Fracton fusion and statistics

We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the nontrivial mobility of fractons and subdimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions of the fusion theory in the X-cube and checkerboard fracton models, as well as for gapped electric and magnetic excitations of some gapless $\mathrm{U}\left(1\right)$ tensor gauge theories. An alternate route to the X-cube model fusion theory is provided by starting with a system of decoupled two-dimensional toric code layers, and giving a description of the $p$-string condensation mechanism within our approach. We discuss examples of statistical processes of fractons and subdimensional excitations in the X-cube and checkerboard models. As an application of the ideas developed, we prove that the X-cube and semionic X-cube models realize distinct translation-invariant fracton phases, even when the translation symmetry is broken corresponding to an arbitrary but finite enlargement of the crystalline unit cell.

Figure 1

Graphical representation of the mapping between excitations in the X-cube model (left) and configurations of ${\mathbb{Z}}_2$ plane charges in the fusion theory (right). In the top panel, a single cube excitation $f$ is a fracton that maps to a configuration of charges on three intersecting planes. In the bottom panel, ${\ell }^z$ is a vertex excitation, and carries the charge on two intersecting planes. It is apparent from the graphical representations that ${\ell }^z$ is a lineon able to move along the vertical axis, while $f$ is immobile. We note that there are two distinct, independent types of plane charges, with cube excitations carrying one type, and vertex excitations carrying the other type (indicated by color online). The map $\pi$ is a map of $\mathbb{Z}\left[T\right]$ modules defined in the text.

Figure 2

Operators in the X-cube model. (Left) $A_v^z$ is a product of $Z_{\ell }$ over the star of four thickened links. (Right) $B_c$ is the product of $X_{\ell }$ over the edges of an elementary cube.

Figure 3

Illustration of two closed $p$-string configurations of $m$ particle excitations in a model of decoupled $d=2$ toric code layers on the simple cubic lattice. Panel (a) shows a finite closed $p$ string, while (b) is an infinite closed $p$ string, which corresponds to an element $f_z^{-}\left(\mathit{r}\right)+f_z^{+}\left(\mathit{r}\right)\in {\mathcal{E}}_m^{\mathrm{ext}}$. The gray-shaded plaquettes are $m$-particle excitations, and a segment of blue $p$ string is shown intersecting transversely to each $m$-particle plaquette. The $p$ strings here are infinite (no endpoints) because every cube has an even number of $m$-particle excitations on its faces.

Figure 4

Illustration of the labeling of cubic lattice plaquettes used in the discussion of $p$-string condensation. Each plaquette is labeled by a pair $(\mathit{r},\mu )$, where $\mu$ is the normal direction, and $\mathit{r}$ is the vertex at the corner of the plaquette as shown.

Figure 5

The shaded cubes form the A sublattice in the checkerboard model. The $A_c$ and $B_c$ terms in the Hamiltonian are products of $Z_v$ and $X_v$, respectively, over the eight vertices at the corners of each A-sublattice cube.

Figure 6

Braiding of an $e$ particle (red dashed circle) around a $m$ particle (black solid circle) in the $d=2$ toric code. First the $m$ particle is moved to the right from its initial position, then the $e$ particle is braided around it, and finally the $m$ particle is moved back to its initial position. Each step is sequence of a large number of local moves.

Figure 7

Deformations of the $e-m$ braiding process in the $d=2$ toric code. The panels of the figure are described in the text.

Figure 8

Fracton-lineon process in the X-cube model. In step (1) the fracton cube excitation $f$ (blue circle) is destroyed, and three cube excitations are created at the other corners of the blue membrane. In step (2), the lineon vertex excitation $\ell$ (red square) is moved through the blue membrane. Step (3) then undoes step (1), and step (4) undoes step (2).

Figure 9

The spatial separation requirement is violated if steps 1 and 3 of the fracton-lineon process are deformed by removing local moves to shrink the membrane to that illustrated on the top-right. At an intermediate stage of the deformation (bottom), a cube excitation is present within the black circle after step 1, and the lineon passes nearby this cube excitation during step 2, violating the spatial separation requirement.

Figure 10

(a) An allowed deformation of step 4 in the fracton-lineon process. (b) Step 2 in a modified fracton-lineon process where step 4 has been deformed away, as described in the text.

Figure 11

Two processes related to the fracton-lineon process by changing the initial state, but with the same sequence of steps. (a) shows the initial state in the lineon-planon process, which consists of a pair of $f$ fractons that together form a planon, and the same lineon $\ell$ as in the fracton-lineon process. (b) shows the initial state for a vacuum process where no particles are present.

Figure 12

Lineon-lineon exchange process in the X-cube model. The initial configuration consists of two vertex excitations ${\ell }^x\left(0\right)$ and ${\ell }^x\left(x_d\widehat{x}\right)$ on the $x$ axis at positions labeled $a$ and $d$, respectively, with $x_d\widehat{x}$ the position vector of $d$ and $x_d\in \mathbb{Z}$. In step 1, the lineon at $a$ is moved to position $c$. Step 2 is the most nontrivial step: the lineon initially at $d$ is moved off the $x$ axis by splitting it into ${\ell }^y$ and ${\ell }^z$ vertex excitations, producing three ${\ell }^x$ excitations off the axis, which are then moved in the $-x$ direction and eventually recombined into a ${\ell }^x$ excitation at position $b$. In the remaining three steps, ${\ell }^x$ excitations are moved along the $x$ axis as shown. If the steps are executed in the order shown, then the two lineons are exchanged during this process. However, if instead the steps are carried out in the order 15243, no exchange occurs.

Figure 13

Fracton-fracton process in the checkerboard model. The initial configuration is shown in (1) and has two pairs of fractons, one pair separated along the $x$ axis and the other pair separated along the $z$ axis. The blue solid circle fractons are excitations of the $A_c$ terms in the checkerboard Hamiltonian, while the red striped circle fractons are excitations of the $B_c$ terms. In step (1), the blue solid circle pair of fractons is moved to the positions indicated by the dashed blue circles. Step (2) proceeds similarly for the red striped circle fractons. Carrying out this process on the lattice, the two membrane operators used to implement steps (1) and (2) intersect on a line segment containing $L$ lattice sites. Step (3) reverses step (1), and step (4) reverses step (2). Multiplying together the operators implementing each step gives a phase of ${(-1)}^L$ for the order of steps shown, but a trivial phase for the order 1324.

Figure 14

(a) and (b) depict the generators of ${\mathcal{L}}_m\subset {\mathcal{E}}_m$, the $\mathbb{Z}\left[T\right]$ module of locally creatable magnetically charged excitations of the (1,1) scalar charge theory. Solid lines represent $q(\mathit{r},\mu )=\pm 1$, with the sign given by the direction of the arrow, while dotted lines represent $q(\mathit{r},\mu )=0$. (c) Shows an element of ${\mathcal{L}}_m$ that can be obtained by adding generators; elements of this form are used in Appendix pp1-s5 in a proof that $ker\pi \subset {\mathcal{L}}_m$. The line with a double arrow represents $q(\mathit{r},\mu )=\pm 2$.

Figure 15

Reducing a configuration of $1d$ particles with trivial plane charges to that with only $s^z$ excitations.

Figure 16

Local moves needed to “clean up” a configuration of trivial plane charges to the vacuum.

Figure 17

The $z=0$ face of the $2\times 2\times 2$ cube with zero charge and dipole moment. The four vertices have charges $q_1,\cdots ,q_4$ as shown, and positions ${\mathit{r}}_1=(0,0,0), {\mathit{r}}_2=(1,0,0)$, and so on.