Fracton topological order via coupled layers

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the ‘four color cube model”. The couplings considered lead to fracton topological orders via mechanisms we dub “p-string condensation” and “p-membrane condensation”, in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

Figure 1

Illustration of cube and vertex terms in the X-cube model related to plaquette and vertex terms from toric code layers.

Figure 2

An elementary p-string which forms the building block of the p-string condensate. The green thick link denotes an action of $Z_{\ell }^x{Z}_{\ell }^y$, which creates four $m$ particles (shown as black x's connected by the dashed lines). The black dot shows the location of the physical spin ${\mathcal{Z}}_{\ell }$($=Z_{\ell }^x=Z_{\ell }^y$). The blue string represents the p-string, which connects the $m$ particles on the perimeter of the membrane.

Figure 3

A larger p-string, obtained by acting with $Z_{\ell }^x{Z}_{\ell }^z$ operators along the links orthogonal to a rectangular membrane (marked in green).

Figure 4

A braiding process between a p-string in the condensate and an $e_{P_0^{\mu }}$ particle. The process is drawn in a “continuum limit,” where we do not show the individual $m$ particles making up the p-string. During the braiding process the right side of the p-string is held fixed, while the left side sweeps out the motion indicated by the gray arrow.

Figure 5

(a) An open p-string (dark blue), created at the edge of a series of $xy$ plane $m$-string operators (which act on the red links) stacked in the $z$-direction. The ends of the $m$-string operators are $m$-particles, and are marked with black crosses. (b) Acting with $Z_{\ell }^x{Z}_{\ell }^z$ on the green link creates a short $m$-string in the $yz$ plane and deforms the p-string.

Figure 6

A membrane operator ${\mathcal{M}}_M$ which creates a pair of open p-strings harboring $\mathfrak{m}$ excitations (purple cubes) at their endpoints, formed by creating a vertical stack of $m$-strings. The red links possess ${\mathcal{X}}_{\ell }$ eigenvalues of $-1$, and the dashed lines represent $m$-strings.

Figure 7

An illustration of the operator ${\mathcal{S}}_m\left(\gamma \right)$ for a section of a generic path $\gamma$. The red links denote links whose ${\mathcal{X}}_{\ell }$ eigenvalue is $-1$, and the blue string denotes a section of the path $\gamma$.

Figure 8

A single $m$-string (dashed line) creates two ${\mathfrak{m}}_c$ excitations (purple cubes) at each of its ends, illustrating the fact that $m$-particles (black crosses) survive in the X-cube topological order as bound states of two ${\mathfrak{m}}_c$ fractons.

Figure 9

(a) Two fractons are transported around a rectangular prism by acting with membrane operators (shaded areas) on the faces. (b) The membrane operator in (a) can be regarded as a stack of $m$ string operators (dashed lines), which transport a stack of $m$ particles (black crosses) around the prism.

Figure 10

Lattice geometry of the coupled X-cube and FCC models, which are defined on four interpenetrating simple cubic lattices that we label by the colors black, red, green and blue. Dots, squares, up triangles, and down triangles indicate the vertices of the four simple cubic lattices. Simple cubic lattice links intersect in mutually perpendicular triples of three different colors; for example, red, green, and blue links intersect at centers of black cubes, as shown.

Figure 11

The p-membrane condensation in FCC model. The big black dot denotes one of the cubes in the black X-cube model. It is surrounded by plaquettes of the other three colors representing the planes perpendicular to the motion of $d=1$ particles, generated at vertices of different cubic lattices denoted by different shapes as shown in Fig. (10).

Figure 12

Illustration of a FCC model $A_c$ operator on a black cube. The operator is a product of $A_i^{\mu }$ over the cube's six faces shown by the red, green, and blue links, whose vertices are represented by corresponding shapes shown in Fig. 10. For each edge of the cube, two $Z_{\ell }^w$ operators contribute to $A_c$.

Figure 13

(a) Acting with $X_{\ell }^k$ on the thick black link (pointing in the $z$ direction) creates eight $A_c=-1$ excitations, four of which are blue (open squares) and four green (solid squares). (b) $X_{\ell }^k$ is dual to $Z_{\ell }^g{Z}_{\ell }^b$, where each $Z$ operator creates four $B_c=-1$ excitations of the same color, again for a total of eight excitations, which illustrates the self-duality. The four excitations created by each $Z_{\ell }^w$ are four fractons in one of the underlying X-cube models. (c) In both cases, the eight excitations can be viewed as created at corners of two perpendicular membrane operators.

Figure 14

(a) Acting with a product of $X_{\ell }^k$ along a straight line oriented along the $z$-axis creates one-dimensional particle excitations at the end of the line, each composed of four $A_c=-1$ excitations (open/solid squares). These excitations are the remnants of one-dimensional $\mathfrak{e}$ excitations in the black X-cube model. (b) Electric-magnetic duality shows that acting with a product of $Z_{\ell }^g{Z}_{\ell }^b$ along a line in the $z$ direction also creates one-dimensional particle excitations. These one-dimensional excitations can be thought of as bound states of two-fracton two-dimensional particles from green and blue X-cube models. The blue bound state moves in a $yz$ plane, while the green bound state moves in a $xz$ plane, so the bound state of both of them is constrained to move in the $z$ direction.

Figure 15

(a) Diamond pattern in a $\left(001\right)$ plane that comprises a single layer of the skyscraper operator. Each circle represents a black or red link pointing in the $z$ direction, and a product of $X_{\ell }^w$ is taken over the filled circles. $A_c=-1$ excitations are created at the corners at the top and bottom of the skyscraper, as indicated by the blue open squares and green solid squares. The filled circles also indicate the positions of $\mathfrak{e}$ particles created at the top and bottom of the skyscraper, when we act with the skyscraper operator in the limit of decoupled X-cube models. (b) A three-dimensional view of the skyscraper operator. Each green layer represents a diamond pattern as shown in (a), and the locations of the eight $A_c=-1$ excitations are shown by blue squares.

Figure 16

Acting with the operator $\mathcal{B}$ can be viewed as implementing a process where three one-dimensional particles are created at each corner of a rectangular prism, and these particles are then moved along edges to annihilate. This process has a nontrivial statistical phase of $\pi$ when a single $B_c=-1$ excitation is present inside the prism, where $c$ and the edges of the prism all belong to the same color cubic lattice.

Figure 17

(a) A trivalent 2D lattice from decorating 2D square lattice with diamond shapes at each vertex. (b) Stacks of such trivalent lattices in $x$ (solid lines), $y$ (dash lines), and $z$ (dotted lines) planes; the edges in $x$, $y$, and $z$ directions overlap in pairs.

Figure 18

Illustration of the properties of excitations in the modified X-cube model on a decorated lattice. (a) Excitations in diamond plaquettes can move freely in 2D with a string of $\mathcal{Z}$ operators as shown; (b) Excitations at vertices can move in 1D with a string of $\mathcal{X}$ operators as shown.

Figure 19

Illustration of a type 2 XLO running along the $\left[110\right]$ direction. A product is taken over a line of thick-shaded black and red links as shown. The string extends in the $\left[110\right]$ direction, while the links point in the $z$ direction.