Topological defect networks for fractons of all types

Fracton phases exhibit striking behavior which appears to render them beyond the standard topological quantum field theory (TQFT) paradigm for classifying gapped quantum matter. Here, we explore fracton phases from the perspective of defect TQFTs and show that topological defect networks—networks of topological defects embedded in stratified $3+1$-dimensional $(3+1\mathrm{D})$ TQFTs—provide a unified framework for describing various types of gapped fracton phases. In this picture, the subdimensional excitations characteristic of fractonic matter are a consequence of mobility restrictions imposed by the defect network. We conjecture that all gapped phases, including fracton phases, admit a topological defect network description and support this claim by explicitly providing such a construction for many well-known fracton models, including the X-cube and Haah's B code. To highlight the generality of our framework, we also provide a defect network construction of a fracton phase hosting non-Abelian fractons. As a byproduct of this construction, we obtain a generalized membrane-net description for fractonic ground states as well as an argument that our conjecture implies no topological fracton phases exist in $2+1$-dimensional gapped systems. Our paper also sheds light on techniques for constructing higher-order gapped boundaries of $3+1\mathrm{D}$ TQFTs.

Figure 1

A stratification of three-dimensional space into 3-strata (white regions), 2-strata (red squares), 1-strata (blue lines), and 0-strata (black vertex). A $3+1\mathrm{D}$ TQFT lives on each 3-strata, and they are coupled together along the lower-dimensional 2- and 1-strata defects. This defect TQFT could be described by a lattice model with qubits on, e.g., the black cubic lattice.

Figure 2

A 2-stratum defect layer (black line) in the string-membrane-net model, which is embedded in a $3+1\mathrm{D}$ toric code. The three-dimensional (3D) toric code has charge (labeled $e$) and flux string (red lines in figure) excitations. The defect layer also supports two-dimensional (2D) toric code charge (labeled $e^{\prime }$) and pointlike flux (labeled $m^{\prime }$) excitations. The following excitations can be created near a defect layer using local operators: (a) a 2D toric code charge $e^{\prime }$ and a 3D toric code charge on each side of the defect layer; (b) a pair of 2D toric code fluxes $m^{\prime }$ at the end points of a 3D toric code flux string; and (c) a closed 3D toric code flux string.

Figure 3

Stacks of 2-strata defect layers (defined in Fig. 2) result in the fracton and lineon mobility constraints of the X-cube model. (a) The defect layers cause the three-dimensional (3D) toric code charge (labeled $e$) to obtain fractonlike mobility. To see this, consider (i) trying to move a 3D charge $e$ to move through a bunch of layers. (ii) This can only be done by creating the triplets of charges in Fig. 2 on the defect layers. (iii) Pairs of 3D charges $e$ can annihilate, but a string of two-dimensional (2D) charges $e^{\prime }$ is left behind. This string of excitations linearly confines the 3D charge $e$ to its initial location, just like an X-cube fracton. (b) A lineon results from a pair of 2D toric code fluxes $m^{\prime }$ (which are bound to the end points of a 3D toric code flux) on two intersecting defect layers.

Figure 4

When the 2-stratum condenses all fluxes, the 1-strata boundary conditions can be understood using a dimensional reduction trick. In (a) we depict how membranes near a 1-strata effectively behave as strings in a dimensionally reduced $2+1\mathrm{D}$ toric code. In (b) we show how the 3-strata excitations map onto the dimensionally reduced toric code excitations.

Figure 5

(a) The edge condensation [Eq. (8)] allows four charges to be created in four neighboring 3-strata, analogous to X-cube fractons. (b) A lineon results from a flux string between two neighboring 2-cells. (c) Three lineons can be created from the vacuum by creating a small flux loop (light red) near a 0-strata, and then stretching it to the nearby 2-strata. This is analogous to the X-cube model where three lineons can be created at a vertex. (d) The condensation at 1-strata defects [Eq. (8)] allows the flux loop to move through the 1-strata. (e)–(f) A sequence of four configurations which show how the lineon can move past a 2-strata by making use of the moves in (c) and (d).

Figure 6

A representative set of allowed membrane configurations near a 1-cell as viewed from above (i.e., down a 1-cell). The remaining allowed configurations come from rotating the two diagrams on the right.

Figure 7

Terms in the Hamiltonian for Haah's B code, which has four qubits per site of the cubic lattice. $X$ and $Z$ are Pauli operators, $I$ is the identity, and the shorthand $XXXX$ (for example) means a product of Pauli-$X$ operators on all four qubits on the site in question.

Figure 8

Excitations in Haah's B code. (a) Excitations created by some local operators (spheres) on a ground state of the B code. Colored cubes indicate elementary cubes where the labeled Hamiltonian term has eigenvalue $-1$ (AB means both $A_c$ and $B_c$ have eigenvalue $-1$). (b) An operator with fractal-like support (the product of local operators indicated by the colored spheres) creates widely separated excitations (cubes) in Haah's B code.

Figure 9

Labeling scheme for 3-strata used in defining condensed objects on the colored 1-strata for the defect construction of Haah's B code.

Figure 10

Electric excitations in the defect network construction and their correspondence to $A$ and $B$ excitations in the B code. Orange (blue) lines are electric string operators in the A (B) $3+1\mathrm{D}$ toric code layer, and correspondingly colored spheres are pointlike electric excitations in the appropriate layer. Perspective is the same as Fig. 9. (a) Correspondence between the action of local Pauli-$Z$ operators in the B code [see Fig. 8(a)] and the condensation of electric charges on 1-strata in the defect network construction. (b) Network of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ $3+1\mathrm{D}$ toric code string operators which create isolated electric excitations (spheres) in the defect network construction of the B code. The choice of condensations in Eq. (15) ensures that no additional excitations are created at the 1-strata. Compare the excitation configuration to that in the B code in Fig. 8.

Figure 11

Magnetic excitations in the defect network construction for the B code. (a) A B-layer magnetic membrane operator (blue sheet) which creates excitations only at three of its corners (colored spheres). (b) A magnetic membrane operator with fractal-like support which creates isolated, widely separated excitations. Orange (blue) is a membrane operator in the A (B) toric code layer, and purple is a product of membrane operators in both layers. (c) The B code $C_c$ and $D_c$ operators on the blue and yellow cubes, respectively, represented as $3+1\mathrm{D}$ toric code string operators (orange and blue are A and B layer string operators, respectively) using the correspondence of local operators with condensed objects shown in Fig. 10. The excitations in (a) are labeled by considering which membrane operators commute and anticommute with these string operators.

Figure 12

Allowed membrane configurations on the $x$-oriented 1-strata for the defect construction of Haah's B code. The perspective is looking down the 1-strata from the positive $x$ axis towards the origin. Orange (blue) membranes are nets from the $A$ (resp. $B$) toric code layers.

Figure 13

A 2-stratum defect layer (black line) in the non-Abelian $D_4$ model described in the main text. The defect allows $s^2$ membranes to terminate on the 2-strata. The following excitations can be created near a defect layer using local operators: (a) pairs of Abelian charges labeled $ij$ on opposite sides of the defect, (b) a flux arc labeled by the conjugacy class $s^2$, and (c) a flux loop on opposing sides of the defect labeled by the same conjugacy class $c$.

Figure 14

A representative set of 1-stratum condensations described by Eq. (22). (a) Four non-Abelian $\sigma$ charges can simultaneously condense to the vacuum on a 1-stratum. (b) The Abelian charges labeled $ij$ are allowed to freely pass through the 2- and 1-strata. (c) Any pair of $s^2$ flux strings can condense at the 1-strata. (d) Any quadruple of flux strings (labeled by a conjugacy class $c$) meeting at a 1-stratum can condense into vacuum.

Figure 15

(a) A topological defect network in $2+1$ dimensions. (b) Gapped boundary to vacuum domain walls that prevent particle mobility and a topological charge (red) being measured by a conjugate string operator near a corner. (c)–(e) Applying the inflation trick to a 2-stratum from (b).

Figure 16

Folding a $2+1\mathrm{D}$ topological defect network to transform a column into a gapped boundary to vacuum.

Figure 17

Inflation trick: Certain 2-strata defects are effectively gapped boundaries to vacuum. In this case, we can deform the 2-strata bounding a given 3-stratum into the center of the 3-stratum. The $3+1\mathrm{D}$ topological order in each 3-stratum undergoes a dimensional reduction to sheets of (effectively) $2+1\mathrm{D}$ topological order, and one finds a purely $2+1\mathrm{D}$ defect network for the same phase.

Figure 18

A 2D slice of an $XY$ plane showing electric and magnetic lineon excitations created at corners of fractal operators (blue and red). Each triangle represents a triangular prism of 3D toric code that extends into the $Z$ axis.

Figure 19

Particles that condense at the 1-strata of the ${\mathbb{Z}}_3$ defect network. Blue spheres denote a 1 particle and red spheres with a bar denote its antiparticle 2. The arrows indicate the chirality of ${\mathbb{Z}}_3$ layers.

Figure 20

Excitations in the ${\mathbb{Z}}_3$ defect network corresponding to a fracton (a), lineon (b), and planon (c).

Figure 21

We choose a right-handed frame such that all links of the dual lattice (in blue) are oriented with respect to these directions. This defines a natural orientation for the plaquettes $p$ (in orange) of the original lattice.

Figure 22

Pairs of $D_4$ gauge theory meet at 2-strata defects (green plaquettes) with four of them meeting at a 1-strata defect (red links). Plaquettes belonging to the two $\overline{D_4}$ 3-strata have orientations opposite to those belonging to the $D_4$ 3-strata.

Figure 23

(a) Cubes, plaquettes, and links sharing a face ($f$), link ($\ell$), or vertex ($v$) with a 2-strata are denoted by $c_f,p_{\ell }$, and ${\ell }_v$ respectively. (b) Cubes and plaquettes sharing a link ($\ell$) or vertex ($v$) with a 1-strata are labeled $c_{\ell }$ and $p_v$, respectively. The 3-strata index $\alpha$ has been suppressed for clarity.

Figure 24

Terms in $H_2$ Eq. (F13) act as follows: (a) $A_2^K$ acts on two cubes sharing a 2-strata face $f$, (b) $T_2^K$ and $L_2^K$ act on two plaquettes sharing a 2-strata link $l$, and (c) $B_2^K$ acts on two links sharing a 2-strata vertex $v$.

Figure 25

Terms in $H_1$ Eq. (F16) act as follows: (a) $A_1^K$ acts on four cubes sharing a 1-strata link $l$, and (b) $T_1^K$ and $L_1^K$ act on four plaquettes sharing a 1-strata vertex $v$.

Figure 26

Eight $D_4$ 3-strata meeting at a 0-strata defect. The 3-strata are indexed as shown (with the hidden cube labeled $c_v^8$). The first term in $H_0$ Eq. (F18) acts on all eight cubes meeting at $v\in S_0$ while the remaining terms act on four adjacent cubes as shown in (b). There are six such terms generated by $H_0\forall v\in S_0$, given by $90{}^{\circ }$ rotation symmetry.

Figure 27

(a) An operator ${\mathcal{O}}_{\ell }$ that isolates potentially fractonic excitations (red) and a local operator that may not commute with it (dotted black line). (b) The restriction of ${\mathcal{O}}_{\ell }$ to a thin slice of the excitation-free column indicated. This restriction creates particle-antiparticle pairs of topological charges $b_i$ (pale blue) in the 2-strata. If all strip operators on the black dotted line commute with ${\mathcal{O}}_{\ell }$, then ${\overline{b}}_2$ and $b_2$ are $\widehat{y}$ lineons. (c) The operator ${\mathcal{O}}_{\ell }^{\prime }$ in the case that ${\mathcal{O}}_{\ell }$ had no energy penalty in the columns that now support $\widehat{y}$ lineons $b_L$ and $b_R$ (orange). (d) The operator ${\mathcal{O}}_{\ell }^{″}$ in the case that ${\mathcal{O}}_{\ell }^{\prime }$ had no energy penalty in the row that now supports an $\widehat{x}$ lineon $b_D$ (purple). $b_U$ is a trivial vacuum excitation in this example.