Classification of symmetry enriched topological phases with exactly solvable models

Andrej Mesaros and Ying Ran

Phys. Rev. B 87, 155115 – Published 4 April 2013

Abstract

Recently, a new class of quantum phases of matter—symmetry protected topological states, such as topological insulators—has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. 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 the 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 $S G$ , and with topological order described by a finite gauge group $G G$ , are classified by the cohomology group $H^{3} (S G \times G G, U (1))$ . Generally, in $d + 1$ dimensions, such quantum phases are classified by $H^{d + 1} (S G \times G G, U (1))$ . 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 [Chen et al., Phys. Rev. B 87, 155114 (2013)]. 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.

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Received 11 December 2012

DOI:https://doi.org/10.1103/PhysRevB.87.155115

Authors & Affiliations

Andrej Mesaros and Ying Ran

Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA

Symmetry protected topological orders and the group cohomology of their symmetry group

Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen

Phys. Rev. B 87, 155114 (2013)

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Vol. 87, Iss. 15 — 15 April 2013

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Figure 1
Illustration of the basic assumption of symmetry fractionalization Eq. (20): Under a symmetry transformation $U (g)$ with $\forall g \in S G$ , an excited state is transformed by the product of local transformation operators $U_{i} (g)$ , with each operator only acting on one quasiparticle locally.Reuse & Permissions
Figure 2
A bilayer system in which the topological order and the global symmetry decouple.Reuse & Permissions
Figure 3
The 3-cocycle $ω (h_{1}, h_{2}, h_{3})$ 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 $1 \to 2 \to 3 \to 4$ . 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 $g_{i j}$ to $j \to i$ edge, with $g_{j i} = g_{i j}^{- 1}$ . (Four of six elements are shown explicitly.) The shown tetrahedron $1 \to 2 \to 3 \to 4$ is assigned the phase $ω {(g_{43}, g_{32}, g_{21})}^{ε} = ω (g_{43}, g_{32}, g_{21})$ . The exponent $ε = \pm 1$ is determined by (b) chirality. For tetrahedron $1 \to 2 \to 3 \to 4$ , looking from vertex 1, (counter-)clockwise loop 234 means $ε = - 1$ ( $+ 1$ ), which is realized in the right (left) tetrahedron. (c) The zero-flux rule applies to all tetrahedron faces, i.e., triangles. Generally, $g_{k i} \cdot g_{i j} \cdot g_{j k} = 1$ , the group identity element. Recall that $g_{j k} = g_{k j}^{- 1}$ . Choosing an ordering and assigning elements to ordered bonds, like shown, leads to the constraint $z = y \cdot x$ .Reuse & Permissions
Figure 4
The 1-4 move (three dimensions) changes triangulation but not total product of phases $\prod_{I} W {(σ_{I})}^{ε_{I}}$ of 3-simplices $σ_{I}$ . (Left) The initial tetrahedron is assigned the phase $W_{0} \equiv ω {(g_{43}, g_{32}, g_{21})}^{- 1}$ (see Fig. 3). (Right) The vertex $a$ is added, and we choose the ordering such that $1 \to 2 \to 3 \to 4 \to a$ [obvious from chosen orientations of new (red) edges]. There are now four smaller tetrahedrons, with phases 1, tetrahedron $1 \to 2 \to 4 \to a$ , $W_{1} \equiv ω (g_{a 4}, g_{42}, g_{21})$ ; 2, tetrahedron $2 \to 3 \to 4 \to a$ , $W_{2} \equiv ω (g_{a 4}, g_{43}, g_{32})$ ; 3, tetrahedron $1 \to 3 \to 4 \to a$ , $W_{3} \equiv ω {(g_{a 4}, g_{43}, g_{31})}^{- 1}$ ; 4, tetrahedron $1 \to 2 \to 3 \to a$ , $W_{4} \equiv ω {(g_{a 3}, g_{32}, g_{21})}^{- 1}$ . The 3-cocycle condition, Eq. (9), says the total phase does not change by the move: $W_{0} = W_{1} W_{2} W_{3} W_{4}$ . Note that only one independent new group element is introduced (we marked the $g_{a 4}$ ), and from zero-flux rule $g_{a 3} = g_{a 4} \cdot g_{43}$ . Changing our choice of ordering for $a$ relative to $1, 2, 3, 4$ would lead to an equivalent 3-cocycle condition.Reuse & Permissions
Figure 5
The 2-3 move (three dimensions) changes triangulation but not total product of phases $\prod_{I} W {(σ_{I})}^{ε_{I}}$ . (Left) Two initial tetrahedrons, 1234 and 1235, are assigned the total phase $W_{0} \equiv ω (g_{53}, g_{32}, g_{21}) ω {(g_{43}, g_{32}, g_{21})}^{- 1}$ (see Fig. 3). (Right) One edge (red) is added, and we choose the ordering $4 \to 5$ . The volume is now divided into three smaller tetrahedrons, with phases 1, tetrahedron $1 \to 2 \to 4 \to 5$ , $W_{1} \equiv ω (g_{54}, g_{42}, g_{21})$ ; 2, tetrahedron $2 \to 3 \to 4 \to 5$ , $W_{2} \equiv ω (g_{54}, g_{43}, g_{32})$ ; 3, tetrahedron $1 \to 3 \to 4 \to 5$ , $W_{3} \equiv ω {(g_{54}, g_{43}, g_{31})}^{- 1}$ . The 3-cocycle condition, Eq. (9), says the total phase does not change by the move: $W_{0} = W_{1} W_{2} W_{3}$ . Note that the new group element $g_{54}$ is not independent, e.g., $g_{54} = g_{52} \cdot g_{42}^{- 1}$ .Reuse & Permissions
Figure 6
The DW topological invariant $Z_{M}$ and TQFT for manifold $M$ with boundary. (a) Two copies of 2-manifold $B$ : $B_{1}, B_{2}$ , with colorings $τ_{1}, τ_{2}$ , form the boundary of 3-manifold $M = B \times [0, 1]$ . The triangulation and ordering in $B_{1}$ and $B_{2}$ are chosen identical, as sketched. The triangulation and ordering in the bulk of $M$ (example red vertices and edges) does not influence the value of DW invariant $Z_{M} (τ = τ_{1} \cup τ_{2})$ . A coloring $τ_{i}$ of $B$ represents a quantum state $|τ_{i}〉$ , and then $Z_{M} (τ) = 〈τ_{2}| P |τ_{1}〉$ is a quantum amplitude of operator $P$ in the DW TQFT associated with $B$ . The image of $P$ is the ground-state manifold of the TQFT. (b) The operator $P$ is a projector: The quantum amplitude $〈τ_{2}| P^{2} |τ_{1}〉$ is sketched, representing a product of amplitudes of $P$ from $τ_{1}$ to $τ_{3}$ , and $τ_{3}$ to $τ_{2}$ , with a sum over colorings $τ_{3}$ . This becomes a sum over internal colorings in space $M = B \times [0, 2]$ with boundary $B_{1} \cup B_{2}$ . The latter is equal to the amplitude $〈τ_{2}| P |τ_{1}〉$ from panel (a).Reuse & Permissions
Figure 7
The 2-cocycle $ω (h_{1}, h_{2})$ assigns a $U (1)$ complex number (i.e., a phase) to a 2-simplex (triangle). (a) Left to center: the “ordering” of triangle's three vertices; we choose here $1 \to 2 \to 3$ . 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 $g_{i j}$ to $j \to i$ edge, with $g_{j i} = g_{i j}^{- 1}$ . The shown triangle $1 \to 2 \to 3$ is assigned the phase $W = ω {(g_{32}, g_{21})}^{ε} = ω (g_{32}, g_{21})$ . The exponent $ε = \pm 1$ is determined by (b) chirality. For triangle $1 \to 2 \to 3$ , (counter-)clockwise loop 123 means $ε = + 1$ ( $- 1$ ), which is realized in the left (right) triangle. (c) The zero-flux rule applied to a triangle. Generally, $g_{k i} \cdot g_{i j} \cdot g_{j k} = 1$ , the group identity element. Recall that $g_{j k} = g_{k j}^{- 1}$ . Choosing an ordering and assigning elements to ordered bonds as shown leads to the constraint $z = y \cdot x$ .Reuse & Permissions
Figure 8
The 1-3 and 2-2 moves (in two dimensions) change triangulation but not total product of phases $\prod_{I} W {(σ_{I})}^{ε_{I}}$ of 2-simplices $σ_{I}$ . (a) The 1-3 move. (Left) The initial triangle is assigned the phase $W_{0} \equiv ω (g_{32}, g_{21})$ (see Fig. 7). (Right) The vertex $a$ is added, and we choose the ordering such that $1 \to 2 \to 3 \to a$ [obvious from chosen orientations of new (red) edges]. There are now three smaller triangles, with phases 1, triangle $1 \to 2 \to a$ , $W_{1} \equiv ω (g_{a 2}, g_{21})$ ; 2, triangle $2 \to 3 \to a$ , $W_{2} \equiv ω (g_{a 3}, g_{32})$ ; 3, triangle $1 \to 3 \to a$ : $W_{3} \equiv ω {(g_{a 3}, g_{31})}^{- 1}$ . The 2-cocycle condition, Eq. (6), says the total phase does not change by the move: $W_{0} = W_{1} W_{2} W_{3}$ . Note that only one independent new group element is introduced (we marked the $g_{a 3}$ ), and from zero-flux rule $g_{a 2} = g_{a 3} \cdot g_{32}$ . Changing our choice of ordering for $a$ relative to $1, 2, 3$ would lead to an equivalent 2-cocycle condition. (b) The 2-2 move. (Left) Two initial triangles, 123 and 124, are assigned the total phase $W_{0} \equiv ω (g_{32}, g_{21}) ω {(g_{42}, g_{21})}^{- 1}$ . (Right) The area is divided into two triangles by the other possible edge (red), and we choose the ordering $3 \to 4$ . The phases of new triangles: 1, triangle $1 \to 3 \to 4$ , $W_{1} \equiv ω {(g_{43}, g_{31})}^{- 1}$ ; 2, triangle $2 \to 3 \to 4$ , $W_{2} \equiv ω (g_{43}, g_{32})$ . The 2-cocycle condition gives $W_{0} = W_{1} W_{2}$ . Note that the group element $g_{43}$ is not independent; e.g., $g_{43} = g_{42} \cdot g_{32}^{- 1}$ .Reuse & Permissions
Figure 9
Action of the operator ${\hat{B}}_{p}^{s}$ , $s = h_{s} \cdot \tilde{s}$ , with $h_{s} \in G G$ , $\tilde{s} \in S G$ , on plaquette $p$ centered on site $i$ : For six elements $h_{i l} \in G G$ on edges $i l$ it resembles a local gauge transformation preserving zero flux through all triangles, e.g., $h_{i j}^{'} = h_{s} \cdot h_{i j}$ and $h_{k i}^{'} = h_{k i} \cdot h_{s}^{- 1}$ , while the on-site element transforms as $u_{i}^{'} = \tilde{s} u_{i}$ , leading to, e.g., $g_{i j}^{'} = s \cdot g_{i j}$ and $g_{k i}^{'} = g_{k i} \cdot s^{- 1}$ ; additionally, there is an overall phase factor which is the product of complex numbers assigned by the 3-cocycle $ω \in Z^{3} (S G \times G G, U (1))$ to the six 3-simplices (tetrahedrons) forming the “pyramid” [see Fig. 3 and Eq. (51)]. Note that fixing the initial and final state (i.e., the values $u_{i}, u_{i}^{'}$ , etc.) leads to a unique value of $s$ for which the operator matrix element does not vanish. In that sense, this picture also describes the action of the plaquette operator ${\hat{B}}_{p}$ .Reuse & Permissions
Figure 10
Overlapping plaquette operators commute. As in Fig. 9, the action and matrix elements of ${BB}_{1} = 〈f| {\hat{B}}_{p} {\hat{B}}_{p^{'}} |i〉$ (right) and ${BB}_{2} = 〈f| {\hat{B}}_{p^{'}} {\hat{B}}_{p} |i〉$ (left) are shown. Note that fixing $|f〉$ and therefore $g_{i j}^{''}$ in both cases is consistent, giving $g_{i j}^{''} = s \cdot g_{i j} \cdot {s^{'}}^{- 1}$ . Concerning the overall phase factor due to the 3-cocycle factors in Eq. (51), the two images differ only in the choice of the single internal edge (dark blue on left, pink on right) of the triangulation of the tent-shaped object. The summation over elements $s, s^{'}$ inherent in the plaquette operators, Eq. (52), amounts to the sum over internal colorings, while $|i〉, |f〉$ fix the surface coloring of the “tent.” The phase factor in both quantities ${BB}_{1, 2}$ becomes just the DW topological invariant [Eq. (43)] of the “tent” calculated with different triangulation choices, therefore having the same value.Reuse & Permissions
Figure 11
Creating a loop operator ${\hat{L}}_{P}^{s} = \prod_{p \in A} {\hat{B}}_{p}^{s}$ by multiplying plaquette operators centered on sites within area $A$ bounded by a lattice loop $P$ . This leads to two types of phase factors in the bulk of $A$ , coming from (a) left, down triangles; right, up triangles. The ${\hat{B}}_{p}$ 's in the product are chosen to be ordered according to order of vertices $i$ on which they are centered ( $p \equiv i$ ), with highest $p$ being rightmost (i.e., applied first). For an Abelian group and the choice $s \equiv h_{s} \cdot 1_{S G}$ the elements within $A$ and $P$ do not change, i.e., $u_{4}^{''} = u_{4}$ , $g_{32}^{''} = g_{32}$ , etc., and phase factors from all tetrahedrons cancel up to a total phase due to a 2-cocycle evaluated on edges along $P$ [Eq. (70)]. (b) The loop operator only acts on triangle edges (thicker black) lying outside $A$ but sharing one site with $P$ (red edges).Reuse & Permissions
Figure 12
(a) The loop operator ${\hat{L}}_{P}^{s}$ , with $P$ the loop of lattice edges bounding area $A$ , has total contribution from triangles within $A$ which equals the product of 2-cocycle $c_{s}$ [Eq. (67)] evaluated on the triangles as oriented 2-simplices (compare to Figs. 7 and 11). (b) An example loop $P$ (red edges). (c) The phase can be calculated by retriangulating the area $A$ inside the loop $P$ . The resulting expression for the phase $Θ_{P}^{s}$ , Eq. (66), is given in terms of elements along edges on $P$ with clockwise ordered vertices $i = 0, ..., N$ . The expression is used for the definition of an open string; see Sec. 4b.Reuse & Permissions
Figure 13
The phase contribution $w_{s}^{Γ} (g) = Ψ_{Γ}^{s} Θ_{Γ}^{s} δ (\prod_{n = 1}^{N} a_{n}, g)$ to matrix element of ribbon operator $F^{(s, g)} (Γ)$ , with $s \in G G$ , on a length $N = 5$ example. (Left) The operator acts nontrivially only on elements adjacent to path $Γ$ (thick black edges). Note that elements $a_{i} \equiv g_{i, i - 1} \equiv h_{i, i - 1} u_{i} u_{i - 1}^{- 1}$ are defined to be directed along the path, so for $n = 4, 5$ they are opposite to the standard definition, which is along the edge orientation. Elements at ends $A$ , $B$ of the ribbon $Γ$ , i.e., vertex $i_{A}$ and vertex $i_{B}$ , are $u_{5} \in S G$ and $u_{0} \in S G$ , respectively. The phase factor $Ψ_{Γ}^{s}$ is the product of 3-simplex phases $W {(σ_{I})}^{ε (σ_{I})}$ shown on top of shaded triangles. (Right) The phase $Θ_{Γ}^{s}$ is the product of 2-simplex phases $W {(σ_{t})}^{ε (σ_{t})}$ shown, where $1$ is the identity element, and the parameter $g = a_{1} \dots a_{N} \in G$ .Reuse & Permissions
Figure 14
Inner segment of ribbon operator $F^{(s, g)} (Γ)$ , with $s \in G G$ , commutes with plaquette operator ${\hat{B}}_{p}^{s^{'}}$ , having $p$ centered on inner edge of $Γ$ , and therefore also with the Hamiltonian. (a) Only elements on two edges (thick black) are changed, but in the same way for $F \hat{B}$ and $\hat{B} F$ due to Abelian $G$ . (b) Quantum phase for $F \hat{B}$ (top row) and $\hat{B} F$ (bottom row) differ only due to shown 3-simplices which are on top of three blue shaded triangles in (a). (c) The phase contribution due to string phase $Θ$ is shown. (d) The 3-simplices contribution to phase factor ratio $WB/BW$ (from panel b) equals precisely the phase $W {(σ_{t})}^{ε (σ_{t})}$ due to two blue 2-simplices [see Eq. (78)]. The $WB$ (top) and $BW$ (bottom) cases are therefore graphically seen to be the same, as interior points of the polygon can be removed in a 1-3 move (Fig. 8).Reuse & Permissions
Figure 15
(a) Ribbon operator with both ends (red dots), shown schematically at the bottom. It only acts on the thick black edges, consistent with zero-flux rule on all 2-simplices above and within the ribbon. The transition amplitude of the ribbon is given in Fig. 13 without the phase factor $f_{A} f_{B} f_{A B}$ local at the string ends; $f$ is involved (Appendix B), but includes the edges of two triangles $t_{A}, t_{B}$ at the ends (shaded light orange). Dashed violet lines indicate edge elements that are not acted on by the operator. (b) The local operator at position $B \equiv (i, t)$ , with $i$ the lattice site and triangle $t$ (dark blue) having vertices $i, j, k$ (shaded dark blue), has the transition amplitude $〈f| D_{(h, g)} (B) |i〉 = δ (g_{i j} \cdot g_{j k} \cdot g_{k i}, h) W_{h} W_{6}$ . It acts as the plaquette operator ${\hat{B}}_{p}^{g}$ (Fig. 9) with plaquette $p$ centered on $i$ (changing only thick black edge elements), except that it also projects flux in $t$ to value $h$ , and also has an additional phase factor $W_{h}$ , which is the phase of the 2-simplex (shaded light blue). [For operator $D (A)$ , the triangle $i k i^{'}$ is used instead.] The phase factor $W_{6}$ depends on six drawn 3-simplices (tetrahedrons) having altered elements $o$ (red edges) such that it is well defined even if the zero-flux rule is violated in the plane (as occurs at ribbon string ends); see after Eq. (79).Reuse & Permissions
Figure 16
(a) Ribbon operators for two-particle excited state and braiding two excitations at $A_{1}, A_{2}$ (see also Fig. 15). The common end point $B$ contains no excitation, but “topological” operators of $i$ th ribbon $D^{(i)} (B)$ , $i = 1, 2$ , local at $B$ , can be used to represent braiding. (b) Applying different ribbons (blue) than in original state shown in (a) leads to a braided state (see Sec. 4d). (c) The resulting counterclockwise braid of particles 1 and 2.Reuse & Permissions
Figure 17
Multiple ribbons sharing end $B$ , describing quasiparticles at their end $A$ 's. (a) Example of ordering multiple (five) ribbons, with strings $Γ_{1}, ..., Γ_{5}$ , in a “counterclockwise sense.” The ribbons are allowed to have overlaps only over a finite length starting from end $B$ . The figure shows a realization possible on a triangular lattice. (b) The counterclockwise $180^{\circ}$ braiding of particles $A_{n}$ and $A_{n + 1}$ in an $N$ -particle state, a generalization of Fig. 16. The (blue) strings $Γ^{'}$ apply to the braided state.Reuse & Permissions
Figure 18
A general F move.Reuse & Permissions
Figure 19
Multiple ribbons sharing end $A$ , describing quasiparticles at their end $B$ 's. (a) Example of ordering multiple (five) ribbons, with strings ${\bar{Γ}}_{1}, ..., {\bar{Γ}}_{5}$ , in a “counterclockwise sense.” The ribbons are allowed to have overlaps only over a finite length starting from end $A$ . The figure shows a realization possible on a triangular lattice. (b) The counterclockwise $180^{\circ}$ braiding of particles $B_{n}$ and $B_{n + 1}$ in an $N$ -particle state. The (blue) strings ${\bar{Γ}}^{'}$ apply to the braided state.Reuse & Permissions
Figure 20
Braiding quasiparticle $A_{1}$ with the fused quasiparticle $A_{2} A_{3}$ can be calculated using the shown formula for a two-step process. The resulting braiding operator is determined by the action of two topological operators [Eq. (B35)], analogously to the simple case of two-particle braiding; however, in this case one topological operator relates to particle $A_{1}$ , but the other is a “comultiple” [Eqs. (B32) and (B33)] of topological operators acting on particles $A_{2}$ and $A_{3}$ .Reuse & Permissions
Figure 21
Quasiparticle statistics from ribbon commutation. (a) Commutation of two strings (ribbon operators, black and red) at the intersection point (blue square) gives the braiding statistics. (Top) Strings representing tunneling of particle-antiparticle pairs across the system (the surface of torus) can give the self-statistics. (Bottom) Braiding setup independent of system shape. (b) Near the intersection of two ribbon operators ${\hat{w}}_{h_{1}}^{Γ_{1}} (g_{1})$ , ${\hat{w}}_{h_{2}}^{Γ_{2}} (g_{2})$ , the phase difference occurs due to 3-simplices on top of two blue shaded triangles, detailed in (c) with phases $W_{m n} = 〈f| {\hat{w}}_{h_{m}}^{Γ_{m}} (g_{m}) {\hat{w}}_{h_{n}}^{Γ_{n}} (g_{n}) |i〉$ shown in top ( $W_{12}$ ) and bottom ( $W_{21}$ ) rows. Note that elements $b_{i}$ are defined to be oriented oppositely to our lattice edge orientations, but consistent with definition of ribbon operator phase, Fig. 13, hence the appearance of $b_{i + 1}^{- 1}$ in Eq. (C3).Reuse & Permissions
Figure 22
Recovering the two $Z_{2}$ topological phases from the exactly solvable models: toric code (TC) and double semion theory (DS). The dual lattice is honeycomb (dashed). Red dots mark the positions of group elements $g_{i j}$ assigned to edges $i j$ , becoming “spin- $\frac{1}{2}$ ” states in TC and DS. A honeycomb lattice plaquette $p h$ is shaded. The plaquette operator ${\hat{B}}_{p}$ (Fig. 9) acts on the six “spins” of $p h$ . The DS model differs from TC by having an additional phase in its plaquette operator. This phase depends on spins on the six outer legs of $p h$ , one of which is marked by a green line. These six “outer” spins belong to the six tetrahedrons in our ${\hat{B}}_{p}$ ; see Fig. 9. The $Q_{t}$ operator in both models just acts on the three spins nearest to a site (spins on blue lines).Reuse & Permissions

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