Bosonic anomalies, induced fractional quantum numbers, and degenerate zero modes: The anomalous edge physics of symmetry-protected topological states

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of (2+1)D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to $G={\prod }_i{Z}_{N_i}=Z_{N_1}\times Z_{N_2}\times Z_{N_3}\times \cdots )$. We demonstrate that some classes of SPTs (termed “Type II”) trap fractional quantum numbers (such as fractional $Z_N$ charges) at the 0D kink of the symmetry-breaking domain walls, while some classes of SPTs (termed “Type III”) have degenerate zero energy modes (carrying the projective representation protected by the unbroken part of the symmetry), either near the 0D kink of a symmetry-breaking domain wall, or on a symmetry-preserving 1D system dimensionally reduced from a thin 2D tube with a monodromy defect 1D line embedded. More generally, the energy spectrum and conformal dimensions of gapless edge modes under an external gauge flux insertion (or twisted by a branch cut, i.e., a monodromy defect line) through the 1D ring can distinguish many SPT classes. We provide a manifest correspondence from the physical phenomena, the induced fractional quantum number, and the zero energy mode degeneracy to the mathematical concept of cocycles that appears in the group cohomology classification of SPTs, thus achieving a concrete physical materialization of the cocycles. The aforementioned edge properties are formulated in terms of a long wavelength continuum field theory involving scalar chiral bosons, as well as in terms of matrix product operators and discrete quantum lattice models. Our lattice approach yields a regularization with anomalous non-onsite symmetry for the field theory description. We also formulate some bosonic anomalies in terms of the Goldstone-Wilczek formula.

Figure 1

The boundary of 2D SPT state harbors 1D gapless edge modes if the global symmetry is preserved (not broken spontaneously or explicitly). The global symmetry transformation $S$ of 1D edge mode acts in a non-onsite manner, where $S$ cannot be written as a tensor product form on each site (i.e., the symmetry operator $S$ acts on more than a single site for its tensor operators, where we show schematically $S$ acts on two neighbor sites.)

Figure 2

The intuitive way to view the bulk-boundary correspondence for edge modes of SPTs (or intrinsic topological order) under the flux insertion, or equivalently the monodromy defect/branch cut (blue dashed line) modifying the bulk and the edge Hamiltonians. SPTs located on a large sphere with two holes with flux-in and flux-out is analogous to a Laughlin-type flux insertion through a cylinder, inducing anomalous edge modes (red arrows) moving along the opposite directions on two edges.

Figure 3

The illustration of 1D lattice model with $M$ sites on a compact ring.

Figure 4

We expect some fractional charge trapped near a single kink around $x=0$ (i.e., $x=0+\epsilon )$ and $x=L$ (i.e., $x=0-\epsilon )$ in the domain walls. For $Z_{N_1}$-symmetry-breaking domain wall with a kink jump $\mathrm{\Delta }{\varphi }_1=2\pi \frac{n_{12}}{N_{12}}$, we predict that the fractionalized $(n_{12}/N_{12})p_{21}$ units of $Z_{N_2}$ charge are induced.

Figure 5

A nontrivial winding ${\int }_0^{L}dx{\partial }_x\varphi \left(x\right)=2\pi$. This is like a soliton (or particle) insertion. For $N_{12}$ number of $Z_{N_1}$-symmetry-breaking domain walls, we predict that the integer $p_{21}$ units of total induced $Z_{N_2}$ charge on a 1D ring. On average, each kink captures a $p_{21}/N_{12}$ fractional units of $Z_{N_2}$ charge.

Figure 6

A profile of several domain walls, each with kinks and antikinks (in blue color). For $Z_{N_1}$-symmetry-breaking domain wall, each single kink can trap fractionalized $Z_{N_2}$ charge. However, overall there is no nontrivial winding ${\int }_0^{L}dx{\partial }_x{\varphi }_1\left(x\right)=0$ (i.e., no net soliton insertion), so there is no net induced charge on the whole 1D ring.

Figure 7

In the fermionized language, one can capture the anomaly effect on induced (fractional) charge/current under soliton background by the one-loop diagram [43]. The solid line represents fermions, the wavy line   represents the external (gauge) field coupling to the induced current $J^{\mu }$ (or charge $J^0)$, and the dashed line represents the scalar soliton (domain walls) background. Here in Sec. 4b, instead of fermionizing the theory, we directly address in the bosonized language to capture the bosonic anomaly.

Figure 8

(a) The induced 2-cocycle from a 2+1D $M^3=M^2\times I^1$ topology with a symmetry-preserving $Z_{N_u}$ flux $A$ insertion. (b) Here $M^2=S^1\times I^1$ is a 2D spatial cylinder, composed by $A$ and $B$, with another extra time dimension $I^1$. Along the $B$ line we insert a monodromy defect of $Z_{N_1}$, such that $A$ has a nontrivial group element value $A=g_{1^{\prime }}{g}_1^{-1}=g_{2^{\prime }}{g}_2^{-1}=g_{3^{\prime }}{g}_3^{-1}\in Z_{N_1}$. The induced 2-cocycle ${\beta }_A(B,C)$ is a nontrivial element in ${\mathcal{H}}^2(Z_{N_v}\times Z_{N_w},\text{U}\left(1\right))$ $={\mathbb{Z}}_{N_{vw}}$ (here $u,v,w$ cyclic as ${\epsilon }^{uvw}=1)$, thus which carries a projective representation. (c) A monodromy defect can be viewed as a branch cut induced by a ${\mathrm{\Phi }}_B$ flux insertion (both modifying the Hamiltonians). (d) This means that when we do dimensional reduction on the compact ring $S^1$ and view the reduced system as a 1D line segment, there are $N_{123}$ degenerate zero energy modes (due to the nontrivial projective representation).

Figure 9

The triangulation of a $M^3=M^2\times I^1$ topology (here $M^2$ is a spatial cylinder composed by the $A$ and $B$ directions, with a $I^1$ time) into three tetrahedra with branched structures.

Figure 10

The $Z_{N_1}$-symmetry-breaking domain wall along the red $\times$ mark and/or orange + mark, which induces $N_{123}$-fold degenerate zero energy modes. The situation is very similar to Fig. 8 (however, there was $Z_{N_1}$-symmetry-preserving flux insertion). We show that in both cases the induced 2-cochain from calculating path integral ${\mathbf{Z}}_{\text{SPT}}$ renders a nontrivial 2-cocycle of ${\mathcal{H}}^2(Z_{N_2}\times Z_{N_3},\text{U}\left(1\right))$ $={\mathbb{Z}}_{N_{23}}$, thus carrying nontrivial projective representation of symmetry.

Figure 11

(a) Thread a gauge flux ${\mathrm{\Phi }}_B$ through a 1D ring (the boundary of 2D SPT). (b) The gauge flux is effectively captured by a branch cut (the dashed line in the blue color). Twisted boundary condition is applied on the branch cut. The (a) and (b) are equivalent in the sense that both cases capture the equivalent physical observables, such as the energy spectrum.

Figure 12

The illustration of an effective 1D lattice model with $M$ sites on a compact ring under a discrete $Z_N$ flux insertion. Effectively the gauge flux insertion is captured by a branch cut located between the site $M$ and the site 1. This results in a $Z_N$ variable $\omega$ insertion as a twist effect modifying the lattice Hamiltonian around the site $M$ and the site 1.

Figure 13

For topological phases, the anomalous current $J_b$ of the boundary theory along the $x$ direction leaks to $J_y$ along the $y$ direction in the extended bulk system. ${\mathrm{\Phi }}_B$-flux insertion $d{\mathrm{\Phi }}_B/dt=-\oint EdL$ induces the electric $E_x$ field along the $x$ direction. The effective Hall effect dictates that $J_y={\sigma }_{xy}{E}_x={\sigma }_{xy}{\varepsilon }^{\mu \nu }{\partial }_{\mu }{A}_{\nu }$, with the effective Hall conductance ${\sigma }_{xy}$ probed by an external U(1) gauge field $A$.

Figure 14

In the fermionic language, the 1+1D chiral fermions (represented by the solid line) and the external U(1) gauge field (represented by the wavy curve) contribute to a one-loop Feynman diagram correction to the axial current $j_A^{\mu }$. This leads to the nonconservation of $j_A^{\mu }$ as the anomalous current ${\partial }_{\mu }{j}_A^{\mu }={\varepsilon }^{\mu \nu }(q{K}^{-1}q/2\pi )F_{\mu \nu }.$