A classification of symmetry enriched topological phases with exactly solvable models

TL;DR

This work proposes a unifying classification of bosonic gapped quantum phases with or without topological order in the presence of on-site symmetry, using group cohomology $H^{d+1}(SG\times GG,U(1))$ to encode SPT, DW, and SET data. It presents exactly solvable lattice models that realize each class, linking DW gauge theories and SPT phases via a geometric interpretation of group cohomology and 3-cocycle twists, and derives the twisted extended ribbon algebra describing elementary excitations and their braiding via a quasi-quantum double $\mathbf{D}^{\tilde{\omega}}(GG)$. The paper further analyzes symmetry fractionalization, projects out the distinct SET components $SFC(SG,GG)$ and $EXTRA(SG,GG)$, and provides concrete examples showing SF, symmetry-protected degeneracy, and species-interchanging phenomena, illustrating the richness of SET physics and guiding numerical/experimental probes. It also discusses higher-dimensional generalizations, continuous/anti-unitary symmetries, and limitations, highlighting that the classification, while powerful, is not fully complete and may miss phases outside the DW/SPT framework. The results establish measurable signatures and a solid theoretical toolkit for exploring SET phases across condensed-matter systems.

Abstract

Recently a new class of quantum phases of matter: symmetry protected topological states, such as topological insulators, attracted much attention. In presence of interactions, group cohomology provides a classification of these [X. Chen et al., arXiv:1106.4772v5 (2011)]. These phases have short-ranged entanglement, and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement, in the presence or absence of on-site global symmetries. In 2+1 dimensions, the quantum phases in the presence of a global symmetry group SG, and with topological order described by a finite gauge group GG, are classified by the cohomology group H^3(SGxGG,U(1)). Generally in d+1 dimensions, such quantum phases are classified by H^{d+1}(SGxGG,U(1)). Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [X. Chen et al., arXiv:1106.4772v5 (2011)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases, and generalizations of our classification are discussed.

Figures (24)

• Figure 1: Illustration of the basic assumption of symmetry fractionalization Eq.(\ref{['eq:s_f_assumption']}): Under a symmetry transformation $U(g)$ with $\forall g\in SG$, an excitated state is transformed by the product of local transformation operators $U_i(g)$, with each operator only acting on one quasiparticle locally.
• Figure 2: A bilayer system in which the topological order and the global symmetry decouple.
• Figure 3: The 3-cocycle $\omega(h_1,h_2,h_3)$ assigns a $U(1)$ complex number (i.e. a phase) to a 3-simplex (tetrahedron). (a) Left to Center: the "ordering" of tetrahedron's four vertices, we choose here $1\rightarrow 2\rightarrow 3\rightarrow 4$. An edge is oriented from lower to higher vertex, so no triangle forms an oriented loop. (Alternatively, one orients all edges without forming oriented triangle loops, and a consistent underlying vertex ordering is guaranteed.) Center to Right: "Coloring" assigns group element $g_{ij}$ to $j\rightarrow i$ edge, with $g_{ji}=g_{ij}^{-1}$. (Four out of six elements are shown explicitly.) The shown tetrahedron $1\rightarrow 2\rightarrow 3\rightarrow 4$ is assigned the phase $\omega(g_{43},g_{32},g_{21})^\epsilon=\omega(g_{43},g_{32},g_{21})$. The exponent $\epsilon=\pm1$ is determined by: (b) Chirality. For tetrahedron $1\rightarrow 2\rightarrow 3\rightarrow 4$, looking from vertex 1, (counter-)clockwise loop 234 means $\epsilon=-1$ ($+1$), which is realized in the right (left) tetrahedron. (c) The zero-flux rule applies to all tetrahedron faces, i.e. triangles. Generally, $g_{ki}\cdot g_{ij}\cdot g_{jk}=\openone$, the group identity element. Recall that $g_{jk}=g_{kj}^{-1}$. Choosing an ordering and assigning elements to ordered bonds, like shown, leads to the constraint$z=y\cdot x$.
• Figure 4: The 1-4 move (3 dimensions) changes triangulation but not total product of phases $\prod_I W(\sigma_I)^{\epsilon_I}$ of 3-simplices $\sigma_I$. Left: The initial tetrahedron is assigned the phase $W_0\equiv\omega(g_{43},g_{32},g_{21}) ^{-1}$ (see Fig. \ref{['fig:order_color']}). Right: The vertex $a$ is added, and we choose the ordering such that $1\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow a$ (obvious from chosen orientations of new (red) edges). There are now four smaller tetrahedrons, with phases: 1. Tetrahedron $1\rightarrow 2\rightarrow 4\rightarrow a$: $W_1\equiv\omega(g_{a4},g_{42},g_{21})$; 2. Tetrahedron $2\rightarrow 3\rightarrow 4\rightarrow a$: $W_2\equiv\omega(g_{a4},g_{43},g_{32})$; 3. Tetrahedron $1\rightarrow 3\rightarrow 4\rightarrow a$: $W_3\equiv\omega(g_{a4},g_{43},g_{31})^{-1}$; 4. Tetrahedron $1\rightarrow 2\rightarrow 3\rightarrow a$: $W_4\equiv\omega(g_{a3},g_{32},g_{21})^{-1}$. The 3-cocycle condition, Eq. \ref{['eq:3-cocycle']}, says the total phase does not change by the move: $W_0=W_1W_2W_3W_4$. Note that only one independent new group element is introduced (we marked the $g_{a4}$), and from zero-flux rule $g_{a3}=g_{a4}\cdot g_{43}$. Changing our choice of ordering for $a$ relative to $1,2,3,4$ would lead to an equivalent 3-cocycle condition.
• Figure 5: The 2-3 move (3 dimensions) changes triangulation but not total product of phases $\prod_I W(\sigma_I)^{\epsilon_I}$. Left: Two initial tetrahedrons, $1234$ and $1235$, are assigned the total phase $W_0\equiv\omega(g_{43},g_{32},g_{21})\omega(g_{43},g_{32},g_{21})^{-1}$ (see Fig. \ref{['fig:order_color']}). Right: One edge (red) is added, and we choose the ordering $4\rightarrow 5$. The volume is now divided into 3 smaller tetrahedrons, with phases: 1. Tetrahedron $1\rightarrow 2\rightarrow 4\rightarrow 5$: $W_1\equiv\omega(g_{54},g_{42},g_{21})^{-1}$; 2. Tetrahedron $2\rightarrow 3\rightarrow 4\rightarrow 5$: $W_2\equiv\omega(g_{54},g_{43},g_{32})^{-1}$; 3. Tetrahedron $1\rightarrow 3\rightarrow 4\rightarrow 5$: $W_3\equiv\omega(g_{54},g_{43},g_{31})$. The 3-cocycle condition, Eq. \ref{['eq:3-cocycle']}, says the total phase does not change by the move: $W_0=W_1W_2W_3$. Note that the new group element $g_{54}$ is not independent, e.g. $g_{54}=g_{52}\cdot g_{42}^{-1}$.