ sin (2θp) 3 ( ) iΩ( 2 ) sin(2θp) cos(2θp) 3 ( ) M 1 1 2F2 2πl0 p 3πl0 2πl0 p χij = ij− = − 2 − − 1 iΩ κ 2− 2 1 iΩ (6.3) L C(Ω, p) iΩ( 2 ) sin(2θp) cos(2θp) 3 ( ) p δ cos (2θp) 3 ( ) − 3πl0 − 2πl0 p − 2F2 − − 2πl0 p ! where

κ 2 2 κ 2 1 iΩ 2 2 2 2 1 iΩ C(Ω, p) = ( p + δ) + ( p + δ) 3 ( ) Ω ( 2 ) δ + δ 3 . (6.4) 2F2 2F2 2πl0 p − 3πl0 ≈ 2πl0 p

δ The nematic susceptibility is finite in the isotropic phase matic order is in the M direction (M = | | ). The 1 iΩ 1 1 2λ where 3 1. As we approach the quantum criti- 2πl0 p ≪ nematic susceptibility in the symmetry brokenq phase can cal point, δ 0, the nematic susceptibility diverges, as be obtained, expected. In→ the nematic phase, we assume the the ne-

κ 2 2 1 iΩ 2 1 iΩ p δ sin (2θp) 3 ( ) iΩ( 2 ) i sin(2θp) cos(2θp) 3 ( ) M 1 2F2 2πl0 p 3πl0 2πl0 p χij (Ω, p)= − 2 − | |− 1 iΩ κ− 2 2 1 iΩ (6.5) C′(Ω, p) iΩ( 2 ) i sin(2θp) cos(2θp) 3 ( ) p cos (2θp) 3 ( ) − 3πl0 − 2πl0 p − 2F2 − 2πl0 p ! where

κ 2 1 iΩ 2 C′(Ω, p) ( p + 3 sin (2θp)) δ , (6.6) ≈ 2F2 2πl0 p | |

B. Mode mixing in the Nematic phase and at modes are mixed, leading to the following modified dis- quantum criticality persions for these collective modes

In the theory of the nematic transition in a Fermi liquid κV (0)l0 2 l0V (0) 3 ω1 q + i q , of OFK, when approaching the criticality, δ 0, the dif- ∼ √2F2π 2 → p ference in the dynamics of the two polarizations becomes κ 3 ω2 i 2 3/2 q (6.7) more noticeable. At criticality there is an underdamped ∼ 2F2(π/l0) 2 longitudinal mode ωT q and an overdamped trans- 3 ∼ verse mode ωL iq . In the nematic phase this mode Thus, the transverse mode, ω2, remains overdamped (as becomes the (overdamped)∼ Goldstone mode. In our ne- in the OKF theory). In contrast, the longitudinal mode, matic criticality in the half-filled LL, due to the existence ω1, is now underdamped (with z = 2) only in the deep of the Berry phase term, the transverse and longitudinal asymptotic long-wavelength regime q 0, crossing over → 13 to an overdamped regime at larger values of q. This of the Wen-Zee term. However, upon the inclusion of the crossover can happen at long wavelengths if the range of gauge field fluctuations, which are parity-odd, we will the quadrupolar interaction is small. find the parity-odd part of the Wen-Zee term, as well as a modification of the parity-even part. We start by calculating the first part of the Wen-Zee C. Electromagnetic Response and Spectral Peak term given by a Feynman diagram that has one gauge leg and one nematic leg shown below in Fig. 3. The In the nematic phase, the electromagnetic response coupling between gauge-invariant current, generated by (and the conductivity tensor) can be obtained after inte- the probe electromagnetic gauge fields Aµ, and nematic grating out the gauge fluctuations. The calculation de- order parameter M is given by tails of the electromagnetic response is presented in Ap- pendix B. Here we propose a possible experimental test δ ( i) S (p, Ω) MlJk (p, Ω) of the nematicity by measuring the spectrum of the con- − δMlδAk ≡ h i ductivity σ . To this end, let us assume that we are in xx 1 2 1 2 the deep nematic phase with a nonzero nematic order, =i tr Πij− (ω, q)ω Π0tlTmi− (ω +Ω, q + p)ω Π0Qk , q,ω say in the M1 component. The conductivity σxx as a Z h i 1 Λ¯m p Ω p Ω function of momentum is, = ( )1/3 x y , (7.1) √ ρ¯ Ωpy pxΩ 6π 3 − lk i (1 + M1) σxx = JxJx = ωΠ0(q,ω) , ω h i − 4D′(q,ω) where, as before, t1 = σ3 and t2 = σ1, and where we

V (q) 2 2 2 D′(q,ω) = Π (q,ω) Π (q,ω) (q + M (q q )) 0 2 − 16π2 1 x − y   1 + Π2(q,ω)ω2 (6.8) − 4 0 In the limit of ω/q 1, the conductivity is approxi- mately given by ≪

ω(1 + M1)/V (0) σxx 2 (6.9) ω ω 2 ∼ (1 + M cos(2θ ))i + 2 2 q /4 1 q qV (0)l0 V (0) q − FIG. 3. Leading parity-even contribution to the Wen-Zee Deep nematic phase where M1 O(1), the damping term on the denominator will nearly∼ vanish at θ = π/2. Thus, term. The wavy lines are gauge field propagators and the q broken line is a nematic propagator. The blobs are composite by measuring σ as a function of the angle θ of the xx q fermion loops. direction of propagation measured form the nematic axis, one will find a resonance, i.e. a peak in the spectrum, at θ = π/2. q 1 used the notation Qx = I and Qy = iσ2, and Tmi− is given by − VII. THE WEN-ZEE TERM IN THE CFL 1 1 qxω qyω Tmi− = amMi = (7.2) h i Π mq2ω2 ωqy qxω From the Hall viscosity term, we expect that there may 2 −  be the Wen-Zee type terms for the dynamical metric asso- 41,45 ciated with the nematic order parameters. The Wen- where we used the temporal gauge, a0 = 0. This is the Zee like term consists of two parts: a parity-even term parity-even linear coupling between the nematic order linear in nematic order parameter and in the gauge field, parameter and the probe electromagnetic gauge field. and a second term that is parity-odd, and is quadratic To calculate the parity-odd contributions to the Wen- in nematic order parameter and linear in gauge field. At Zee term, we calculate a Feynman diagram with one ex- the mean field level of the CFL, which is parity-even ex- ternal probe gauge field leg and two nematic order pa- cept for the Chern-Simons term, the composite fermions rameter legs. Once again we choose the gauge a0 = 0, on the Fermi surface generate only the parity-even part and find the result 14

FIG. 4. Parity-odd contribution to the Wen-Zee term at cubic level. The wavy lines are gauge field propagators and the broken line is a nematic propagator.

δ ( i) S (Ω1, Ω2; p1, p2) MhMlJk (Ω1, Ω2; p1, p2) − δMhδMlδAk ≡ h i 1 2 1 2 1 2 = tr Πij− (ω, q)ω Π0σhΠij− (ω +Ω1, q + p1)ω Π0σlTmi− (ω +Ω1 +Ω2, q + p1 + p2)ω Π0Qk q,ω Z h i 1 Λ¯m Λ¯m = [( )1/3 + ]ǫhlǫνµkpν (pµ + pµ) (7.3) 6π√3 ρ¯ ρ¯ 1 1 2

which is manifestly odd under parity and time reversal incompressible FQH states. In the incompressible states, symmetries. the universal coefficients s of the Wen-Zee term is directly sρ¯ 51,52,87,91 By combining the two contributions, we finally find a related to the Hall viscosity ηH , i.e., ηH = 2 . Wen-Zee term of the form However, in the compressible CFL state, the coefficients ¯ 1/3 ¯ of the Wen-Zee term and the Hall viscosity do not have 1 Λm Λm µνρ W Z = + ǫ ωµ∂ν Aρ (7.4) such relation because of the non-local nature of such re- L 6π√3 ρ¯ ρ¯   sponses in a compressible state. The coefficients appear- h i ing in the expressions are seemingly numeric constants, where Λ¯ is the frequency UV cutoff of the CFL. Further ¯ 2 but it is important to remember that the coefficients are reduce the expression by taking Λm = l0− , we have, Ω in fact the functions of the ratio s = | | (not to be vF q | | 2 µνρ W Z = ǫ ωµ∂ν Aρ (7.5) confused with the coefficient of the Wen-Zee term!), and L 3π√3 are in general non-local in space and time. Instead, in the incompressible states where we can perform the well- Here we have denoted by ωµ the spin connection associ- ated with the nematic fields,41 defined gradient expansions due to the energy gap to all the excitations. Hence we find that in the CFL, the co- ij ω0 = ǫ Mi∂0Mj (7.6) efficients of the (seemingly local) Wen-Zee term and Hall ω = ǫij M ∂ M (∂ M ∂ M ) viscosities are not universally related to each other. x i x j − x 2 − y 1 ij ωy = ǫ Mi∂yMj + (∂xM1 + ∂yM2) Similar to what we did for the calculation of the Berry phase term using a Ward Identity, the Wen-Zee term can A. Nematic-Electric field correlator also be derived from the Ward Identity, and the result is consistent with the diagrammatic calculation present here. The details of the calculation of Wen-Zee term from The Wen-Zee-type term Eq.(7.5), which couples the Ward Identity is presented in the Appendix E. nematic field and the electromagnetic field, implies that We should stress that the connection between the Hall the change in the nematic field will induce the electron viscosity (given by the Berry phase term) and the Wen- quadrupole moment. Thus, one can measure the nematic Zee term in the CFL is not as straightforward as in the susceptibility with respect to the external electric field 15

χ = ∂Ti as a function of momentum, The composite fermion has orbital spin s = 151 so the M,E ∂Ej covariant derivative of the composite fermion contains 2 δ ∂Ti the spin connection with coefficient 1 dictated by the χM,E = S (q)= , orbital spin s = 1. At the mean field level, the Chern- δMiδEj ∂Ej Simons flux cancels the external magnetic fields so we 2 q q = − x y . (7.7) only need to consider the gauge fluctuation δa. 3π√3 qy qx − −  We can now perform the functional bosonization pro- cedure, following Refs.[50, 93–95], of our theory and in- This susceptibility indicates that the energy-momentum 96 current will be induced in the presence of the spatially- troduce the hydrodynamic gauge field bµ, modulating electric field. 2 1 gij = dr dt Ψ†(r,t)(iD + D D + D D )Ψ(r,t) S t 2m i i 2m i j Z 1 2 2 ′ VIII. GENERAL RELATION BETWEEN THE dr′ dr dt V (r − r )δb(r)δb(r′) HALL VISCOSITY AND THE HYDRODYNAMIC − 32π2 Z GAUGE THEORY IN HALF-FILLED LANDAU 2 1 dr2dt ǫµνρb ∂ b + dr2dt ǫµνρδa ∂ b LEVELS − 4π µ ν ρ 2π µ ν ρ Z Z (8.3) We have shown that, unlike the case of the incompress- ible FQH states, in the CFL the gauge field fluctuations where Dµ = ∂µ + i(δaµ + ωµ) is the covariant deriva- tive , and δb(r)= ∂ δa (r) ∂ δa (r) is the fluctuating contribute to the Hall viscosity of the half-filled Landau x y − y x level. In order to prove that the corrections to the Hall magnetic flux of the gauge field δaµ. viscosity come from the fluctuation of the Chern Simons Solving the saddle-point equation of δaµ, we find gauge fields, we start from CFL and dualize the CFL into ∂ b ∂ b =Eb = J CF , a hydrodynamic gauge theory. x 0 − 0 x x y b CF If we couple our CFL to a smooth deformation of the ∂yb0 ∂0by =Ey = Jx , background geometry (instead of to the nematic order − − b CF DxDy parameter), we find that the composite fermions at the ∂ E =T =Ψ† Ψ, x x 12 2m mean field level receive an orbital spin from the flux at- 51 b CF DyDx tachment procedure. Beyond this, if we include the ∂yEy = T21 = Ψ† Ψ (8.4) gauge field fluctuation, the viscoelastic response of the − − 2m CFL is related with the parity-odd electromagnetic re- Now we turn on a momentum current of the composite 87,92 CF sponse of the hydrodynamic gauge theory. Since the fermion T12 in the system. Different from the parity- original CFL is coupled to the fluctuating Chern-Simons even Fermi surface where the metric only couples with gauge boson, the time-reversal odd electromagnetic re- the momentum current, the composite fermions carry in- sponse of the hydrodynamic gauge theory is always non- trinsic orbital spin so the spin connection appears in the zero. Consequently, the CFL contains an additional con- covariant derivative. Hence, tribution to the Hall viscosity arising from the fluctuating gauge field. δ CF 1 CF S2 = T12 + (∂µe1)Jµ . (8.5) To see this, we start from the Chern-Simons theory δe1 of CFL where we attach two flux quanta of the Chern- CF The composite fermion current T12 is bound with the Simons gauge field to the electrons to turn them into the b spatial derivative of electric field of bµ as ∂xEx. Once we composite fermions, b have a nonzero ∂xEx, a polarized charge density appears. 2 1 2 gij If the hydrodynamic gauge theory of the gauge field bµ = dr dt Ψ† r,t)(iDt + D + DiDj Ψ(r,t) contains a term like β( Eb)Bb, the polarized charge S 2m 2m ∇ · Z   density associated with the field bµ acts as a magnetic 1 2 2 ′ dr′ dr dt V (r − r )ρ(r)ρ(r′) moment which couples with magnetic flux of b . Thus, − 2 µ Z we have 1 + dr2dt ǫµνρa ∂ a (8.1) 8π µ ν ρ δ Z S = β∂x(∂xb0 ∂0bx) (8.6) δ(∂xby) − where Dµ = ∂µ + i(Aµ + aµ + ωµ) is the covariant deriva- tive. Here, ωµ is the spin connection of the background By solving the equation of motion for the hydrodynamic metric gij , which is related to the local frame fields by gauge field, a a a a gij = (ej + δj )(ei + δi ). We only keep the leading orders a δ δ in ei so that the distorted spatial metric is defined as, S = ∂0 S δ(∂xby) δ∂0(∂xby) e1 e2 δ δ δ δg = 1 1 . (8.2) S = S = S = e1 (8.7) ij e1 e2 δ∂ (∂ b ) δ∂ (∂ b ) δT 1  2 2 0 x y x 0 y 11 16

We finally have compressible phase of the 2DEG with a strong anisotropy (for a review see Ref. [1].) This interpretation is further δ 1 supported by comparing the anisotropy in transport data S = ∂0e1 = β∂x(∂xb0 ∂0bx). (8.8) δ(∂xby) − with a simple model of a nematic, a two-dimensional clas- sical XY model for a director order parameter. By menas Thus the Hall viscosity can be read off from of Monte Carlo calculations it was found that indeed this 17,18 δ 1 1 model fits really well the transport anisotropy data. S = ∂ e1 + (∂ e1)J CF = (ρ + )∂ e1, (8.9) δe2 β 0 1 µ 1 µ β 0 1 Furthermore, these fits show that there is a very low en- 1 ergy scale for the native anisotropy is of the order of i.e., the Hall viscosity is (ρ + 1 ) in which ρ is due to 3 5mK, which is presumably related to the coupling of β the− 2DEG to the underlying lattice. Although the ex- the flux attachment,51 and 1/β is due to the parity-odd periments cannot exclude the possibility that the ground fluctuations of the hydrodynamic gauge field. The parity- state at T = 0 is actually a possible striped state, which odd viscosity has two contributions: one is the intrinsic will be melted thermally into a nematic state at finite orbital spin that the composite fermion carries, and the temperature, the absence of any evidence of stripiness at other is through the term ( Eb)Bb of the hydrodynamic the lowest temperatures seems to imply that the ground gauge field. Hence, if the dual∇· hydrodynamic gauge the- state is a nematic (which, most likely, should be regarded ory has the parity-odd β( Eb)Bb term, we expect that as a quantum melted striped phase.) A detailed review there should be additional∇ Hall · viscosity from the gauge on these experiments (prior to 2010) and their interpre- fluctuation. The theory of the b field can be obtained by µ tation can be found in Ref. [1]. integrating out the composite fermion surface and gauge fluctuation of the gapless aµ. However, if the effective In this section we focus on a relatively recent set of ex- theory of aµ is gapless and nonlocal, the coefficient β for periments on radio-frequency conductivity measurements the term ( Eb)Bb will be non-universal (and depend on in the N = 2 Landau level near filling fraction ν 9/2 ≈ the UV cutoff∇· in a singular way). This is in accordance by Sambandamurthy et.al.97 which, from our perspec- with our result for the CFL. tive, can be naturally interpreted as the following. In this experiment,97 they observe an anisotropy in the lon- gitudinal conductivities as well as a resonant peak of IX. CONNECTION TO EXPERIMENTS the radio-frequency longitudinal conductivity along the hard direction of transport, say σxx, with a frequency In this section, we discuss the connections between our of 100 MHz. This 100 MHz resonance was originally results with the experiments12,13,21,97 on half-filled Lan- interpreted as evidence for the existence of a pinning mode of a stripe state.31 Given that the I V curves dau levels with N > 1. These experiments show that in − half-filled Landau levels with N 2, there is a spectac- are linear, and hence that there is no evidence of transla- ular anisotropy in the longitudinal≥ transport with ratios tion symmetry breaking, it is natural to seek a nematic explanation for this energy gap. In a nematic state in of the resistances as large as Rxx/Ryy 3, 500 at the lowest temperatures. The anisotropy in the∼ longitudinal the continuum there would not be a gap. On the other hand, the transport anisotropy experiments show that resistivities, expressed in the difference ρxx ρyy, raises very rapidly below a critical temperature −T 65mK the anisotropy saturates below T 20mK. Tilted-field c 16,18 ∼ (for N = 2). These results were originally interpreted∼ experiments and the fits to the XY model with a 17 as the signature of a striped phase,24–26 an unidirec- weak symmetry-breaking field, show that the energy scale is of the order of 3 5mK, which is quite comparable tional charge-density-wave state which breaks transla- − tion symmetry (and, necessarily, rotational symmetry.) with a resonant frequency of 100nHz as seen by Samban- 97 However, further transport experiments showed that the damurthy et.al. Thus, we are led to the interpretation I V (current-voltage) curves were metallic and showed that the radio-frequency conductivity measurements are a− linear behavior at low bias voltages.12 In contrast, a detecting a nematic Goldstone mode gapped out by the charge-density-wave (CDW) would have exhibited non- coupling to the lattice which induces a term that breaks linear I V curves with a sharp onset at a critical voltage. the continuous rotational symmetry to the C4 symmetry Extremely− sharp onset behavior has been seen indeed in of the lattice (i.e. of the surface on which the 2DEG is the reentrant integer quantum Hall regime away from the confined.) It is also important to note that the resonant center of the Landau level and has been interpreted as ev- peak seen in by Sambandamurthy et.al. behaves remark- idence for a “bubble phase” (i.e. a bidirectional CDW.) ably close to what is seen in the transport anisotropy in Moreover, in the reentrant IQH regime the experiments the DC experiments. show broadband noise in the current, which are observed Finally, we should note that transport experiments in in many CDW phases, but which is absent the anisotropic the N = 1 Landau level have shown a close connection half-filled Landau levels. between the nematic state and the paired FQH state at For these reasons, the experiments in the center of the ν = 5/2. Indeed, earlier experiments22 showed that by Landau levels (with N 2) have been interpreted instead tilting the magnetic field the FQH state at ν =5/2 is de- as evidence for a nematic≥ phase phase, i.e. a uniform and stroyed and that the resulting compressible state behaves 17 much like an anisotropic HLR state. A relatively recent of composite fermions coupled to a gauge field. This ap- experiment23 has also given evidence of a nematic state proach has been known for a long time to give the correct in the ν = 7/3 FQH plateau (presumably a Laughlin- universal properties of the FQH states,49 including subtle type state) and was interpreted as such in several the- responses to changes in the external geometry.51,52 These ory papers.39,41,45,98 Very recent transport experiments theories are also known to give a qualitative description by Samkharadze et.al.,21 found that, by applying a suf- of the compressible phases, i.e. the HLR theory.53 How- ficiently large external hydrostatic pressure, the incom- ever, it is also well known that the mean field approxi- 5 pressible isotropic FQH state at ν = 2 spontaneously mations based on this mapping involve a large amount gives a way to the compressible anisotropic phase. This of Landau level mixing (even in the limit of a very large is possible since the pressure tunes the width of the quan- magnetic field) which becomes extreme in the compress- tum well and thus tunes the effective interaction between ible states. A symptom of these problems is the lack of the electrons, as the form and spread of the electron particle-hole symmetry even in the limit in which all ex- wavefunction will be varied as the width is changed. At cited Landau levels are projected out. These difficulties the critical pressure, it is found that the rotational sym- have been the focus of intense recent work,70–74 already metry is spontaneously broken and the anisotropy in con- noted in the Introduction. This approach proposes to ductivities develops. Though this transition is between describe the half-filled Landau level, instead, as proxi- incompressible QH state and compressible nematic CFL mate to a theory of Dirac fermions “dually” coupled to a instead of the transition from the compressible isotropic dynamical gauge field. The problem of the possible con- state, namely CFL, to the compressible anisotropic state, nection between the “more conventional” Chern-Simons this result gives a strong hint that tuning an external approach and these recent proposals is at present unclear, parameter, such as pressure, the effective electron inter- including how they may relate to the nematic and paired action such as F2 in this paper can be tuned to find the states. We will consider these connections in a separate transition that we have studied in this paper. publication. Finally we note that there have not been systematic numerical studies of the nematic transition in 2DEGs in X. CONCLUSIONS AND OUTLOOK large magnetic fields. Most of the existing studies of rota- tional (and translational) symmetry breaking have been 99 In this work, we considered the problem of a 2DEG in done either by means of finite-size diagonalizations a half-filled Landau level with strong quadrupolar inter- (done on small systems with toroidal boundary con- ditions which break rotational invariance explicitly) or actions. We mapped the fermion theory into a composite 64 Fermi liquid coupled to a gauge field with a Chern Simons with projected wave functions or with variational wave functions ,48 mostly done on relatively small systems on term, extended by a quadrupolar interaction. Both the nematic fluctuations and gauge fluctuations are found to the sphere (which also poses problems for an order that breaks rotational invariance). A more careful numerical soften the Fermi surface and drive the system into a non- Fermi liquid state. We started from the nematic Fermi study of this problem is clearly needed. liquid theory corrected by the fluctuations of the Chern- Simons gauge boson fluctuations and looked at the ne- matic instability of the non-Fermi liquid. The nematic ACKNOWLEDGMENTS fluctuations are Landau-damped by the non-Fermi liquid state and thus the dynamical critical exponent is z = 3. This work was supported in part by the Na- The nematic theory was also shown to contain a Berry tional Science Foundation through grants DMR-1408713 phase term arising from the gauge field fluctuations, (Y.Y,E.F) at the University of Illinois, and PHY11-25915 which correct the nematic correlator. This non-zero at the Kavli Institute for Theoretical Physics (KITP) Berry phase term suggests the gauge fluctuation would (Y.Y.), and the Brain Korea 21 PLUS Project of Korea also contribute to the Hall viscosity of the half-filled Lan- Government (G.Y.C). Y.Y thanks the KITP Graduate dau level. The resulting odd viscosity in this gapless fellowship program for support and G.Y.C thanks ICMT system is non-universal. In addition, inside the nematic for partial support and hospitality. phase, the nematic vortex current couples with the gauge field. This also demonstrates that the half-filled Landau level orbital spin is not exactly equal to s = 1 as both Appendix A: The isotropic-Nematic quantum phase gauge fluctuation and orbital spin of the CFL in the mean transition in a Fermi Liquid field level have separate contribution to Wen-Zee term. The computation of the Hall viscosity was confirmed by The problem of the isotropic-nematic quantum phase an argument based on a set of Ward identities. transition by a Pomeranchuk instability in a Fermi The theory of the nematic composite Fermi liquid pre- liquid (without a background lattice) was studied by sented here is based on the concept of flux attachment, Oganesyan, Kivelson and Fradkin56 whose work we fol- i.e. on the equivalency between different theories of in- low in detail. The lattice version of this problem was teracting fermions in two space dimensions to a theory studied by several authors.100–102 For a review of elec- 18 tronic nematic phases see Ref.[1]. Here we will use a of the form56 perturbative approach, following the standard work of 103 104 1 2 2 Hertz and Millis (for a review see Ref.[105]). The Q = dtd rd r′ F2( r r′ )tr Qˆ(r,t) Qˆ(r′,t) , full non-perturbative behavior of Fermi fluid is not fully S 2 | − | · Z   understood and has been the focus of considerable work, (A2) both analytic106 and, more recently, numerical.107 The where F ( r r′ ) is a short-ranged quadrupolar interac- problem of the quantum phase transition to an electron 2 | − | nematic state from a charge stripe state has not been tion, studied as much (see, however, Ref.[108].) 2 d q F2 F ( r r′ )= (A3) We start from the isotropic FL described by the free- 2 | − | (2π)2 1+ κq2 Z fermion action 0 in two space dimensions for spinless S 1 fermions (spin will play no role here), where κ− is the range of the quadrupolar interaction and F2 is the quadrupolar coupling. In Eq.(A2) denoted by Qˆ(r) the electronic quadrupolar density defined in the ∇2 2 OFK paper, = d rdtΨ†(r,t) i∂ + + µ Ψ(r,t), (A1) S0 t 2m Z 2 2 h i ˆ 1 ∂x ∂y 2∂x∂y Q(r)= 2 Ψ†(r) − 2 2 Ψ(r). (A4) kF 2∂x∂y ∂y ∂x in which µ is the chemical potential and Ψ is the spin-  −  less fermionic field. This theory 0 is invariant under Note that the electronic quadrupolar density in the OFK arbitrary spatial rotations. It is knownS that this Fermi paper is different from our definition in Section II up to 2 liquid is stable to all infinitesimal interaction except the kF . superconducting (BCS) channel.109,110 Hence, by exclud- It is clear by dimensional counting that the interac- ing the pairing-channel, the only way to introduce any tion Eq.(A2) is irrelevant at the Fermi liquid fixed point qualitative and quantitative change is to turn on the of Eq.(A1), and thus we will need the finite strength interaction beyond some finite strength. Hereafter, we of F2 in Eq.(A2) to be large enough (and attractive) to will ignore the pairing instability and concentrate on drive a phase transition out of the isotropic Fermi liquid phase transitions only in the particle-hole channel (al- Eq.(A1). though the nematic quantum criticality can lead to a su- To understand the quantum phase transition better, perconducting state.111,112) In addition, Oganesyan and we decouple the interaction term of Eq.(A4) by means of coworkers,56 found that in order to stabilize the nematic a Hubbard-Stratonovich transformation, and replace the ground state it is necessary to include in the free fermion quartic form of the action Q by another one in which Hamiltonian terms in the dispersion relation that are at the nematic order parametersS are coupled linearly to two least cubic in the momentum relative to the Fermi mo- real Hubbard-Stratonovich fields, M1 and M2. After this mentum. Such terms are explicitly irrelevant in the Lan- is done, Eq.(A4) becomes dau FL phase. Although in this section we will not in- clude these terms explicitly, we will make them explicit M1(r,t) 2 2 Q = 2 Ψ†(r,t)(∂x ∂y )Ψ(r,t) in the theory of the nematic CFL of Section II. L kF − We are interested in the process of the spontaneous M2(r,t) + 2 Ψ†(r,t)(2∂x∂y)Ψ(r,t) breaking of the rotational symmetry in a FL. The most kF obvious way to break the rotational symmetry is the 1 2 ∇ 2 spontaneous distortion of the Fermi surface. In this pa- M(r,t) + κ ( M(r,t)) (A5) − 2F2 | | per, we are mainly interested in the distortion in the d- h i wave channel, i.e., the quadrupolar channel. The sponta- where we introduced the director field M = (M1,M2). neous symmetry breaking transition is thus induced nat- Hence we obtain a theory of the fermion nematic order urally by turning on the strength of the Landau param- parameter coupled to the Hubbard-Stratonovich fields eter with an attractive coupling F2 for the quadrupolar M1 and M2. In momentum and frequency space the ac- interaction, represented by a term in the full action tion becomes SQ

2 2 k kF 1 2 2 = Ψ†(k,ω) ω − Ψ(k,ω) 1+ κk M(k,ω) S k,ω − 2m − 2F2 | | Z h      i M1(q, Ω) 2 2 M2(q, Ω) + Ψ†(k + q,ω + Ω) 2 (kx ky)+ 2 (2kxky) Ψ(k,ω), (A6) k,ω q,Ω kF − kF Z Z h i 19

2 dωd q 3 Goldstone mode is Landau damped. Oganesyan et al. Here we used the the short-hand notation q,ω = (2π) showed that this overdamped Goldstone mode leads, to and have set the Fermi momentum to beRkF = R√2mµ, and the chemical potential µ is the Fermi energy. lowest order in perturbation theory, to a quasiparticle 2/3 self-energy whose imaginary part scales as Σ′′(ω) ω After the Hubbard-Stratonovich transformation, we ∝ | | proceed to integrate out the fermions to obtain the effec- and, consequently, to the breakdown of the quasiparticle picture and to non-Fermi liquid behavior. However, in tive action for the order parameters M1 and M2. Close to the quantum phase transition to the nematic state we the case of the 2DEG, the continuous rotations symme- can approximate the effective action by a Landau expan- try is broken down to the C4 point group symmetry of sion in powers of the nematic order parameter fields. To the surface. Although this explicit symmetry-breaking is quadratic and quartic order one finds very weak, it results in a finite (but small) energy gap for the nematic Goldstone mode. The results summa- 1 rized above hole above this energy scale. S = dωd2q M (q,ω) (q,ω)M (q,ω) n 2N i Lij j F Z It is important to emphasize that in this picture the κ quadrupolar interaction of Eq.(A2) drives the quantum d2rdt (∇M)2 + λM 4 (A7) − 2F phase transition to the nematic state, if the coupling con-  2  Z stant F2 exceeds the critical value. We will see in Sec- 56 3 2 where λ = (3αNF F2 )/(8EF ), and α is the coeffi- tion II that the same interaction will induce the nematic cient of the quartic term| | in the single-particle dispersion. quantum phase transition in the CFL. The inverse of the propagator of the order parameters, ij (q,ω), contains the information about quantum criti- calL dynamics of the order parameters. The analytic form of ij (q,ω) can be obtained up to one-loop correction in Appendix B: The HLR Composite Fermi Liquid theL fermions. The result is56

κ 2 ij (q,ω)= δij ( q + δ)+ ij (s, φ), (A8) Now we review briefly the physics of the composite L 2F2 M Fermi liquid (CFL) of Halperin, Lee and Read,53,54,85,86 where δ = 1 1 parametrizes the distance from which is another key ingredient of our analysis. Before − 2 − NF F2 NF flux attachment, the action for the HLR compressible the nematic quantum critical point at F ∗ = (the 2 − 2 CFL is Pomeranchuk transition), s = ω , v = kF is the Fermi vF q F m velocity, and φ is the polar angle of the momentum q. 2 2 D The matrix kernel (s, φ) is given by = dtd rΨ†(r,t) iDt + + µ Ψ(r,t), Mij S 2m Z h i s B(s)+ A(s) cos(4φ) A(s) sin(4φ) 1 2 2 (s, φ)= , dtd rd r′ V ( r r′ )δρ(r,t)δρ(r′,t) (B1) Mij 2 A(s) sin(4φ) B(s) A(s) cos(4φ) − 2 | − |  −  Z (A9) where in which we have introduced the covariant derivative Dµ = ∂µ + iAµ, with µ = t,x,y, where Aµ is the ex- 1 2 4 B(s)= , A(s)= B(s)( s 1 s) , ternal electromagnetic gauge field. Here V ( r r′ ) rep- √s2 1 − − | − | − p (A10) resents the density-density interaction between the elec- trons, i.e., δρ(r)=Ψ†(r)Ψ(r) ρ¯ is the local deviation Since we are interested in the dynamics of the asymp- − totic regime s = ω 1, we further expand the func- of the electronic density from the averageρ ¯. In Section II vF q ≪ we will also include a quadrupolar interaction discussed tions A(s) and B(s) for small s around s = 0 and take in Section A. φ = 0 to find For a half-filled Landau level, the average densityρ ¯ i ω + κ q2 + δ 0 of the electrons and the uniform magnetic field B, are vF q 2F2 ij = − ω 2 κ 2 , ρ¯ 1 L 0 ( ) + q + δ related by the filling factor ν = 2π B = 2 . The HLR  − vF q 2F2  (A11) theory also applies to the other compressible states at filling factors ν = 1/2n. In this paper we will focus on where q = q . This result of Oganesyan et al.56 shows the case ν =1/2. We will also include a probe (and un- that the quantum| | dynamical exponent is z = 3. The quantized) component of the electromagnetic gauge field, finite density of states at the Fermi surface is the origin of which we denote by δAµ. The total electromagnetic field ¯ this strong Landau damping of this longitudinal critical is Aµ = Aµ + δAµ. mode. Now we perform the flux attachment transformation 1 Provided the rotational symmetry of the system is not suitable for ν = 2 , i.e. we will attach two flux quanta explicitly broken, inside the nematic phase there is a to each fermion. The flux attachment is implemented by Goldstone mode associated with the spontaneously bro- coupling the fermions to a statistical gauge field aµ whose ken rotational invariance. In this metallic system, the action is a Chern-Simons term. The total action of the 20 transformed, composite, fermion is49,53 δa δa δA , we can write the action in the form µ → µ − µ 2 2 D = dtd rΨ†(r,t) iD + + µ Ψ(r,t), S t 2m 2 Z 2 D h i = dtd r Ψ†(r,t) iDt + + µ Ψ(r,t) 1 2 2 1 dtd rd r′ V ( r r′ )δb(r,t)δb(r′,t), S 2m − 2 16π2 | − | Z h i Z 1 2 2 1 dtd rd r′ V ( r r′ )δρ(r,t)δρ(r′,t), + dtd2r εµνλ(δa δA )∂ (δa δA ), (B5) − 2 | − | 8π µ − µ ν λ − λ Z Z 2 1 µνλ + dtd r ε aµ∂ν aλ, (B2) 8π where now the covariant derivative is Dµ = ∂µ + iδaµ. Z Here we have used the Gauss law constraint of Eq.(B4), to write the density-density interaction in terms of the fluctuations of the gauge flux δb(r,t), resulting in a flux- 49,82 Here Dµ = ∂µ+iAµ+iaµ (where µ = t,x,y), is the covari- flux coupling for the gauge fields. We will see in Sec- ant derivative required by a gauge-invariant (minimal) tion II that for the quadrupolar interaction this identity coupling of the fermions to the electromagnetic gauge does not apply and the form of the interaction is more field Aµ and to the Chern-Simons gauge field aµ. complex. Therefore, the flux attachment transformation maps a We now perform the average field approximation in half-filled Landau level to a system of composite fermions which the average part A¯µ of the electromagnetic gauge at finite density coupled to a dynamical gauge field (with field is cancelled by the average parta ¯µ of the Chern- vanishing average) with a Chern-Simons term (and a Simons gauge field, A¯µ +¯aµ = 0. After this approxima- Maxwell-like term as well). The Fermi momentum of 1/2 1/2 1 tion, we end up with the effective theory the composite fermions is kF = (4πρ¯) = B = ℓ0− , where ℓ0 is the magnetic length. Since the composite fermions do not experience the magnetic field (on aver- age), the composite fermions form a FL, provided that D2 2 the coupling to the gauge field δaµ can be neglected. This = dtd rΨ†(r,t) iDt + + µ Ψ(r,t) S 2m is a gapless state, the HLR composite Fermi liquid.53 The Z 1 h i crucial question is what tis he fate of the FL if the cou- + dtd2r εµνλδa ∂ δa , 8π µ ν λ pling is included. In fact, it turns out that the coupling Z of the composite fermions to the fluctuating gauge field 1 2 2 dtd rd r′ V ( r r′ )δρ(r,t)δρ(r′,t), (B3) destroys completely the well-defined composite fermion − 2 | − | Z quasiparticles on the Fermi surface and leads to a non- Fermi liquid state . Since the CFL action of Eq.(B5) is quadratic in the with Dµ = ∂µ +iδAµ +iδaµ, where we set aµ =¯aµ +δaµ. composite fermion fields, we can proceed to integrate out the composite fermions, and obtain an effective action The Chern-Simons gauge theory is a topological field for the fluctuating gauge fields δaµ. HLR showed that, theory.113 At the local level, its content is a a set of com- at the one loop (RPA) level, the fluctuation δaµ of the mutation relations between the spatial components of the gauge field experiences a strong Landau damping due to gauge field, and a constraint on the space of states (a the finite density of states of electron-like excitations at Gauss law) which in this case reduces to a constraint be- the Fermi surface. The damping manifests in the po- tween the charge density, δρ(r,t) and the local flux of larization bubble of the gauge field when the fermion is the Chern-Simons gauge fields, integrated out at the one-loop level (with the free fermion propagator). To quadratic order in the gauge fields, their effective action in the CFL state is,

1 tij 1 CFL[δaµ,δAµ]= δρ(r,t)= ε ∂iδaj (r,t)= δb(r,t), (B4) S 4π 4π 1 δa (q,ω)Π (q,ω)δa ( q, ω) −2 µ µν ν − − Zq,ω 1 + dtd2rεµνλ(δa δA )∂ (δa δA ), as an operator identity in the Hilbert space of gauge- 8π µ − µ ν λ − λ invariant states. Z (B6)

Finally, upon a shift of the Chern-Simons gauge field, where Πµν (q,ω) is the polarization tensor of the CFL, 21

2 q Π0(q,ω) qxωΠ0(q,ω) qyωΠ0(q,ω) 1 q ωΠ (q,ω) ω2Π (q,ω) q2 Π (q,ω)+ V (q) q q Π (q,ω)+ V (q) Πµν (q,ω)=  x 0 0 y 2 y x 2  , (B7) 2π −   2  2  qyωΠ0(q,ω) qyqx Π2(q,ω)+ V (q) ω Π0(q,ω) qx Π2(q,ω)+ V (q)   −       where V (q) is the Fourier transform of the interaction. given by The functions Π0(q,ω) and Π2(q,ω) are Π (q,ω) K (q,ω)= 0 , 0 − 4D(q,ω) m i ω 1 i ω ρ¯ 2 √ 1 1 V (q)Π0(q,ω)q Π0(q,ω)= 2 | | 3 , Π2(q,ω)= + γ | | 3 , K (q,ω)= + + , q − √ρ¯ q m q 1 2 8D(q,ω) 16π2D(q,ω) | | | | (B8) Π (q,ω) V (q)ω2Π (q,ω)2 whereρ ¯ is the electron density, m is the electron bare 2 0 K2(q,ω)= + 2 (band) mass, and γ =2√3 is a numerical constant. Here, 4D(q,ω) 16π D(q,ω) we have temporarily switched off the external probe elec- V (q)Π (q,ω)Π (q,ω)q2 + 0 2 , tromagnetic gauge field δAµ for clarity of the presenta- 16π2D(q,ω) tion. Here the momentum q = q carried by the gauge | | 2 2 1 2 field is much smaller than the Fermi momentum k of the D(q,ω) =Π0(q,ω) ω ( ) F − 2 composite fermions, i.e., q k , and ω v q, where ≪ F ≪ F V (q) the quantum critical dynamics is manifest. + Π (q,ω)(Π (q,ω) )q2. (B11) 0 2 − 16π2 In contrast to the incompressible FQH states, where the Chern-Simons term is the most relevant term and From these expressions, we can calculate the spectrum dominates the low-energy physics, in the CFL the damp- of collective modes. In the incompressible FQH state the ω lowest energy collective mode is the Girvin-MacDonald- ing term i 3 is the most relevant term in the low- ∝ q Platzman (GMP), which in Ref.[41] we showed condenses energy regime. Furthermore, by gauge invariance, the at the nematic quantum phase transition. From the po- gauge fields must remains gapless, which hence also play larization tensor of the external electromagnetic field, a key role in the low-energy physics of the nematic phase Eq.(B10). we can extract the analog of the GMP mode transition inside the composite Fermi liquid state of sec- for the HLR state. The pole in the polarization tensor tion II. component K00(q,ω) determines the collective mode. In By restoring the probe field δAµ and integrating out the limit of q k and ω v q, the gapless collective ≪ F ≪ F the low-energy fluctuation δaµ in Eq.(B6), we can also excitation of the half-filled Landau level is, derive effective action for the probe external field δAµ, which encodes de correlation functions of the densities i√3 q 3 1/2 ω 1 1 | 2| 3 (1 + mV (q)) (B12) and currents of the CFL, ± ∼ q ± − 12π kF | | h   i From the residue of the polarization tensor for this mode 1 we find a structure factor S(q) q 4, in analogy of the eff [δAµ]= δAµ(q,ω)Kµν (q,ω)δAν ( q, ω) ∝ | | S − 2 − − Girvin-MacDonald-Platzman mode in FQHE states. Zq,ω (B9) Now we turn our attention to the composite fermions, where Kµν (q,ω) is the Fourier transform of the polariza- and calculate the self-energy correction Σf (k,ω) for the tion tensor of the CFL. Its components are given by53,82 composite fermion propagator by calculating the one- loop diagram shown in Fig. 5. The inverse fermion prop- 1 agator g− (k,ω) is 1 2 1 1 K00(q,ω)= q K0(q,ω), g− (k,ω)= g− (k,ω) Σ (k,ω) (B13) 2π 0 − f 1 K0i(q,ω)= (ωqiK0(q,ω)+ iǫikqkK1(q,ω)), where g0(k,ω) is the free fermion propagator. The imag- 2π inary part of the self-energy Σ (k,ω) is 1 ′′ Ki0(q,ω)= (ωqiK0(q,ω) iǫikqkK1(q,ω)), 2π − β ω 2/3 1 Σ′′(q,ω) 2√3 sign(ω)( | |) (B14) K (q,ω)= (ω2δ K (q,ω) iǫ ωK (q,ω) ≃− 4π ij 2π ij 0 − ij 1 2 kF + (q δij qiqj )K2(q,ω)), (B10) where β = 4√m + V (0)kF √m, and where we have − dropped the bare term ω which is much smaller than the correction ω 2/3. Here we have assumed the density- where the functions K (q,ω), K (q,ω) and K (q,ω) are density interaction| | is short-ranged, and hence only V (q 0 1 2 ≈ 22

0) appears in the expression. Here and after, we only con- sider the short-ranged density-density interaction. This result shows that the singular forward scattering interac- tion with the fluctuating gauge boson softens the Fermi surface and that the composite fermion quasiparticle is no longer well defined. The gauge boson drives the com- posite fermion into a non Fermi liquid.53

FIG. 5. The gauge boson correction to the fermion propaga- tor.

Appendix C: Coupling between the Gauge field and nematic field

We start from the theory of the Fermi surface coupled to the Chern-Simons gauge field,

2 1 2 M1 2 2 M2 = d rdt Ψ†(r,t)[D + D + (D D )+ (2D D )]Ψ(r,t) S t 2m 2m x − y 2m x y Z 1 2 µνρ 2 2 ′ ′ + d rdt ǫ δa ∂ δa + d rd r′dtV (r − r )δρ(r,t)δρ(r ,t) (C1) 8π µ ν ρ Z Z

where Dµ = ∂µ + δaµ. The external magnetic field is polarization for the gauge field screened by the Chern-Simons flux while δai represent the gauge fluctuation. In this mean-field level, the aver- 1 a = δaµ(q,ω)Πµν (q,ω)δaν (−q, ω) (C2) age flux felt by the composite fermion is zero so the CF S −2 q,ω − is in the Fermi liquid state. If we consider gauge fluc- Z tuation to the quadratic order, we obtain the effective where

2 q Π0 qxωΠ0 + iqy/2 qyωΠ0 iqx/2 1 2 2 − Πµν = qxωΠ0 iqy/2 ω Π0 qy(Π2 + V (q)) qyqx(Π2 + V (q)) iω/2 (C3) 2π q ωΠ +− iq /2 q q (Π−+ V (q)) + iω/2 ω2Π q2(Π +−V (q))  y 0 x y x 2 0 − x 2   with field level has the form,

1 [M,δa ]= M (q,ω)T (q,ω)δa (q,ω) m i ω Sa,M µ −2 i iν ν Π (q,ω)= | | , (C4) Zq,ω 0 q2 − √ρ¯ q 3 (C6) | | 1 i ω √ρ¯ where Π2(q,ω)= + γ | | , (C5) 3 2 2 m q Π2(q,ω)m q q q ω q ω | | T (q,ω)= x − y x − y , 2π 2qxqy ωqy qxω   i ω √ρ¯ whereρ ¯ is the electron density. Π (q,ω)=γ | | (C7) 2 q 3 The linear, parity-even, coupling between the nematic | | field and the gauge field, i.e. the quadrupolar coupling where we used the overdamped form of Π2(q,ω), for 1 ω 2 ≫ between nematic fluctuations and gauge fields, at mean q . qvF ≫ 23

We also have the coupling between the nematic order parameter and the Chern-Simons gauge fields, 1 [δa , M]= δa (q,ω)V [M]δa (−q, ω) Sa,M,a µ −2 µ µν ν − Zq,ω (C8)

M1 2 2 (qx qy) (M1qx + M2qy)ω ( M2qy + M2qx)ω Π0(q,ω) 2 V [M]= (M q + M− q )ω M ω2 − M ω2 (C9) µν 2π 1 x 2 y 1 2 ( M q + M q )ω M ω2 M ω2  − 2 y 2 x 2 − 1  

This parity-even term represents the coupling of the Fig.2. We obtained a similar term in the case of the FQH gauge fields to the nematic fluctuations as a local fluctu- in Ref.[41]. The Berry phase term is obtained once we in- ation of the spatial metric. tegrate out the gauge fluctuation and expand the theory in quadratic level. Here, we start from the gauge-nematic order parameter theory. Here we choose the temporal Appendix D: Details of calculation of the Berry gauge a0 = 0, and set m = 1 (which we will restore back phase term later).

In this section, we show the detailed calculation of the Berry phase term whose Feynman diagram is shown in

1 = δa (q,ω)[Π + t1 (ω2Π )M + t2 (ω2Π )M )]δa (−q, ω) S − 2 i ij ij 0 1 ij 0 2 j − Zq,ω 2 2 ω Π0 qxΠ2 qxqyΠ2 iω/2 − − 2 −2 1 qxqyΠ2 + iω/2 ω Π0 qyΠ2 Π− = − −  (D1) ω4Π2 q2ω2Π Π ω2/4 0 − 0 2 −

ω Since we are looking at the limit < 1, we only where t1 = σ3 and t2 = σ1. | vF q | keep the order of O( ω ) for the damping part in the Thus the anti-symmetric part of the effective theory of vF q gauge boson propagator and drop terms like ( ω )2. Af- the nematic order parameters is vF q terwards, the inverse of the polarization tensor is, M ij 2 ij = χǫ Mi(Ω, Q)(iΩ)Mj ( Ω, Q) (D4) qxΠ2 qxqyΠ2 iω/2 L − − q q−Π + iω/2 − q2Π− 1 x y 2 y 2 2 Π− = − −  where χ = Λ,¯ and Λ¯ is the frequency cut off. After − q2ω2Π Π + ω2/4 3π 0 2 putting back in the electron mass, we have χ = 2 Λ¯m. m i ω √ρ¯ 3π Π (q,ω)= , Π (q,ω)= γ | | , (D2) 0 q2 2 q 3 | | Here we temporarily use units in which m = 1 for Appendix E: Details of calculations of the Wen-Zee convenience. Then the effective theory for the nematic term order parameters can be obtained as δ ( i) S (Ω, Q)= First, we calculate the parity-even part of the Wen- − δMiδMj Zee term where the diagram has one gauge leg and one nematic leg (see Fig.VII) We use the temporal gauge a = iTr Π 1(ω, q)ω2Π t Π 1(ω +Ω, q + Q)ω2Π t 0 − 0 i − 0 j 0 and take units in which m = 1, and find Zq,ω Π q2 1 ij 2 δ = iΩ/2ǫ 2 2 2 2 (D3) 2 mn 2 mi q,ω Π2q + q /4 Π2q + q /4 ( i) S = amω Π0tl anamω Π0Qk Mi (E1) Z − δMlδak h i 24 where with the notation Qx = I,Qy = iσ2, and q ω q ω x − y ωqy qxω 1  mi Tmi− = amMi (q,ω)= 2 2 , (E2) h i Π2q ω

δ 2 2 1 1 i S (p, Ω) = i Tr (ω Π0) Π− (ω Ω/2, q − Q/2)tlT − (ω +Ω/2, q + Q/2)Qk − δMlδak q,ω − Z h i p Ω p Ω 1 = x − y Ωpy pxΩ 2q(Π q3 + q3/4)  lk Zp,ω 2 1 Λ¯ p Ω p Ω = ( )1/3 x − y (E3) 6π√3 ρ¯ Ωpy pxΩ  lk Restoring the mass back, we have

δ 1 Λ¯m p Ω p Ω i S (p, Ω) = ( )1/3 x − y (E4) − δMlδak 6π√3 ρ¯ Ωpy pxΩ  lk To calculate the cubic level of the Wen-Zee term where the diagram has one gauge field external leg and two nematic fields external legs (shown in Fig. 4).We will choose the axial gauge where a0 = 0 (and use units which the effective mass is m=1), to find

δ 2 mn 2 mn 2 mi i S = amω Π0σh anamω Π0σl anamω Π0Qk Mi (E5) − δMhδMlδak −h i Similarly, δ ( i) S (Ω1, Ω2; p1, p2) − δMhδMlδak 1 2 1 2 1 2 = Tr Π− (ω, q)(ω Π0σh)Π− (ω +Ω1, q + p1)(ω Π0σl)T − (ω +Ω1 +Ω2, q + p1 + p2)(ω Π0Qk) − q,ω Z h i ω q2 Π q3 = ǫhlǫνµkpν (pµ + pµ) | | + 2 1 1 2 8q(Π q3 + q3/4)2 8q(Π q3 + q3/4)2 Zq,ω 2 2 1/3 h i 1 Λ¯ Λ¯ = + ǫhlǫνµkpν (pµ + pµ) (E6) 6π√3 ρ¯ ρ¯ 1 1 2 h   i

Restoring the mass back, we find the effective theory as from the Ward Identity. The nematic order parameter modifies the local geometry metric of the Maxwell term δ and couples to the bilinear of the Chern-Simons gauge ( i) S (Ω1, Ω2; p1, p2) − δMhδMlδak fields. At this level, we can integrate out the gauge fields 1/3 1 Λ¯m Λ¯m and perform the loop expansion of the nematic fields at = + ǫhlǫνµkpν (pµ + pµ) (E7) the cubic level. 6π√3 ρ¯ ρ¯ 1 1 2 h   i Similar to the calculation of the Berry phase term by Ward Identity, the Wen-Zee term can also be derived

T T T (p , p , Ω , Ω )= h 1 2 xyi 1 2 1 2 1 2 1 2 1 2 Tr Π− (k,ω)(Π0ω )σzΠ− (k + p1,ω +Ω1)(Π0ω )σzΠ− (k + p1 + p2,ω +Ω1)(Π0ω )σ+ (E8) − k,ω Z h i 25

By taking advantage of the Ward Identity, we find

m∂ J + ∂ T + ∂ T = δb J 0 x x xx y yx · y m∂ J + ∂ T + ∂ T = δb J (E9) 0 y x xy y yy − · x

We can set up several relations between the Wen-Zee term and the terms cubic in nematic order parameters,

m T (r )T (r )∂ J (r ) + T (r )T (r )∂ T (r ) + T (r )T (r )∂ T (r ) =2 T (r )T (r )δρ(r )J (r ) h 1 1 2 2 0 x 3 i h 1 1 2 2 x xx 3 i h 1 1 2 2 y yx 3 i h 1 1 2 2 3 y 3 i m T1(r1)T2(r2)∂0Jy(r3) + T1(r1)T2(r2)∂xTxy(r3) + T1(r1)T2(r2)∂yTyy(r3) = 2 T1(r1)T2(r2)δρ(r3)Jx(r3) h i h i h i − h (E10)i

It is straightforward to see that the correlators on the right hand sides are actually zero. Also, the further calculation shows that T1(r1)T2(r2)∂xTxx(r3) and T1(r1)T2(r2)∂yTyy(r3) does not generate any anti-symmetric term at the leading orderh (O(q3)). Now the identitiesi h become, i

m T (r )T (r )∂ J (r ) = T (r )T (r )∂ T (r ) h 1 1 2 2 0 x 3 i −h 1 1 2 2 y yx 3 i m T (r )T (r )∂ J (r ) = T (r )T (r )∂ T (r ) (E11) h 1 1 2 2 0 y 3 i −h 1 1 2 2 x xy 3 i

We have

ij y y ij y T T ∂ T (p1,p2,ω ,ω )= sǫ ω (ω + ω )( p p ) sǫ ω (ω + ω )p h i j y yxi 1 2 1 1 2 − 1 − 2 − 1 1 2 1 ij x x ij x T T ∂ T (p1,p2,ω ,ω )= sǫ ω (ω + ω )( p p )+ sǫ ω (ω + ω )p (E12) h i j x xyi 1 2 − 1 1 2 − 1 − 2 1 1 2 1

Thus, the Wen-Zee term has the form

s ij y y ij y T T J (p1,p2,ω ,ω )= ǫ ω ( p p ) sǫ ( ω ω )p h i j xi 1 2 m 1 − 1 − 2 − − 1 − 2 1 s ij x x ij x T T J (p1,p2,ω ,ω )= ǫ ω ( p p )+ sǫ ( ω ω )p (E13) h i j yi 1 2 −m 1 − 1 − 2 − 1 − 2 1

By taking advantage of the current conservation relation, Thus, we finally obtain that the Wen-Zee term W Z of the effective Lagrangian is L

¯ 1/3 ¯ ∂µJµ =0 (E14) 1 Λm Λm µνρ Q W Z = + ǫ ω ∂ν Aρ (E17) L 6π√3 ρ¯ ρ¯ µ h   i We obtain Appendix F: Nematic correlators

s ij y x x TiTjJ0 (p1,p2,ω1,ω2)= ǫ p1( p1 p2 ) The self-energy of the composite fermion close to the h i m − − Fermi surface is, ij y y x sǫ ( p1 p2)p1 (E15) − − − βω Σ (ω)= i2√3 sign(ω)( )2/3 (F1) s − 4π s The Fermi surface is softened by the gauge boson and Where the coefficient m is obtained to be thus no longer well defined. Here we check the correction from the self-energy to of the nematic correlator. The nematic correlator of the Fermi surface without s 1 Λ¯m 1/3 Λ¯m = + (E16) self-energy correction is shown in the Feynman diagram m 6π√3 ρ¯ ρ¯ of Fig.1, and it is given by h   i 26

(a) (b)

(c)

FIG. 6. Leading corrections from gauge field fluctuations to the nematic correlators: a) and b) composite fermion self-energy corrections, and c) vertex correction.

δS i = T1T1 0(p, Ω) − δM1δM1 h i (k2 k2) (k2 k2) = i g(k + p,ω + Ω) x − y g(k,ω) x − y − 2m 2m Zk,ω 1 (k2 k2) (k2 k2) = i x − y x − y − 4m2 ω +Ω ((k + p)2 + k2 )/2m + iη sign(ω + Ω) ω (k2 + k2 )/2m + iη sign(ω) Zk,ω − F − F k3 k2 m = k4 /(4πm) cos2(2θ ) F (iΩ/p) sin2(2θ ) F (Ω/p)2 (F2) F − p 2π − p 2π

If we only look into the leading self-energy correction, it is the sum of the diagrams in Fig. 6,

T T (p, Ω) = h 1 1i1 i (k2 k2)2/(4m2)[g(k + p,ω + Ω)Σ (ω)g2(k,ω)+ g2(k + p,ω + Ω)Σ (ω + Ω)g(k,ω)] (F3) − x − y s s Zk,ω

Here the self-energy Σs(ω) only affects the fermions near the Fermi surface where k kF , Take advantage of the identity ∼

Σ (ω) g(k + p,ω + Ω)Σ (ω)g2(k,ω)= s [g2(k,ω) g(k + p,ω + Ω)g(k,ω)] s −Ω k p/2m − − · Σ (ω + Ω) g2(k + p,ω + Ω)Σ (ω + Ω)g(k,ω)= s [g2(k + p,ω + Ω) g(k + p,ω + Ω)g(k,ω)] (F4) s Ω k p/2m − − · We can rewrite the integral as

Σ (ω) Σ (ω + Ω) T T (p, Ω) = i (k2 k2)2/(4m2) s − s g(k + p,ω + Ω)g(k,ω) (F5) h 1 1i1 − x − y Ω k p/2m Zk,ω − · 27

In the same way, the other nematic correlators can be written as

Σ (ω) Σ (ω + Ω) T T (p, Ω) = i (2k k )2/(4m2) s − s g(k + p,ω + Ω)g(k,ω) h 2 2i1 − x y Ω k p/2m Zk,ω − · Σ (ω) Σ (ω + Ω) T T (p, Ω) = i (k2 k2)(2k k )/(4m2) s − s g(k + p,ω + Ω)g(k,ω) (F6) h 2 1i1 − x − y x y Ω k p/2m Zk,ω − · Here we first calculate T T , h 1 1i1 T T (p, Ω) = h 1 1i1 k4 cos2(2θ) Σ (ω) Σ (ω + Ω) 1 1 = i s − s − 4m2 Ω kp cos(θ θ )/2m ω +Ω (E(k)+ kp cos(θ θ )/2m)+ iη sign(ω + Ω) ω E(k)+ iη sign(ω) Zk,ω − − p − − p − (F7) where E(k) = (k2 k2 )/2m. − F Since the self energy correction Σ(ω) only appears near the Fermi surface, we can replace k/2m = vF = kF /2m in the calculation to simplified the problem. We can replace the integral dk = dE(k)dθ 2m,

T T (p, Ω) = h 1 1i1 k4 cos2(2θ) Σ (ω) Σ (ω + Ω) 1 1 i dωdE(k)dθ F s − s − 4m Ω v p cos(θ θ ) ω +Ω (E(k)+ v p cos(θ θ )) + iη sign(ω + Ω) ω E(k)+ iη sign(ω) Z − F − p − F − p − k4 cos2(2θ) Σ (ω) Σ (ω + Ω) = dωdθ F s − s ( sign(ω) sign(ω + Ω)) (F8) 4m (Ω v p cos(θ θ ))2 − Z − F − p

First, we look at the static limit Ω = 0. Then we have Σs(ω) Σs(ω +Ω) = 0 and therefore the integral is zero. This indicates the leading order self-energy correction does not gener− ate any constant term, and thus the mass term of the nematics remains unchanged by the imaginary part of self-energy. Now we turn to the case when Ω is nonzero and take the limit p kF and Ω p. The integral of θ can be done by contour integral, we take the leading order in regard of Ω/p. Since≪ we are only≪ interested in the leading order behavior of the damping term, we calculate (T T + T T ) to simplify the calculation. h 1 1 2 2 i1 T T (p, Ω) = h i ii1 k4 Σ (ω) Σ (ω + Ω) = dωdθ F s − s ( sign(ω) sign(ω + Ω)) 4m (Ω v p cos(θ θ ))2 − Z − F − p k4 Σ (ω) Σ (ω + Ω) Ω 1 = dω F s − s ( sign(ω) sign(ω + Ω)) 4m (v p)2 − v p (1 + (Ω/v p)2)3/2 Z F F F k4 Σ (ω) Σ (ω + Ω) Ω dω F s − s ( sign(ω) sign(ω + Ω)) ≈ 4m (v p)2 − v p Z F F 4 2/3 5/3 kF √3 β Ω Ω = 2 (F9) 2m (vF p) vF p

The leading order damping upon the self-energy correction is (Ω)8/3/q3, which is highly irrelevant so we can ignore it. However, one might worry that T1T1 1 could contain a singular damping term which could cancel when added to T T . If this cancellation happened,h i then the damping coefficient T T would be a function of cos(4θ ). We h 2 2i1 h 1 1i1 p can then choose θp = 0 and calculate T1T1 1 to check the leading damping term. The leading order of damping for 5/3 2 h i T1T1 1(θp =0) is of order Ω /q , which is still irrelevant compared to the damping term. h In conclusion,i the leading order self-energy correction does not change the leading order behavior of the nematic transition. In sum, the nematic polarization tensor is

TiTj (p, Ω) = h i 3 2 3 4 2 kF 2 kF m 2 kF kF /(4πm) cos (2θp) (iΩ/p) sin (2θp) (Ω/p) sin(2θp) cos(2θp) (iΩ/p) − 2π 3− 2π 3 2π 2 kF 4 2 kF 2 kF m 2 sin(2θp) cos(2θp) (iΩ/p)) k /(4πm) sin (2θp) (iΩ/p) cos (2θp) (Ω/p) ! 2π F − 2π − 2π (F10) 28

Appendix G: Vertex correction for nematic polarization tensor

In this section, we show the vertex correction for the nematic polarization tensor is irrelevant in our theory (shown in Fig.6c)

T T (p, Ω) + T T (p, Ω) = h 1 1iv h 2 2iv k6 i g(k + p,ω + Ω)g(k,ω)g(k + p + q,ω +Ω+ ν)g(k + q,ω + Ω)D (q,ν) (G1) − 4m4 11 Zk,q,ω,ν To proceed, we first calculate the vertex,

Γ(k, p, ω, Ω) =

i g(k + p + q,ω +Ω+ ν)g(k + q,ω + Ω)D (q,ν) − 11 Zq,ν q i | v| − (ω +Ω+ ν (E(k)+ v q + v p cos(θ θ )+ q p sin(θ θ )) + iη sign(ω +Ω+ ν)) Zq,ν − F l F − p v − p 1 1 (G2) × (a ν + b q 3) ω + ν (E(k)+ v q )+ iη sign(ω + ν) | | | v| − F l where q = q sin(θ θ ), q = q cos(θ θ ). a = 2m√3 and b = 1/4 are constants which we will take as unity for v − q l − q simplicity since we are only interested in scaling behaviors. Taking the integral ql first, we find

Γ(k, p, ω, Ω) = q ( sign(ω +Ω+ ν) sign(ω + ν)) i | v| − − (Ω v p cos(θ θ )+ q p sin(θ θ ))(a ν + b q 3) Zqv ,ν − F − p v − p | | | v| q ω +Ω ω i | v| ln(1 + | |) sign(ω + Ω) ln(1 + | | ) sign(ω) ∼− Ω v p cos(θ θ )+ q p sin(θ θ ) q 3 − q 3 Zqv − F − p v − p | v| | v|  (G3)

It is obvious that when Ω = 0, this expression is vanishes. Thus the vertex correction does not contribute to the mass term of the polarization tensor. Now we turn to the damping term in the vertex corrections to the polarization tensor,

T T (p, Ω) = h i iiv qv ω +Ω ω i | | ln(1 + | 3 |) sign(ω + Ω) ln(1 + | |3 ) sign(ω) − q ,k,ω Ω vF p cos(θ θp)+ qvp sin(θ θp) qv − qv Z v − − −  | | | |  6 1 k 1 − ω +Ω E(k) v p cos(θ θ )+ iη sign(ω + Ω))− (ω E(k)+ iη sign(ω) (G4) × 4m4 − − F − p −   6 6 k kF As the gauge field correction only affects the fermion propagator near the Fermi surface, we can take 4m4 = 4m4 and take this constant aside. We can then replace the integral over dk = dE(k)dθ 2m and integrate over dE(k) first,

qv ω +Ω ω TiTi v(p, Ω) | | ln(1 + | 3 |) sign(ω + Ω) ln(1 + | |3 ) sign(ω) h i ∼ q ,θ,ω Ω vF p cos(θ θp)+ qvp sin(θ θp) qv − qv Z v − − −  | | | |  sign(ω + Ω) sign(ω) − × Ω v p cos(θ θ ) − F − p Ω q ω 1 dω | v| ln(1 + | | ) (G5) ∼ Ω v p cos(θ θ )+ q p sin(θ θ ) q 3 Ω v p cos(θ θ ) Zqv ,θ Z0 − F − p v − p | v| − F − p By the change of variables l = ω/Ω, we get

1 Ω q l 1 T T (p, Ω) dl | v| ln(1 + Ω | | ) (G6) h i iiv ∼ Ω v p cos(θ θ )+ q p sin(θ θ ) q 3 Ω v p cos(θ θ ) Zqv ,θ Z0 − F − p v − p | v| − F − p 29

Performing the θ integral (and take vF = 1 to simplify the expression), we find

1 Ω qv Ω l 1 TiTi v(p, Ω) dl | | ln(1 + Ω | | ) (G7) 2 3 2 2 2 h i ∼ q 0 p p qv (1 (Ω/p) ) 1 + (q /v ) (Ω/p) Z v Z | | − v f −

Since we are looking at the fermion near the Fermi surface, qv shall have a cutp off which is smaller than vf . Once we assume qv/vF < 1,

Λ 1 Ω qv Ω l 1 TiTi v(p, Ω) dqv dl | | ln(1 + Ω | | ) (G8) 2 3 2 2 h i ∼ 0 0 p p qv (1 (Ω/p) ) 1 (Ω/p) Z Z | | − − where Λ is the UV momentum cut off for qv, integrate over qv and ignore the numericalp constant,

1 Ω Ω 2/3 1 TiTi v(p, Ω) dl (Ωl) 2 2 2 h i ∼ 0 p p (1 (Ω/p) ) 1 (Ω/p) Z − − 1 Ω8/3 dl (l)2/3 p ∼ p3 Z0 (G9)

From the above calculation, it is obvious that the damp- the damping coefficient T T must be a function of h 1 1iv ing term from the vertex correction is of higher order cos(4θp). We can then choose θp = 0 and calculate Ω compared to p . One might worry that T1T1 v con- T1T1 v to check the leading damping term. The leading h i h i 5/3 2 tains some singular damping term which cancels when order of damping for T1T1 v(θp = 0)is of order Ω /q , h i sum with T2T2 v. If this cancellation happens, then which is still irrelevant compared to the damping term. h i Thus, the vertex correction to the overdamped mode is irrelevant.

1 E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, 11 R. A. Borzi, S. A. Grigera, J. Farrell, R. S. Perry, S. J. S. and A. P. Mackenzie, Annual Review of Condensed Mat- Lister, S. L. Lee, D. A. Tennant, Y. Maeno, and A. P. ter Physics 1, 153 (2010). Mackenzie, Science 315, 214 (2007). 2 P. M. Chaikin and T. C. Lubensky, Principles of Con- 12 M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, densed Matter Physics (Cambridge University Press, and K. W. West, Phys. Rev. Lett. 82, 394 (1999). Cambridge, UK, 1995). 13 R. R. Du, D. C. Tsui, H. L. St¨ormer, L. N. Pfeiffer, K. W. 3 P. G. de Gennes and J. Prost, The Physics of Liquid Crys- Baldwin, and K. W. West, Solid State Comm. 109, 389 tals (Oxford Science Publications/Clarendon Press, Ox- (1999). ford, UK, 1993). 14 K. B. Cooper, M. P. Lilly, J. P. Eisenstein, L. N. Pfeiffer, 4 S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, and K. W. West, Phys. Rev. B 65, 241313 (2002). 550 (1998). 15 K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, and K. W. 5 E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065 West, Phys. Rev. Lett. 90, 226803 (2003). (1999). 16 J. Pollanen, K. B. Cooper, S. Brandsen, J. P. Eisenstein, 6 Y. Ando, K. Segawa, S. Komiya, and A. N. Lavrov, Phys. L. N. Pfeiffer, and K. W. West, Phys. Rev. B 92, 115410 Rev. Lett. 88, 137005 (2002). (2015). 7 V. Hinkov, D. Haug, B. Fauqu´e, P. Bourges, Y. Sidis, 17 E. Fradkin, S. A. Kivelson, E. Manousakis, and K. Nho, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Sci- Phys. Rev. Lett. 84, 1982 (2000). ence 319, 597 (2008). 18 K. B. Cooper, M. P. Lilly, J. P. Eisenstein, T. Jungwirth, 8 R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choiniere, F. Lal- L. N. Pfeiffer, and K. W. West, Sol. State Commun. 119, iberte, N. Doiron-Leyraud, B. J. Ramshaw, R. Liang, 89 (2001). D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature 463, 19 M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfeiffer, 519 (2010). and K. W. West, Phys. Rev. Lett. 83, 824 (1999). 9 M. J. Lawler, K. Fujita, J. Lee, A. R. Schmidt, 20 W. Pan, R. R. Du, H. L. St¨ormer, D. C. Tsui, L. N. Y. Kohsaka, C. K. Kim, H. Eisaki, S. Uchida, J. C. Davis, Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. J. P. Sethna, and E.-A. Kim, Nature 466, 347 (2010). Lett. 83, 820 (1999). 10 J.-H. Chu, H.-H. Kuo, J. G. Analytis, and I. R. Fisher, 21 N. Samkharadze, K. A. Schreiber, G. C. Gardner, M. J. Science 337, 710 (2012). Manfra, E. Fradkin, and G. A. Csathy, Nature Phys. 12, 30

191 (2016). 57 J. K. Jain, Advances in Physics 41, 105 (1992). 22 J. Xia, V. Cvicek, J. P. Eisenstein, L. N. Pfeiffer, and 58 R. L. Willett, M. A. Paalanen, R. R. Ruel, K. W. West, K. W. West, Phys. Rev. Lett. 105, 176807 (2010). L. N. Pfeiffer, and D. J. Bishop, Phys. Rev. Lett. 65, 112 23 J. Xia, J. Eisenstein, L. Pfeiffer, and K. West, Nature (1990). Phys. 7, 845 (2011). 59 W. Kang, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, 24 R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006 and K. W. West, Phys. Rev. Lett. 71, 3850 (1993). (1996). 60 V. J. Goldman, B. Su, and J. K. Jain, Phys. Rev. Lett. 25 A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys. 72, 2065 (1994). Rev. Lett. 76, 499 (1996). 61 J. K. Jain, Composite Fermions, 1st ed. (Cambridge Uni- 26 M. Fogler, A. Koulakov, and B. Shklovskii, Phys. Rev. B versity Press, Cambridge U.K., 2007). 54, 1853 (1996). 62 K. Park, V. Melik-Alaverdian, N. E. Bonesteel, and J. K. 27 A. H. MacDonald and M. P. A. Fisher, Phys. Rev. B 61, Jain, Phys. Rev. B 58, R10167 (1998). 5724 (2000). 63 N. Read and D. Green, Phys. Rev. B 61, 10267 (2000). 28 T. D. Stanescu, I. Martin, and P. Phillips, Phys. Rev. 64 K. Lee, J. Shao, E. Rezayi, and E.-A. Kim, (2015), in Lett. 84, 1288 (2000). preparation. 29 D. G. Barci, E. Fradkin, S. A. Kivelson, and 65 C. Nayak and F. Wilczek, Nuclear Physics B 417, 359 V. Oganesyan, Phys. Rev. B 65, 245319 (2002). (1994). 30 M. M. Fogler, Int. J. Mod. Phys. B 16, 2924 (2002). 66 A. Stern and B. I. Halperin, Phys. Rev. B 52, 5890 (1995). 31 R. Cˆot´eand H. A. Fertig, Phys. Rev. B 62, 1993 (2000). 67 Y. B. Kim, A. Furusaki, X.-G. Wen, and P. A. Lee, Phys. 32 A. Lopatnikova, S. H. Simon, B. I. Halperin, and X.-G. Rev. B 50, 17917 (1994). Wen, Phys. Rev. B 64, 155301 (2001). 68 R. Shankar and G. Murthy, Phys. Rev. Lett. 79, 4437 33 M. J. Lawler and E. Fradkin, Phys. Rev. B 70, 165310 (1997). (2004). 69 S. A. Kivelson, D.-H. Lee, Y. Krotov, and J. Gan, Phys. 34 V. J. Emery, E. Fradkin, S. A. Kivelson, and T. C. Luben- Rev. B 55, 15552 (1997). sky, Phys. Rev. Lett. 85, 2160 (2000). 70 D. T. Son, Phys. Rev. X 5, 031027 (2015). 35 A. Vishwanath and D. Carpentier, Phys. Rev. Lett. 86, 71 M. Barkeshli, M. Mulligan, and M. P. A. Fisher, Phys. 676 (2001). Rev. B 92, 165125 (2015). 36 S. L. Sondhi and K. Yang, Phys. Rev. B 63, 054430 72 C. Wang and T. Senthil, Phys. Rev. X 5, 041031 (2015). (2001). 73 S. D. Geraedts, M. P. Zaletel, R. S. K. Mong, M. A. Metl- 37 H. A. Fertig, Phys. Rev. Lett. 82, 3693 (1999). itski, A. Vishwanath, and O. I. Motrunich, Science 352, 38 E. H. Rezayi and F. Haldane, Phys. Rev. Lett. 84, 4686 197 (2016). (2000). 74 M. A. Metlitski and A. Vishwanath, “Particle-vortex du- 39 M. Mulligan, C. Nayak, and S. Kachru, Phys. Rev. B 82, ality of 2d Dirac fermion from electric-magnetic duality 085102 (2010). of 3d topological insulators,” (2015), arXiv:1505.05142. 40 Y. You and E. Fradkin, Phys. Rev. B 88, 235124 (2013). 75 S. Kachru, M. Mulligan, G. Torroba, and H. Wang, Phys. 41 Y. You, G. Y. Cho, and E. Fradkin, Phys. Rev. X 4, Rev. B 92, 235105 (2015). 041050 (2014). 76 M. Mulligan, S. Raghu, and M. P. A. Fisher, “Emergent 42 F. D. M. Haldane, ““Hall viscosity” and intrinsic metric particle-hole symmetry in the half-filled Landau level,” of incompressible fractional Hall fluids,” (2009), unpub- (2016), arXiv:1603.05656. lished, arXiv:0906.1854. 77 D. F. Mross, J. Alicea, and O. I. Motrunich, “Ex- 43 F. D. M. Haldane, Phys. Rev. Lett. 107, 116801 (2011). plicit derivation of duality between a free Dirac cone and 44 K. Yang, Phys. Rev. B 88, 241105 (2013). quantum electrodynamics in (2+1) dimensions,” (2015), 45 J. Maciejko, B. Hsu, S. Kivelson, Y. Park, and S. Sondhi, arXiv:1510.08455. Phys. Rev. B 88, 125137 (2013). 78 G. Murthy and R. Shankar, Phys. Rev. B 93, 085405 46 L. Radzihovsky and A. T. Dorsey, Phys. Rev. Lett. 88, (2016). 216802 (2002). 79 S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Phys. 47 C. Wexler and A. T. Dorsey, Phys. Rev. B 64, 115312 Rev. Lett. 99, 236807 (2007). (2001). 80 M. Levin, B. I. Halperin, and B. Rosenow, Phys. Rev. 48 Q. M. Doan and E. Manousakis, Phys. Rev. B 75, 195433 Lett. 99, 236806 (2007). (2007). 81 A. L´opez and E. Fradkin, Phys. Rev. Lett. 69, 2126 49 A. L´opez and E. Fradkin, Phys. Rev. B 44, 5246 (1991). (1992). 50 E. Fradkin, Field Theories of Condensed Matter Systems, 82 A. L´opez and E. Fradkin, Phys. Rev. B 47, 7080 (1993). Second Edition (Cambridge University Press, Cambridge, 83 V. Pasquier and F. Haldane, Nucl. Phys. B. 516, 719 UK, 2013). (1998). 51 G. Y. Cho, Y. You, and E. Fradkin, Phys. Rev. B 90, 84 N. Read, Phys. Rev. B 58, 16262 (1998). 115139 (2014). 85 E. Rezayi and N. Read, Phys. Rev. Lett. 72, 900 (1994). 52 A. Gromov, G. Y. Cho, Y. You, A. G. Abanov, and 86 Y. B. Kim and X.-G. Wen, Phys. Rev. B 50, 8078 (1994). E. Fradkin, Phys. Rev. Lett. 114, 016805 (2015). 87 N. Read and E. H. Rezayi, Phys. Rev. B 84, 085316 53 B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, (2011). 7312 (1993). 88 B. Bradlyn, M. Goldstein, and N. Read, Phys. Rev. B 54 N. Read, Semicond. Sci. Technol. 9, 1859 (1994). 86, 245309 (2012). 55 J. K. Jain, Phys. Rev. Lett. 63, 199 (1989). 89 A. Gromov and A. G. Abanov, Phys. Rev. Lett. 113, 56 V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev. 266802 (2014). B 64, 195109 (2001). 90 B. Bradlyn and N. Read, Phys. Rev. B 91, 125303 (2015). 31

91 N. Read, Phys. Rev. B 79, 045308 (2009). 103 J. A. Hertz, Phys. Rev. B 14, 1165 (1976). 92 C. Hoyos and D. T. Son, Phys. Rev. Lett. 108, 066805 104 A. J. Millis, Phys. Rev. B 48, 7183 (1993). (2012). 105 S. Sachdev, Quantum Phase Transitions (Cambridge Uni- 93 E. Fradkin and F. A. Schaposnik, Physics Letters B 338, versity Press, Cambridge, UK, 1999). 253 (1994). 106 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 94 J. C. Le Guillou, E. Moreno, F. A. Schaposnik, and (2010). C. N´u˜nez, Phys. Lett. B 409, 257 (1997). 107 Y. Schattner, S. Lederer, S. A. Kivelson, and E. Berg, 95 A. Chan, T. L. Hughes, S. Ryu, and E. Fradkin, Phys. “Ising nematic quantum critical point in a metal: a Monte Rev. B 87, 085132 (2013). Carlo study,” (2015), arXiv:1511.03282. 96 X. G. Wen, Adv. Phys. 44, 405 (1995). 108 K. Sun, B. M. Fregoso, M. J. Lawler, and E. Fradkin, 97 G. Sambandamurthy, R. M. Lewis, H. Zhu, Y. P. Chen, Phys. Rev. B 78, 085124 (2008). L. W. Engel, D. C. Tsui, L. N. Pfeiffer, and K. W. West, 109 R. Shankar, Rev. Mod. Phys. 66, 129 (1994). Phys. Rev. Lett. 100, 256801 (2008). 110 J. Polchinski, in Recent directions in particle theory: from 98 M. Mulligan, C. Nayak, and S. Kachru, Phys. Rev. B 84, superstrings and black holes to the standard model (TASI 195124 (2011). - 92), edited by J. Harvey and J. Polchinski, Theoretical 99 F. D. M. Haldane, E. H. Rezayi, and K. Yang, Phys. Rev. Advanced Study Institute in High Elementary Particle Lett. 85, 5396 (2000). Physics (TASI 92), Boulder, Colo., 1-26 Jun, 1992, (World 100 C. J. Halboth and W. Metzner, Phys. Rev. Lett. 85, 5162 Scientific, Singapore, 1993). (2000). 111 M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, 101 W. Metzner, D. Rohe, and S. Andergassen, Phys. Rev. Phys. Rev. B 91, 115111 (2015). Lett. 91, 066402 (2003). 112 S. Raghu, G. Torroba, and H. Wang, Phys. Rev. B 92, 102 I. Khavkine, C.-H. Chung, V. Oganesyan, and H.-Y. Kee, 205104 (2015). Phys. Rev. B 70, 155110 (2004). 113 E. Witten, Commun. Math. Phys. 121, 351 (1989).