Exact topological quantum order in $D=3$ and beyond: Branyons and brane-net condensates

We construct an exactly solvable Hamiltonian acting on a three-dimensional lattice of spin-$\frac{1}{2}$ systems that exhibits topological quantum order. The ground state is a string net and a membrane-net condensate. Excitations appear in the form of quasiparticles and fluxes, as the boundaries of strings and membranes, respectively. The degeneracy of the ground state depends on the homology of the 3-manifold. We generalize the system to $D⩾4$, where different topological phases may occur. The whole construction is based on certain special complexes that we call colexes.

Figure 1

In (a), the two curves are homologous because they form the boundary of a deformed disk. In (b) and (c), the 3-torus is represented as a cube in which opposite sides must be identified. In (b), a basis for $1\text{−}\text{cycles}$ is shown and in (c), a basis for $2\text{−}\text{cycles}$.

Figure 2

(a) A vertex, (b) an edge, (c) a face, and (d) a polyhedral solid.

Figure 3

The neighborhood of a vertex in a 3-colex. Four edges, six faces, and four cells meet at each vertex.

Figure 4

(Color online) Neighborhoods in a 3-colex of (a) a vertex, (b) a $g$ edge, (c) a $by$ face, with the yellow side visible and the blue one hidden, and (d) a $b$ cell. Faces are colored according to the color of the cell at their visible side.

Figure 5

(Color online) The projective space $P^3$ can be obtained starting with a solid sphere and identifying opposite points in its surface. Here, we use such a representation to show a 3-colex in $P^3$.

Figure 6

(Color online) (a) The unit cell of a colex that can fill either the infinite space or a three-dimensional torus. Forgetting about colors, which are unphysical, its symmetries are those of the cube, with a single $r$ cell at its core. Here, colors have been given both to cells and to links. A yellow cell was removed to show the interior of the unit cell. (b) Eight unit cells put together. Recall that color is not reflected in the Hamiltonian, so that all unit cells are equal although they look different due to the coloring.

Figure 7

(Color online) (a) A representation of the space $S^2\times S^1$. Each section of the solid tube is a sphere, and both ends of the tube are identified. (b) A 3-colex in $S^2\times S^1$. It consists of 24 vertices, twelve edges of each color, four $br$ faces, eight $by$ faces, six $rg$ faces, four $ry$ faces, four $gy$ faces, two $b$ cells, one $r$ cell, three $g$ cells, and two $y$ cells. (c) The $r$-shrunk complex of the previous colex. The vertex corresponds to a $r$ cell, and edges to $r$ edges. An example of a closed string is the edge marked with an $s$. It has nontrivial homology. (d) The $gy$-shrunk complex of the previous colex. Vertices correspond to $b$ and $r$ cells, edges to $rb$ faces, face to $gy$ faces, and cells to $g$ and $y$ cells. An example of a closed membrane is a combination of the faces marked with an $m$. This membrane has nontrivial homology.

Figure 8

(Color online) In this figure, the top represents part of a colex, and the bottom the corresponding portion of the $b$-shrunk complex. Vertices in the $b$-shrunk complex come from $b$-cells in the colex, edges from $b$-edges, faces from $ry$-, $rg$-, and $gy$-faces, and cells from $r$-, $y$-, and $g$-cells. (a) A $b$-string. In the colex, it is a collection of $b$-edges linking $b$-cells. In the $b$-shrunk complex, the path of edges can be clearly seen. (b) A $ry$-face corresponds to a face in the $b$-shrunk complex, and thus its boundary can be viewed as a $b$-string.

Figure 9

(Color online) In this figure, the top represents part of a colex, and the bottom the corresponding portion of the $ry$-shrunk complex. Vertices in the $ry$-shrunk complex come from $g$ and $b$ cells in the colex, edges from $gb$ faces, faces from $ry$ faces, and cells from $r$ and $y$ cells. (a) An $ry$ membrane. In the colex, it is a collection of $ry$ faces linked by $gb$ faces. In the $ry$-shrunk complex, the brane can be clearly seen. (b) An $r$ cell corresponds to a cell in the $ry$-shrunk complex, and thus its boundary can be viewed as a $ry$ membrane.

Figure 10

(Color online) When an $r$ string $s$ crosses an $ry$ membrane $m$, they meet at a vertex. In terms of string and membrane operators, this means that $B_m^X$ and $B_s^Z$ act in a common site.

Figure 11

(Color online) Here, we represent $S^2\times S^1$ as in Fig. 7. In (a), a basis for nontrivial closed strings is shown. The other possible such string is green, but it is a combination of these ones (10). (b) A membrane basis in $S^2\times S^1$. We have chosen a $gb$, a $gr$, and a $gy$ membrane. There are three other nontrivial membranes, in particular, a $br$, a $by$ and a $yr$ membrane, but they are combinations of these ones (11).

Figure 12

(Color online) The ground state of the system is a string-net condensate. This picture represents in a 3-torus a typical element of the summation (13).

Figure 13

(Color online) In a membrane net, different membranes can connect along common boundaries. This is related to the preservation of flux, as shown in this figure.

Figure 14

(Color online) There are two ways in which quasiparticles can be created locally. We can create them either (a) by pairs of the same color forming a string or (b) in groups, one of each color forming a string net.

Figure 15

(Color online) The flux excitation created with the membrane operator $B_m^X$ of an $ry$ membrane made up of a single $ry$ face.

Figure 16

(Color online) (a) The border of a $gy$ membrane is a $gy$ flux. (b) A $gy$ flux can split into a $gb$ flux and a $by$ flux when it goes across an $r$ cell (16). (c) The microfluxes at a given site, as explained in the text.

Figure 17

(Color online) When a $g$ charge winds around a loop of a $gy$ flux, the system gets a global $-1$ phase. This is because the membrane operator giving rise to the flux and the string operator associated with the winding anticommute.

Figure 18

(Color online) An example of a red surface, located at the boundary between the 3-colex and an erased big $r$ cell. At a red surface, $r$ strings can have end points without quasiparticles, and $bg$, $gy$, and $by$ membranes can have borders without fluxes. These quasiparticles and fluxes would live precisely in the red cell that is missing. Note that a $y$ cell was removed in the figure.

Figure 19

(Color online) A $c$ graph that yields a 4-sphere. It is the simplest possible 4-colex, with only two vertices.

Figure 20

(Color online) A $bry$ cell belonging to some $D$ colex with $D⩾3$. We show superimposed in black thick line the structure of its dual boundary, which plays an important role when constructing the $bry$-shrunk complex.

Figure 21

(Color online) This figure shows the two possible kinds of shrunk complex in a 2-colex. The shrunk complexes appear superimposed in black thick line to the original colex. In (a), the $y$-shrunk complex is shown, and in (b) the $by$-shrunk complex.

Figure 22

(Color online) This figure explains a process that converts an arbitrary 2-complex on a surface into a 2-colex. In (a), green color is given to all the 2-cells of the 2-complex. In (b), 1-cells are inflated to give blue 2-cells. Finally, in (c), 0-cells are inflated to give red 2-cells, and 1-cells are accordingly colored.