Symmetric-Gapped Surface States of Fractional Topological Insulators
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Symmetric-Gapped Surface States of Fractional Topological Insulators
Gil Young Cho,
1, 2
Jeffrey C. Y. Teo, and Eduardo Fradkin School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea Department of Physics, University of Virginia, Charlottesville, VA 22904 USA Department of Physics and Institute for Condensed Matter Theory,University of Illinois, 1110 W. Green St., Urbana, Illinois 61801-3080, USA (Dated: July 23, 2018)We construct the symmetric-gapped surface states of a fractional topological insulator with elec-tromagnetic θ -angle θ em = π and a discrete Z gauge field. They are the proper generalizations ofthe T-pfaffian state and pfaffian/anti-semion state and feature an extended periodicity comparedwith their of “integer” topological band insulators counterparts. We demonstrate that the surfacestates have the correct anomalies associated with time-reversal symmetry and charge conservation. Introduction:
The three-dimensional topologicalband insulator[1–4] is an electronic topological phase.Its discovery embodies the remarkable progresses in ourunderstanding of the interplay between symmetry andtopology of quantum states of matter. Its topologicalnature manifests spectacularly as a single gapless Diracfermion living at its boundary, which is otherwise impos-sible to exist. It has been thought for some time thata single gapless Dirac fermion is the only allowed sur-face state respecting time-reversal and charge conserva-tion symmetries. However, surprisingly, it turns out thatthere is another option [5]: the surface can be gappedwhile respecting the symmetries, at the cost of introduc-ing topological order, resulting in the T-pfaffian state andthe pfaffian/anti-semion state [6–9].In this paper, we will consider the surface of a frac-tional topological insulator (FTI). The fractional topo-logical insulator[10–15] is a symmetry-enriched topologi-cally ordered state of matter in three spatial dimensions,which supports anomalous surface states protected bytime-reversal symmetry and charge conservation. Thesimplest 3D FTI [12] contains fractional excitations suchas gapped charge- fermions and Z gauge fluxes. It ischaracterized by a term in the effective action for the elec-tromagnetic response of the bulk with a fractional axionangle θ em = π [12] L θ = θ em π ε µνλρ F µν F λρ = 196 π ε µνλρ F µν F λρ . (1)This state can be constructed by fractionalizing theelectron into the three charge- fermionic partons, i.e.,Ψ e = ψ ψ ψ , which, at the mean field theory level, is de-scribed by the topological band insulator. The fractional-ization of the physical electron into the multiple fermionicpartons introduces unphysical states in the Hilbert spaceand those states need to be projected out. This pro-jection is done efficiently by introducing a Z gauge fieldand make the partons ψ j carry the unit charge under thisgauge field, i.e., under the gauge transformation, the par-ton transforms as Z : ψ j → ωψ j with ω = 1. On theother hand, the electron is locally gauge-invariant, i.e., Z : Ψ e → Ψ e , as it should be. Here we will asume thatthe Z gauge theory is realized in its deconfined phases[16, 17]. Several works on theoretical constructions of 3DFTIs have been written, and some of the physics of thebulk states is by now reasonably well understood.Compared to the bulk, the surface states of FTIs havebeen less studied and are not well understood, largely be-cause of the strong interactions required for these statesto occur. The surface of a FTI is intrinsically strongly-correlated and thus the fate of the surface Dirac fermions,which result in the mean-field description, is not a pri-ori clear. In the presence of the strong interactions, thereare several scenarios possible to happen. The surface maybreak the symmetries protecting the gapless-ness spon-taneously and be gapped. A more interesting possibil-ity is to have a transition to a phase which is gappedwhile respecting all the symmetries. This phase is thesymmetric-gapped surface state, which lives only on this(3+1)-dimensional state with symmetry-enriched topo-logical order. Such a surface state should realize the sym-metries in an anomalous fashion which cannot be realizedwithin strictly two space dimensions.In this paper we construct a gapped state on thesurface of a 3D FTI. This state is invariant under the Z gauge symmetry and respects global electric chargeconservation and time-reversal invariance. Since it isgapped, this state should be stable against interactionswith moderate strength. This state is the generalizationof the T-pfaffian state of the 3D time-reversal-invarianttopological insulator [8] (see also Ref. [18]) to the moregeneral problem of the surface of a 3D FTI. More pre-cisely, we show that the generalization of the symmetric-gapped surface states of the topological insulator havethe extended periodicity, which are forced by the Z gauge invariance. This extended periodicity makes thesurface of the FTI to have the correct parity anomaly.The symmetries of symmetric surface states of the 3DFTI, at the quantum level, are realized anomalously,which implies that this state can only occur on the sur-face of a 3D systems this the correct bulk anomaly. Theanomaly of the surface that we are mainly concerned inthis paper is a fractional parity anomaly with an asso-ciate surface Hall conductivity σ xy = . This anomalymust either be cancelled by the bulk or by another sur-face state [19–21]. For example, the T-pfaffian state[8]has the same parity anomaly as the single Dirac fermion,i.e., σ xy = [22, 23]. To see this clearly, we note thatthe single Dirac fermion alone is not invariant under largegauge transformations, and we need to regularize the the-ory properly to restore the gauge symmetry at the cost ofbreaking the time-reversal symmetry, i.e., the properly-regularized theory comes along with a half-level of theChern-Simons term, − π ε µνλ A µ ∂ ν A λ , which explicitlybreaks the time-reversal symmetry [22].However, when coupled to the bulk of the topologicalinsulator, time-reversal symmetry at the surface is re-stored [23] by the axion term in the bulk effective electro-magnetic action, L em = π ε µνλρ F µν F λρ , whose bound-ary action cancels the half-level Chern-Simons term gen-erated from the regularization of the Dirac fermion. TheT-pfaffian state also has the same parity anomaly σ xy = which exactly matches this bulk contribution [8]. Inthe fractional topological insulator case, the axion angleEq.(1) is θ em = π implies that the correct boundary stateshould have a parity anomaly with σ xy = . Hence, welook for states with Z gauge symmetry, global electriccharge conservation, and time-reversal symmetry, and aparity anomaly σ xy = .Here we construct such symmetric-gapped states withthe help of the recently-developed fermionic dualities in(2+1) space-time dimensions [24–27]. One of the statesthat we construct is the generalization of T-pfaffian state,that exactly matches the topological order that two of usfound previously in an anyon-theoretic construction [28].Various heterostructures of FTI thin films were consid-ered and constrained the possible structures to derive asymmetric-gapped state. Here, we present a field theo-retic derivation of this state, and construct other classesof the symmetric-gapped states for the FTI. Generalization of the T-pfaffian State:
At thelevel of mean field theory, the surface state of the 3DFTI consists of the three partons, electric charge- Diracfermions L = X j =1 ¯ ψ j i /D A/ ψ j (2)where A is the background electromagnetic gauge field.Note that there are no Chern-Simons terms for the A and the Z gauge fields [29]. As noted above, this theoryis incomplete: the partons must also be coupled to a Z dynamical gauge field to reproduce the correct Hilbertspace. The fluctuations of Z gauge field are gappedin the deconfined phase and we have suppressed theirexplicit contribution to the low-energy effective theory.Nevertheless, the requirement of Z gauge invariance willplay a key role. On the other hand, two of the three Dirac fermions can become massive without breaking any ofthe symmetries of the theory and, generically, we are leftwith only one massless Dirac fermion, whose mean-fieldeffective action is L = ¯ ψi /D A/ ψ (3)We should recall that the Dirac fermion ψ carries theunit Z charge, even though the gauge field is not shownexplicitly in this low-energy theory.Another surface state can be obtained from the dualtheory of Eq.(3). Duality has provided a direct way togap out the Dirac fermion without breaking symmetriesin the topological band insulator case [24, 25], and wewill follow the same strategy here. Upon the dualitytransformation, the dual theory of Eq.(3) becomes L = ¯ χi /D a χ + 112 π ε µνλ a µ ∂ ν A λ (4)This duality is a short-hand representation which is suf-ficient for present purposes [30]. Here we would like tointroduce the gap while preserving the symmetries. Wefirst introduce an s-wave pairing field, i.e., a singlet pair-ing in the spinor index, to χ fermions. Note that herethe pairing is dynamical and originates from the strongcorrelations, contrary to the proximity effect in the Fuand Kane model [31]. Because the χ fermion is explic-itly electrically charge neutral, the s-wave paired state ofEq.(4) respects time-reversal symmetry and charge con-servation. This is the T-pfaffian state of the parton ψ .We review a few facts about the T-pfaffian state,needed for our construction. The effective low energytheory of the T-pfaffian has a charge sector and a neu-tral (Ising) sector [8]. The excitations of the charge sectorare labelled by their vorticity k mod 8, and are charge- k anyon excitations of the filling ν = state of the charge-2boson. The excitations of the Ising sector are the abelianboson I , fermion f , and the non-abelian anyon σ . In theT-pfaffian state, excitations with even vorticity, k = 2 n ,are bound with the abelian anyons I and f of the Isingsector, and excitations with odd vorticity k = 2 n + 1 arebound to the non-abelian anyon σ . The resulting statesare respectively denoted below as I k , f k and σ k . I isa boson braiding trivially with all the other anyons. Itspresence truncates the spectrum of the theory to 12 ex-citations [8, 24]. The main difference from the T-pfaffianstate of the topological band insulator is in the electriccharge carried by the excitations: The vorticity k excita-tion carries the electric charge k instead of k as in thecharge of excitations of the topological band insulator.Without the Z gauge field, this T-pfaffian state ofthe parton could have been a legitimate symmetric sur-face state. However, here we need to be more carefulbecause of the internal Z gauge field. To see this, wefirst identify the Z gauge charge of the excitations. Wefirst assign the Z gauge charge q to the smallest excita-tion, σ . Then, the excitation of the vorticity k carriesthe Z gauge charge k × q mod 3. Since the fermion f has the same quantum numbers as the parton ψ [8, 24],i.e., electric charge- and unit Z charge, we obtain theconstraint 4 q = 1 mod 3 , (5)One solution to Eq.(5) is q = 1. (We will come backbelow to the other solution q = mod 3.) From this, weread how the excitation V k of the vorticity k transformsunder the Z gauge transform, i.e., Z : V k → ω k V k .This has a striking effect on the anyon theory: theT-pfaffian of the parton ψ breaks Z gauge symmetry because the supposedly-‘transparent’ boson I [8] trans-forms non-trivially under the Z gauge transformations,i.e., Z : I → ω I . The boson I is non-local because ithas a non-trivial braiding phase with the Z flux. Hence,the anyon contents can no longer have period 8 if theinternal gauge invariance Z is to be respected. If weenforce the periodicity to be 8, then we need to break Z gauge symmetry completely and remove the Z fluxfrom the excitation spectrum. Physically, the boson I corresponds to the pair field of the fermion ψ , which car-ries charge-2 under Z gauge group and electric charge . Thus, the T-pfaffian of the parton is not compatiblewith the internal Z gauge symmetry.There are two options to restore the Z gauge symme-try to this state. One is simply to remove the pairing inthe χ fermions and to go back to the metallic state ofEq.(3). The other option is to enter into a new topolog-ical state, and this is the direction that we pursue. Thenew topological state features an extended periodicity ofthe anyon contents enforced by Z gauge symmetry.We start with noting that the triple of I , i.e., I ∼ ( I ) , is neutral under the Z gauge field, and, thus, ithas trivial braiding phases with all the anyons, includingthe Z gauge fluxes. Thus, we can truncate the anyoncontents at k = 24 instead of at k = 8. Therefore, the Z gauge symmetry can be restored simply by extendingthe periodicity of the anyon content of the vorticity from8 to 24. For this state, time-reversal symmetry as well ascharge conservation are inherited directly from the “par-ent” T-pfaffian state of the parton ψ . Hence, this staterespects all the required symmetries to be the legitimatesurface state of the fractional topological insulator.We now investigate the consequences of the extendedperiodicity, k ∼ k + 24. In this theory, the topologicalspins and the action of time-reversal symmetry T stillrepeat with period 8. The charges are assigned as follows Q em,k = k , and Z : V k → ω k V k , (6)where V k represents the anyons with vorticity k , with k ∼ k + 24. Also, I k and f k with k ≡ V k , i.e., I and f , carrying electric charge , and are exchanged under the time-reversal symmetry, asin the usual T-pfaffian state. There are two excitationsto which we pay a particular attention. The first is theelectron quasiparticle Ψ, i.e., f , which carries electriccharge 1, is neutral under Z , and has T = −
1. Thesecond is the (singlet) Cooper pair of electrons, which isidentified with I : a boson that has electric charge 2 andis a Kramers singlet.We now come to the parity anomaly. Without referringback to the field theoretic derivation, we can read off theanomaly directly from the anyon content of the theory.This way of reading off the anomaly will be useful whendiscussing the generalization of the pfaffian/anti-semionstate. In the case of the T-pfaffian state of the topo-logical band insulator, the period is 8 and the vacuumis identified with the charge-2 boson I . This impliesthat σ xy = ν × Q = × = , where ν is the in-verse of the periodicity (more precisely, it is the “filling”of the bosonic charge sector, or the level of the charge-sector Chern-Simons term) and Q is the charge of thetransparent boson. Hence, we see that the T-pfaffianstate has the correct parity anomaly σ xy = . Now, forthe surface state of the fractional topological insulator ,the period 24 with charge-2 boson I implies that thesurface has the charge response with Hall conductivity σ xy = ν × Q = × = . This is precisely theexpected parity anomaly of the FTI! A more accuratestatement is that the charge sector of this anyon theoryis U (1) , as was shown explicitly in Ref. [28].It is now clear that the other solution q = to Eq.(5)generates the same anyon content as the q = 1 solutionbecause, in the above analysis, only the Z gauge chargeof I is important. Obviously, the Z gauge charge of I is the same in both the solutions. The fractionaliza-tion of Z gauge charge by q = , which extends the Z gauge theory to the Z gauge theory only at the surface ,does not break the Z gauge symmetry, charge conser-vation, and time-reversal symmetry. Hence, this stateis also another legitimate surface state of the fractionaltopological insulators. The two solutions, q = mod 3and q = 1 mod 3, can be distinguished by the braidingwith the Z flux and the surface excitations because thesurface excitations carry the Z gauge charge and thushave non-trivial statistics with the flux.In our system, the Z gauge theory is not coupled withthe electromagnetic field. For instance, a Z flux doesnot necessarily carry a non-trivial magnetic flux. Fur-thermore, because of the time-reversal symmetry, whenit intersects the symmetric surface, it does not carry anelectric or gauge charge.We now discuss the topological degeneracy on the open3-manifold D × S of this fractional topological insula-tor with the symmetric-gapped topologically ordered sur-face, where D × S is the filled spatial torus. We notethat there are only one non-contractible loop along S along which we have three possible degeneracies labelledby the Z -charge Wilson loop around this S . On theother hand, given a Z charge, there are six possible any-onic loops living purely on the surface of D × S . So,the total degeneracy is 18. Relation with the paired FQH state at filling ν = : It has been conjectured from the link between thehalf-filled composite Fermi liquid and the surface of thetopological band insulator [32, 33] that the particle-holesymmetric version of the T-pfaffian state, the PH-pfaffian[32], can be realized in a half-filled Landau level. Thisstate is essentially equivalent to the T-pfaffian in terms ofsymmetries and excitations but time-reversal symmetryis replaced by the particle-hole symmetry of the half-filled Landau level (in the large cyclotron energy limit).We can ask if our exotic surface state of the FTI canbe realized in a Landau level. From the charge response σ xy = of the surface state, it is natural to comparethis state with a putative paired FQH state at ν = .However, contrary to the half-filled case, we do not havean obvious particle-hole symmetry at ν = . This impliesthat the surface state of the FTI does not have a naturalpartner in a fractionally-filled Landau level. In fact, theexcitations of the paired FQH state at the filling ν = can be generated by tensoring of the charge sector U (1) and the neutral Ising sector, quotiented by an extendedsymmetry. However, as discussed in the work [28], torestore the time-reversal symmetry another neutral Z sector is needed, which is absent in the paired FQH state. Other Symmetric-Gapped Surface States:
Onestablishing the generalization of the T-pfaffian state atthe surface of FTIs, we now address if we can constructthe generalization of another symmetric-gapped state oftopological band insulator, i.e., a pfaffian/anti-semionstate.[6, 8] For this, we note that the essential step in ourgeneralization of the T-pfaffian state is to identify f , theelectron in the topological band insulator case, with theminimal parton ψ carrying unit Z gauge charge. Thisstate breaks the internal Z gauge symmetry which canbe restored by extending the periodicity of the anyoncontent from 8 to 24. This strategy, “extending periodic-ity” to restore the internal gauge symmetry, straightfor-wardly generalizes to the other symmetric-gapped sur-face order, e.g., the pfaffian/anti-semion state. Thisstate [6, 8], which is realized at the surface of the topo-logical band insulator, respects time-reversal symmetryand charge conservation. Its excitations are labelled by { I k , I k s, σ k , σ k s, f k , f k s } with vorticity k ∼ k + 8 (here s is the anti-semion), and carry electric charge k . In thisstate, f is the electron, i.e., a Kramers doublet charge-1fermion, and the singlet Cooper pair I is “transparent”to all the anyons, as in the T-pfaffian state.On the surface of the fractional topological insula-tor, we imagine to put the parton ψ first into thepfaffian/anti-semion state. Temporarily ignoring the Z gauge symmetry, we find a theory respecting all the sym-metries. The only difference is the electric charge carried by the anyons. Now the excitation with vorticity k car-ries electric charge k since the “elementary” excitation ψ has fractional electric charge . Obviously, on bring-ing the Z gauge symmetry back into the discussion, wesee that I , which is supposed to be local in the usualpfaffian/anti-semion state, is no longer local and braidswith the Z fluxes since it carries charge 2 under the Z gauge field. However, we can restore the Z gauge sym-metry by extending the periodicity once again from 8 to24, i.e., k ∼ k + 24. In this state, the charge sector hasperiod 24 and the identity boson carries electric charge2. Hence the parity anomaly associated with this state isagain σ xy = , the correct anomaly to be on the surfaceof the fractional topological insulator. Furthermore, itobeys time-reversal symmetry and charge conservation,inherited from the original pfaffian/anti-semion theory. Conclusions and Outlook:
In this paper, with thehelp of the fermion-fermion duality we constructed thesymmetric-gapped surface states of FTIs with electro-magnetic axion angle θ em = π , whose excitations are thefractional parton and the discrete Z gauge flux. Thesymmetric-gapped surface states are generalizations ofthe T-pfaffian state and the pfaffian/anti-semion statebut with an extended periodicity . We showed that thesurface states respect the required symmetries of chargeconservation, time-reversal symmetry, and the Z gaugesymmetry, and that they have the correct parity anomaly,i.e., σ xy = , which matches the axion angle θ em = π .At the heart of finding these surface states, the identifi-cation of the electron in the symmetric-gapped surfacesof topological band insulator by the non-local fermionicparton plays an essential role. This requirement forcedthe extension the periodicity of the anyon content so as torestore the internal Z gauge symmetry. Here we focusedon the case of FTIs with θ em = π , but it is straightfor-ward to generalize our construction to the other FTIswith angle θ em = π n +1 and a Z n +1 gauge field. We endby noting that structure we used (“extending periodic-ity”) will arise in the construction the symmetric-gappedsurface states for other various bosonic and fermionicFTIs. Generically, we expect that they will inherit theirglobal symmetries from their counterparts in the “inte-ger” bosonic and fermionic topological insulators.We thank F. Burnell, M. Cheng, T. Faulkner, H. Gold-man, M. Metlitski, S. Ryu, N. Seiberg, H. Wang, and P.Ye, for helpful discussions and comments. This work issupported by the Brain Korea 21 PLUS Project of KoreaGovernment (G.Y.C.), Grant No. 2016R1A5A1008184under NRF of Korea (G.Y.C.), NSF Grant No. DMR-1653535 at the University of Virginia (J.C.Y.T.), andNSF grant No. DMR 1408713 at the University of Illi-nois (E.F). G.Y.C. also acknowledges the support fromthe Korea Institute for Advanced Study (KIAS) grantfunded by the Korea government (MSIP). G.Y.C andJ.C.Y.T thank S. Sahoo and A. Sirota for collaborationin a previous work, EF to P. Ye and M. Cheng. [1] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics , 1057 (2011).[3] M. Z. Hasan and J. E. Moore, Annual Review of Con-densed Matter Physics , 55 (2011).[4] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Rev. Mod. Phys. , 035005 (2016).[5] A. Vishwanath and T. Senthil, Physical Review X ,011016 (2013).[6] C. Wang, A. C. Potter, and T. Senthil, Physical ReviewB , 115137 (2013).[7] M. A. Metlitski, C. Kane, and M. P. Fisher, PhysicalReview B , 125111 (2015).[8] X. Chen, L. Fidkowski, and A. Vishwanath, Physical Re-view B , 165132 (2014).[9] P. Bonderson, C. Nayak, and X.-L. Qi, Journal of Statis-tical Mechanics: Theory and Experiment , P09016(2013).[10] J. 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Incontrast to [25], the Dirac fermion action as written heredoes not contain any ‘hidden’ half-level Chern-Simonsterm (or η -invariant term) emerging from the regulariza-tion, because the term is already cancelled by the bulkaxion term Eq.(1).[30] A more correct version of the duality [25] yields the sameanswer. See the section 5 of reference [25] for the details.[31] L. Fu and C. L. Kane, Physical Review Letters ,096407 (2008).[32] D. T. Son, Physical Review X , 031027 (2015).[33] C. Wang and T. Senthil, Physical Review B , 245107(2016). Supplemental Material for “Symmetric-Gapped Surface States of FractionalTopological Insulators”
Gil Young Cho, Jeffrey C. Y. Teo, and Eduardo Fradkin
DIRAC FERMION AND CHERN-SIMONS TERM
In this supplemental material, we clarify the meaning of the Dirac fermion path integral in comparison with Ref.[25]. In Ref. [25], when the Dirac fermion action is written down, L = i ¯Ψ /D A Ψ , (1)it is understood that this action is well-defined theory by a Pauli-Villars regularization. The contribution from theregularization is hidden in the integration measure of the fermionic field Ψ and does not appear explicitly in theaction. The contribution is the η -invariant term, which is essentially equivalent to the half-level Chern-Simons term, − π ǫ µνλ A µ ∂ ν A λ [18, 23]. Hence, the Dirac fermion action appearing in Ref. [25] is not time-reversal symmetric, or,equivalently, is time-reversal symmetric up to the anomaly π ǫ µνλ A µ ∂ ν A λ .In our convention (which is more familiar with condensed matter community, e.g., this is the convention used inRef. [24]), when we write the Dirac fermion action, L = i ¯Ψ /D A Ψ , (2) we do not assume a ‘hidden’ contribution from the Pauli-Villars regularization and this action lives only on the surfaceof a three-dimensional topological band insulator. Hence, this action is time-reversal symmetric. When regularizedproperly, this action must be modified as L reg. = i ¯Ψ /D A Ψ − π ǫ µνλ A µ ∂ ν A λ . (3)The Chern-Simons term for A µ originates from the regularization. To restore the time-reversal symmetry for thisregularized Dirac fermion action, we need to attach the bulk of the topological band insulator whose effective actionis L bulk = 132 π ǫ µνλρ F µν F λρ , (4)which contributes another Chern-Simons term π ǫ µνλ A µ ∂ ν A λ on the surface, which exactly cancels the contributionof the Pauli-Villars regularization fields.Here we note that, in the convention of Ref. [25], the time-reversal symmetric regularized Dirac fermion correspondsto L = i ¯Ψ /D A Ψ + 18 π ǫ µνλ A µ ∂ ν A λ , (5)in which only the Chern-Simons term from the bulk is explicitly written out. This bulk term is explicitly cancelledby the regularization contribution on the surface.Keeping this in mind, we now show that there is no Chern-Simons term for the background electromagnetic gaugefield A µ and dynamical Z gauge field in Eq. (2) of the main text. Here we take the Z gauge field to be obtainedfrom the dynamical U(1) gauge field α µ by condensing the charge-3 scalar field [17]. Hence, we show that there is noChern-Simons terms for α µ and A µ on the surface.We start with the theory in which the charge-3 scalar field does not condense but the partons ψ j form the topologicalband insulator and the U(1) gauge field α µ is in the deconfined Coulomb phase. Then the bulk action for thistopological phase is L = 332 π ǫ µνλρ ( 13 F µν + f µν )( 13 F λρ + f λρ ) , (6)in which f µν = ∂ µ α ν − ∂ ν α µ and F µν is the field strength of A µ . On the surface, this bulk action contributes theChern-Simons term L = 38 π ǫ µνλ ( 13 A µ + α µ ) ∂ ν ( 13 A λ + α λ ) . (7)This contribution, however, is exactly cancelled by the regularization contribution of the Dirac fermions living on thesurface of this topological phase. To see explicitly, we note that the regularized theory of the surface Dirac fermionsis L = X i =1 i ¯ ψ j /D A + α ψ j − π ǫ µνλ ( 13 A µ + α µ ) ∂ ν ( 13 A λ + α λ ) , (8)where the second Chern-Simons term is the contribution from the regularization term, which is exactly the oppositeof the bulk contribution. Hence, on the surface of this topological phase with the bulk contribution and regularizationcontribution, we end up with the following theory L = X i =1 i ¯ ψ j /D A + α ψ j . (9)Now, we condense the charge-3 scalar field, which will break the U(1) gauge symmetry of the field α µ to its discretesubgroup Z . This condensation does not break time-reversal invariance nor changes the topological band structureof the fermionic partons. Thus, we do not expect to have any topological terms to appear on the surface after con-densation. Because the condensation will gap out the fluctuations of the gauge field α µ , we can ignore its fluctuationsat the lowest energies, and find L = X i =1 i ¯ ψ j /D A ψ j , (10)which is the Eq. (2) in the main text. Note that there is no Chern-Simons term for the A and the Z3