Quantum spin liquid with seven elementary particles

We present an exactly solvable model of a quantum spin liquid with Abelian anyons in $d=2$ spatial dimensions. With spins 1/2 on a triangular lattice and six-body interactions, our model has zero spin correlation length and localized elementary excitations like the toric codes of Kitaev and Wen. In contrast to those earlier models, it has more elementary particles—four bosons and three fermions—and higher topological degeneracy of 64 on a torus. Elementary excitations are boson-fermion pairs that come in 12 distinct flavors. We use string operators to expose the topological nature of the model.

Figure 1

(a) The six-body spin interaction (2). (b) Its alternative representation (3).

Figure 2

Basic properties of strings. (a) Multiplication of two closed strings sharing a segment. (b) Concatenation of two open strings sharing a point. (c) Deformation of a string by the attachment of a closed loop. (d) In a ground state, a small deformation does not change the value of a string. (e) Reversal of the string direction may change its sign. (f) Strings intersecting once either commute or anticommute.

Figure 3

Fermionic Wilson loops live on honeycombs. Loops of the same color can be deformed into one another by attaching or removing hexagons.

Figure 4

Three honeycombs—red (R), green (G), and blue (B)—with links on the original triangular lattice.

Figure 5

The shaded hexagon contains an elementary excitation $W=-1$. The little red loop on its perimeter and the big red loop on the same honeycomb return a nontrivial Aharonov-Bohm phase $W=-1$. The blue loop belongs to a different honeycomb and thus has $W=+1$.

Figure 6

Four dual honeycombs—cyan (C), magenta (M), yellow (Y), and black (K).

Figure 7

Bosonic Wilson loops on dual honeycombs.

Figure 8

Fourteen topologically distinct global loops winding in the two directions of the torus.

Figure 9

(a) Seven overlapping strings cancel out as at each intersection we have a product of two operators ${\sigma }^x$ and two operators ${\sigma }^y$. (b) Global loops $W_{Rx}$ and $W_{Ry}$ commute, whereas $W_{Rx}$ and $W_{Cy}$ anticommute. Note that strings $R_x$ and $R_y$ are considered to have a single intersection, even though they share two lattice sites.

Figure 10

Open fermionic string $S_{\mathrm{T}}=\left({\sigma }_7^x{\sigma }_6^x\right)...\left({\sigma }_1^y{\sigma }_0^y\right)$ creates excitations within the shaded areas.

Figure 11

Two string operators $S_{{\mathrm{T}}_1}$ and $S_{{\mathrm{T}}_2}$ with heads at 0 and tails at 1 and 2, respectively, create excitations near the tails (shaded areas).

Figure 12

An open bosonic string $S_{\tau }={\sigma }_7^z{\sigma }_6^y...{\sigma }_2^y{\sigma }_1^z$ creates excitations within the shaded areas.

Figure 13

(a)–(c) Braiding of two red fermions (the ends of a red string). The initial (a) and final (c) states are physically equivalent to each other. 1, 2, 3, and 4 are sites of the original lattice. (d)–(f) Braiding of two black bosons (the ends of a black string). The initial (d) and final (f) states are physically equivalent to each other. $A,B,C$, and $D$ are sites of a dual honeycomb. (g)–(i) A blue fermion (the end of a blue string) goes around a black boson (the end of a black string).

Figure 14

Creation of a single elementary excitation $W=-1$ at the ends of two open strings detected by large Wilson loops.

Figure 15

The action of two open strings, bosonic $S_{\tau }$ (18) and fermionic ${\tilde{S}}_{\mathrm{T}}$ (29), creates two elementary excitations $W=-1$ at the ends (shaded hexagons).

Figure 16

Each hexagon, labeled by a circle at its center, is assigned two flavors, fermionic $(R,G$, or $B)$ and bosonic $(C,M,Y$, or $K)$. The resulting pattern of hexagons has a period $\sqrt{12}\times \sqrt{12}$.

Figure 17

(a) Short bosonic strings at the edge (30) commute with Wilson loops but may anticommute with one another. (b) A Majorana zero mode (31). (c) An equivalent combination of two bosonic strings (18).

Figure 18

The action of a ${\sigma }_n^x$ operator creates different numbers of excitations (shaded areas) depending on the location of site $n$ (black dot): (a) one hexagon for a site at a horizontal edge, (b) four hexagons for a site in the bulk (excluding the hexagon centered on site $n)$, (c) three hexagons for a site near a horizontal edge.

Figure 19

(a) and (b) Clusters with periodic boundary conditions (opposite edges identified) containing $N=12$ (a) and 24 (b) sites. Some of the global Wilson loops are shown. (c) A cluster with $N=24$ sites and two open edges (top and bottom) containing $L=4$ sites each. Left and right edges are identified.