Bosonic Analogue of Dirac Composite Fermi Liquid
BBosonic Analogue of Dirac Composite Fermi Liquid
David F. Mross, Jason Alicea,
1, 2 and Olexei I. Motrunich
1, 2 Department of Physics and Institute for Quantum Information and Matter,California Institute of Technology, Pasadena, CA 91125, USA Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
We introduce a particle-hole-symmetric metallic state of bosons in a magnetic field at odd-integer filling. Thisstate hosts composite fermions whose energy dispersion features a quadratic band touching and corresponding π Berry flux protected by particle-hole and discrete rotation symmetries. We also construct an alternativeparticle-hole symmetric state—distinct in the presence of inversion symmetry—without Berry flux. As in theDirac composite Fermi liquid introduced by Son [1], breaking particle-hole symmetry recovers the familiarChern-Simons theory. We discuss realizations of this phase both in 2D and on bosonic topological insulatorsurfaces, as well as signatures in experiments and simulations.
Introduction.
The last year has seen numerous exciting de-velopments in our understanding of electronic quantum-Hallstates that resolved long-standing puzzles regarding particle-hole (PH) symmetry. At filling factor ν = , electrons fillexactly half of the available single-particle orbitals in the low-est Landau level (LLL). Within that subspace the system en-joys PH symmetry that is conspicuously absent in the clas-sic Halperin-Lee-Read (HLR) theory [2–8]. There, compos-ite fermions , obtained by attaching two fictitious flux quantato electrons that cancel the applied field on average, fill aparabolic band and form a Fermi surface—i.e., a compositeFermi liquid (CFL). The corresponding Lagrangian densityreads L CS = f † (cid:34) i ( D − iA ) − ( (cid:126)D − i (cid:126)A ) m ∗ (cid:35) f − k π (cid:15) κµν a κ ∂ µ a ν , (1)where f is the composite fermion field, A µ (with µ = 0 , , )is the electromagnetic vector potential, D µ = ∇ µ − ia µ de-notes the covariant derivative with a µ an emergent gauge field,and k = 1 is the level of the Chern-Simons term that attachesflux. Despite the absence of PH symmetry, HLR theory is re-markably successful in predicting experimental results at andaround ν = .To incorporate PH symmetry, Son proposed that compos-ite fermions are Dirac particles at finite density coupled to anemergent gauge field [1] without a Chern-Simons term: L QED = i ¯Ψ D µ γ µ Ψ − k π (cid:15) κµν A κ ∂ µ a ν . (2)Here Ψ and ¯Ψ = Ψ † γ are two-component spinors while γ µ are Dirac matrices. Equation (2) implements two importantfeatures of the half-filled Landau level: ( i ) the Dirac com-posite fermions are neutral under the external vector potential A µ [9–12] and ( ii ) the theory preserves the anti-unitary PHtransformation C Ψ C − = iσ Ψ , C ( a , (cid:126)a ) C − = ( a , − (cid:126)a ) . (3)Several subsequent works support Son’s theory and the pres-ence of PH symmetry at ν = [13–20]. These developments prompt us to revisit the CFL formedby bosons at filling factor ν = 1 , where flux attachment yieldsEq. (1) with k = 2 . An important conceptual difference be-tween bosonic quantum-Hall states and their fermionic coun-terparts is the absence of a single-particle description of in-teger quantum-Hall (IQH) states, which in turn obscures aprecise definition of PH symmetry even when restricting tothe LLL. To access PH-symmetric CFLs of ν = 1 bosons,we therefore follow a two-pronged approach: First, we studybosons at a ‘plateau transition’ [21] between a ν = 2 IQHstate [22–24] and the vacuum. Upon fine-tuning, the criticaltheory exhibits a PH symmetry analogous to the electroniccase, in addition to microscopic inversion symmetry. Second,we consider the surface of a particular 3D symmetry protectedtopological phase (SPT) of bosons [25], where both symme-tries can be realized microscopically.These methods suggest a natural bosonic analogue ofEq. (2) given by L CFL π = Ψ † (cid:32) iD ( D + iD ) ( D − iD ) iD (cid:33) Ψ − π (cid:15) κµν A κ ∂ µ a ν . (4)Note that the composite-fermion density is dynamically fixedto n CF = ( ∂ A − ∂ A ) / π . The first line of Eq. (4) alsodescribes electronic excitations of bilayer graphene near oneof the two valleys [26]. In the present context, the compositefermions analogously exhibit a single quadratic band touch-ing that is protected against weak perturbations that respectboth PH symmetry C and fourfold rotation symmetry R ( π/ (see Fig. 1). When only inversion symmetry I is present,the spectrum remains gapless but the band touching generi-cally splits into two Dirac cones, similar to the effect of trig-onal warping in bilayer graphene [26]. At suitable doping,either case features a single Fermi surface enclosing π Berryflux. To distinguish different kinds of PH-symmetric CFLs weadopt notation CFL kπ , where kπ is the Berry flux enclosedin the composite-fermion Fermi surface. Thus, the fermionic ν = state described by L QED corresponds to CFL π whilethe bosonic state described by L CFL π is CFL π . As we willsee, an alternative C - and I -symmetric state for bosons, CFL , a r X i v : . [ c ond - m a t . s t r- e l ] M a y CFL CFL CFL CFL CFL CFL distinct distinct not distinct (a) Rotation symmetric (c) No spatial symmetries(b) Inversion symmetric FIG. 1. Composite-fermion band structures for CFL π and CFL states with particle-hole symmetry and varying spatial symmetries. Hor-izontal planes indicate the chemical potentials of interest. (a) With four-fold rotation symmetry CFL π exhibits a protected quadratic bandtouching, in sharp contrast to the gapped spectrum of CFL . (b) Breaking rotation down to inversion symmetry allows the band touching tosplit into two protected Dirac cones but preserves the π Berry flux. (c) Lifting inversion symmetry generically produces a gapped spectrumwith non-zero Berry curvature indicated by color (blue and red for opposite sign). The sharp distinction between CFL π and CFL thendisappears. Away from degeneracy points, particle-hole and inversion symmetries respectively constrain the Berry curvature B at momentum k as B ( k ) = −B ( − k ) and B ( k ) = B ( − k ) ; the presence of both symmetries [i.e., cases (a) and (b)] thus implies zero Berry curvature. with zero Berry flux is also possible. For any CFL kπ , weakPH symmetry breaking yields a gapped spectrum, and subse-quently integrating out the negative energy states generates alevel- k Chern-Simons term as in L CS (i.e., no Chern-Simonsterm for k = 0 ). Absent inversion symmetry, the sharp dis-tinction between CFL π and CFL disappears. Bosons at ν = 1 as a plateau transition. Consider a sys-tem composed of narrow strips of width d along the y direc-tion and infinite along x ; see Fig. 2(a). The boson density ρ for adjacent strips alternates between ρ and , and a uniformperpendicular magnetic field B = chρ e yields filling factor ν = chρeB = 2 for the ρ strips. At length scales much largerthan d we thus obtain bosons with average filling ν = 1 . QSH 𝑦𝑦 − QSH 𝑦𝑦 QSH 𝑦𝑦 − b) 𝑎𝑎 𝑎𝑎 † 𝑎𝑎 ~ 𝕀𝕀 𝑑𝑑 𝑑𝑑 † 𝑑𝑑𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 σ 𝑥𝑥𝑥𝑥 = ⁄ 𝑒𝑒 ℎσ 𝑥𝑥𝑥𝑥 = 0 σ 𝑥𝑥𝑥𝑥 = 0 σ 𝑥𝑥𝑥𝑥 = ⁄ 𝑒𝑒 ℎσ 𝑥𝑥𝑥𝑥 = ⁄ 𝑒𝑒 ℎ Boson IQHBoson IQHBoson IQH
3D Boson SPT2D Plateau Transition σ 𝑥𝑥𝑥𝑥 = ⁄ + 𝑒𝑒 ℎσ 𝑥𝑥𝑥𝑥 = ⁄ + 𝑒𝑒 ℎσ 𝑥𝑥𝑥𝑥 = ⁄ + 𝑒𝑒 ℎσ 𝑥𝑥𝑥𝑥 = ⁄−𝑒𝑒 ℎσ 𝑥𝑥𝑥𝑥 = ⁄−𝑒𝑒 ℎ (a) N=2 QED (a) (b) 𝑦𝑦𝑦𝑦 + 1 𝑦𝑦 + 3 𝑦𝑦 + 5 𝑦𝑦 + 2 𝑦𝑦 + 4 FIG. 2. (a) Alternating strips of bosons at ν = 2 and ν = 0 cor-respond to an average filling factor ν = 1 . Edges contain a chiralcharge mode and a counter-propagating neutral mode, described bythe K-matrix σ x . (b) The same edge-state network arises on the sur-face of a 3D bosonic SPT with U (1) × C symmetry when the antiu-nitary PH-symmetry C is broken oppositely for neighboring strips. We require the ν = 2 strips to form bosonic IQH states [22–24]. It is useful to view a given ν = 2 strip as composedof quantum wires [27] labeled by j = 1 , . . . , N (not to beconfused with the domain-wall labels y in Fig. 2). Each wirehosts charge- e bosons ∼ e iϕ j described by L wire = ∂ x θ j π ( ∂ t ϕ j − A ) + v π [( ∂ x ϕ j − A ) + ( ∂ x θ j ) ] where ∂ x θ j π is conjugate to ϕ j . When a boson hops betweenneighboring wires at non-zero magnetic field—convenientlytaken in the gauge A = Bx —it acquires an Aharonov-Bohmphase e i πedBhc x that prevents condensate formation. These os-cillating phases can be compensated, however, when a phaseslip ∼ e iθ + i πdρ x accompanies boson hopping. For ν = 2 this occurs for second-neighbor hopping described by L boson IQH = g IQH (cid:88) j cos ( ϕ j +1 − ϕ j − + 2 θ j ) . (5)A bosonic IQH state with σ xy = 2 e /h [22–24] emergeswhen g IQH flows to strong coupling. The ν = 2 strip thenhosts edge states with two flavors α = ± of charge- e bosons b y =1(2) ,α ∼ e iφ y =1(2) ,α at the lower (upper) edge, where φ y =1 , + ≡ ϕ , φ y =1 , − ≡ ϕ + 2 θ ,φ y =2 , + ≡ ϕ N − − θ N , φ y =2 , − ≡ ϕ N . (6)The Lagrangian density for the lower and upper edges is suc-cinctly written as L edge = ( − y K αα (cid:48) π ∂ x φ y,α ∂ t φ y,α (cid:48) + u π ( ∂ x φ y,α ) . (7)Here K = σ x and α, α (cid:48) are implicitly summed here and be-low. Equation (7) generalizes straightforwardly to the full 2Dsystem in Fig. 2(a) when edges are enumerated by integers y .To access 2D CFL’s we allow tunneling between neighbor-ing edges. For example, flavor-conserving tunnelings read L hop = w y,α (cid:16) e i heBhc dNx b † y +1 ,α b y,α + c.c. (cid:17) , (8)i.e., the edge bosons b y,α experience a uniform magnetic field.Our setup preserves a microscopic inversion symmetry, I b y,α ( x ) I − = b − y, − α ( − x ) (inversion) , (9)[see Eqs. (6)] that constrains the hopping amplitudes via w y,α = w − y, − α . In the case of translation invariance y → y +2 , only w y = even (hopping across vacuum) and w y = odd (hop-ping across IQH strips) are independent. When these are fine-tuned to be equal, the low energy theory L hop + L edge addi-tionally exhibits an emergent anti-unitary PH symmetry, C b y,α C − = b † y +1 ,α (particle-hole) . (10)
3D boson SPT surface.
Closely related physics can ap-pear at the surface of a 3D bosonic SPT [28] with a conservedcharge that is odd under a local anti-unitary PH symmetry,i.e., U (1) × C local [29] [30]. This symmetry pins the chem-ical potential associated with the conserved charge to zerobut permits an orbital magnetic field. Breaking C local gener-ates a gapped, unfractionalized surface with Hall conductance σ xy = ± e /h . At boundaries between domains with oppo-sitely broken C local , the Hall conductance changes by e /h —implying the existence of gapless edge states described byEq. (7). Consider now a situation where the SPT surface hostsalternating ± e /h strips of equal width [Fig. 2(b)]. Such asurface breaks C local but retains this symmetry when com-posed with a translation T y by one strip width, i.e., Eq. (10)with C = C local T y . Note that unlike the 2D plateau transition, C is a true microscopic symmetry on the SPT surface. Bosonic CFL’s.
Both the plateau transition and the SPTsurface realization lead us to study the theory L edge + L hop .Our analysis is facilitated by an explicit duality mapping fornetwork models of this kind [18] relating the surface statesof 3D topological insulators to quantum electrodynamics in (2 + 1) dimensions (QED ) described by L QED (see alsoRefs. 13, 31–33). The density of dual composite fermions isproportional to the physical magnetic field B , while the num-ber of flavors N f depends on the statistics of the microscopicparticles forming the 3D TI:3D electron TI surface ↔ N f = 1 QED ,
3D boson TI surface ↔ N f = 2 QED . Either bosonic setup from Fig. 2 thus maps to L QED in Eq. (2)with k = 2 and N f = 2 fermion flavors.We will package the two flavors into a single four-component spinor Ψ and use Dirac matrices γ = τ σ z , γ = iτ σ y , γ = − iτ σ x , where σ µ and τ µ are respectivelyintra- and inter-flavor Pauli matrices. Following the mappingfrom Ref. 18, the symmetries in Eqs. (9) and (10) act as [34] C Ψ C − = σ y τ y Ψ , I Ψ I − = σ z τ z Ψ . (11)The continuum dual QED theory also preserves continuousrotations R (Φ)Ψ ( (cid:126)x ) R − (Φ) = e i Φ2 ( τ z + σ z ) Ψ [ R (Φ) (cid:126)x ] (12) (b) CFL (d) CFL 𝑘𝑘 (a) 𝑁𝑁 𝑓𝑓 =2 QED 𝐸𝐸 𝑘𝑘 𝑘𝑘 𝑘𝑘 (c) transition µ CF FIG. 3. (a) Massless Dirac band structure for ‘pure’ N f = 2 QED that is dual to the bosonic models studied here (shown withdifferent velocities to emphasize N f = 2 ). Magnetic field for thebosons maps to a non-zero composite fermion density that yieldstwo Fermi surfaces. (b) PH and rotation-symmetric perturbationswith | g | > | ∆ | yield the CFL π with a quadratic band touching anda single Fermi surface. (c) The transition between CFL π and CFL occurs at | g | = | ∆ | where three bands meet at k = 0 . (d) For | g | < | ∆ | one obtains CFL featuring a gapped spectrum and a sin-gle Fermi surface at appropriate density. Note that the nature of thepartially filled positive-energy band changes qualitatively, with Berryflux jumping from π to . with R ( π ) = I . While R is not a microscopic symmetryof the network-model, we expect it to be relevant for CFLrealizations in isotropic systems.We thus first analyze a composite-fermion Hamiltonian H R = Ψ † h R Ψ containing general momentum-independentbilinears preserving both C and R ( π/ [35], h R = iσ x D + iσ y D − ∆ τ z σ z + g ( τ x σ x + τ y σ y ) . (13)At mean-field level (neglecting a µ ) the spectrum contains fourbands which we label as positive and negative according totheir large- k asymptotics, i.e., E positive ( (cid:126)k ) = ± g + (cid:113) (cid:126)k + ( g ± ∆) , (14) E negative ( (cid:126)k ) = ± g − (cid:113) (cid:126)k + ( g ± ∆) . (15)Several distinct regimes are accessible depending on g, ∆ assketched in Figs. 3(a)-(d):(a) For g = ∆ = 0 we recover two massless Dirac cones.(b) At finite | g | > | ∆ | a quadratic band-touching emerges;the CFL π state then appears when the chemical potential in-tersects only one of the central bands as in Fig. 3(b). In thiscase one can project onto states close to the band touching(see Appendix), yielding Eq. (4) with two-component spinors Ψ that transform as C Ψ C − = σ x Ψ , R (Φ)Ψ R − (Φ) = e i Φ σ z Ψ . (16)The only perturbation allowed by R ( π/ up to O ( k ) is themass term Ψ † σ z Ψ —which is odd under C . Thus the CFL π with quadratic band touching is stable with these symmetries.Relaxing R ( π/ → R ( π ) = I allows the terms Ψ † σ x,y Ψ ,which split the band touching into two Dirac cones with-out opening a gap [see Fig. 1(b)]. Upon breaking inversion,the PH-symmetric terms Ψ † σ z i∂ , Ψ gap out the two Diraccones [Fig. 1(c)].(c) At | g | = | ∆ | the spectrum hosts a three-fold band touch-ing.(d) For | g | < | ∆ | a gap opens and the conduction band‘detaches’ from the valence bands (see Appendix for details).The special point (c) thus marks the transition at which thetopological winding associated with the quadratic band touch-ing transfers to the bottommost bands (for ∆ > ; with ∆ < the band order reverses). Integrating out these filled negative-energy bands does not generate a Chern-Simons term. At suit-able doping one thus obtains a single Fermi surface with nei-ther a Chern-Simons term nor Berry curvature, correspondingto CFL .We emphasize that the distinction between CFL π andCFL requires both PH and inversion symmetries, sincebreaking either generically produces a gapped spectrum.When PH is broken, generic Fermi surfaces enclose a non-universal non-quantized Berry flux (Fig. 4). On the otherhand, breaking inversion symmetry while preserving PH al-ways yields zero enclosed Berry flux [Fig. 1(c)].
Properties.
It is instructive to analyze how both CFL π andCFL reduce to the conventional HLR state upon breaking PHsymmetry. A useful quantity in this context is the composite-fermion Hall conductance ˜ σ xy = k − γ π , where k ∈ Z is thelevel of the Chern-Simons term for the emergent gauge fieldand γ ∈ R is the Berry flux enclosed in the Fermi surface thatyields an anomalous Hall effect [1]. In the bosonic HLR state[Eq. (1)] k = 2 and γ = 0 so that ˜ σ xy = 1 . PH symmetry,however, demands ˜ σ xy = 0 in both CFL π and CFL . Forthe former, breaking PH symmetry via m Ψ † σ z Ψ splits thebands [Fig. 4(a)], whereupon integrating out the negative en-ergy states generates a k = 2 Chern-Simons term. In contrast,weakly breaking PH symmetry in CFL does not produce aChern-Simons term but induces non-zero Berry curvature inthe partially filled band; see Fig. 4(c) and the Appendix fordetails. The two cases can be summarized as: k CFL π = 2 , γ CFL π = 2 π − g − ∆) m (cid:113) (2( g − ∆) m ) + K F ,k CFL = 0 , γ CFL = − π mK F g ∆ + O (cid:18) K F ∆ , m g (cid:19) , (17)where K F is the Fermi wavevector. As Fig. 4(b) illustrates,when m → ∞ we asymptotically recover ˜ σ xy = 1 startingfrom CFL π or CFL , consistent with the HLR result. While ˜ σ xy is not directly observable, Ref. 1 suggested that it may bedetermined from detailed Hall-effect measurements; Ref. 36suggested the Nernst effect as a more sensitive ˜ σ xy probe.A useful device for determining the presence of PH sym-metry in numerical studies is via K F oscillations in thecomposite-fermion density ˜ ρ . In the electronic case, ˜ ρ k ≈ K F is PH-odd and thus generically contributes to the physicalcharge density [15]. In the bosonic CFL π and CFL , by con-trast, ˜ ρ k ≈ K F is PH-even and does not contribute to the bosondensity ρ . Any K F oscillations in the boson density thus CFL CFL �σ 𝑥𝑥𝑥𝑥 = 0 �σ 𝑥𝑥𝑥𝑥 = 1 1 ⁄Δ 𝑔𝑔 m (a) (c)(b) FIG. 4. Both CFL π and CFL exhibit vanishing composite-fermionHall conductance ˜ σ xy = 0 due to PH symmetry C . Upon breaking C by the term m Ψ † σ z Ψ , ˜ σ xy = 0 crosses over to the HLR value ˜ σ xy = 1 (center). For ∆ /g < the CFL π band touching is liftedby g , resulting in bands with opposite Berry-curvature (left). For ∆ /g > the partially filled CFL band develops Berry curvature asa function of m and changes sign between small and large momenta(right). directly probe PH-symmetry breaking.The distinction between CFL π and CFL in the presenceof both I and C is more subtle; operators constructed fromcomposite fermions near the Fermi surface do not distinguishbetween the two. Still, the two clearly differ in the limit oflow composite-fermion density K F (cid:28) ∆ , g in the same waythat bilayer graphene is distinct from 2D electron gases withparabolic dispersion [37] (see also the Appendix). Gapped phases.
It is well known that Cooper-pairing com-posite fermions generates a quantum-Hall insulator of the mi-croscopic particles. As with the HLR theory, it is natural inCFL π or CFL to consider chiral, odd-angular-momentumpairing between composite fermions—which permits a fullgap for spinless fermions. Such states always (spontaneously)break C [29]. An alternative gapped phase arose in the studyof time-reversal-symmetric surfaces of 3D bosonic SPTs inRef. 28. In our network-model this phase arises from the edge-boson interaction L σ x = (cid:88) α,y u α cos ( φ y − ,α − φ y,α + φ y +1 ,α ) , (18)where u + (cid:54) = u − results in C -symmetric—but not I -symmetric—topological order with K = 2 σ x [38]. ( I in-terchanges u + and u − , with the gap closing when u + = u − .)A fully symmetric gapped state is nevertheless readily con-structed as a composite-fermion superconductor driven by L Pair ∼ Ψ † σ y τ x Ψ † + H.c.. We expect that this state corre-sponds to a ‘larger’ topological order that can be reduced to K = 2 σ x by condensing an I -odd boson. Edge-boson inter-actions that generate such a state may be obtained followingRefs. 18, 38, and 39, but that is not our focus. We simply notethat in CFL π and CFL with a single Fermi surface, L σ x and L Pair are both absent in the projected Hilbert space. Access-ing either C -symmetric gapped state requires a finite couplingstrength. Conversely, any gapped state emerging from a weak-coupling instability of CFL π or CFL necessarily breaks C . Conclusions.
We constructed two PH-symmetric metal-lic states, dubbed CFL π and CFL , for bosons at ν = 1 .These phases are distinct provided PH and inversion symme-tries are present. In either case, K F oscillations in the phys-ical boson density are absent but appear when PH symme-try is broken. Furthermore, once PH symmetry is (weakly)broken the crossover between these states and conventionalHLR theory may be observed in transport measurements. Wealso elucidated the relationship between PH-symmetric CFLsand gapped quantum Hall states, such as the bosonic Moore-Read state which breaks PH symmetry, and the K = 2 σ x statewhich does not.A recent study by Wang and Senthil [29] considers bosonsat ν = 1 in the LLL with PH symmetry and proposes a CFLwith Berry phase − π . We believe that the states introducedhere are closely related; we emphasize however that inversionsymmetry is crucial in our setup to define a Berry phase of π . Acknowledgments.
We gratefully acknowledge T. Senthil,C. Kane, M. Metlitski, and D. T. Son for valuable discussions.This work was supported by the NSF through grants DMR-1341822 (JA) and DMR-1206096 (OIM); the Caltech Insti-tute for Quantum Information and Matter, an NSF PhysicsFrontiers Center with support of the Gordon and Betty MooreFoundation; and the Walter Burke Institute for TheoreticalPhysics at Caltech. [1] D. T. Son, Phys. Rev. X , 031027 (2015).[2] B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B , 7312(1993).[3] R. L. Willett, M. A. Paalanen, R. R. Ruel, K. W. West, L. N.Pfeiffer, and D. J. Bishop, Phys. Rev. Lett. , 112 (1990).[4] W. Kang, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, andK. W. West, Phys. Rev. Lett. , 3850 (1993).[5] V. J. Goldman, B. Su, and J. K. Jain, Phys. Rev. Lett. , 2065(1994).[6] S. D. Sarma and A. Pinczuk, Perspectives in quantum Hall ef-fects : novel quantum liquids in low-dimensional semiconduc-tor structures (Wiley, Chichester, 1996).[7] R. L. Willett, Advances in Physics , 447 (1997),http://dx.doi.org/10.1080/00018739700101528.[8] J. K. Jain, Composite Fermions (Cambridge University Press,Cambridge, 2007).[9] N. Read, Semiconductor Science and Technology , 1859(1994).[10] R. Shankar and G. Murthy, Phys. Rev. Lett. , 4437 (1997).[11] D.-H. Lee, Phys. Rev. Lett. , 4745 (1998).[12] V. Pasquier and F. Haldane, Nuclear Physics B , 719 (1998).[13] M. A. Metlitski and A. Vishwanath, “Particle-vortex duality of2d Dirac fermion from electric-magnetic duality of 3d topolog-ical insulators,” (2015), unpublished, arXiv:1505.05142 [cond-mat.str-el].[14] C. Wang and T. Senthil, Phys. Rev. B , 085110 (2016).[15] S. D. Geraedts, M. P. Zaletel, R. S. K. Mong, M. A. Metlitski,A. Vishwanath, and O. I. Motrunich, Science , 197 (2016).[16] G. Murthy and R. Shankar, Phys. Rev. B , 085405 (2016).[17] S. Kachru, M. Mulligan, G. Torroba, and H. Wang, Phys. Rev.B , 235105 (2015). [18] D. F. Mross, J. Alicea, and O. I. Motrunich, ArXiv e-prints(2015), arXiv:1510.08455 [cond-mat.str-el].[19] M. Mulligan, S. Raghu, and M. P. A. Fisher, ArXiv e-prints(2016), arXiv:1603.05656 [cond-mat.str-el].[20] A. C. Balram and J. K. Jain, ArXiv e-prints (2016),arXiv:1604.03911 [cond-mat.str-el].[21] Technically, the model studied here requires disorder or a su-perlattice to describe a plateau transition.[22] T. Senthil and M. Levin, Phys. Rev. Lett. , 046801 (2013).[23] Y.-M. Lu and A. Vishwanath, Phys. Rev. B , 125119 (2012).[24] S. D. Geraedts and O. I. Motrunich, Annals of Physics , 288(2013).[25] A. Vishwanath and T. Senthil, Phys. Rev. X , 011016 (2013).[26] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,and A. K. Geim, Rev. Mod. Phys. , 109 (2009).[27] J. C. Y. Teo and C. L. Kane, Phys. Rev. B , 085101 (2014).[28] A. Vishwanath and T. Senthil, Phys. Rev. X (2013),10.1103/PhysRevX.3.011016.[29] C. Wang and T. Senthil, ArXiv e-prints (2016),arXiv:1604.06807 [cond-mat.str-el].[30] One can alternatively view PH as a time-reversal symmetry, asdone earlier in Ref. 28 where this SPT was labeled U (1) × Z T .[31] C. Wang and T. Senthil, Phys. Rev. X , 041031 (2015).[32] C. Wang and T. Senthil, Phys. Rev. X , 011034 (2016).[33] C. Xu and Y.-Z. You, Phys. Rev. B , 220416 (2015).[34] We performed a change of basis relative to Ref. 18 that resultsin τ z σ z → τ x σ x .[35] An additional allowed term g (cid:48) ( τ x σ y − τ y σ x ) can be rotatedonto g , see Appendix.[36] A. C. Potter, M. Serbyn, and A. Vishwanath, ArXiv e-prints(2015), arXiv:1512.06852 [cond-mat.str-el].[37] K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I.Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim,Nature Physics , 177 (2006).[38] D. F. Mross, A. Essin, and J. Alicea, Phys. Rev. X , 011011(2015).[39] D. F. Mross, A. Essin, J. Alicea, and A. Stern, Phys. Rev. Lett. , 036803 (2016). General composite-fermion band structure
In the absence of any symmetry, the most gen-eral momentum-independent bilinear perturbations to thecomposite-fermion Hamiltonian H = Ψ † ( σ x k + σ y k )Ψ (19)are given by δH = (cid:88) α,β =0 ,x,y,z c αβ ˆ h αβ , (20) ˆ h αβ ≡ Ψ † σ α τ β Ψ , (21)where c αβ are real constants. We now discuss these termsaccording to their symmetries:1. Particle-hole symmetry broken.
The six terms ˆ h i and ˆ h i with i = x, y, z , are odd under particle-holesymmetry. Among these, ˆ h x/y, and ˆ h ,x/y break ro-tation symmetry; the former moves the Dirac cones inmomentum while the latter splits them in energy. Theterms ˆ h z and ˆ h z are rotation symmetric; the formeropens a gap while the latter is a chemical potential withopposite sign for the two cones.2. Particle-hole and rotation symmetries.
The combina-tion of particle-hole and (fourfold) rotation symmetryallows only the terms ˆ h zz , ˆ h xx + ˆ h yy , and ˆ h xy − ˆ h yx .The latter two can be turned into each other using a uni-tary transformation e iθτ z which commutes with C and R (Φ) . One may therfore without loss of generality re-strict the analysis to ˆ h zz and ˆ h xx + ˆ h yy only, as we didin the main text. Depending on the relative magnitudeof ˆ h zz and ˆ h xx + ˆ h yy one either finds a quadratic bandtouching (CFL π ), or a gapped spectrum (CFL ).3. Particle-hole and inversion symmetries.
When therotation symmetry is broken down to twofold rota-tions (i.e., inversion), additional terms ˆ h xx − ˆ h yy and ˆ h xy +ˆ h yx are allowed. In the CFL π regime, their effectis to split the quadratic band touching into two Diraccones at different momenta. Inversion and particle-holesymmetries map these cones onto one another and pro-tect them from opening a gap.4. Particle-hole symmetry only.
When particle-hole isthe only symmetry present, ˆ h zx , ˆ h zy , ˆ h xz and ˆ h yz arealso allowed and generically give rise to a gapped spec-trum. Diagonalization of C - and R ( π/ -symmetricHamiltonian We consider the Hamiltonian H R = Ψ † h R Ψ with h R ( (cid:126)k ) = − ∆ ke − iφ k ke iφ k ∆ 2 g
00 2 g ∆ ke − iφ k ke iφ k − ∆ , (22)where ke iφ k = k + ik . The eigenvalues of h R ( (cid:126)k ) are E ( k ) = (cid:113) k + g + g,E ( k ) = (cid:113) k + g − − g,E ( k ) = − (cid:113) k + g + g,E ( k ) = − (cid:113) k + g − − g, where g ± ≡ g ± ∆ and E > E > E > E for g > . Thecorresponding normalized eigenvectors are given by u / ( (cid:126)k ) = 1 (cid:114) g + ± √ k + g ) k e − iφ k g + ± √ k + g kg + ± √ k + g k e iφ k , (23) u / ( (cid:126)k ) = 1 (cid:114) g − ∓ √ k + g − ) k − e − iφ k g − ∓ √ k + g − k − g − ∓ √ k + g − k e iφ k . (24) Evolution of bands from CFL π to CFL Figure 5 shows the composite-fermion band structure for g = cos α , ∆ = sin α over a range of α . (i)-(iii) As α in-creases from zero, the curvatures of the positive and negativeenergy bands that meet at the quadratic band touching becomeunequal. (iv) The transition between CFL π and CFL occursat | ∆ | = | g | where three bands meet at one point. (v-viii) For | ∆ | > | g | the spectrum is gapped, and the positive (negative)energy bands become degenerate at g = 0 . Regarding π Berry flux
One may be tempted to argue that π Berry phases are notmeaningfully distinct from zero Berry phases. Indeed, in a (ii) 𝛼𝛼 = 𝜋𝜋10 (i) 𝛼𝛼 = 0 (iii) 𝛼𝛼 = (iv) 𝛼𝛼 = 𝜋𝜋4 (vi) 𝛼𝛼 = (v) 𝛼𝛼 = (vii) 𝛼𝛼 = . (viii) 𝛼𝛼 = 𝜋𝜋2 𝐸𝐸𝐸𝐸 µ CF µ CF FIG. 5. Composite-fermion band structure shown for parameters g =cos α , ∆ = sin α . rotationally symmetric system the Berry phase may be com-puted as γ Berry = − i (cid:90) π dθ (cid:104) u | ∂ θ | u (cid:105) , (25)and a gauge transformation | u (cid:105) → e inθ | u (cid:105) (26)changes γ Berry by πn . In this section we explain the sharpdistinction between CFL π and CFL in the presence of in-version symmetry.First, we note that the above ambiguity only arises in thecase of degeneracies; non-degenerate bands always feature awell-defined (smooth) local Berry curvature which can be in-tegrated to a well-defined finite value for γ Berry ∈ R withoutany compactification. For this reason we refer to γ Berry hereand in the main text as ‘Berry flux’. In Eq. (25), the gaugetransformation, Eq. (26), is singular at the origin and is thusnot meaningful for a non-degenerate band. A useful, physi-cal way to resolve this question in the present context is thusto infinitesimally break particle-hole symmetry, while keep-ing inversion symmetry intact. This procedure results in ± π Berry flux in the partially filled positive energy band in theCFL π regime and zero in the CFL regime; cf. Eq. (17) inthe main text.Alternatively, one may sharply distinguish between CFL π and CFL band structures via a pseudospin ˆ n with ˆ n = (cid:32) ˆ h xx − ˆ h yy ˆ h xy + ˆ h yx (cid:33) , (27)where ˆ h αβ are defined in Eq. (21). Evaluating ˆ n for the eigen-states | u j ( (cid:126)k ) (cid:105) of the C - and R ( π/ -invariant Hamiltonian, Eq. (22), we find (cid:104) u j ( (cid:126)k ) | ˆ n | u j ( (cid:126)k ) (cid:105) = f j (cid:16) (cid:126)k (cid:17) (cid:32) cos 2 φ k sin 2 φ k (cid:33) . (28)In the CFL π regime, g > | ∆ | , lim (cid:126)k → ( f , f , f , f ) = (0 , − , , , (29)while in the CFL regime, ∆ > g > , lim (cid:126)k → ( f , f , f , f ) = (0 , , , − . (30)Consequently, the partially filled band, j = 2 , features a wind-ing of the pseudospin in CFL π that is absent in CFL . Projection onto quadratically touching bands