Bosonic topological insulator in three dimensions and the statistical Witten effect
BBosonic topological insulator in three dimensions and thestatistical Witten effect
Max A. Metlitski, C. L. Kane, and Matthew P. A. Fisher Kavli Institute for Theoretical Physics, UC Santa Barbara, CA 93106 Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104 Physics Department, UC Santa Barbara, CA 93106 (Dated: February 27, 2013)
Abstract
It is well-known that one signature of the three-dimensional electron topological insulator is theWitten effect: if the system is coupled to a compact electromagnetic gauge field, a monopole inthe bulk acquires a half-odd-integer polarization charge. In the present work, we propose a corre-sponding phenomenon for the topological insulator of bosons in 3d protected by particle numberconservation and time-reversal symmetry. We claim that although a monopole inside a topologicalinsulator of bosons can remain electrically neutral, its statistics are transmuted from bosonic tofermionic. We demonstrate that this “statistical Witten effect” directly implies that if the surfaceof the topological insulator is neither gapless, nor spontaneously breaks the symmetry, it necessarilysupports an intrinsic two-dimensional topological order. Moreover, the surface properties cannotbe fully realized in a purely 2d system. We also confirm that the surface phases of the bosonictopological insulator proposed by Vishwanath and Senthil (arXiv:1209.3058) provide a consistenttermination of a bulk exhibiting the statistical Witten effect. In a companion paper, we will providean explicit field-theoretic, lattice-regularized, construction of the 3d topological insulator of bosons,employing a parton decomposition and subsequent condensation of parton-monopole composites. a r X i v : . [ c ond - m a t . s t r- e l ] F e b . INTRODUCTION The discovery of electronic topological insulators has fueled a growing interest in so-called symmetry-protected topological (SPT) phases of matter. These are phases with a fullygapped bulk spectrum, whose stability, as their name suggests, is guaranteed by a globalsymmetry. When the symmetry is present, one cannot continuously deform a non-trivialSPT state to a trivial product state without a bulk phase transition. On the other hand,when the symmetry is absent, an SPT phase may be continuously connected to a trivialproduct state. Thus, we may say that SPT phases have no “intrinsic” topological order:they have a unique ground state on any closed manifold and possess no fractional bulkexcitations. Although their bulk spectrum is trivial, SPT phases display highly unusualedge properties. Namely, in one dimension, the zero-dimensional edge is gapless, in twodimensions, the one-dimensional edge is either gapless or spontaneously breaks a symmetry,finally, in three dimensions, the two-dimensional edge is either gapless, spontaneously breaksa symmetry or carries intrinsic topological order. Moreover, in all these cases, the propertiesof the edge of a d -dimensional non-trivial SPT phase cannot be realized in a purely d − U (1) (cid:110) Z T ,with U (1) - the charge conservation symmetry and Z T - time reversal. The bulk spectrumis gapped and trivial, however, the surface supports an odd number of gapless Dirac conesand preserves the full U (1) (cid:110) Z T symmetry. It is well known (and will be reviewed below)that such properties cannot be realized in a purely two-dimensional system. Moreover,one can distinguish a topological insulator from a trivial one even in the bulk by consideringits response to an external compact electromagnetic field. The topological insulator exhibitselectromagnetic response with a topological θ angle, θ = π , while the trivial insulator has θ = 0. The value of the θ angle is manifested in the Witten effect : if one inserts a magneticmonopole into a topological insulator, it acquires a half-odd-integer charge. Over the past few years remarkable progress has been made in understanding SPT phases.In particular, all symmetry protected phases of non-interacting fermions in arbitrary dimen-sion have been classified.
Likewise, it is believed that all possible interacting gappedphases of bosons and fermions in one dimension have been constructed.
The phases ofbosons in d = 1 are classified by the set of projective representations of the symmetry group G , which is identical to the second cohomology group of G . A Jordan-Wigner transforma-tion can be used to extend this result to one-dimensional fermionic systems. A generalization The semi-direct product (cid:110) means that the anti-unitary time reversal operator T and the U (1) rotations g do not commute, T − g T = g − . Moreover, time-reversal is realized projectively here, T = −
1. Note thatthis corresponds to the standard transformation properties of spinfull fermions f α under time-reversal: T : f α → (cid:15) αβ f β .
2f the one-dimensional cohomology classification to higher-dimensional bosonic systems hasbeen recently proposed. However, the physical properties of the resulting higher dimen-sional SPT phases are just starting to be exposed. In the case of two dimensions, it has beenshown that a K -matrix construction - a multi-component Abelian Chern-Simons theory -analogous to the one used to describe topologically ordered Abelian quantum Hall states,reproduces the cohomology classification of SPT phases and gives a direct description oftheir gapless edge modes.In the case of three dimensions, an important advance has been made by Vishwanath andSenthil (VS), who have suggested effective bulk and surface field theories for a number ofbosonic SPT phases. In particular, VS have proposed an effective theory for the bosonicSPT phase with the symmetry group U (1) (cid:110) Z T , which is the direct bosonic analogue ofthe three-dimensional fermionic topological insulator. In the present paper we will focuson this particular phase and refer to it simply as the bosonic topological insulator.VS have claimed that the bosonic topological insulator is characterized by an electromag-netic response with the θ angle, θ = 2 π . This was interpreted to mean that one possibleway to terminate the bulk topological insulator at the surface is to have a fully gapped sur-face state with no intrinsic topological order, which spontaneously breaks the time-reversalsymmetry and has a Hall conductivity σ xy = ±
1. A general physical argument of Ref. 22,complemented by the explicit K -matrix construction of Ref. 20, indicates that purely two-dimensional bosonic SPT phases with a global U (1) symmetry always have an even integer σ xy , so the surface proposed by VS cannot be realized in a strictly two-dimensional model.Moreover, one can go between the time-reversal conjugate surface states with σ xy = 1 and σ xy = − σ xy = 2 on the surface,which is consistent with time-reversal symmetry being broken only on the surface.By driving surface phase transitions, VS were then able to construct alternative surfacephases of the bosonic topological insulator, where the time-reversal symmetry is restored.These include:i) a gapped state which preserves the full U (1) (cid:110) Z T symmetry, but carries an intrinsic Z topological order. This state has the unusual property that both the electric and magneticanyons carry charge 1 / U (1) symmetry.ii) a superfluid state, which spontaneously breaks the global U (1) symmetry, but leavesthe Z T symmetry intact. This state is characterized by unusual vortex properties that willbe discussed in more detail below.iii) certain (multi)-critical (gapless) states preserving both the U (1) and the Z T symmetry,which will not be discussed in the present paper.The work of VS has left three questions open:I. How to distinguish a non-trivial bosonic topological insulator from a trivial one inthe bulk? In the case of the fermionic topological insulator with θ = π , the Witten effect In the bosonic case, time-reversal is realized non-projectively T = 1, and boson operators transformsimply as, T : b → b . θ = 2 π , the charge carried by a magnetic monopole inside the topological insulator isquantized in integer units, just as in a trivial insulator with θ = 0. In particular, in bothcases monopoles can carry zero electric charge.II. What is the fundamental physical reason why the surface of a 3d bosonic topologicalinsulator, independent of what phase it is in, cannot be realized in a purely 2d system? Forthe case of the fermionic topological insulator, such a reason exists: a putative 2d modelexhibiting the same properties as the surface cannot be consistently coupled to a compact electromagnetic gauge field. III. Why is the surface of a bosonic topological insulator necessarily either gapless, sym-metry broken or topologically ordered?In the present paper we resolve the above three questions and demonstrate that theyare intimately linked. We show that one can distinguish between the trivial and non-trivialbosonic topological insulators in the bulk by coupling them to a weakly fluctuating compact electromagnetic field. It is, indeed, appropriate to think of the trivial and non-trivial bosonictopological insulators as having θ = 0 and θ = 2 π . Although magnetic monopoles in bothinsulators can be electrically neutral, their statistics are different. In the trivial insulator( θ = 0), monopoles are bosons, while in the non-trivial insulator ( θ = 2 π ), monopoles arefermions. We call this phenomenon the “statistical Witten effect.” Once we dial θ to 4 π ,monopoles are again bosons. Hence, the θ angle in a bosonic insulator is periodic modulo 4 π .This is in sharp contrast to a fermionic insulator, where monopoles have bosonic statisticsboth at θ = 0 and at θ = 2 π , so the θ variable is periodic modulo 2 π in accordance to thecommon lore.We note that the idea of distinguishing different SPT phases by “gauging” the globalsymmetry group has been utilized before. An example is provided by phases with a global Z symmetry in two dimensions. In this case, the cohomology classification gives one non-trivial SPT phase. As shown in Ref. 23, if one starts with the trivial phase and couples itto a weakly fluctuating Z gauge field, one obtains a system with a “toric code” topologicalorder. The Z fluxes (visons) in this state have bosonic statistics. On the other hand, if onestarts with the non-trivial SPT phase and gauges the Z symmetry, one obtains a systemwith a “double-semion” topological order. The fluxes in this state have semionic statistics.The statistical Witten effect in a three-dimensional topological insulator is a direct analogueof this phenomenon. We would like to mention that a general duality between SPT phasesand “weakly fluctuating” gauge theories was discussed in Ref. 24. It is believed that theduality transformation can be physically interpreted as gauging the global symmetry of theSPT.Returning to the 3d bosonic topological insulator, the statistical Witten effect in the bulkgives a clue as to why the surface physics cannot be fully realized in a 2d system. It turnsout that, as in the fermionic case, a putative 2d model exhibiting the same properties as thesurface of a bosonic topological insulator cannot be consistently coupled to a two-dimensionalcompact electromagnetic gauge field. Such a coupled theory would have instanton events4n space-time, where the magnetic flux through the 2d surface changes by 2 π . We willshow that these events are accompanied by the creation of a single fermion excitation outof the 2d vacuum. This is impossible in a local theory. Hence, the surface cannot existwithout the bulk! When the bulk is present, the above “anomaly” of the surface theory iscured as follows. In the three-dimensional world, the instanton tunneling event of the 2dsurface theory corresponds to the passage of a magnetic monopole from the trivial vacuumoutside into the topological insulator. In the process, the monopole changes its statisticsfrom bosonic to fermionic and leaves another fermion on the surface . Thus, fermions arecreated in pairs and no violation of locality occurs.Besides demonstrating that the surface of a bosonic topological insulator cannot existpurely in 2d, we will use the statistical Witten effect to argue that if the surface stateis neither gapless, nor spontaneously breaks the global symmetry it must possess intrinsictopological order.We now give a more detailed exposition of the above arguments. In this paper, our dis-cussion of the bulk and surface properties of the bosonic topological insulator will be guidedonly by general considerations of symmetry and locality, without reference to a constructionof this phase. In a companion paper, we will supplement the present conclusions by anexplicit field-theoretic construction of the bosonic topological insulator, which displays thestatistical Witten effect in the bulk and realizes the phases proposed by VS at its surface.This paper is organized as follows. In section II, we briefly reveiw the θ -parameter, theWitten effect and its role in fermionic topological insulators. Section III is devoted to thestatistical Witten effect in the bulk of a bosonic topological insulator. Section IV discussesthe link between the statistical Witten effect in the bulk and the physics at the surfaceof the bosonic topological insulator. Section V discusses a potential place of the bosonictopological insulator phase, to which this paper is devoted, within the general cohomologyclassification of Ref. 19. II. THE WITTEN EFFECT AND FERMIONIC TOPOLOGICAL INSULATORS
We begin with a brief review of the θ -parameter, the Witten effect and its role in fermionictopological insulators.Consider a fully gapped fermion insulator with no intrinsic topological order, i.e. with aunique ground state on any closed manifold and with excitations carrying an integer electriccharge. It is useful to study the response of the insulator to an external electromagneticfield, A µ . In 3d, in addition to the standard Maxwell term describing the electric polar-izability/magnetic permeability of an insulator, the electromagnetic response involves thetopological θ -term, S θ = iθ π (cid:90) d xdτ (cid:15) µνλσ F µν F λσ (2.1) This picture is slightly simplified; we will give a more general discussion below. F µν = ∂ µ A ν − ∂ ν A µ . For smooth configurations of the electromagnetic field on afour-dimensional space-time torus, the imaginary time action S θ evaluates to S θ = iθn , withinteger n . In this sense, the θ -parameter is periodic modulo 2 π . Moreover, under timereversal, θ → − θ . Hence, naively, the distinct time-reversal invariant values of θ are θ = 0and θ = π . Below, we will see that this conclusion is correct for fermionic insulators, butnot for bosonic ones. It turns out that both θ = 0 and θ = π , can, indeed, be realizedalready in non-interacting time-reversal invariant fermionic systems. Trivial non-interactingfermionic insulators have θ = 0. On the other hand, non-interacting fermionic topologicalinsulators have θ = π . This provides a formal distinction between the two classes offermionic insulators with time-reversal symmetry which does not rely on surface properties.Another bulk manifestation of the θ term is the so-called Witten effect. Since we arediscussing lattice systems here, the insulator with a global U (1) symmetry can be consistentlycoupled to a compact electromagnetic gauge field. Hence, monopole configurations of themagnetic field are allowed. As shown by Witten, in the presence of the θ -angle, a magneticmonopole with flux 2 πm also carries an electric charge q = n + θm π , with integer n . Theinteger n corresponds to the freedom of adding extra particle excitations of the insulatoron top of the monopole. We note that the allowed combinations of electric charge q andmagnetic charge m are invariant under θ → θ + 2 π , reaffirming the interpretation of θ as a periodic variable. Hence, if a single magnetic monopole is placed inside a fermionictopological insulator, it acquires a half-odd-integer polarization charge. Thus, the Witteneffect gives a clear way to distinguish trivial and topological fermionic insulators.The Witten effect is directly connected to the reason why the surface of a 3d fermionictopological insulator cannot be realized in a two-dimensional model. Imagine an interfacebetween a topological insulator and the vacuum (or a trivial insulator). Consider takinga magnetic monopole in vacuum and dragging it across the surface with the non-trivialinsulator. The monopole in vacuum carries no charge but acquires a half-odd-integer chargeonce inside the topological insulator. Since electric charge is conserved, this means that ahalf-odd-integer charge must be left on the surface.Now suppose the surface of the topological insulator could be realized purely in a 2dlattice model. Then we can couple this 2d model to a compact U (1) gauge field. Imaginestarting with a configuration with no magnetic flux through the surface. This configurationis equivalent to one where flux 2 π passes through a single plaquette of the surface. Nowlet this flux 2 π slowly expand to form a smooth flux distribution over some portion of thesurface. This process is an instanton event in space-time, where the flux changes from 0 to2 π . Now, from the point of view of the 3d system, the instanton event corresponds to thepassage of a monopole through the surface of the topological insulator. As noted above, thisprocess must deposit a half-odd-integer charge on the boundary. Thus, in a pure 2d theory We would like to stress that in the presence of monopole configurations, S θ , Eq. (2.1), like the rest of theterms in the continuum field theory needs to be regularized. The Witten effect, however, is independentof this regularization.
6f the boundary, local instanton events violate charge conservation. The boundary theorytaken by itself is, therefore, inconsistent. We may say that the U (1) symmetry on the surfaceis “anomalous.” In the three-dimensional world, this surface anomaly is compensated by theWitten effect in the bulk and so charge is conserved.The above argument indicates that the surface of the topological insulator exhibits aparticular kind of anomaly, independent of what phase the surface is in. Let us now demon-strate the anomaly for a particular realization of the surface. The most commonly discussedsurface phase is a time-reversal invariant state with a single gapless 2d Dirac cone. Althoughit is possible to show that a putative 2d system with such properties cannot be consistentlycoupled to a 2d compact U (1) gauge-field, the demonstration is much simpler for a dif-ferent realization of the surface. Imagine spontaneously (or explicitely) breaking the Z T symmetry on the surface. This gaps out the Dirac cone and results in the surface having aHall conductivity σ xy = ± /
2, i.e. a background gauge field A µ induces an electromagnetic“polarization” current on the surface, J polµ = σ xy π (cid:15) µνλ ∂ ν A λ . (2.2)We focus on one of the time-reversal conjugate states, say the one with σ xy = 1 /
2. The onlyexcitations of this state are gapped fermions with charge 1. We now show that such a statecannot exist purely in 2d.Imagine first coupling the putative 2d model to a non-compact 2 + 1 dimensional U (1)gauge field A µ . Upon integrating the gapped fermions out, we obtain an effective Chern-Simons action for the electromagnetic field, S = ik π (cid:90) d xdτ (cid:15) µνλ A µ ∂ ν A λ , (2.3)with k = σ xy = 1 /
2. If the system is realized in a 2d lattice model, we can also couple it toa compact U (1) gauge field, so we should be able to promote A µ in Eq. (2.3) to a compactfield. It is well-known, that the level k of the Chern-Simons action (2.3) must be an integerif the U (1) gauge field is compact and the charged excitations are fully gapped and haveinteger charge, so a level k = 1 / π is nucleated througha single plaquette and then allowed to expand to a smooth distribution. According toEq. (2.2), this flux distribution carries a polarization charge 1 /
2. If the theory was purelytwo-dimensional, a local charge − / However,the only excitations of the surface in this σ xy = 1 / This is just the standard Laughlin argument which states that a 2d system with a Hall conductivity σ xy should possess excitations with charge q = σ xy .
7. So, a purely two-dimensional theory is not consistent: instanton events necessarily violatecharge conservation by a half-odd-integer. Of course, this is precisely the property that thesurface of a topological insulator should satisfy: charge − / − / − n is transferredonto the monopole and n charge 1 fermions are locally excited on the surface are likewiseallowed.We can use a variation of the above argument to show that if the surface of a fermionictopological insulator is neither gapless, nor spontaneously breaks the time-reversal symmetrythen it is necessarily topologically ordered. Indeed, suppose the surface is fully gapped andpreserves the time-reversal symmetry. The surface Hall conductivity must then be 0. Now,imagine dragging a magnetic monopole through the surface of the topological insulator.Since σ xy = 0, the magnetic flux through the surface induces no surface polarization charge.Hence, the half-odd-integer charge left by the monopole on the surface must be a localizedsurface quasiparticle excitation. So, the surface supports excitations with a fractional chargeand, therefore, by the standard flux threading argument, the ground state of the system ona 3d solid torus is degenerate. QED. III. THE STATISTICAL WITTEN EFFECT AND THE BOSONIC TOPOLOGI-CAL INSULATOR
Let us turn our attention to 3d bosonic insulators with no intrinsic topological order.Now the bulk excitations are bosons b with integer charge. We may again consider theresponse of the system to a compact U (1) gauge field: a θ -term (2.1) in the effective actionwill generally be induced. Our discussion of the θ -term and the Witten effect in the previoussection was independent of the statistics of the bulk excitations. In particular, for smoothconfigurations of the electromagnetic field on a four-torus, we still have S θ = iθn , with aninteger n . Moreover, magnetic monopoles with flux 2 πm still carry electric charge q = n + θm π ,with n an integer. Thus, we may naively conclude that θ is periodic modulo 2 π and thatthe time-reversal invariant points are θ = 0 and θ = π , corresponding to bosonic trivial andtopological insulators, respectively. It turns out that this conclusion is incorrect. In reality,in a bosonic insulator, the θ -variable is periodic modulo 4 π and the distinct time-reversalinvariant values of θ are θ = 0 (trivial insulator) and θ = 2 π (topological insulator), while θ = π always breaks the time-reversal symmetry.To reach the above conclusion we need a somewhat finer grating than the ordinary Witteneffect discussed above. Namely, we need to consider the statistics of monopole excitations.Let us, again, couple our bosonic insulator to a weakly fluctuating compact U (1) gaugefield. The gauge field in the bulk of the insulator will be in a Coulomb phase - i.e. there willbe a gapless photon excitation, described by Maxwell electrodynamics. In addition to thegapless photon, there will be gapped bosonic excitations b with integer electric charge. Thephoton mediates a standard Coulomb interaction V ( r ) = e q q c/ (4 πr ) between two staticexcitations with charges q and q . Here, e is a dimensionless coupling constant and c is the8 (cid:45) (cid:45) Π Π Π Π Θ q FIG. 1: Electric charge q of dyons with magnetic flux 2 π as a function of the θ -angle in a bosonicinsulator. Red lines denote dyons with bosonic statistics and blue lines - dyons with fermionicstatistics. Although the allowed values of electric charge are invariant under θ → θ + 2 π , thecorresponding statistics is periodic only modulo 4 π . velocity of the photon.Besides the electric charge excitations, the theory will possess gapped magnetic monopoleexcitations with flux 2 πm . For now, let us consider the trivial bosonic insulator with θ = 0. The monopoles then have bosonic statistics and carry electric charge 0. Twostatic monopoles with fluxes 2 πm and 2 πm experience a Coulomb interaction, V ( r ) =(2 π ) m m c/ (4 πe r ).The photon also mediates a “statistical” interaction between the charges and themonopoles. It is known that this statistical interaction results in the bound state of n charges and m monopoles having statistics ( − nm , where +1 corresponds to bosonic statis-tics and − θ variable. As we start tuning θ away from 0, a monopolewith flux 2 πm acquires an electric “polarization” charge θm/ π . Thus, a dyon which hadelectric/magnetic charges ( n, m ) at θ = 0, now carries charges ( n + θm/ π, m ). For θ = 2 π ,both the total (physically observable) electric charge, q = n + θm/ π , and the magneticcharge m are integer. However, the polarization charge does not affect the dyon statistics. This is consistent with the fact that statistics in three-dimensions are either bosonic orfermionic. Thus, when θ is finite, the statistics of a dyon with charges ( n + θm/ π, m )remain at ( − nm . Let us express the statistics of a dyon at a finite value of θ in terms of itstotal electric charge q = n + θm/ π . This yields ( − qm − θm / π for the statistics of a ( q, m )9yon at a general θ . Thus, when θ = 0, the statistics of a ( q, m ) dyon is ( − qm , but when θ = 2 π , the statistics of a ( q, m ) dyon ( q and m - both integer) is ( − qm + m . This meansthat although the allowed electric/magnetic charges at θ = 0 and θ = 2 π are the same, thedyon statistics are in general different! In particular, excitations with total electric charge 0and magnetic flux 2 π are bosons at θ = 0 and fermions at θ = 2 π . Below, we refer to theseexcitations simply as single monopoles. Note that once θ = 4 π , the statistics of a ( q, m )dyon, again with q and m - both integer, return to ( − qm . Thus, we conclude that in abosonic insulator, the θ variable is periodic only modulo 4 π . Therefore, the only possibledistinct time-reversal invariant points are θ = 0 and θ = 2 π . In particular, the point θ = π manifestly breaks time-reversal invariance: here the (1 / ,
1) dyon has bosonic statistics, butits putative time-reversal partner (1 / , −
1) has fermionic statistics.We identify the values θ = 0 and θ = 2 π with trivial and topological insulators ofbosons. The bulk signature of a bosonic topological insulator is the statistical Witten effect:monopole excitations carry fermionic statistics. Below, we will show that this identificationis consistent with the surface phases of the bosonic topological insulator proposed by VS.Before we proceed to the surface of a bosonic topological insulator, let us reconsiderfermion insulators from the point of view of dyon statistics. Let us first take the triv-ial fermion insulators with θ = 0. Here, unit electric charges have fermionic statistics,while monopoles have bosonic statistics. The statistical interaction between charges andmonopoles, therefore, endows the ( n, m ) dyon with statistics ( − nm + n . In particular, dyonswith flux 2 π and arbitrary electric charge are bosons. Now, let us turn on a finite θ an-gle. The statistics of an excitation with total electric charge q and magnetic flux m is now,( − qm + q − θm ( m +1) / (2 π ) . The above expression is invariant under θ → θ +2 π . So the θ variablein a fermionic insulator is, indeed, periodic modulo 2 π , as is commonly assumed, and thetime reversal invariant points are θ = 0 and θ = π . In particular, at θ = π , the (1 / , ± , ±
2) dyons are both fermions.Before we conclude this section, we briefly mention a more formal field-theoretic wayto deduce the periodicity of the θ angle in bosonic and fermionic systems, which considersonly smooth configurations of the electromagnetic field without monopole defects. Here,one places the system on a generic closed four-manifold, instead of the four-torus, which weconsidered so far. It turns out that on a most general four-manifold, S θ = iθn , with n - a half -integer. An example of a manifold where a gauge field configuration with half-odd n exists is CP , which allows for n = 1 /
2. Thus, the θ variable is periodic at least modulo4 π . However, theories with fermion degrees of freedom require that the space-time manifoldbe endowed with a spin structure. Not all manifolds admit a spin structure; those that do,have integer n . Thus, in theories with fermions, θ is quantized modulo 2 π .10 V. SURFACE OF THE BOSONIC TOPOLOGICAL INSULATOR AND THESTATISTICAL ANOMALY
The statistical Witten effect displayed by the bulk of a bosonic topological insulatorstrongly constrains the properties of its surface. Indeed, as usual, let’s couple the system toa weakly fluctuating compact U (1) gauge field. As we drag a monopole through the surface,it changes its statistics from bosonic to fermionic. Since in a local theory fermions are createdin pairs, this means that a fermion must be left on the surface. Thus, from the point of viewof the 2d surface, instanton events create single fermion excitations, (apparently) violatinglocality. Therefore, the surface cannot be realized in a local two-dimensional model. We willcall this property of the surface, “the statistical anomaly.” We note that this “anomaly,”must be manifested by the surface, independent of what phase it is in, in order to have aconsistent three-dimensional theory.It turns out that the discussion above is slightly simplified. In reality, the weakly fluctu-ating U (1) gauge theory in the bulk is in the Coulomb phase and long-range statistical inter-actions between charged excitations on the surface and monopoles in the bulk are present.As we will see in an explicit example below, this means that in the full 3d theory, the pointexcitation left by the monopole on the surface need not be a fermion. Nevertheless, a pu-tative 2d model with the properties of the surface must still exhibit the statistical anomaly,when coupled to a 2d compact U (1) gauge field.We now turn to the various possible surface phases of a bosonic topological insulatorproposed by VS. We will show that all of these states exhibit the statistical anomaly in 2dand discuss how the statistical Witten effect resolves the anomaly in the full 3d theory. Wewill then present an argument showing that if the surface of a bosonic topological insulatoris neither gapless, nor spontaneously breaks the symmetry it must carry topological order. A. Surface phases with σ xy = ± . The simplest surface phase proposed by VS is an insulator with no intrinsic topologicalorder and a Hall conductivity σ xy = ±
1. This state spontaneously breaks the time-reversalsymmetry. Its excitations are gapped bosons b with integer charge. Let us show that apurely 2d system with such properties would exhibit a statistical anomaly and so cannotbe realized. Imagine first coupling the putative 2d model to a weakly fluctuating 2d non-compact U (1) gauge field A µ . We again obtain a 2d Chern-Simons effective action but nowwith level k = 1, S = (cid:90) d xdτ (cid:18) e F µν + ik π (cid:15) µνλ A µ ∂ ν A λ + iA µ J bµ (cid:19) (4.1) Our argument is very similar to one given in Ref. 22. The only difference is that we treat the electromag-netic field as a weakly fluctuating dynamical field, instead of an external field. F µν in the action,as well as an explicit coupling of the gauge field to the gapped boson excitations, whose cur-rent is denoted by J bµ . Thus, the total electric current is given by the sum of the polarizationcurrent (2.2) and the boson current J bµ , J EMµ = k π (cid:15) µνλ ∂ ν A λ + J bµ (4.2)As is well known, the 2+1 dimensional theory (4.1) is fully gapped. The effect of theChern-Simons term with k = 1 is to dynamically attach magnetic flux − π to the b exci-tations, which have intrinsic electric charge 1 as seen in (4.1). As a result, the statisticsof b is transmuted from bosonic to fermionic. Moreover, since the magnetic flux Φ carriesan electric polarization charge Φ / π , the b particle with flux − π attached is electricallyneutral.Now, if the phase under consideration can be realized in a 2d lattice model, we should beable to promote A µ to a compact gauge field. Imagine an instanton event in the putative 2dtheory (4.1). Start with a configuration with no flux and no charge, nucleate flux 2 π througha single plaquette and let this flux expand into a continuous distribution. The flux will thencarry a polarization charge 1. Therefore, in order for the total charge to be conserved, a localcharge − b (anti)-particle. Thus, an instanton event creates a b anti-particle together with flux 2 π out of the vacuum. As noted above, this composite objecthas fermionic statistics. Thus, single fermion excitations are created locally by instantonevents: the surface theory has a statistical anomaly, and therefore, cannot be realized in apurely two-dimensional model.Note that the above argument explains why σ xy must be an even integer for a purely two-dimensional bosonic insulator with no topological order, in accordance with the K -matrixconstruction of Ref. 20, as well as general constraints on the consistency of Chern Simonsterms on 3 manifolds with and without spin structures . Indeed, for a general integerlevel k of the Chern-Simons theory (4.1), b bosons have flux − π/k bound to them and aretransmutted to anyons with a statistical angle θ = − π/k . An instanton tunnelling eventinvolves the creation of flux 2 π (and the corresponding polarization charge k ) together with k b -anti-particles. This composite object has statistics θ = − πk and so k must be an eveninteger in order to prevent single fermions from being created out of the vacuum.We also briefly note that the above argument is consistent with the fact that a two-dimensional fermionic insulator with no topological order can have any integer σ xy . Indeed,suppose the charge carriers in the theory (4.1) are fermions f with unit charge. Theseexcitations again have a flux − π/k dynamically bound to them and their statistics istransmutted to θ = π − π/k . An instanton event involves the creation of flux 2 π togetherwith k f -anti-particles. This composite object has statistics θ = πk − πk , which is bosonic The 2d statistical angle should not be confused with the θ -angle in 3d.
12s long as k is an integer.Let us now return to the surface of the bosonic topological insulator with σ xy = 1 anddiscuss how the anomaly in the 2d model is resolved in three dimensions. Let’s place atrivial vacuum in the region z >
0, a bosonic topological insulator in the region z < σ xy = 1 at z = 0. We now couple the system to a fully three-dimensionalcompact U (1) gauge field. The simplest thought experiment to carry out is to start witha neutral monopole in vacuum and let it pass through the surface of a bosonic topologicalinsulator, picking up charge − − b -anti-particle on the surface together with apolarization charge 1. Now, the neutral monopole in the bulk of the topological insulator isa fermion. However, the statistical interaction between the b -boson on the surface and themonopole in the bulk allows us to view this pair together as a boson. Thus, statistics areagain conserved.We can also look at the full 3d problem from the perspective of surface physics. It turnsout that the electromagnetic response of the surface with σ xy = 1 endows the excitationsnear the surface with an effective 2d statistical interaction. Processes where two particlesare exchanged in the plane of the surface over distances much larger than their separationfrom the surface receive a Berry’s phase, which mimics two-dimensional statistics. Labellingthe monopoles on the vacuum side of the interface as m + and monopoles on the topologicalinsulator side of the interface as m − , we obtain the following effective 2d exchange statistics(see appendix A) θ ( b, b ) = − πα α , θ ( m + , m + ) = π α ) , θ ( m − , m − ) = π α ) + π (4.3) θ ( b, m ± ) = ∓ π α , θ ( m + , m − ) = − π α ) (4.4)Here, Eq. (4.3) summarizes the self-statistics and Eq. (4.4) - the mutual statistics of ex-citations. We note that the b -particle preserves its self and mutual statistics as it passesthrough the interface, so there is no need to distinguish b excitations on the two sides of theinterface. Note that all the statistical angles depend on the three-dimensional fine-structureconstant α = e π (we assume, for simplicity, that the dielectric constant and permeabilityof the topological insulator are the same as of vacuum). Note, in particular, that unlike ina theory where the electromagnetic field is purely two-dimensional, the effective statistics θ ( b, b ) of the b -particles near the surface is generally not fermionic; θ ( b, b ) → − π only when α → ∞ . We also note that unlike in a purely two-dimensional theory, static b -particles willexperience long-range 1 /r Coulomb interactions mediated by the gapless bulk photon. Fi-nally, the extra π in the self-statistics of m − compared to m + reflects the intrinsic fermionic13ature of monopoles in the topological insulator.Let us repeat the thought experiment above. Take a neutral monopole in vacuum ( m + )and let it tunnel through the interface turning into a neutral monopole m − and creatinga b -anti-particle. Now, the 2d self-statistics of an anti- b - m − composite is θ = θ ( b, b ) − θ ( b, m − ) + θ ( m − , m − ) = π α ) , which turns out to be exactly the same as the self-statisticsof m + . Hence, statistics are again conserved! B. Surface superfluid.
Another surface phase proposed by VS is a superfluid where the global U (1) symmetry isspontaneously broken but the time-reversal symmetry is preserved. The excitations of thisphase include a 2d gapless Goldstone mode and gapped superfluid vortices on the surface.An effective theory of this surface phase is most easily expressed in dual variables. Thesurface action S = (cid:82) d xdτ L with L = 18 π ρ s ( (cid:15) µνλ ∂ ν a λ ) + i ( a µ + A µ j + µ + i ( a µ − A µ j − µ − i π (cid:15) µνλ A µ ∂ ν a λ . (4.5)Here, the superfluid current J sµ is expressed in terms of the dual gauge field a µ as J sµ = − π (cid:15) µνλ ∂ ν a λ . The dual gauge field a µ should not be confused with the electromagnetic gaugefield A µ , which we treat at this stage as a non-compact external probe field. Here ρ s denotesthe superfluid stiffness. The effective 2d theory involves two types of gapped superfluidvortices, ψ ± ; j ± are the corresponding vortex currents. The vortices are minimally coupledto a µ . Note that under time-reversal, ψ ± → ψ †∓ .The only unusual property of the surface superfluid described by the action (4.5), com-pared to an ordinary two-dimensional superfluid, is that the vortices ψ ± formally carry aglobal U (1) charge ± /
2. Thus, the total electromagnetic current, J EMµ = − π (cid:15) µνλ ∂ ν a λ + 12 ( j + µ − j − µ ) , (4.6)involves both the superfluid current and the vortex currents. Since the global U (1) sym-metry in a superfluid is spontaneousy broken, it is inappropriate to label its excitation bytheir global charge; we will discuss a more physical distinction between the present surfacesuperfluid and an ordinary purely two-dimensional superfluid shortly.As noted, we temporarily switch off the fluctuations of the external electromagnetic field A µ . Then the dual gauge field a µ will be in the Coulomb phase, i.e. the dual photon will begapless and will directly correspond to the superfluid Goldstone mode. The gapless 2 + 1dimensional photon mediates a logarithmic Coulomb interaction between static 2d surfacevortices. We note that since flux φ of a µ corresponds to a global charge − φ/ π , pure flux-tunneling events of a µ are prohibited by charge conservation. As a result, the dual gaugefield does not go into a confined phase: the superfluid is stable. However, events where flux φ = 2 π is created together with a vortex ψ † + and an antivortex ψ − are allowed. Indeed, such14vents conserve both the global U (1) charge and the vorticity of the superfluid. We canwrite the corresponding term in the Lagrangian as,∆ L = λm † ψ † + ψ − + h.c., (4.7)where the monopole operator m † creates flux 2 π of a µ and λ is a coupling constant. It iswell-known that m † acquires a finite expectation value in the 2d Coulomb phase. In fact,since m † carries charge − U (1) symmetry, it serves as the superfluid orderparameter and may be identified with the physical boson operator b . Moreover, we mayschematically replace m † in Eq. (4.7) by its expectation value,∆ L → λ (cid:104) m † (cid:105) ψ † + ψ − + h.c.. (4.8)Hence, the term (4.7) induces tunneling between the gapped ψ + and ψ − vortices; energyeigenstates will be superpositions of ψ + and ψ − . Note that time-reversal maps the vortex ψ † + to an antivortex ψ − . Hence, in the absence of an extra particle-hole symmetry, there isno reason for ψ + and ψ − vortices to have the same energy and so a finite tunneling betweenthem generally will not give rise to an equal weight superposition.Let us now switch on the fluctuations of A µ , first treating it as a non-compact two-dimensional gauge-field. The effective 2d theory (4.5) then becomes fully gapped - indeed,when the global U (1) symmetry of a superfluid is gauged, the Goldstone mode disappearsand the external electromagnetic field becomes Higgsed. The effect of the mutual Chern-Simons term i π (cid:15) µνλ A µ ∂ ν a λ in Eq. (4.5) is to attach flux 2 π of A to ψ †± : the superfluidvortices become magnetic flux tubes. Moreover, a flux ± π of a is attached to ψ †± ; Eq. (4.6)indicates that this flux corresponds to an electric polarization charge ∓ / A . Thus,the ψ ± flux tubes, which couple minimally to A µ with charges Q = ± /
2, as in Eq. (4.5),become overall electrically neutral, in accordance with the fact that electric charge is Debyescreened in a superconductor.As in an ordinary 2d superconductor, the interaction between two static flux tubes isexponentially screened - the flux tubes are local excitations. However, the flux attachmentdiscussed above endows the flux-tubes with a statistical interaction: both ψ + and ψ − acquirefermionic self-statistics. The mutual statistics between ψ + and ψ − are bosonic. Indeed, wecan think of the action (4.5) as a Chern-Simons theory for the two-component gauge field( a µ , A µ ) with the K -matrix, K = (cid:32) (cid:33) . The ψ ± vortices carry charges (1 , ± /
2) underthis two-component gauge field; the aforementioned statistics immediately follow.Note that the tunelling operator (4.7) converts a ψ − flux-tube to a ψ + flux-tube: thestatistics is preserved in the process. Hence, the statistics of the energy eigenstates, whichare superpositions of ψ + and ψ − will likewise be fermionic.We conclude that the key signature of the superfluid on the surface of a topologicalinsulator is that upon coupling to a 2d electromagnetic field, its flux-tubes possess fermionicstatistics. This is in contrast to an ordinary time-reversal invariant purely two-dimensional15uperfluid, whose flux tubes are bosons. The fermionic statistics of flux-tubes directly implythat a putative purely 2d system with the same properties suffers from a statistical anomalyand so cannot exist. Indeed, if the effective theory (4.5) emerges from a purely 2d latticemodel, we should be able to promote the electromagnetic field A µ to a compact gaugefield. An instanton of A µ creates a flux 2 π excitation out of the vacuum. The only suchexcitations in the theory are the fermionic flux-tubes. Thus, single fermions are created outof the vacuum during instanton events and a purely two-dimensional model is inconsistent.How is the statistical anomaly resolved in the full 3d theory in the present case? Theelectromagnetic field in the bulk of the system is in the Coulomb phase. Nevertheless, themagnetic field must still penetrate the surface in the form of tubes with quantized flux.The “demagnetization effects” in the bulk lead to a V ( r ) = αr interaction between staticflux-tubes. Moreover, the magnetic field profile of a single flux-tube on the surface now hasan algebraic 1 /r tail, instead of falling off exponentially, as in a purely 2d system. Likewise,the electric charge cloud that screens the ± / ψ ± vortex has a density profilewith a 1 /r tail. It turns out that these tails fall off sufficiently quickly for the statistics offlux-tubes to remain well-defined and fermionic.Thus, as a monopole passes from the vacuum outside the topological insulator throughthe interface, it changes its statistics from bosonic to fermionic and excites a fermionicflux-tube on the surface. So, the statistics are conserved!Before we conclude this section, we would like to stress that the fermionic statisticsacquired by flux tubes upon gauging the U (1) symmetry are not directly related to thestatistics of global superfluid vortices in the absence of an external gauge field. VS haveclaimed that it is also appropriate to think of the statistics of global superfluid vortices onthe surface of a topological insulator as fermionic. As we discuss in appendix B, this claimis correct only when the system has an additional particle-hole symmetry. On the otherhand, the fermionic statistics of flux-tubes are completely robust to particle-hole symmetrybreaking. Moreover, as emphasized in appendix B, the two effects have different dynamicalorigins. C. Surface phase with Z topological order. Yet another surface phase proposed by VS is fully gapped, respects the U (1) (cid:110) Z T symme-try and carries an intrinsic 2d Z (toric code) topological order. The excitations of this phaseare e and m anyons carrying electric and magnetic charge respectively under the emergent Z gauge field. e and m have bosonic self-statistics and are mutual semions. The boundstate of e and m is a fermion. The e and m anyons transform trivially under time-reversal,but both carry charge 1 / U (1) symmetry. This property is somewhatunusual; standard two-dimensional bosonic states with Z topological order, global U (1)and time-reversal symmetries, have just one (or none) of the anyons carrying charge 1 / U (1) gauge field. Let us imaginea flux-tunneling event in such a coupled model. We start with an initial state with nomagnetic flux and no charge, nucleate flux 2 π through a single plaquette at the origin andlet it expand to a smooth distribution. Since the system is time-reversal invariant, σ xy = 0,and so the flux carries no electric charge. Thus, any local quasi-particle created during theinstanton tunneling event must be neutral. Now, both e and m charge 1 / π upon encircling the flux in the final state, but acquire no Berry’sphase upon encircling the origin in the initial state. In a local model an instanton tunnelingevent cannot change the Berry’s phase acquired by a distant quasiparticle upon encirclingthe spatial location of the event. The only possible resolution of the above puzzle is thatthe instanton event must create a quasiparticle, which has mutual semionic statistics withboth e and m anyons, compensating the π Berry’s phase due to the magnetic flux. Thisquasiparticle must be identified with the neutral fermion f = em † . Hence, the instantonevent creates a neutral f fermion together with flux 2 π . Since f is neutral, its statistics arenot affected by the 2 π flux and remain fermionic. Thus, single fermions are created out ofthe vacuum during such instanton tunneling events. This is not possible in a local 2d model,which is thus seen to possess a statistical anomaly.We note in passing that, as pointed out by VS, a toric code with both e and m anyonscarrying charge 1 / σ xy = 1. This 2d system breaks time reversal symmetry and is described by a K matrix construction, with K = (cid:32) (cid:33) and the charge vector t = (1 , σ xy = 0 surface state can be understood by considering a slabof topological insulator of a large but finite thickness, with the σ xy = 0 toric code state onthe top surface, and the non topological σ xy = 1 state on the bottom surface. The entireslab viewed as a 2d system realizes the toric code with σ xy = 1. It follows that an interfacebetween the toric code state and the σ xy = 1 state on the surface of the topological insulatorwill exhibit edge states identical to those of the 2d σ xy = 1 state.Since a K -matrix construction exists, the 2d theory should be free of anomalies. Let usexplicitely check this. As usual, let’s couple the 2d model to a compact U (1) gauge field andimagine an instanton event which creates flux 2 π out of the vacuum. Since σ xy = 1, thisflux carries a polarization charge 1. As for the toric code on the surface of the topologicalinsulator, the instanton also creates an em bound state, whose intrinsic semionic mutualstatistics with the charge 1 / e and m anyons cancels the π Berry’s phase acquired by thelatter upon encricling the 2 π flux. However, in the present case, the em bound state carriescharge − em boundstate with charge − π . The attached flux transmutes the intrinsicfermionic statistics of em to bosonic. Thus, the statistics are conserved in the process andthe theory is anomaly free.Returning to the bosonic topological insulator with the time-reversal invariant toric code17urface, in the present case, the anomaly is resolved very directly in three dimensions:the monopole simply passes through the surface of the topological insulator changing itsstatistics and leaving an f fermion on the surface. Thus, the fermion parity is conserved.Note that both the monopole and the f particle are electrically neutral, so there is nolong-range statistical interaction between them mediated by the 3d photon. D. General constraints on the surface.
The discussion in the previous section immediately generalizes to a proof of the statementthat if the surface of a bosonic topological insulator is neither gapless, nor spontaneouslybreaks the symmetry it must possess topological order. Indeed, suppose the surface is fullygapped and does not break the time-reversal symmetry. Then the surface Hall-conductivity σ xy = 0. Now, let’s tunnel a neutral monopole across the interface between the vacuumand the topological insulator. Since σ xy = 0, no polarization charge is induced on thesurface. This means that any excitation created on the surface during the tunneling processis neutral. Thus, this excitation will possess no statistical interaction with the monopolein the bulk. Now, the neutral monopole in the bulk is a fermion. Thus, the quasiparticleleft on the surface is likewise a fermion. Hence, the gapped surface state must supportfermionic excitations. Since we are dealing with a system made out of bosons, the presenceof fermionic quasiparticles implies that the surface phase has intrinsic 2d topological order.QED.The above argument not only proves that the gapped, symmetry respecting surface ofthe topological insulator must support intrinsic topological order, but also places someconstraints on the order allowed. In general, it may be convenient to label SPT phases bytheir gapped, symmetry respecting topologically ordered surface states. Such a label willcarry the information about the intrinsic topological order (fusion rules, braiding statistics),as well as the quantum numbers of anyon excitations under the global symmetry. Of course,many different topologically ordered surface states may be realized; in fact, one can always“paint” an additional layer of a purely two-dimensional topologically ordered phase on top ofthe surface. A possible way to arrange the different topologically ordered states is by theirquantum dimension D . Thus, we may use the surface state(s) with the lowest quantumdimension to label an SPT phase. (There may still be several such states allowed).We note that any surface state of an SPT that has intrinsic topological order, apartfrom the realization of global symmetry, has to be identical to one of the allowed strictly 2dtopological orders. This follows immediately from the slab construction mentioned above, inwhich the top surface is the state in question and the bottom surface is in a gapped globalsymmetry violating topologically trivial state. Since both the bulk of the SPT phase andthe bottom surface support no topologically non-trivial excitations, the topological order ofthe slab as a whole is just given by the topological order on the top surface. This proves thatthe braiding statistics on the surface of an SPT phase must be realizable strictly in 2d. Notethat since the bottom surface in the above construction breaks the symmetry, this argument18akes no statement about the realization of symmetry in the topologically ordered state onthe top surface.Let us now identify the symmetry respecting topologically ordered surface state(s) of thebosonic topological insulator with the lowest quantum dimension. One allowed topologicallyordered surface state is the toric code with charge 1 / D = 2. It is known that there are no time-reversalinvariant topological orders with a smaller quantum dimension. Moreover, the only othertime-reversal invariant topological order with D = 2 is the double-semion theory. However,as argued above, the topologically ordered state on the surface must posses fermionic quasi-particles. The double-semion theory supports no fermionic excitations. Thus, we concludethat the toric code with charge 1 / V. RELATION TO THE COHOMOLOGY CLASSIFICATION
In this section, we briefly comment on the potential place of the bosonic topological insu-lator phase considered in the present paper within the cohomology classification of Ref. 19.A similar discussion has been given by VS.The cohomology classification predicts a Z structure for 3d phases with the bulk sym-metry group U (1) (cid:110) Z T . This means that there are three non-trivial phases, g , g , g wherethe third one, “ g = g + g ,” can be thought of as a weakly interacting “mixture” of the firsttwo. Moreover, the cohomology technology predicts a Z classification for phases with justthe Z T symmetry and no nontrivial phases with just the U (1) symmetry. This implies thatone of the three non-trivial phases with U (1) (cid:110) Z T symmetry, say g , is just the non-trivial Z T phase, i.e it does not involve the U (1) symmetry in any interesting way. In particular,if we start with g and explicitely break the U (1) symmetry, the resulting state is still pro-tected by Z T and cannot be smoothly connected to a product state. On the other hand,since g = g + g , out of the remaining two phases, g and g , one (say g ) must becometrivial under the Z T classification if U (1) is explicitely broken, while the other ( g ) remainsnon-trivial. The phase that we refer to as the bosonic topological insulator in this paperis g : it is unstable whenever either the U (1) or the Z T symmetry is broken in the bulk.VS have also constructed an effective theory for the pure Z T phase g , and thereby, for thephase g . However, we do not consider these phases in the present work.We also note that VS have proposed an additional phase with just the Z T symmetry,which falls outside the cohomology classification of Ref. 19. The time-reversal symmetrybreaking surface state of this phase has a thermal Hall response with κ xy = 4, so that adomain wall on the surface between two time-reversal conjugate regions with κ xy = ± c L − c R = 8). This gapless domain wall is identical to theedge state of a 2d E integer quantum Hall state of bosons with κ xy = 8. Recently, severalexplicit constructions of this 3d phase have appeared (including an exactly solvable modelin Ref. 34). Again, we do not consider this phase (and its mixtures with “conventional”19ohomology phases) in this paper. VI. CONCLUSION
In the present paper we have identified the statistical Witten effect as the bulk signa-ture of three-dimensional bosonic topological insulators, protected by the symmetry group U (1) (cid:110) Z T . We have shown that this effect immediately implies that the surface physicsof the topological insulator cannot be fully realized in a local two-dimensional model. Thestatistical Witten effect also implies that if the surface is neither gapless, nor spontaneouslybreaks the symmetry, it must be topologically ordered. Moreover, we have demonstratedthat the surface phases of the bosonic topological insulator inferred by Vishwanath andSenthil are consistent with the statistical Witten effect in the bulk.A reader may ask, how do the authors know that time reversal respecting insulatorsexhibiting the statistical Witten effect can exist? In a companion paper, we will provide anexplicit field-theoretic, lattice-regularized, construction of a time-reversal invariant bosonicinsulator, which displays the statistical Witten effect in its bulk and realizes the surfacephases of VS. The first step of our construction involves a parton decomposition of themicroscopic bosons, which yields a phase with an emergent gapless u (1) gauge field anddeconfined parton and monopole excitations. (The emergent gauge field and its monopolesshould not be confused with the external electromagnetic gauge field). In the second step, weform certain dyon bound states of monopoles and partons, and by dyon condensation drivea confinement transition to a fully gapped phase with no intrinsic topological order. Thisphase has all the bulk and surface properties of a bosonic topological insulator discussed inthe present paper.In this work we have pursued the line of attack, where symmetry protected topolog-ical phases are differentiated by “weakly gauging” their global symmetry. As the presentpaper was being completed, significant broad progress on the same front was reported inRef. 35, which utilized this general approach to distinguish SPT phases within the coho-mology classification with various global symmetries. However, Ref. 35 did not considerany phases with time-reversal symmetry, and so the U (1) (cid:110) Z T topological insulator phase,to which the present paper is devoted, fell outside its scope. In fact, we do not know howto “gauge” the time-reversal symmetry. Thus, we do not have a completely general tool todiagnose SPT phases whose global symmetry group contains time-reversal. Fortunately, forthe topological insulator phase considered in this paper, it was sufficient to gauge just the U (1) part of the symmetry to expose the statistical Witten effect; time-reversal remainedas a global symmetry of the resulting weakly fluctuating U (1) gauge theory. However, thereare phases within the cohomology classification, which appear to involve the time-reversalsymmetry in a more “active” manner. For instance, the classification predicts a single non-trivial SPT phase in three dimensions with just the time-reversal symmetry. Although asurface theory for this phase was proposed by VS, a bulk signature is currently lacking. Yeta bulk diagnostic would be greatly desirable, since the surface termination is generally not20nique. Moreover, it is often not immediately obvious why a given surface phase cannot berealized in a purely two-dimensional system. As we have seen with the example of the ordi-nary/statistical Witten effect in fermionic/bosonic topological insulators, a bulk signaturemay be the key to identifying (and resolving) the surface “anomaly.” Acknowledgments
We would like to thank A. Vishwanath, T. Senthil, X. G. Wen, C. Xu , T. Grover, G. Cho,C. Nayak and E. Witten for stimulating discussions. Thanks also to the Aspen Centerfor Physics where some of this work was initiated. This research was supported in partby the National Science Foundation under Grant No. NSF PHY11-25915, DMR-1101912(M.P.A.F.) and DMR 0906175 (C.L.K.), by the Caltech Institute of Quantum Informationand Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty MooreFoundation (M.P.A.F.) and by a Simons Investigator award from the Simons Foundation(C.L.K.).
Appendix A: Statistical interactions near the σ xy = 1 surface of a bosonic topologicalinsulator. In this appendix, we study the surface of the bosonic topological insulator with σ xy = 1when the system is coupled to a weakly fluctuating 3d compact electromagnetic gauge field A µ . We show that bosons b and monopoles m moving in the plane of the surface experiencean interaction, which mimics two-dimensional fractional exchange statistics.We begin with the following action, S = 14 e (cid:90) d x ( F µν − πM µν ) + i π (cid:90) z =0 d x (cid:107) (cid:15) ijk A i ∂ j A k + i (cid:90) d xA µ J bµ + S B (A1)Here, the region z > z < z = 0 plane. The Greek indices run over τ, x, y, z ,while the Latin indices run over τ, x, y ; x (cid:107) = ( τ, x, y ) labels the coordinates in the plane ofthe interface. The Chern-Simons term for A encodes the σ xy = 1 electromagnetic responseof the surface. J bµ is the 3+1 dimensional boson current. M µν represents the Dirac stringsof magnetic monopoles, so that the 3+1 dimensional monopole current is given by J mµ = − (cid:15) µνλσ ∂ ν M λσ . S B is a Berry’s phase term, which endows the monopoles in the topologicalinsulator with fermionic statistics. We will give an explicit expression for this term below.Note that we have not “compactified” the Chern-Simons term; here, for simplicity, wewill not expicitely consider events where monopoles tunnel through the surface. We can thenseparate monopoles into those in the trivial vacuum ( m + ) and in the topological insulator( m − ), and write M µν = M + µν + M − µν , with the corresponding monopole currents J m ± µ = − (cid:15) µνλσ ∂ ν M ± λσ . We choose a gauge where the Dirac strings of m + and m − do not pass21hrough the interface. More specifically, we take the Dirac string of a static m + monopoleto run along the + z direction and the Dirac string of a static m − monopole to run alongthe − z direction: M ± ij = − (cid:15) ijk (cid:90) d x (cid:48) S ± ( x − x (cid:48) ) J m ± k ( x (cid:48) ) , M ± iz = 0 (A2) S ± = ± θ ( ± z ) δ ( x ) δ ( y ) δ ( τ ) (A3)Then, M + , J m + ( M − , J m − ) vanish as long as z < z > S B : S B = πi (cid:90) d xd x (cid:48) J m − i ( x ) (cid:15) ijk ∂ j D ( x (cid:107) − x (cid:48)(cid:107) ) J m − k ( x (cid:48) ) (A4)Here, D ( x (cid:107) ) = (4 π | x (cid:107) | ) − , is the 2 + 1 dimensional propagator. Eq. (A4) can also berewritten as S B = πi (cid:90) d x (cid:107) d x (cid:48)(cid:107) J m −(cid:107) i ( x (cid:107) ) (cid:15) ijk ∂ j D ( x (cid:107) − x (cid:48)(cid:107) ) J m −(cid:107) k ( x (cid:48)(cid:107) ) (A5)where J m ±(cid:107) i ( x (cid:107) ) = (cid:90) dzJ m ± i ( x (cid:107) , z ) (A6)is the projection of the 3 + 1 dimensional monopole current onto the z = 0 plane. Eq. (A5)gives the familiar Berry’s phase for a 2d fermion with current J m −(cid:107) i . Now, if we consider aprocess where two 3d monopoles m − are exchanged, their projections onto the z = 0 planeare also exchanged. Hence, Eq. (A5) accurately captures the full 3+1 dimensional “intrinsic”fermionic exchange statistics of m − monopoles. (Here, we do not consider configurationswhere two m − monopoles simultaneously have the same x, y coordinates and so have thesame projection onto the z = 0 plane. Such configurations form a set of measure zero.)We next integrate out the gauge field A µ in the action (A1) to determine the full statisticalinteraction between the b and m ± particles. To do so, it is convenient to introduce anauxilliary 2 + 1 dimensional gauge field α i living on the interface and rewrite (A1) as, S = 14 e (cid:90) d x ( F µν − πM µν ) + i (cid:90) d xA µ J bµ + S B + (cid:90) d x (cid:107) (cid:18) − i π (cid:15) ijk α i ∂ j α k + i π A i (cid:15) ijk ∂ j α k (cid:19) (A7)The integral over A µ is now easy to perform, giving S eff = S + S B + S α (A8)22here S = 12 (cid:90) d xd x (cid:48) (cid:18) J bµ ( x ) D ( x − x (cid:48) ) J bµ ( x (cid:48) ) + (2 π ) e J mµ ( x ) D ( x − x (cid:48) ) J mµ ( x (cid:48) ) (cid:19) + 2 πi (cid:90) d xd x (cid:48) J bµ ( x ) D ( x − x (cid:48) ) ∂ ν M µν ( x (cid:48) ) (A9)and S α = − i π (cid:90) d x (cid:107) α i (cid:15) ijk ∂ j α k + e π ) (cid:90) d x (cid:107) d x (cid:48)(cid:107) α i ( x (cid:107) )( − ∂ (cid:107) δ ij + ∂ i ∂ j ) D ( x (cid:107) − x (cid:48)(cid:107) ) α j ( x (cid:48)(cid:107) )+ i (cid:90) d x (cid:48)(cid:107) α i ( x (cid:107) ) j αi ( x (cid:107) ) (A10)with j αi ( x (cid:107) ) = (cid:90) d x (cid:48) (cid:15) ijk ∂ j D ( x (cid:107) − x (cid:48) ) (cid:18) ∂ l M kl ( x (cid:48) ) − ie π J bk ( x (cid:48) ) (cid:19) (A11)Here, D ( x ) = (4 π | x | ) − , is the 3 + 1 dimensional propagator. The effective action S represents the standard bulk interaction between charges and monopoles mediated by thegapless photon, while S α captures the modification of the interaction due to the interface.Let us first discuss S . The first two (real) terms in Eq. (A9) give rise to the 1 /r Coulomb interaction between static charges/monopoles. The last (imaginary) term inEq. (A9), which we denote as S B , encodes the bulk statistical interaction between chargesand monopoles. Explicitely, S B = − πi (cid:90) d xd x (cid:48) J bi ( x ) (cid:15) ijk (cid:0) ∂ j K + ( x − x (cid:48) ) J m + k ( x (cid:48) ) + ∂ j K − ( x − x (cid:48) ) J m − k ( x (cid:48) ) (cid:1) (A12)where K ± ( x (cid:107) , z ) = ± π | x (cid:107) | (cid:18) π ± tan − (cid:18) z | x (cid:107) | (cid:19)(cid:19) (A13)We note that the seeming difference between K + and K − corresponds to a gauge choice;a replacement of K − by K + in Eq. (A12) modifies the action by a multiple of 2 πi . Here,we are interested in configurations where the separation between the excitations along thesurface, | x (cid:107) | , is much larger than their distance to the surface z . In this “2d” limit, wemay set z in Eq. (A13) to zero. (A finite value of z gives corrections to the “2d” statisticalinteraction, which are less relevant in the RG sense.) We then obtain, S B = − πi (cid:90) d x (cid:107) d x (cid:48)(cid:107) J b (cid:107) i ( x (cid:107) ) (cid:15) ijk ∂ j D ( x (cid:107) − x (cid:48)(cid:107) )( J m + (cid:107) k ( x (cid:48)(cid:107) ) − J m −(cid:107) k ( x (cid:48)(cid:107) )) (A14)with J b (cid:107) i defined analogously to Eq. (A6). Eq. (A14) implies that in the absence of a σ xy = 1surface, charges and monopoles behave as mutual semions, when moving in a 2d plane. Thisis the expected result. 23e now turn our attention to the modification of the bulk statistics by the interface,described by the S α term (A10). Integrating the gauge field α i out, we obtain, S α eff = 12 (cid:90) d x (cid:107) d x (cid:48)(cid:107) j αi ( x (cid:107) ) D αij ( x (cid:107) − x (cid:48)(cid:107) ) j αj ( x (cid:48)(cid:107) ) (A15)with D α - the propagator of the α field, explicitely given by D αij ( x (cid:107) ) = 2 π α ( − i(cid:15) ijk ∂ k D ( x (cid:107) ) + 2 αδ ij D ( x (cid:107) )) (A16)Here, we work in the gauge ∂ i α i = 0. After some algebra, we find that in the “2d” limitdescribed above, the imaginary (Berry’s phase) part of Eq. (A15) simplifies to, S αB = i (cid:90) d x (cid:107) d x (cid:48)(cid:107) (cid:126)J Ti ( x (cid:107) )Θ α (cid:15) ijk ∂ j D ( x (cid:107) − x (cid:48)(cid:107) ) (cid:126)J k ( x (cid:48)(cid:107) ) (A17)with (cid:126)J = ( J b (cid:107) , J m + (cid:107) , J m −(cid:107) ) andΘ α = π α − α α / − α / α / / − / − α / − / / (A18)After combining Eq. (A17) with the bulk Berry’s phases, Eqs. (A5), (A14), we find that thetotal Berry’s phase for 2d exchange processes is given by, S B = i (cid:90) d x (cid:107) d x (cid:48)(cid:107) (cid:126)J Ti ( x (cid:107) )Θ (cid:15) ijk ∂ j D ( x (cid:107) − x (cid:48)(cid:107) ) (cid:126)J k ( x (cid:48)(cid:107) ) (A19)with Θ = π α − α − / / − / / − / / − / / α ) (A20)The effective statistical angles in Eqs. (4.3), (4.4) immediately follow. We can now imaginea process discussed in the main text, where an m + monopole tunnels through the interface,turning into an m − monopole and a b anti-particle, and leaving a polarization charge 1 onthe surface. Such a process can be thought off as a creation of an excitation with quantumnumbers ( − , − , (cid:126)J . One can use Eq. (A20) tocheck that the ( − , − ,
1) excitation has bosonic self-statistics and trivial mutual statisticswith all the other excitations. Therefore, monopole tunneling events preserve the statisticsand the theory is consistent. 24 ppendix B: Statistics of superfluid vortices on the surface of a bosonic topologicalinsulator.
This appendix is devoted to the statistics of vortices on the superfluid surface of a 3dbosonic topological insulator in the absence of a fluctuating electromagnetic gauge field.As discussed in section IV B, the superfluid surface is described by the effective theory inEqs. (4.5),(4.7). Here, we turn off the external electromagnetic field A µ .As superfluid vortices have a long-range logarithmic interaction, the notion of statisticshere is formal. For the present purposes, we define statistics as the Berry’s phase in theimaginary time path integral accumulated during an adiabatic exchange. With this formaldefinition, as long as the tunneling between the ψ + and ψ − vortices in Eq. (4.7) is switchedoff, the statistics of ψ ± are clearly bosonic. Any “statistical interaction” between superfluidvortices must, therefore, come from this tunneling term (4.7). In contrast, once fluctuationsof the external electromagnetic field are switched on, vortices acquire fermionic statisticseven when the tunneling term (4.7) is absent. Moreover, as already noted, once the theory isgauged, a weak tunneling λ between the ψ + and ψ − flux-tubes does not affect their statistics.Hence, the fermionic statistics of flux-tubes and any potential statistics of superfluid vorticeshave different dynamical origins.Let us now turn on a weak tunneling term (4.7) and carefully examine the statistics ofresulting superfluid vortex excitations. We can think of an isolated vortex as a two-levelsystem ( ψ + , ψ − ) with an energy splitting 2∆ related to the difference of bare energies of ψ + and ψ − vortices. As we already pointed out, in the absence of an additional particle-holesymmetry, ∆ will generally be non-vanishing. Our two-level system also has a tunnelingamplitude λ (cid:104) m † (cid:105) related to the local expectation value of the monopole operator m . Hence,we can think of the vortex as a spin in a magnetic field (cid:126)b = ( λRe (cid:104) m † (cid:105) , λIm (cid:104) m † (cid:105) , ∆).As we already noted, m † may be interpreted as the local order parameter of the superfluid.According to this interpretation, the phase of (cid:104) m † (cid:105) should wind by 2 π around a superfluidvortex. This can be confirmed by an explicit calculation of the monopole expectation valuein our dual gauge theory description. Now, imagine starting with vortex 1 at (cid:126)x = ( a,
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