Topological Edge and Interface states at Bulk disorder-to-orderQuantum Critical Points

Yichen Xu,1 Xiao-Chuan Wu,1 Chao-Ming Jian,2 and Cenke Xu1

1Department of Physics, University of California, Santa Barbara, CA 93106, USA2Station Q, Microsoft, Santa Barbara, California 93106-6105, USA

We study the interplay between two nontrivial boundary effects: (1) the two dimensional (2d)edge states of three dimensional (3d) strongly interacting bosonic symmetry protected topologicalstates, and (2) the boundary fluctuations of 3d bulk disorder-to-order phase transitions. We thengeneralize our study to 2d gapless states localized at an interface embedded in a 3d bulk, whenthe bulk undergoes a quantum phase transition. Our study is based on generic long wavelengthdescriptions of these systems and controlled analytic calculations. Our results are summarized asfollows: (i.) The edge state of a prototype bosonic symmetry protected states can be driven to anew fixed point by coupling to the boundary fluctuations of a bulk quantum phase transition; (ii.)the states localized at a 2d interface of a 3d SU(N) quantum antiferromagnet may be driven to anew fixed point by coupling to the bulk quantum critical modes. Properties of the new fixed pointsidentified are also studied.

PACS numbers:

I. INTRODUCTION

The most prominent feature of topological insulators(TI)1–7 and more generally symmetry protected topolog-ical (SPT) states8,9 is the contrast between the boundaryand the bulk of the system. In particular the 2d edge of3d SPT states hosts the most diverse zoo of exotic phe-nomena that keep attracting attentions and efforts fromtheoretical physics. It has been shown that many exoticphenomena such as anomalous topological order10–16, de-confined quantum critical points17, self-dual field theo-ries18–21 can all occur on the 2d edge of 3d SPT stats.Sometimes the symmetry of the system is secretly real-ized as a self-dual transformation of the field theories atthe boundary22,23. All these suggest that the 2d bound-ary of a 3d system is an ideal platform of studying physicsbeyond the standard frameworks of condensed mattertheory.

On the other hand, even the boundary of an ordinaryLandau-Ginzburg type of quantum phase transition canhave nontrivial behaviors. It was studied and understoodin the past that the boundary of a bulk conformal fieldtheory (CFT) follows a very different critical behaviorfrom the bulk24–29, due to the strong boundary condi-tion imposed on the CFT. The boundary fluctuations (orthe boundary CFT) of the Landau-Ginzburg phase tran-sitions were studied through the standard ε−expansion,and it was shown that the critical exponents are verydifferent from the bulk. Hence if experiments are per-formed at the boundary of the system, one should referto the predictions of the boundary instead of the bulkCFT. These two different boundary effects were studiedseparately in the past. In this work we will study the in-terplay of these two distinct boundary effects. Our goalis to seek for new physics, ideally new fixed points underrenormalization group (RG) flow due to the coupling ofthe two boundary effects.

For our purpose we give the system under study a vir-

FIG. 1: We view the system under study as a two layer sys-tem. Layer-1 is a SPT or TI with nontrivial edge states;layer-2 is an ordinary disorder-to-order phase transition whoseorder parameter at the boundary follows the scaling of bound-ary CFT. The boundary of the entire system may flow to newfixed points due to the coupling between the two layers.

tual two-layer structure Fig. 1: layer-1 is a SPT statewith nontrivial edge states, and it is not tuned to a bulkphase transition; layer-2 is a topological trivial systemwhich undergoes an ordinary Landau-Ginzburg disorder-to-order phase transition. Then as a starting point weassume a weak coupling between the boundary of the twolayers, and study the RG flow of the coupling. Besidesthe edge state localized at the boundary of a SPT state,we will also consider symmetry protected gapless stateslocalized at a 2d interface embedded in a 3d bulk. Wewill demonstrate that in several cases, including the edgestate of a prototype bosonic SPT state, the 2d boundaryor interface will flow to a new fixed point due to the bulkquantum phase transition.

Previous works have explored related ideas with differ-ent approaches. Exactly soluble 1d and 2d Hamiltonianshave been constructed for gapless systems with protected

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edge states30; fate of edge states was also studied for 1dand 2d SPT states31–35. But the 2d edge of 3d bosonicSPT systems coupled with boundary modes which origi-nate from bulk quantum critical points, i.e. the situationthat potentially hosts the richest and most exotic phe-nomena, have not been studied to our knowledge. Wenote that the interaction between bulk quantum criti-cal modes and the boundary of free or weakly interact-ing fermion topological insulator (or topological super-conductor) was studied in Ref. 36, but the coupling inthat case was strongly irrelevant hence will not lead tonew physics in the infrared (we will review the interplaybetween the bulk quantum critical modes and the edgestates of free fermion topological insulator in the nextsection). We will focus on bosonic SPT state with intrin-sic strong interaction in this work. We use the genericlong wavelength field theory description of both the bulkbosonic SPT states and the edge states. Due to the lackof exact results of strongly interacting (2 + 1)d field the-ories, we seek for a controlled calculation procedure thatallows us to identify new fixed points under RG flow. In-deed, in several scenarios we will explore in this work,new fixed points are identified based on controlled calcu-lations.

II. EDGE STATES OF 3d SPT AT BULK QCP

A. Edge states of noninteracting 3d TIs

We first consider the edge state of 3d topological in-sulator (TI) and symmetry protected topological states.The edge state of free fermion TI is described by theaction

S =

∫d2xdτ

Nf∑α=1

ψαγµ∂µψα, (1)

with γ1 = σ2, γ2 = −σ1, γ0 = σ3, ψ = ψ†γ0. Basedon the “ten-fold way classification”5–7, for the AIII class,at the noninteracting level the TI is always nontrivialand topologically different from each other for arbitraryinteger−Nf ; while for the AII class the TI is nontrivialonly for odd integer Nf , and they are all topologicallyequivalent to the simplest case with Nf = 1. In bothcases the fermion mass term

∑α ψαψα is forbidden by

the time-reversal symmetry. Hence let us consider thedisorder-to-order phase transition in the 3d bulk associ-ated with a spontaneous time-reversal symmetry break-ing, which is described by an ordinary (3 + 1)d Landau-Ginzburg quantum Ising theory:

Sb =

∫d3xdτ (∂φ)2 + uφ4. (2)

Because u is a marginally irrelevant coupling at the(3 + 1)d noninteracting Gaussian fixed point, the scal-ing dimension of φ in the bulk is precisely [φ] = 1.

Here we stress that the disorder-to-order transition isdriven by the physics in the bulk. Without the bulk,the boundary alone does not support an ordered phase.To study the fate of the edge state when the bulk istuned to the quantum critical point, we view the bulk asa “two layer” system (Fig. 1): layer-1 is a 3d TI which isnot tuned to the quantum phase transition; while layer-2 is at the disorder-to-order bulk quantum phase tran-sition between a time-reversal invariant trivial insula-tor and a spontaneous time-reversal symmetry breakingphase. Now both layers have nontrivial physics at theedge. The quantum critical fluctuation (from layer-2) atthe 2d boundary must satisfy the boundary scaling law.When we impose the most natural boundary conditionφ(z ≥ 0) = 0, the leading field at the boundary whichcarries the same quantum number as φ is Φ ∼ ∂zφ. Sinceφ has scaling dimension 1, Φ should have scaling dimen-sion [Φ] = 2, i.e.

〈Φ(x, z = 0)Φ(0, z = 0)〉 ∼ 1/|x|4, (3)

where x = (τ, x, y). Eq. 3 is a much weaker correlationthan φ in the bulk (more detailed derivation of boundarycorrelation functions can be found in Ref. 24,26–28).

Now we turn on coupling between the 2d boundariesof the two layers. The edge state of the TI in layer-1 isaffected by the boundary fluctuations of layer-2 throughthe “proximity effect”. The coupling between the twolayers at the 2d boundary is described by the followingterm in the action:

Sc =

∫d2xdτ

∑α

gΦψαψα. (4)

Since Φ ∼ ∂zφ has scaling dimension 2, g will have scal-ing dimension [g] = −1, i.e. it is strongly irrelevant. Thisconclusion is consistent with previous study Ref. 36. Anegative “mass term” Φ2 will be generated through thestandard fermion loop diagram, but since Φ has scalingdimension 2, this mass term will be irrelevant. Hence theedge state of a 3d TI is stable even at the bulk quantumcritical point where the time-reversal symmetry is spon-taneously broken, and the properties of the edge states(such as electron Green’s function) should be identical tothe edge state of TI in the infrared. To make the cou-pling g relevant, the quantum critical modes also need tolocalize on the boundary, which is one of the situationsstudied in Ref. 36.

B. Edge states of bosonic SPT states

The situation of bosonic SPT phases can be much moreinteresting. The bosonic SPT state can only exist instrongly interacting systems. We use the prototype 3dbosonic SPT phase with (U(1) × U(1)) × ZT2 symmetryas an example, since this phase can be viewed as the par-ent state of many 3d bosonic SPT phases by breaking thesymmetry down to its subgroups, without fully trivializ-ing the SPT phase. The topological feature of this phase

3

can be conveniently captured by the following nonlinearsigma model in the (3 + 1)d bulk17,37:

S =

∫d3xdτ

1

g(∂n)2 +

i2π

Ω4εabcden

a∂xnb∂yn

c∂znd∂τn

e,(5)

where n is a five component vector field with unit length,and Ω4 is the volume of the four dimension sphere withunit radius. (n1, n2), and (n3, n4) transform as a vectorunder the two U(1) symmetries respectively, and the ZT2changes the sign of all components of the vector n. Thenonlinear sigma model Eq. 5 is invariant under all thetransformations.

The 2d edge state of this SPT phase can be describedby the following (2 + 1)d action:

S =

∫d2xdτ

∑α=1,2

|(∂ − ia)zα|2 + r|zα|2 + u|zα|4

+1

e2(da)2, (6)

where aµ is a noncompact U(1) gauge field. The theoryEq. 6 is referred to as the “easy-plane noncompact CP1”(EP-NCCP1) model. We are most interested in the pointr = 0. The term

∑α r|zα|2 would be forbidden if there is

an extra Z2 self-dual symmetry that exchanges the twoU(1) symmetries38, while without the self-duality sym-metry r needs to be tuned to zero, and the point r = 0becomes the transition point between two ordered phasesthat spontaneously breaks the two U(1) symmetries re-spectively39,40. At r = 0, starting with the UV fixedpoint with noninteracting zα and aµ, both u and e areexpected (though not proven) to flow to a fixed pointwith u = u∗, e = e∗.

The putative conformal field theory at r = 0 and itsfate under coupling to the boundary fluctuations (bound-ary modes) of the bulk quantum critical points is the goalof our study in this section. As was discussed in previousliteratures, it is expected that there is an emergent O(4)symmetry in Eq. 6 at r = 0, when we fully explore all theduality features of Eq. 618–21,23,38,41. In the EP-NCCP1

action, the following operators form a vector under O(4):

(n1, n2, n3, n4) ∼ (z†σ1z, z†σ2z, Re[Ma], Im[Ma]), (7)

where Ma is the monopole operator (the operator thatannihilates a quantized flux of aµ). In the equationabove, (n1, n2) and (n3, n4) form vectors under the twoU(1) symmetries respectively. The emergent O(4) in-cludes the self-dual Z2 symmetry of the EP-NCCP1, i.e.the operation that exchanges the two U(1) symmetries.

Now we consider the 3d bulk quantum phase tran-sition between the SPT phase and the ordered phasesthat break part of the defining symmetries of the SPTphase. We first consider two order parameters: φ0, φ3.φ0 is the order parameter that corresponds to the self-dual Z2 symmetry; and φ3 is a singlet under the emer-gent SO(4) but odd under the improper rotation of theemergent O(4), and also odd under ZT2 . Again we view

our system as a two layer structure: layer-1 is a SPTphase with solid edge states described by Eq. 6; layer-2is a topological-trivial system that undergoes the tran-sition of condensation of either φ0 or φ3. Both orderparameters have an ordinary mean field like transitionin the bulk of layer-2. Again at the boundary, both or-der parameters will have very different scalings from thebulk. We assume that system under study fills the en-tire semi-infinite space at z < 0, then at the boundaryplane z = 0, the most natural boundary condition is thatφ0(z ≥ 0) = φ3(z ≥ 0) = 0, hence all order parame-ters near but inside the bulk should be replaced by thefollowing representations: Φ0 ∼ ∂zφ0, Φ3 ∼ ∂zφ3. Bothorder parameters have scaling dimensions 2 at the (2+1)dboundary of layer-2.

Now we couple Φ0 and Φ3 to the edge states of layer-1.The coupling will take the following form:

Lc0 =∑α

g0Φ0|zα|2, Lc3 = g3Φ3z†σ3z. (8)

The RG flow of coupling constants g0,3 can be systemat-ically evaluated in certain large−N generalization of theaction in Eq. 6:

S =

∫d2xdτ

∑α=1,2

N/2∑j=1

|(∂ − ia)zj,α|2 + u(∑j

|zj,α|2)2.(9)

The large−N generalization facilitate calculations of theRG flow, but the down side is that the duality structureand emergent symmetries no longer exist for N > 2. Inthe large−N limit of Eq. 9, the scaling dimension of theoperators under study is

N → +∞ : [z†σ3z] = [|z|2] = 2. (10)

In the equation above, each operator has a sum of indexj, which was not written explicitly. Apparently couplingconstants g0,3 are both irrelevant with large−N due tothe weakened boundary correlation of Φ0 and Φ3.

We are seeking for more interesting scenarios when theboundary is driven to a new fixed point due to the bulkquantum criticality. For this purpose we consider another

order parameter ~φ which transforms as a vector under oneof the two U(1) symmetries. Here we no longer assumethe Z2 self-dual symmetry on the lattice scale. Again at

the boundary ~φ should be replaced by ~Φ ∼ ∂z~φ. At the

2d boundary, the coupling between ~Φ and the edge stateof layer-2 reads

Lcv = gv(Φ1z

†σ1z + Φ2z†σ2z

). (11)

In the large−N limit of Eq. 9, the scaling dimension ofthe operators under study is

N → +∞ : [z†σ1z] = [z†σ2z] = 1. (12)

Hence gv is marginal in the large−N limit, and there isa chance that gv could drive the system to a new fixedpoint with 1/N corrections.

4

We introduce the following action in order to computethe RG flow of gv with finite but large N :

S =

∫d2xdτ

∑α=1,2

N/2∑j=1

|(∂ − ia)zj,α|2 + iλ+|zj,α|2

+ iλ−z†jσ

3zj + igv~Φ · z†j~σzj +1

2~Φ · 1

|∂|~Φ. (13)

The λ± are two Hubbard-Stratonovich (HS) fields intro-duced for the standard 1/N calculations42,43. The scalingof |z|2 and z†σ3z in Eq. 9 are replaced by the HS fieldsλ+, λ− in the new action Eq. 13 respectively. A coeffi-cient “i” is introduced in the definition of gv by redefiningΦ→ iΦ for convenience of calculation.

FIG. 2: (a, b) the 1/N contribution to z†σ1,2z and ψτ1,2ψfrom the gauge field fluctuation, the solid lines represent ei-ther the propagator of zα or ψα, the wavy line represents thepropagator of the photon; (c, d) the 1/N contribution to z†~σzfrom λ± in Eq. 13; (e, f) the contribution to B in Eq. 14.

The schematic beta function of gv reads

dgvd ln l

= (1−∆v)gv −Bg3v +O(v5). (14)

∆v is the scaling dimension of z†j~σzj in the large−N gen-

eralization of the EP-NCCP1 model Eq. 9, with ~σ =(σ1, σ2). The standard 1/N calculation leads to

∆v = 1− 56

3π2N+O(

1

N2). (15)

FIG. 3: The two diagrams at g3v order which cancel each otherfor arbitrary gauge choices.

The 1/N correction of ∆v comes from diagram Fig. 2(a−d), where the wavy line is the gauge boson propagator,and the dashed line represents propagators of both λ±.The first term of Eq. 15 implies that gv is indeed weaklyrelevant with finite but large−N .

The constant B in the beta function arises from theoperator product expansion of the coupling term Eq. 11,which is equivalent to the diagrams Fig. 2e, f . This com-putation leads to B = 1/(3π2). The two diagrams inFig. 3 which are also at g3v order cancel each other forarbitrary gauge choices. Similar two-loop diagrams atthe same order of 1/N do not enter the RG equation dueto lack of logarithmic contribution, as was explained in

Ref. 43. ~Φ does not receive a wave function renormaliza-tion due to the singular form of its action. Hence withfinite but large−N , gv indeed flows to a new fixed point:

g2v∗ =56

N+O(

1

N2). (16)

We stress that this result is drawn from a controlled cal-culation and it is valid to the leading order of 1/N .

FIG. 4: The g2v diagrams that contributes to the scaling di-mension of [λ+]. Here the solid line represents the propagator

of zj,α, the dotted line represents the vector operator ~Φ, andthe dashed line represents λ+.

As we explained before, the point r = 0 is a direct tran-sition between two ordered phases that spontaneouslybreak the two U(1) symmetries. This transition will bedriven to a new fixed point by coupling to the boundaryfluctuations of bulk critical points as we demonstratedabove. At this new fixed point, the critical exponent νfollows from the relation

ν−1 = 3− [λ+]. (17)

To evaluate the scaling dimension [λ+] we have to incor-porate the contributions of g2v from the diagrams shown

5

in Fig. 4, and combined with 1/N calculations performedpreviously43,44. Then in the end we obtain

ν−1∗ = 1 +160

3π2N+

4g2v∗3π2

+O(1

N2)

= 1 +128

π2N+O(

1

N2). (18)

Again, there are other loop diagrams which appear to beat the same order of 1/N but do not make any logarith-mic contributions43.

III. INTERFACE STATES EMBEDDED IN 3dBULK

A. Interface states of noninteracting electronsystems

FIG. 5: We consider a SU(N) antiferromagnet with self-conjugate representation on each site. The system forms abackground VBS pattern, with opposite dimerizations be-tween semi-infinite spaces z > 0 and z < 0. There is a 2dantiferromagnet localized at the interface z = 0, and the en-tire bulk can undergo phase transition simultaneously due tothe mirror (reflection) symmetry that connects the two sidesof the domain wall.

In previous examples we studied topological edgestates at the boundary of a 3d system. In this sectionwe will consider the 2d states localized at an interface(z = 0) in a 3d space, when the entire 3d bulk (for bothz > 0 and z < 0 semi-infinite spaces) undergoes a phasetransition simultaneously. Without fine-tuning, we needto assume an extra reflection symmetry z → −z thatconnects the two sides of the interface, which guaranteesa simultaneous phase transition in the entire system. In

this case there is no physical reason to impose the strongboundary condition at the interface embedded in the 3dspace, hence the quantum critical modes at the interfacefollow the ordinary bulk scalings, instead of the weakenedcorrelation of boundary CFT.

Again we will consider free fermion systems first. Letus first recall that the AIII class TI has a Z classificationwhich is characterized by a topological index nT . nT willappear as the coefficient of the electromagnetic responseof the TI: L ∼ iπnTE · B. nT must change sign underspatial reflection transformation Mz : z → −z. To con-struct the desired system, we assume the semi-infinitespace z < 0 is occupied with the AIII class TI withHamiltonian H, whose topological index is nT ; and its“reflection conjugate” M−1z HMz fills the semi-infinitespace z > 0. Then there are Nf = 2nT flavors of mass-less Dirac fermions localized at the 2d plane z = 0, whichare still protected by time-reversal symmetry. Now weassume the entire bulk undergoes a quantum phase tran-sition with a spontaneous time-reversal symmetry break-ing, whose order parameter couples to the domain wallDirac fermions as

S =

∫d2xdτ

Nf∑α=1

ψαγµ∂µψα + gφψαψα

+1

2φ(−∂2)1/2φ. (19)

The last term in the action is still defined in the (2 + 1)dinterface, and it reproduces the correlation of φ in thebulk: 〈φ(0)φ(r)〉 ∼ 1/r2. We stress that, since now theorder parameter resides in the entire bulk, φ no longerobeys the boundary scaling as we discussed in previousexamples. A negative boson mass term −rφ2 can be gen-erated through the standard fermion mass loop diagram,hence we need to tune an extra term at the interface tomake sure the mass term of φ vanishes.

In this case the coupling constant g is a marginal per-turbation based on simple power-counting. But g willflow under renormalization group (RG) with loop correc-tions in Fig. 2(e, f):

β(g) =dg

d ln l= − 2

3π2g3 +O(g5). (20)

Hence even in this case, the coupling between the do-main wall states and the bulk quantum critical modes isperturbatively marginally irrelevant.

So far we have assumed that the velocity of the inter-face state is identical with the bulk. Now let us tune thevelocity of the domain wall Dirac fermions slightly differ-ent, which can be captured by the following term in theLagrangian:∑

α

δψα(γ1∂x + γ2∂y − 2γ3∂3)ψα. (21)

δ defined above is an eigenvector under the leading orderRG flow. With the loop diagrams in Fig. 6, we obtain

6

the leading order beta function of δ:

β(δ) =dδ

d ln l= − 1

5π2g2δ. (22)

Together with β(g), the velocity anisotropy is also per-turbatively irrelevant.

FIG. 6: The Feynman diagrams that renormalizes the extravelocity δ in Eq. 21. The box represents the vertex δ, andall three diagrams contributes to the fermion self-energy andrenormalize δ.

FIG. 7: The extra diagrams that contribute to the scalingdimension of

∑α ψαψα at the leading order of 1/Nf in QED3.

Again the wavy lines are photon propagators.

B. Interface states of quantum antiferromagnet

We now consider a SU(N) quantum antiferromagneton a tetragonal lattice with a self-conjugate representa-tion on each site (we assume N is an even integer). Withlarge−N , an antiferromagentic Heisenberg SU(N) modelhas a dimerized ground state45,46 where the two SU(N)spins on two nearest neighbor sites form a spin singlet(valence bond). We consider the following backgroundconfiguration of valence bond solid (VBS): the spins formVBS along the z direction which spontaneously break thetranslation symmetry, while there is a domain wall be-tween two opposite dimerizations at the 2d XY planez = 0, namely z = 0 is still a mirror plane of the system(Fig. 5). In each 1d chain along the z direction, thereis a dangling self-conjugate SU(N) spin localized on thesite at the domain wall. Hence the 2d domain wall iseffectively a SU(N) antiferromagnet on a square lattice.

One state of SU(N) antiferromagnet which is the “par-ent” state of many orders and topological orders on thesquare lattice, is the gapless π−flux U(1) spin liquid47,48.At low energy this spin liquid is described by the follow-ing action of (2 + 1)d quantum electrodynamics (QED3):

S =

∫d2xdτ

Nf∑α=1

ψαγµ(∂µ − iaµ)ψα + · · · (23)

ψα is Nf = 2N flavors of 2−component Dirac fermions,and they are the low energy Dirac fermion modes of the

slave fermion fj,α defined as Sbj = f†j,αTbαβfj,β , T b with

b = 1 · · ·N2 − 1 are the fundamental representation ofthe SU(N) Lie Algebra. Besides the spin components,there is an extra two dimensional internal space whichcorresponds to two Dirac points in the Brillouin zone.There is an emergent SU(Nf ) flavor symmetry in QED3

which includes both the SU(N) spin symmetry and dis-crete lattice symmetry.

It is known that when Nf is greater than a critical in-teger, the QED3 is a conformal field theory (CFT). Wewill consider the fate of this CFT when the three di-mensional bulk is driven to a quantum phase transition.We will first consider a disorder-to-order quantum phasetransition, where the ordered phase spontaneously breaksthe time-reversal and parity symmetry of the XY plane.Notice that due to the reflection symmetry z → −z ofthe background VBS configuration, the two sides of thedomain wall will reach the quantum critical point simul-taneously. The bulk transition is still described by Eq. 2.When we couple the Ising order parameter φ to the do-main wall QED3, the total (2 + 1)d action reads

S =

∫d2xdτ

Nf∑α=1

ψαγµ(∂µ − iaµ)ψα

+ gφψαψα +1

2φ(−∂2)1/2φ. (24)

If the gauge field fluctuation is ignored, or equivalently inthe large−Nf limit, the scaling dimension of ψψ is [ψψ] =2, and hence the scaling dimension of g is [g] = 0, i.e. g isa marginal perturbation. The 1/Nf correction to the RGflow arises from the Feynman diagrams (Fig. 2(a, b) andFig. 7) which involves one or two photon propagators:

Gaµν(~p) =16

Nfp

(δµν −

pµpνp2

). (25)

Again in this case the fermions will generate a mass termfor the order parameter at the interface, which we need totune to zero. At the leading order of 1/Nf the correctedbeta function for g reads

β(g) =dg

d ln l= − 128

3π2Nfg − 2

3π2g3 +O(g3). (26)

But this beta function does not lead to a new unitaryfixed point other than the decoupled fixed point g = 0.Hence in this case the domain wall state is decoupledfrom the bulk quantum critical modes in the infraredlimit.

A more interesting scenario is when the bulk undergoesa transition which spontaneously breaks the translationand C4 rotation symmetry by developing an extra VBSorder within the XY plane. The inplane VBS order pa-rameters are Vx ∼ ψτ1ψ, and Vy ∼ ψτ2ψ, where τ1,2 arethe Pauli matrices operating in the Dirac valley space.

7

The coupling between the VBS order parameter and thedomain wall QED3 reads

Sc =

∫d2xdτ g

(φ∗ψτ−ψ + φψτ+ψ

)+ φ∗(−∂2)1/2φ.(27)

Here τ± = (τ1 ± iτ2)/2. The scaling dimension of theVBS order parameter at the QED3 fixed point has beencomputed previously48–50: [ψτaψ] = 2−64/(3π2Nf ), andthe beta function of g to the leading order of 1/Nf reads

β(g) =64

3π2Nfg − 1

6π2g3 +O(g3). (28)

In the large−Nf limit, the coupling g is marginally irrel-evant; but with finite and large−Nf , g is weakly relevantat the noninteracting fixed point, and it will flow to aninteracting fixed point

g2∗ =128

Nf+O(

1

N2f

). (29)

This new fixed point will break the emergent SU(Nf )flavor symmetry down to SU(N)×U(1) symmetry, whereU(1) corresponds to the rotation of the Dirac valleyspace. The following gauge invariant operators receivedifferent corrections to their scaling dimensions from cou-pling to the bulk quantum critical modes:

[ψψ] = 2 +128

3π2Nf+

2

3π2g2∗ +O(

1

N2f

);

[ψT bψ] = 2− 64

3π2Nf+

2

3π2g2∗ +O(

1

N2f

);

[ψτ3ψ] = 2− 64

3π2Nf− 1

3π2g2∗ +O(

1

N2f

);

[ψτ1,2ψ] = 2− 64

3π2Nf+

1

6π2g2∗. (30)

The operators ψτ1,2ψ have exactly scaling dimension 2,the Feynman diagram contributions from Fig. 2 canceleach other for operator ψτ1,2ψ as they should. Noticethat the last three operators in Eq. 30 should have thesame scaling dimension in the original QED3 fixed pointdue to the large SU(Nf ) flavor symmetry, but at this newfixed point they will acquire different corrections.

Another interesting scenario is that the bulk is at acritical point whose order parameter couples to the Isinglike operator ψτ3ψ, which breaks the inplane parity butpreserves the time-reversal:

Sc =

∫d2xdτ gφψτ3ψ +

1

2φ(−∂2)1/2φ. (31)

The microsopic representation of the operator ψτ3ψ canbe found in Ref. 48. The beta function of the coupling greads

β(g) =64

3π2Nfg − 2

3π2g3 +O(g3), (32)

and once again there is new stable fixed point g2∗ =32/Nf +O(1/N2

f ). And at this fixed point,

[ψψ] = 2 +128

3π2Nf+

2

3π2g2∗ +O(

1

N2f

);

[ψT bψ] = 2− 64

3π2Nf+

2

3π2g2∗ +O(

1

N2f

);

[ψτ1,2ψ] = 2− 64

3π2Nf− 1

3π2g2∗ +O(

1

N2f

);

[ψτ3ψ] = 2− 64

3π2Nf+

2

3π2g2∗. (33)

The domain wall state considered here is formallyequivalent to the boundary state of a 3d bosonic SPTstate with pSU(N)× U(1) symmetry, which can also beembedded to the 3d SPT with pSU(Nf ) symmetry dis-cussed in Ref. 51. This SPT state can be constructed asfollows: we first break the U(1) symmetry in the 3d bulkby driving the bulk z < 0 into a superfluid phase, andthen decorate the vortex loop of the superfluid phase witha 1d Haldane phase with pSU(N) symmetry52–55. Even-tually we proliferate the decorated vortex loops to restoreall the symmetries in the bulk. A 1d pSU(N) Haldanephase can be constructed as a spin-chain with a pSU(N)spin on each site, and there is a dangling self-conjugaterepresentation of SU(N) on each end of the chain. Andthis dangling spin will also exist in the U(1) vortex at theboundary of the pSU(N)×U(1) SPT state. Notice thatthe self-conjugate representation of SU(N) is a projectiverepresentation of pSU(N).

IV. DISCUSSION

In this work we systematically studied the interplayof two different nontrivial boundary effects: the 2d edgestates of 3d symmetry protected topological states, andthe boundary fluctuations of 3d bulk quantum phasetransitions. New fixed points were identified throughgeneric field theory descriptions of these systems and con-trolled calculations. We then generalized our study to the2d states localized at the interface embedded in the 3dbulk.

The last case studied in Eq. 32, 33 is special whenNf = 2, and when the gauge field is noncompact. Thisis the theory that has been shown to be dual to theEP-NCCP1 model21,41 studied in Eq. 6, the operator∑α r|zα|2 is dual to rψτ3ψ, and both theories are self-

dual. By coupling the operator ψτ3ψ to the bulk criticalmodes (rather than the boundary fluctuations of the bulkcritical points), we have shown that this (2 + 1)d theoryis driven to a new fixed point, and the self-duality struc-ture still holds. The self-duality transformation of Eq. 6now is combined with the Ising symmetry of the orderparameter φ. However, the O(4) emergent symmetry nolonger exists at this new fixed point, due to the nonzerofixed point of g in Eq. 31.

8

The methodology used in this work can have manypotential extensions. We can apply the same field the-ory and RG calculation to the 1d boundary of 2d SPTstates (for instance the AKLT state), which was studiedthrough exactly soluble lattice Hamiltonians30 and alsonumerical methods33–35. Also, 1d defect in a 3d topolog-ical state can also have gapless modes56,57, it would beinteresting to investigate the fate of a 1d defect embeddedin a 3d bulk at the bulk quantum phase transition. De-fects of free or weakly interacting fermionic topologicalinsulator and topological superconductor coupled withbulk critical modes was studied in Ref. 36, but we expect

the defect of an intrinsic strongly interacting topologicalstate can lead to much richer physics. Last but not least,the “higher order topological insulator” has nontrivialmodes localized at the corner instead of the boundary ofthe system58. The coupling between the bulk quantumcritical points and corner topological modes is also worthexploration.

This work is supported by NSF Grant No. DMR-1920434, the David and Lucile Packard Foundation, andthe Simons Foundation. The authors thank AndreasLudwig and Leon Balents for helpful discussions.

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