Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements

Symmetry-protected topological (SPT) phases exhibit nontrivial short-ranged entanglement protected by symmetry and cannot be adiabatically connected to trivial product states while preserving the symmetry. In contrast, intrinsic topological phases do not need ordinary symmetry to stabilize them and their ground states exhibit long-range entanglement. It is known that for a given symmetry group $G$, the 2D SPT phase protected by $G$ is dual to the 2D topological phase exemplified by the twisted quantum double model $D^{\omega }\left(G\right)$ via gauging the global symmetry $G$. Recently it was realized that such a general gauging map can be implemented by some local unitaries and local measurements when $G$ is a finite, solvable group. Here, we review the general approach to gauging a $G\text{−}\mathrm{SPT}$ starting from a fixed-point ground-state wave function and applying a $N$-step gauging procedure. We provide an in-depth analysis of the intermediate states emerging during the $N$-step gauging and provide tools to measure and identify the emerging symmetry-enriched topological order (SET) of these states. We construct the generic lattice parent Hamiltonians for these intermediate states and show that they form an entangled superposition of a twisted quantum double (TQD) with an SPT-ordered state. Notably, we show that they can be connected to the TQD through a finite-depth, local quantum circuit which does not respect the global symmetry of the SET order. We introduce the so-called symmetry branch line operators and show that they can be used to extract the symmetry fractionalization classes (SFC) and symmetry defectification classes (SDC) of the SET phases with the input data $G$ and $\left[\omega \right]\in H^3(G,U\left(1\right))$ of the pregauged SPT-ordered state. We illustrate the procedure of preparing and characterizing the emerging SET-ordered states for some Abelian and non-Abelian examples such as dihedral groups $D_n$ and the quaternion group $Q_8$.

Figure 1

On an oriented triangulated plane, two typical simplexes with opposite orientations are shown. Their corresponding cocycles are $\omega (g_3{g}_2^{-1},g_2{g}_1^{-1},g_1)$ and $\omega {(g_{3^{\prime }}{g}_{2^{\prime }}^{-1},g_{2^{\prime }}{g}_{1^{\prime }}^{-1},g_{1^{\prime }})}^{-1}$.

Figure 2

The phase factor $\text{Amp}(\left\{g_v\right\},x)$ can be given by the triangulation of such prisms, where $g_{v^{\prime }}=g_{v}x$.

Figure 3

A positively oriented tetrahedron. The 3-cocycle assigned to it is $\omega (g_4{g}_3^{-1},g_3{g}_2^{-1},g_2{g}_1^{-1})$.

Figure 4

The gauging map $\mathrm{\Gamma }$ maps from vertex DOFs to edge DOFs.

Figure 5

The phase ${\tilde{W}}_v^g$ is defined as the multiplication of the phases corresponding to the tetrahedrons. Here, $g_{v^{\prime }v}=g$.

Figure 6

A negatively oriented tetrahedron. The 3-cocycle assigned to it is ${\omega }^{-1}(g_{43},g_{32},g_{21})$.

Figure 7

The gauging map ${\mathrm{\Gamma }}_N$ maps the normal part from vertex DOFs to edge DOFs.

Figure 8

The phase $W_v^g$ is defined as the multiplication of the phases corresponding to the tetrahedrons. $h_{v^{\prime }v}=x,q_{v^{\prime }}=q_v$.

Figure 9

A negatively oriented tetrahedron. The 3-cocycle assigned to it is ${\omega }^{-1}(g_4{h}_{43}{g}_3^{-1},g_3{h}_{32}{g}_2^{-1},g_2{h}_{21}{g}_1^{-1})$.

Figure 10

The transmutation rule of $z$ factor according to relation $z\left(g_{v^{\prime }}\right)z\left(g_v{g}_{v^{\prime }}^{-1}\right)=z\left(g_v\right)$.

Figure 11

The transmutation rule for $z_n$ factors on step (4).

Figure 12

The transmutation rule for $z_q$ factors on step (7).

Figure 13

One plaquette on a triangulated lattice.

Figure 14

(a) Symmetry action inside of region $\mathcal{R}$ “lifts” $\mathcal{R}$ such that all the simplex in the region correspond to $\tilde{\omega }$. (b) This symmetry action can be equivalently regarded as the insertion of symmetry branch line on $\partial \mathcal{R}$.

Figure 15

$|h_{ij}{\rangle }_e$ is located on edge $e=\langle i,j\rangle ,|g_i{\rangle }_v$ and $|g_j{\rangle }_v$ are located on the two endpoints $i$ and $j$.

Figure 16

An open defect on a triangulated lattice.

Figure 17

The operator ${\left(H_{\partial \mathcal{R}}^x\right)}^2$ applying on curve $\partial \mathcal{R}$ is equivalent to the braiding phase between the $g$-flux of an anyon $b(g$ is the holonomy along $\partial \mathcal{R})$, and the charge of the anyon $\mathcal{w}$.

Figure 18

The braiding phase between a $g$-fluxed ${\mu }_g$-charged anyon $b$ and the anyon $\mathcal{w}(=0_x\times 0_x)$ is written as a product of two braiding phases, as presented in Eq. (132). The second factor in blue is a consequence of Eq. (126). From the phase $B(\mathcal{w},b)$, we infer the unknown charge of the anyon $\mathcal{w}$ (written as “?”), which determines the SFC.

Figure 19

Tetrahedron with orientation.

Figure 20

(a) Symmetry action inside of region $\mathcal{R}$ “lifts” $\mathcal{R}$ such that all the simplex in the region correspond to $\tilde{\omega }$. (b) This symmetry action can be equivalently regarded as the insertion of symmetry branch line on $\partial \mathcal{R}$.

Figure 21

There are three tetrahedrons when one plaquette is lifted.

Figure 22

Geometric diagram illustrating the phase combination in Eq. (D16). Here $g_{v^{\prime }}{g}_v^{-1}=y$ and $g_{v^{″}}{g}_{v^{\prime }}^{-1}=x$ for $v\in V$ (which are enumerated by 1,2,3,4).