Braiding and gapped boundaries in fracton topological phases

We study gapped boundaries of Abelian type-I fracton systems in three spatial dimensions. Using the X-cube model as our motivating example, we give a conjecture, with partial proof, of the conditions for a boundary to be gapped. In order to state our conjecture, we use a precise definition of fracton braiding and show that bulk braiding of fractons has several features that make it insufficient to classify gapped boundaries. Most notable among these is that bulk braiding is sensitive to geometry and is “nonreciprocal”; that is, braiding an excitation $a$ around $b$ need not yield the same phase as braiding $b$ around $a$. Instead, we define fractonic “boundary braiding,” which resolves these difficulties in the presence of a boundary. We then conjecture that a boundary of an Abelian fracton system is gapped if and only if a “boundary Lagrangian subgroup” of excitations is condensed at the boundary; this is a generalization of the condition for a gapped boundary in two spatial dimensions, but it relies on boundary braiding instead of bulk braiding. We also discuss the distinctness of gapped boundaries and transitions between different topological orders on gapped boundaries.

Figure 1

Braiding anyons $a$ and $b$ at a gapped boundary. (a) A pair of open Wilson loops for $a$ and $b$ that leave $a, \overline{a}$ and $b, \overline{b}$ excitations on the boundary. (b) If $a$ and $b$ are condensed at the boundary, one can apply operators $U_a$ and $U_b$ that delete the $a, \overline{a}$ and $b, \overline{b}$ excitations at the boundary and have support only in the vicinity of these excitations.

Figure 2

Terms in the X-cube model Hamiltonian (3.1). Colors are for emphasis and (later) to indicate pictorially which terms are affected at a site. The label $i$ in the star operator $A_i$ denotes the plane perpendicular to the star operator.

Figure 3

Braiding in (2+1)D topological order. The anyons $a$ and $b$ braid nontrivially if the closed Wilson loop operator ${\mathcal{W}}_b$ does not commute with the open Wilson line operator $W_a$, provided $W_a$ creates topological charge $a$ in the area enclosed by the support of ${\mathcal{W}}_b$.

Figure 4

Cage operators for some excitations in the X-cube model. Green (orange) links carry spins on which a $Z$ ($X$) operator acts. The gray planes are a guide to the eye. Coordinate axes in (a) are consistent throughout. (a): Cage for any $e$ particle. (b): Cage for $m_{xy}$; since $m_{xy}$ is 2D, its cage operator is just a Wilson loop. (c) and (d): Two different cage operators for $f_0$.

Figure 5

Two inequivalent ways to braid $f_0$ around an $e_z$. Green (orange) links carry spins on which a $Z$ ($X$) operator acts. (a) Choice of a cage operator which leads to a nontrivial braiding phase. Both the string operator creating $e_z$ excitations and the cage operator for the fracton act on the link indicated by the arrow, causing these operators to anticommute. (b) Choice of a cage operator which leads to a trivial braiding phase. Any translation of this $f_0$ cage operator commutes with the $e_z$ string operator.

Figure 6

Several ways to braid $e_z$ and $m_{xy}$. Green (orange) links carry spins on which a $Z$ ($X$) operator acts. (a) Braiding $e_z$ around $m_{xy}$ in a way that leads to a nontrivial braiding phase; the cage and string operators shown anticommute with each other due to their overlapping support at the link indicated by the black arrow. (b) Displacing the string operator in (a) by a single lattice site in the $z$ direction causes the operators to commute, illustrating the geometric nature of bulk fracton braiding. (c) Braiding $m_{xy}$ around $e_z$; these operators commute no matter where they act relative to one another.

Figure 7

“Fusion” process for $e_x\times e_y\sim e_z$.

Figure 8

Boundary Hamiltonians for each gapped boundary condition for the (001) surface of the X-cube model. (a) $\left(ee\right)$ boundary. (b) $\left(mm\right)$ boundary. (c) $\left(me\right)$ boundary. There are no $A_y$ or $A_z$ operators for the topmost layer of sites, but sites in the next layer down have all three bulk $A_i$ terms. (d) $\left(em\right)$ boundary.

Figure 9

String operator (green dashed line) consisting of $Z$ operators (green circles) that creates isolated $e_x=e_y$ excitations (purple squares) at its ends on an $\left(ee\right)$ boundary of the X-cube model. Acting with $Z$ on a vertically oriented bond fuses an $e_z$ onto the original excitation, transmuting it between $e_x$ and $e_y$. Such operators allow $e_x=e_y$ excitations to move in two dimensions on the surface.

Figure 10

String operator (orange dashed line) consisting of $X$ operators (orange circles) that creates isolated $f_0$ excitations (blue shading) at its ends on an $\left(mm\right)$ boundary of the X-cube model. Such operators allow $f_0$ excitations to move in two dimensions on the surface.

Figure 11

Condensation of $q_{xz}$ on (001) surface of the X-cube model with $\left(me\right)$ boundary conditions. Green links mean $Z$ acts on this link. (a) Operator that creates a single $q_{xz}$ (its directionality can be checked by moving it into the bulk). (b) Operator that takes a bulk $e_z$ excitation (translucent), moves it to the boundary, and moves it in the $y$ direction by fusing it with $q_{xz}$ excitations.

Figure 12

Surface $e$ excitation encircling a surface $f_0$ excitation on the $\left(ee\right)$ boundary of the X-cube model. Orange links are part of the membrane $X$ operator creating the $f_0$ excitations (blue cubes). Green links are part of the surface Wilson loop operator for the surface $e$ excitation. The arrow indicates the only link where both the membrane and Wilson loop have support, causing the operators to anticommute.

Figure 13

Top view of the cubic lattice with (110) boundary, with boundary excitations depicted in Fig. 14. The vertices that live on the boundary can be divided into two sublattices, $A$ and $B$ (dashed lines). The $f_0^A$ excitation is a bound state of three fractons (blue squares) that would be equivalent modulo local operators to a single fracton in the bulk (transparent blue square). The $f_0^B$ excitation is equivalent to a bulk $f_0$. The $e_z^A$ excitation is a bound state of an $e_x$ and an $e_y$ excitation (blue/yellow and blue/red circles) that would be equivalent modulo local operators to an $e_z$ in the bulk (transparent red/yellow circle). The $e_z^B$ excitation is equivalent to a bulk $e_z$.

Figure 14

(a), (c), and (e): Terms in the boundary Hamiltonian for three gapped boundaries at the (110) surface of the X-cube model. Green (orange) spheres represent $Z$ ($X$) operators. For boundary condition (c), there are no spins on the outermost $z$-oriented links of the lattice. (b), (d), and (f): Action of local operators that create condensed excitations for the boundary conditions (a), (c), and (e), respectively. Operators are represented in the same way as in (a), (c), and (e). Excitations are illustrated using the visual representations in Table 1.

Figure 15

Some examples of boundary half cages in the X-cube model. (a) and (b): Boundary half cage for $e_z$ and $m_{yz}$, respectively, on the (001) surface. (c) and (d): Boundary half cage for $e_z^A$ and $e_z^B$, respectively, on the (110) surface. The operators differ by a single lattice site. (e): Boundary half cage for $f_0^A$ on the (110) surface.

Figure 16

Operator that creates isolated $f_0$ excitations on the (001) surface (light gray plane). This operator is not a boundary half cage because it has support on the boundary at points arbitrarily far away from the $f_0$ excitations.

Figure 17

Boundary braiding of $e_z$ and $m_{yz}$ on the (001) surface of the X-cube model. The boundary half-cage operators anticommute because both act on the link indicated by the black arrow. The dashed line is a guide to the eye; placing the $m_{yz}$ excitations at a different position in $x$ may not yield nontrivial braiding.

Figure 18

Boundary braiding on the (110) surface of the X-cube model for (a) $f_0^A$ and $e_z^A$, which is trivial, (b) $f_0^A$ and $e_z^B$, which is nontrivial, (c) $f_0^B$ and $e_z^A$, which is nontrivial, and (d) $f_0^B$ and $e_z^B$, which is trivial. Black arrows point to the links where the boundary half cages have overlapping support.

Figure 19

Terms in the bulk Hamiltonian of the ${\mathbb{Z}}_N$ X-cube model. $N$-state systems live on the links and $X$ and $Z$ are generalized Pauli matrices.

Figure 20

Terms in boundary Hamiltonian of the ${\mathbb{Z}}_4$ X-cube model with (a) $e_z^{1,2,3}$ condensed and (b) with the in-plane $e^2$ excitation condensed on the surface. The bottom face of the open cube operator and the top face of the closed one live in the same layer of the system. All spins are ${\mathbb{Z}}_4$ in (a) and all spins with ${\sigma }_{x,z}$ operators are effective ${\mathbb{Z}}_2$ degrees of freedom in (b).

Figure 21

Top view of a fracton cascading in the (110) direction (the $z$ direction is perpendicular to the page). Orange circles represent $X$ operators applied to the link sticking out of the page. If the fracton was initially present in the bottom-left corner (light blue square), then the arrangement of operators shown cascades the fracton in the direction of the arrow. Note that cascading in a certain direction does not amount to mobility in that direction (although the reverse is true); for example, the depicted cascading process costs finite energy and is thus disallowed at zero temperature.