Space-group symmetry fractionalization in a family of exactly solvable models with ${\mathbb{Z}}_{2}$ topological order

Space-group symmetry fractionalization in a family of exactly solvable models with $Z_{2}$ topological order

Hao Song and Michael Hermele

Phys. Rev. B 91, 014405 – Published 6 January 2015

Abstract

We study square lattice space-group symmetry fractionalization in a family of exactly solvable models with $Z_{2}$ topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types of symmetry fractionalization (symmetry classes) can be realized within this class of models, which are generalizations of Kitaev's $Z_{2}$ toric code to arbitrary lattices. This question is motivated by earlier work of Essin and one of us (M. H.) [Phys. Rev. B 87, 104406 (2013)], where the idea of symmetry classification was laid out, and which, for square lattice symmetry, produces 2080 symmetry classes consistent with the fusion rules of $Z_{2}$ topological order. This approach does not produce a physical model for each symmetry class, and indeed there are reasons to believe that some symmetry classes may not be realizable in strictly two-dimensional systems, thus raising the question of which classes are in fact possible. While our understanding is limited to a restricted class of models, it is complete in the sense that for each of the 2080 possible symmetry classes, we either prove rigorously that the class cannot be realized in our family of models, or we give an explicit model realizing the class. We thus find that exactly 487 symmetry classes are realized in the family of models considered. With a more restrictive type of symmetry action, where space-group operations act trivially in the internal Hilbert space of each spin degree of freedom, we find that exactly 82 symmetry classes are realized. In addition, we present a single model that realizes all $2^{6} = 64$ types of symmetry fractionalization allowed for a single anyon species ( $Z_{2}$ charge excitation), as the parameters in the Hamiltonian are varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models.

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Received 25 September 2014
Revised 11 December 2014

DOI:https://doi.org/10.1103/PhysRevB.91.014405

Authors & Affiliations

Hao Song and Michael Hermele

Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

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Vol. 91, Iss. 1 — 1 January 2015

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Figure 1
Illustration of some geometrical objects important in the square lattice toric code model. The edges in plaquette $p$ are shown as thick dark bonds (blue online), while the edges in $star (v)$ are thick gray bonds (pink online). The two strings $s_{x}$ and $s_{y}$ winding periodically around the system are also shown as thick dark bonds (blue online).
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Figure 2
Depiction of $e$ and $m$ strings in the square lattice toric code. $s$ is an open $e$ string joining vertices $v$ and $v^{'}$ , denoted with thick dark bonds (blue online). $t$ is an open cut joining plaquettes $p$ and $p^{'}$ , shown as a dotted line. The cut $t$ contains the thick gray bonds (pink online) intersected by the dotted line.
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Figure 3
(a) The lattice on which all $2^{6} = 64$ $e$ -particle fractionalization classes can be realized. There are six types of plaquettes not related by symmetries, and the corresponding plaquette terms are assigned independent coefficients $K_{i}^{m}$ $(i = 1, 2, ..., 6)$ . Nearest-neighbor pairs of vertices are joined by two edges (dark and light; blue and red online), drawn curved to avoid overlapping and to be clear about their movement under space-group operations. Plaquettes of type $i = 1, 2, 3$ are each formed by the two edges joining a nearest-neighbor pair of vertices. Two vertices $v_{1}, v_{2}$ and two edges $l_{1}, l_{2}$ are labeled to illustrate the calculation of $σ_{p x}^{e}$ discussed in the main text. (b), (c) Subgraphs of the lattice in (a), each containing all the vertices and half the edges. These subgraphs transform into one another under any improper space-group operation (i.e., reflections). We draw these subgraphs to illustrate the plaquettes of type $i = 4, 5, 6$ .
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Figure 4
$T C_{0} (G)$ models. The shaded square is a unit cell and the origin of our coordinate system is at the center of the square. Below each figure of the lattice is the corresponding TC symmetry class in the form (72). Here, $a_{r}$ is the ground-state eigenvalue of $A_{v}$ for $v$ at special points $r = o, \tilde{o}, κ$ ; and $b, b_{1}, b_{2}$ are the ground-state eigenvalues of $B_{p}$ for the plaquette $p$ , which in these models is picked to be the smallest cycle made with black edges where $b, b_{1}$ , or $b_{2}$ is written, while $b_{3}$ is for the plaquette made of a pair of black and gray edges (black and pink online). These edges are drawn curved to avoid overlapping and to be clear about their movement under space-group operations. The comparison between (a) and (b) gives an explicit example that moving the coordinate system origin by $(\frac{1}{2}, \frac{1}{2})$ results in a transformation (74): $P_{x} \to T_{x} P_{x}, σ_{p x} \leftrightarrow σ_{t x p x}, σ_{p x p x y} \leftrightarrow σ_{p x p x y} σ_{t x t y}$ . The symmetry class differs from (e) by such a transformation can be easily got by moving the coordinate system, so we do not bother drawing a separate lattice for it.
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Figure 5
Two example models in $T C (G)$ that realize TC symmetry classes not possible in $T C_{0} (G)$ . The shaded square is a unit cell and the origin of our coordinate system is at the center of the square. Below each figure of lattice is the corresponding TC symmetry class in the form (72). Here, $a_{r}$ is the ground-state eigenvalue of $A_{v}$ for $v$ at special points $r = o, \tilde{o}, κ$ and $b$ is the ground-state eigenvalue of $B_{p}$ for the plaquette $p$ , defined here to be the smallest cycle enclosing the letter “ $b$ .” We write $α_{i} = c_{l_{i}}^{x} (P_{x}), β_{i} = c_{l_{i}}^{x} (P_{x y}), γ_{i} = c_{l_{i}}^{z} (P_{x})$ , and $δ_{i} = c_{l_{i}}^{z} (P_{x y})$ . (a) A model realizing some TC symmetry classes (and symmetry classes) that cannot be realized without spin-orbit coupling. Here, $h_{1}, h_{2}$ label two positions of an $m$ particle for the calculation of $σ_{p x}^{m} = α_{1}$ in the main text. (b) A model realizing all $2^{6} = 64$ possible $m$ -particle fractionalization classes $[ω_{m}]$ . Here, for simplicity, we make the restriction $γ_{i} = δ_{i} \equiv c_{i}$ .
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Figure 6
Graphical argument that $L_{c}^{e} | ψ_{0} 〉 = | ψ_{0} 〉$ for $c = s_{y} \cup T_{x}^{- 1} s_{y}$ . The dotted lines show the $L \times L$ grid of primitive cells, and the paths $s_{y}$ and $T_{x}^{- 1} s_{y}$ are shown. $c$ encloses a region of area $L$ , which can be broken (dashed lines) into $L$ smaller subregions each of unit area. Let $c^{'}$ be the cycle bounding one of the subregions, then $L_{c}^{e} = \prod_{n = 0}^{L - 1} L_{T_{x}^{n} c^{'}}^{e}$ . In addition, by translation symmetry $L_{c^{'}}^{e} | ψ_{0} 〉 = L_{T_{y} c^{'}}^{e} | ψ_{0} 〉 = \pm | ψ_{0} 〉$ . Since an even number of subregions appear in the decomposition of $L_{c}^{e}$ given above, we have $L_{c}^{e} | ψ_{0} 〉 = | ψ_{0} 〉$ .
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Figure 7
Graphical illustration of the argument that $L_{c}^{e} | ψ_{0} 〉 = | ψ_{0} 〉$ for $c = s_{x} \cup P_{x} s_{x}$ . It is important to note that, in the interest of clarity, this figure is schematic in the sense that it accurately shows the connectivity of the paths involved, and their properties under translation symmetry, but not their properties under $P_{x}$ . The various symbols are defined in the main text. The vertical dashed line is the $P_{x}$ reflection axis, and the vertex $v$ has been chosen to lie near this axis for convenience. $c_{1}$ and $c_{2}$ are the boundaries of the left and right shaded regions, respectively. The most important point is that these two regions are related by $T_{x}^{L / 2}$ translation.
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Figure 8
The calculation of $σ_{p x p x y}^{m}$ . Put an $m$ particle at point $h_{0}$ , let $h_{j} = R^{j} (h_{0}), j = 1, 2, 3$ , let $t_{0} \in \bar{W}$ connecting $h_{0}$ to $h_{1}$ and $t_{j} = R^{j} t_{0}$ . Then, we have $U_{P_{x}}^{m} U_{P_{x y}}^{m} (h_{0}) = f_{0}^{m} L_{t_{0}}^{m}$ with $f_{0}^{m} \in {\pm 1}$ and $U_{P_{x}}^{m} U_{P_{x y}}^{m} (h_{j}) = L_{t_{0} \dots t_{j - 1}}^{m} U_{P_{x}}^{m} U_{P_{x y}}^{m} (h_{0}) R (L_{t_{0} \dots t_{j - 1}}^{m})$ for $j = 1, 2, 3$ . With some calculation, ${(U_{P_{x}}^{m} U_{P_{x y}}^{m})}^{4} (h_{0}) = L_{t}^{m}$ with $t = t_{0} t_{1} t_{2} t_{3}$ . Thus, $σ_{p x p x y}^{e} = a_{V_{t}}$ . If $P (v)$ is enclosed by $t$ with $R^{2} v \neq v$ , then $v, R v, R^{2} v, R^{3} v$ are four different vertices enclosed by $t$ such as the gray vertices shown here. Since $\prod_{i = 0}^{3} a_{R^{i} v} = 1$ , we have $σ_{p x p x y}^{e} = a_{V_{t}} = a_{P^{- 1} (o)} = a_{Γ (R^{2})}$ . The above statements are also true in the cases with spin-orbital coupling using the gauge choice described in Appendix pp4. In the case without spin-orbital coupling, since $a_{v} = a_{P_{x} v} = a_{P_{x y} v}$ , we have $σ_{p x p x y}^{m} = a_{Γ (P_{x}, P_{x y})}$ .
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Figure 9
Illustration of the calculation of $σ_{t y p x}^{m}$ in Lemmas 4 and 13. Solid squares denote the locations of holes $h \in H$ , which are chosen so that $h_{0}$ is arbitrary (but near the $y$ axis), and $h_{1} = P_{x} h_{0}, h_{2} = T_{y} h_{0}, h_{3} = T_{y} h_{1}, h_{4} = P_{y} h_{0}, h_{5} = P_{y} h_{1}$ . $h_{4}$ and $h_{5}$ are not used in Lemma 4. Cuts are represented by solid lines, and $\tilde{h_{0} h_{1}}$ denotes, for example, a cut joining $h_{0}$ to $h_{1}$ . The cuts $t_{0} = \tilde{h_{0} h_{1}}, t_{1} = \tilde{h_{0} h_{2}}, t_{2} = \tilde{h_{1} h_{3}}$ , and $t_{3} = \tilde{h_{2} h_{3}}$ are labeled, and are chosen to have properties described in the text. The points $o = (0, 0)$ and $κ = (0, \frac{1}{2})$ are shown.
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Figure 10
Depiction of the graphical notation used to represent stacking of vertices and edges. The first row shows the connectivity of vertices and edges, and the second row gives the corresponding two-dimensional presentation. It is convenient to imagine the graph of the lattice as first being embedded in three-dimensional space, and then projected into the two-dimensional plane. When these structures are present, we always assume top edges (blue online) are transformed to bottom edges (red online) under improper space-group operations (i.e., reflections), while translations do not swap edges with different colors. Edges parallel to the $x$ axis, $y$ axis, $z$ axis are labeled by symbols $ε, l, ι$ , respectively. We use $ζ$ and $ξ$ to label diagonal edges. For a diagonal edge, we can associate a unit vector $\hat{e}$ running along the direction of the edge, always choosing ${\hat{e}}_{x} > 0$ . Then, $ζ$ $(ξ)$ is used to label edges with ${\hat{e}}_{y} > 0$ $({\hat{e}}_{y} < 0)$ . (a), (d) This configuration is only used in Fig. 11. The two stacking vertices (blue and red online) together with edge $ι_{1}$ connecting them are projected into a point, presented as a ring (blue and red online). Edges $ε_{2}, l_{3}$ pass through the ring but do not end on it. The triple-stacking edges are presented as double lines. (b), (e) A configuration with double-stacking vertices and no stacking edges. We use a darker point (blue online) to represent the upper vertex, and a lighter ring (red online) to represent the lower vertex. The edges linked to the upper vertex are darker (blue online) and the edges linked to the lower vertex are lighter (red online). (c), (f) A situation with double-stacking vertices and edges. The vertices are represented as in (b) and (e). The lower edge is represented by a lighter double line (red online), and the upper edge is a single darker line (blue online) drawn in the center of the double line.
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Figure 11
$T C (G)$ models (Part I). The shaded square is a unit cell and the TC symmetry classes are calculated with the origin $o$ at the center of the shaded square. Below each lattice is the corresponding TC symmetry class in the form (72). The edges are labeled by different letters according to their directions as described in the text and in Fig. 10. Edges that map to a single point under $P$ are labeled by $ι_{o}, ι_{\tilde{o}}, ι_{κ}, ι_{\tilde{κ}}$ with the subscript indicating their position, and $\tilde{o} = (\frac{1}{2}, \frac{1}{2}), κ = (0, \frac{1}{2}), \tilde{κ} = (\frac{1}{2}, 0)$ , in units such that the size of the unit cell is $1 \times 1$ . For short, we define $α_{i} = c_{ɛ_{i}}^{x} (P_{x}), β_{i} = c_{ɛ_{i}}^{x} (P_{x y}), γ_{i} = c_{ɛ_{i}}^{z} (P_{x})$ , and $δ_{i} = c_{ɛ_{i}}^{z} (P_{x y})$ , where $ɛ = l, ε, ξ, ζ, ι$ stands for a generic edge. In addition, $e_{r} = a_{P^{- 1} (r)}$ , and $b$ is the eigenvalue of $B_{p}$ for the plaquette (here meaning smallest cycle) $p$ within which $b$ is written. The values of $e_{r}$ and $b$ are well defined with respect to any local spin frame system satisfying Eqs. (D8)–(D10).
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Figure 12
$T C (G)$ models (Part II). The shaded square is a unit cell and the TC symmetry classes are calculated with the origin $o$ at the center of the shaded square. Below each lattice is the corresponding TC symmetry class in the form (72). The edges are labeled by different letters according to their directions as described in the text and in Fig. 10. Edges that map to a single point under $P$ are labeled by $ι_{o}, ι_{\tilde{o}}, ι_{κ}, ι_{\tilde{κ}}$ with the subscript indicating their position, and $\tilde{o} = (\frac{1}{2}, \frac{1}{2}), κ = (0, \frac{1}{2}), \tilde{κ} = (\frac{1}{2}, 0)$ , in units such that the size of the unit cell is $1 \times 1$ . For short, we define $α_{i} = c_{ɛ_{i}}^{x} (P_{x}), β_{i} = c_{ɛ_{i}}^{x} (P_{x y}), γ_{i} = c_{ɛ_{i}}^{z} (P_{x})$ , and $δ_{i} = c_{ɛ_{i}}^{z} (P_{x y})$ , where $ɛ = l, ε, ξ, ζ, ι$ stands for a generic edge. In addition, $e_{r} = a_{P^{- 1} (r)}$ , and $b$ is the eigenvalue of $B_{p}$ for the plaquette (here meaning smallest cycle) $p$ within which $b$ is written. In panel (l), $b$ is the eigenvalue of $B_{p}$ for the top plaquette. The values of $e_{r}$ and $b$ are well defined with respect to any local spin frame system satisfying Eqs. (D8)–(D10).
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Figure 13
$T C (G)$ models (Part III). The shaded square is a unit cell and the TC symmetry classes are calculated with the origin $o$ at the center of the shaded square. Below each lattice is the corresponding TC symmetry class in the form (72). The edges are labeled by different letters according to their directions as described in the text and in Fig. 10. Edges that map to a single point under $P$ are labeled by $ι_{o}, ι_{\tilde{o}}, ι_{κ}, ι_{\tilde{κ}}$ with the subscript indicating their position, and $\tilde{o} = (\frac{1}{2}, \frac{1}{2}), κ = (0, \frac{1}{2}), \tilde{κ} = (\frac{1}{2}, 0)$ , in units such that the size of the unit cell is $1 \times 1$ . For short, we define $α_{i} = c_{ɛ_{i}}^{x} (P_{x}), β_{i} = c_{ɛ_{i}}^{x} (P_{x y}), γ_{i} = c_{ɛ_{i}}^{z} (P_{x})$ , and $δ_{i} = c_{ɛ_{i}}^{z} (P_{x y})$ , where $ɛ = l, ε, ξ, ζ, ι$ stands for a generic edge. In addition, $e_{r} = a_{P^{- 1} (r)}$ , and $b$ (or $b_{i})$ is the eigenvalue of $B_{p}$ for the plaquette (here meaning smallest cycle) $p$ within which $b$ (or $b_{i})$ is written. In panels (w) and (x), $b_{1}$ and $b$ are the eigenvalues of $B_{p}$ for the top plaquettes. The values of $e_{r}$ and $b$ (or $b_{i})$ are well defined with respect to any local spin frame system satisfying Eqs. (D8)–(D10) except in (w), where a further gauge fixing is needed and we require $c_{l_{1}}^{z} (P_{x y}) = c_{l_{1}^{'}}^{z} (P_{x y}) = c_{l_{2}}^{z} (P_{x}) = 1$ .
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Physical Review B

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Space-group symmetry fractionalization in a family of exactly solvable models with $Z_{2}$ topological order

Hao Song and Michael Hermele

Phys. Rev. B 91, 014405 – Published 6 January 2015

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Physical Review B

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Space-group symmetry fractionalization in a family of exactly solvable models with Z2 topological order

Hao Song and Michael Hermele

Phys. Rev. B 91, 014405 – Published 6 January 2015

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Space-group symmetry fractionalization in a family of exactly solvable models with $Z_{2}$ topological order