Sorting topological stabilizer models in three dimensions

The $S$-matrix invariant is known to be complete for translation invariant topological stabilizer models in two spatial dimensions, as such models are phase equivalent to some number of copies of toric code. In three dimensions, much less is understood about translation invariant topological stabilizer models due to the existence of fracton topological order. Here we introduce bulk commutation quantities inspired by the 2D $S$-matrix invariant that can be employed to coarsely sort 3D topological stabilizer models into qualitatively distinct types of phases: topological quantum field theories, foliated or fractal type-I models with rigid string operators, or type-II models with no string operators.

Figure 1

The foliation structure of the $X$-cube model.

Figure 2

Logical operator segments in the SFSL model [12, 16]. (a) String operator along the $\widehat{z}$ direction. A product of $IX$ operators on adjacent sites along $\widehat{z}$ creates the excitation pattern shown in the dual lattice where the stabilizers live on vertices. (b) A fractal operator moves three excitations apart in the $xy$ plane. Repeating this at longer length scales results in a Sierpinski triangle shaped operator.

Figure 3

Fractal excitation patterns in the cubic code 1 model [13]. (a) The excitation pattern generated by the $XI$ operator on a single site. (b) A small fractal operator that moves excitations apart.

Figure 4

Flat-rod configurations. Open boundary conditions where excitations may be created are marked with black edges. Closed boundary conditions where no excitations are created are marked with blue edges.

Figure 5

The membrane-membrane configuration. Open boundary conditions where excitations may be created are indicated by black edges. Closed boundary conditions where no excitations are created are indicated in green and blue for the two membranes respectively.

Figure 6

Generalized Gauss's laws for the fracton sector of the X-cube model within a finite region. The set of Gauss's laws containing a given stabilizer share only one common intersection point.

Figure 7

Different configurations of excitation pairs (grey cubes) and the minimal boxes containing them.

Figure 8

There are eight different types of corners. The shape of the pair-creation operator doesn't have to cube or cuboid but can always be confined in a cuboidal box as shown. Two corners, if they are of the same type, have identical constraint matrices.

Figure 9

A set of points used in the second order constraint in cleaning of a corner of type $A, D, A^{\prime }$ or $D^{\prime }$. The corner is at position $a$ as shown. Points $a, b$ and $c$ lie in the $yz$ plane.

Figure 10

A set of points used in the third order constraint involving the vertex with type $B, D, B^{\prime }$, or $D^{\prime }$ at position $c$ as shown. Points $a, b, c, d,$ and $e$ lie in the $xy$ plane.

Figure 11

Deformation of $xy$ planar boxes. If the excitations fit within a box along a lattice plane, we refer to it as a planar box.

Figure 12

Steps in deformation of the second $xy$ planar box as shown in Fig. 11. We employ higher-order commutation constraints to deform the box to flat-rods.

Figure 13

Deformation of $xz$ planar boxes.

Figure 14

Deformation of $yz$ planar boxes.

Figure 15

Deformation of 3D configuration of excitations of Fig. 7.

Figure 16

Deformation of the 3D configuration of excitations from Fig. 7.

Figure 17

Deformation of the 3D configuration of excitations in Fig. 7.

Figure 18

Deformation of 3D configuration of excitations of Fig. 7.

Figure 19

Excitation patterns and a string operator for cubic code 5.

Figure 20

Excitation patterns and a string operator for cubic code 6.

Figure 21

Excitation patterns and a string operator for cubic code 9.

Figure 22

Anticommutation relations for planons parallel to the $yz$ plane in CC11. $p_X$ denotes composite planon excitations of $X$ stabilizers, and similarly for $Z$. An edge between a $p_X$ and $p_Z$ planon indicates that they have a braiding phase of $-1$.

Figure 23

Anticommutation relations for planons parallel to the $xz$ plane in CC12. Where $p_X$ denotes composite planon excitations of $X$ stabilizers, and similarly for $Z$. An edge between a $p_X$ and $p_Z$ planon indicates that they have a braiding phase of $-1$.

Figure 24

Anticommutation relations for planons parallel to the $xz$ plane in CC13. Where $p_X$ denotes composite planon excitations of $X$ stabilizers, and similarly for $Z$. An edge between a $p_X$ and $p_Z$ planon indicates that they have a braiding phase of $-1$.

Figure 25

Anticommutation relations for planons parallel to the $xz$ plane in CC14. Where $p_X$ denotes composite planon excitations of $X$ stabilizers, and similarly for $Z$. An edge between a $p_X$ and $p_Z$ planon indicates that they have a braiding phase of $-1$.

Figure 26

Anticommutation relations for planons parallel to the $yz$ plane in CC15. Where $p_X$ denotes composite planon excitations of $X$ stabilizers, and similarly for $Z$. An edge between a $p_X$ and $p_Z$ planon indicates that they have a braiding phase of $-1$.

Figure 27

Anticommutation relations for planons parallel to the $xz$ plane in CC17. The planons can be decomposed into those hopping on a $(B/W)$ checkerboard coloring of the 2D $\widehat{z}$ planes. Where $p_X\left(B\right)$ denotes composite planon excitations of $X$ stabilizers on the black sublattice, and similarly for $Z$ and $\left(W\right)$. An edge between a $p_X$ and $p_Z$ planon indicates that they have a braiding phase of $-1$.

Figure 28

Lineon and planon operators in Chamon's model. (a) The excitation pattern for an $X$ operator, two of the excitations shown on the $yz$ plane form a lineon that moves along the $\widehat{x}$ direction. (b) The excitation pattern for a $Z$ operator, two excitations along $xz$ plane form a lineon that moves along the $\widehat{y}$ direction. (c) The excitation pattern for a $Y$ operator, two excitations along the $xy$ plane form a lineon that moves along the $\widehat{z}$ direction. [(d)–(f)] Excitations that form composite planons in the $xy, yz$ planes and $xz$ plane, respectively.