Gauging fractons: Immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies

The study of gapped quantum many-body systems in three spatial dimensions has uncovered the existence of quantum states hosting quasiparticles that are confined, not by energetics but by the structure of local operators, to move along lower dimensional submanifolds. These so-called “fracton” phases are beyond the usual topological quantum field theory description, and thus require new theoretical frameworks to describe them. Here we consider coupling fracton models to topological quantum field theories in ($3+1$) dimensions by starting with two copies of a known fracton model and gauging the ${\mathbb{Z}}_2$ symmetry that exchanges the two copies. This yields a class of exactly solvable lattice models that we study in detail for the case of the X-cube model and Haah's cubic code. The resulting phases host finite-energy non-Abelian immobile quasiparticles with robust degeneracies that depend on their relative positions. The phases also host non-Abelian string excitations with robust degeneracies that depend on the string geometry. Applying the construction to Haah's cubic code in particular provides an exactly solvable model with finite energy yet immobile non-Abelian quasiparticles that can only be created at the corners of operators with fractal support.

Figure 1

Hamiltonian terms for each layer of the bilayer ($2+1$)D toric code. There are two spins per link of the lattice, and the superscripts refer to which layer the operator acts on.

Figure 2

Gauging procedure for the bilayer toric code. (a) Rewriting of the ungauged sum of plaquette operators in Eq. (4). The sum on the right-hand side is over all choices of signs with an even number of ${\sigma }_z^{(-)}$ operators. (b) Hilbert space for the gauged bilayer toric code. There are two of the original “matter” qubits per link (light grey circles) and four additional “gauge” qubits per site (dark circles), one per orange link. (c) Generator of the gauge symmetry in the gauged bilayer toric code. (d) Gauge-invariant version $B_p$ of (a). Distinct pairs of ${\sigma }_z^{(-)}$ operators are connected by ${\tau }_z$ operators.

Figure 3

Commutation properties of example terms in the gauged bilayer toric code's $A_s$ and $B_p$ operators. Here $s$ labels the same lattice site.

Figure 4

String-net configurations before and after gauging the SWAP symmetry of the bilayer toric code. Orange and blue strings correspond to the original toric code string-net wave functions. (a) Example string-net configuration in the presence of twist defects (grey circles labeled $t$). Dashed purple line is a branch cut for the layer-swap symmetry. (b) Example ground-state string-net configuration for the gauged bilayer toric code. Solid purple strings are the proliferated branch cuts.

Figure 5

String-net configurations in the gauged bilayer toric code in the presence of a symmetry flux through one handle of the torus. Orange (respectively, blue) is a string in the first (respectively, second) layer of toric code and the purple dashed line is the branch cut implementing the symmetry twist due to the symmetry flux. All topologically trivial strings are omitted. (a) Nontrivial Wilson loop around the $x$ handle of the torus. (b) Nontrivial Wilson loop around the $y$ handle of the torus. A nontrivial red string is equivalent to a nontrivial blue string by the process of moving the string all the way around the torus (black arrow).

Figure 6

Abelian excitations in the string-net picture of the gauged bilayer toric code. Orange strings are ${\sigma }_z^{\left(1\right)}=-1$, blue strings are ${\sigma }_z^{\left(2\right)}=-1$, and purple strings are ${\tau }_z=-1$. (a) One term in the superposition for a pair of $e^{\left(1\right)}{e}^{\left(2\right)}$ excitations (green). A blue string and an orange string both end at the excitation. (b) Some terms in the superposition for a pair of $m^{\left(1\right)}{m}^{\left(2\right)}$ excitations (dark grey). (c) Some terms in the superposition for a pair of ${\mathbb{Z}}_2$ charges $\varphi$ (light grey).

Figure 7

Example string-net configurations associated with $\left[e\right]$ excitations (yellow). The outline on $\left[e\right]$ indicates whether it is the end $e^{\left(1\right)}$ of a layer-1 string (orange) or the end $e^{\left(2\right)}$ of a layer-2 string (blue). The first term in each picture is the “reference” configuration with no branch cuts (purple). (a) Two $\left[e\right]$ excitations, demonstrating how $\left[e\right]$ can be thought of as a superposition of $e^{\left(1\right)}$ and $e^{\left(2\right)}$. (b) A basis for the set of four degenerate states associated to a set of four $\left[e\right]$ excitations with overall fusion channel equal to the identity.

Figure 8

Action of $W_{\sigma }$ on the reference configuration for $|OOOO\rangle$.

Figure 9

Constructing string operators for the non-Abelian $\left[e\right]$ particle in the gauged bilayer toric code. (a) Commutation relations between one term in the superposition of string operators for $\left[e\right]$ commuting with one star term in superposition making up $A_s$ in the Hamiltonian Eq. (7). Moving the plaquette operator past the string operator turns the star term into a different term in the superposition forming $A_s$. The new string operator which results should be superposed with the original one. (b) Superposition making up the full string operator. There are four such operators, obtained by choosing either the upper or lower sign on each end of the string. These operators commute with all terms in the Hamiltonian except for $A_s$ (and possibly $C_e$) at their endpoints.

Figure 10

Terms in the expression of string operators in the gauged bilayer toric code as a superposition of products of local operators. (a) Reference operator for a pure charge; each end can have either ${\mathcal{U}}_{+}$ or ${\mathcal{U}}_{-}$. This freedom would move to the outermost $\mathcal{V}$ operators for a pure flux. (b) Another term in the superposition for a string operator that creates pure charges. At each site marked with a purple square, we have interchanged all surrounding subscripts $+\leftrightarrow -$. (c) Another term in the superposition for a string operator that creates pure fluxes. At each plaquette which is shaded purple, we have interchanged all surrounding subscripts $+\leftrightarrow -$. This operator has ${\mathcal{V}}_{+}$ on both ends in its reference configuration.

Figure 11

Operators appearing in each layer of the bilayer X-cube Hamiltonian.

Figure 12

Gauging the bilayer X-cube model. (a) Positions of spins for the Hilbert space of the gauged model. There are two “matter” spins (light) per link of the lattice and one “gauge” spin (dark) per orange line connecting nearest-neighbor matter spins, for a total of 18 spins per site. (b) Generator $C_e$ of local gauge transformations. The operator acts as SWAP on the matter spins $\sigma$ and acts with ${\tau }_x$ on the eight surrounding gauge spins. (c) Gauged cube term $B_c$. The sphere color indicates whether ${\sigma }_z^{(+)}$ (green) or ${\sigma }_z^{(-)}$ (grey) acts on the matter spins, and ${\tau }_z$ acts on the orange gauge spins. Three of the 2048 possible terms (which all have an even number of ${\sigma }_z^{(-)}$ operators) are shown. (d) Flux operators $D_{▵}$, which is a three-spin operator on the faces of the octahedra, and $D_f$, which is a four-spin operator within each face of the cubic lattice.

Figure 13

(a) Example ground-state cage-net wave-function configuration in the ungauged bilayer X-cube model. Orange and blue are strings in different layers. (b) Cage-net configurations in the presence of a twist defect (purple string, branch cut membrane in light purple). Overlapping strings are displaced slightly for visibility. (c) Example ground-state cage-net configuration in the gauged bilayer X-cube model. The purple membrane is one of the proliferated closed branch cut membranes.

Figure 14

Properties of strings in the X-cube model on the 3-torus. Opposite faces of the cube are identified in all subfigures. (a) Constraint on rigid, topologically nontrivial Wilson loops in the cage-net wave-function picture of a single copy of the X-cube model; four Wilson loops (orange) on the edges of a rectangular prism can disappear into the vacuum. The black dashed lines are guides to the eye. (b) Basis for topologically nontrivial Wilson loops oriented in the $z$ direction in a single copy of the X-cube model. Each orange line is an independent Wilson loop.

Figure 15

Topologically nontrivial cage-net wave-function strings in the gauged bilayer X-cube model in the presence of twisted boundary conditions (purple membrane is the branch cut) on the 3-torus. Opposite faces of the cubes are identified. (a) Strings which pass through the branch cut must change layers (colors) and so must circle the 3-torus twice in order to close. (b) Modification of the prism constraint in Fig. 14 in the presence of the branch cut; strings on the opposite side of the branch cut have opposite colors. (c) Strings in layer 1 (orange) which are oriented in the $y$ direction at the same $x$ position can be moved around the torus using the constraint in (b) and end up in the other layer (dashed blue) after passing through the branch cut.

Figure 16

Sample terms in the cage-net wave function for Abelian quasiparticles in the gauged bilayer X-cube model. (a) $e_x^{\left(1\right)}{e}_x^{\left(2\right)}$ excitations. (b) $f_0^{\left(1\right)}{f}_0^{\left(2\right)}$ excitations. (c) ${\mathbb{Z}}_2$ charge $\varphi$.

Figure 17

Cage-net configurations near a single $\left[f_0\right]$ excitation illustrating the interaction of branch membranes and cages. The sphere color indicates whether layer-1 or layer-2 cages (in the absence of branch membranes) introduce minus signs into the wave-function layer in that cage-net configuration. As before, orange (respectively, blue) indicates layer 1 (respectively, layer 2) cages and excitations and purple are branch membranes. The third and fourth terms show the interaction between branch membranes and cages surrounding $\left[f_0\right]$. All other excitations in the system are assumed to be very far from everything in these pictures.

Figure 18

States associated to many $\left[f_0\right]$ excitations. Color coding is the same as Fig. 17, and dashed lines and grey rectangles are guides to the eye. (b) A reference configuration for $4N_q \left[f_0\right]$ fractons created in $N_q$ “layers”; there are $8^{N_q-1}$ degenerate states associated with these fractons in this geometry. [(c) and (d)] Example reference configurations for eight $\left[f_0\right]$ fractons in different geometries from (a). In (c), there are four degenerate states and in (d), there are only two. (e) Example reference configuration in the geometry used to calculate the quantum dimension of $\left[f_0\right]$.

Figure 19

Color coding is the same as Fig. 18. (a) Applying a membrane operator $M_{\sigma }$ which measures the ${\mathbb{Z}}_2$ charge contained by two fractons produces an “illegal” state which violates the Hilbert space constraints. (b) Applying a membrane operator $M_{\sigma }^{\prime }$ which measures the ${\mathbb{Z}}_2$ charge contained by four fractons produces a valid state. (c) States where the top (and bottom) four fractons fuse to two $f_0^{\left(1\right)}{f}_0^{\left(2\right)}$ fractons with (minus sign) and without (plus sign) a $\varphi$.

Figure 20

Constructing membrane operators for $\left[f_0\right]$ fractons. (a) Commutation relation between one term of $B_c$ (involving ${\sigma }_z$ operators) and a small membrane of ${\sigma }_x^{(+)}$ operators. Operator legend is in the bottom left. (b) and (c) Two choices of membrane operator that create $\left[f_0\right]$ fractons (purple cubes) in the gauged bilayer X-cube model. They differ only by the choice of operator (color) in the top right and bottom right corners of the membrane. All terms with an even number of grey spheres appear in each sum. The grey membrane is a guide to the eye representing the membranelike support of the operator. The operator legend in (a) applies to all of the figures.

Figure 21

Local operator locations for membrane operator in the gauged bilayer X-cube model. A superposition is formed from a reference configuration where every operator is chosen to be $+$ by choosing whether or not to flip $+\leftrightarrow -$ on every spin touching a face (for an operator that creates fractons) or on an elementary cube (for an operator that creates a twist string).

Figure 22

(a) Wilson lines in the presence of an extrinsic string defect (purple) in the bilayer X-cube model with open boundary conditions where the $e$ excitations are condensed. The $e$ Wilson line consists of ${\sigma }_z^{\left(i\right)}$ operators, where $i=1$ when the line is orange and $i=2$ when the line is blue, and the $m_{xz}^{\left(1\right)}$ Wilson line consists of ${\sigma }_x^{\left(1\right)}$ operators. (b) Independent $e_z$ Wilson lines in the presence of a twist defect. The colors may be exchanged to find another set of independent Wilson lines. (c) Additional $e$ Wilson lines which appear when the string bends.

Figure 23

Wilson “lines” in the presence of two twist defect strings (purple) when the strings are separated [(a) and (b)] and linked [(c) and (d)]. The implication is that the $e_z^{\left(1\right)}$ particles are brought in from the boundary and annihilated on the opposite boundary, where we have chosen open boundary conditions with $e$ excitations condensed everywhere. (The details of the Wilson operators depend on the boundary conditions, although the number of independent Wilson operators does not.) Operators within a figure anticommute and operators in different figures (with the same defect configurations) commute.

Figure 24

(a) Definition of the operator $\mathcal{C}$. (b) “Braiding” of the cage $\mathcal{C}$ (green) around a pair of $\left[f_0\right]$ particles (purple). The fractons are all taken to be well-separated despite the fact the cubes representing the fractons are large in this figure.

Figure 25

Operators appearing in the ungauged bilayer Haah's code model. The Hilbert space consists of two qubits per layer for a total of four qubits per site. The Pauli operators are written as $X$ and $Z$ for legibility.

Figure 26

Gauging procedure for bilayer Haah's code. (a) Decomposition of the ungauged Hamiltonian terms $A_c^{\left(i\right)}$ into terms even and odd under local SWAPs [of the sort in Eq. (48)]. The sum is over all 64 terms with an even number of $(-)$ operators; three are shown. (b) Hilbert space arrangement for the gauged model. There are four “matter” spins (dark grey) per site of the lattice and one “gauge” spin (orange) per link of the lattice. (c) Generator $C_s$ of local gauge transformations. (d) Gauged cube term $A_c$. The only difference from (a) is the addition of ${\tau }_z$ operators (orange) acting on gauge spins.

Figure 27

Reference cage-net-like configurations for $\left[a\right]$ fractons (spheres) in the gauged bilayer Haah's code. In these configurations which lack branch membranes, fractons can be assigned definite layers (orange and blue for layers 1 and 2, respectively). Grey tetrahedra have fractons at their corners and are guides to the eye [one not shown for legibility in (b)]. Configuration (b) is only possible when the four small tetrahedra of fractons are arranged at the corners of a larger tetrahedron. (c) Geometry used to compute the quantum dimension of $\left[a\right]$. Fractons within each grey prism fuse to the same superselection sector.

Figure 28

(a) Operator which creates Abelian fractons (grey) in Haah's code. (b) Commutation relations between one term in $A_c$ and a small piece of a Sierpinski prism operator. (c) and (d) Operators which create non-Abelian $\left[a\right]$ fractons (purple) in Haah's code in different geometries. The tetrahedra are guides to the fractal structure; the edges of the tetrahedra are not necessarily links of the cubic lattice. Hence in (d) the dashed line consists of several edges of the actual lattice. In (a), (c), and (d), these operators do not commute with the $A_c$ terms in the Hamiltonian at the location of the excitations.

Figure 29

Operator definitions in the alternate gauging procedure for the bilayer toric code. (a) Generator $C_s$ of the gauge symmetry and orientation convention for links. (b) Star operator $A_s$. (c) Plaquette operator $B_p$.

Figure 30

Operator definitions for the ${\mathbb{D}}_4$ quantum double Hamiltonian (A6).

Figure 31

Excitations and operators that create them in the X-cube model. (a) Four fractons $f_0$ (purple cubes), which are cubes where $B_c=-1$, created at the end of a membrane (light grey) of ${\sigma }_x$ operators (blue spheres). (b) Two-dimensional bound state $m_{xy}$ of two fractons, created at the end of a string of ${\sigma }_x$ operators. (c) One-dimensional quasiparticles $e_x$ (yellow spheres) where $A_y=A_z=-1$, created at the end of a string of ${\sigma }_z$ operators (green spheres).

Figure 32

Constraints in the single-twist sector arising from the constraint in Fig. 15 and process in Fig. 15. (a) Two bichromatic strings can mutually annihilate (equivalently, a single bichromatic string can move in the $z=0$ plane). (b) A bichromatic string can annihilate on a monochromatic string at the cost of flipping the color of the monochromatic string.

Figure 33

Sign structure used in creating the operator $\mathcal{O}$ (see text) when $W$ is the product of ${\tau }_x$ over the orange links. If $A_c$, where $c$ is the blue cube in the operator, has been commuted past $R$ to produce $R^{\prime }$, the operator $R^{\prime \prime }$ produced by commuting a $B_{c^{\prime }}$ operator through $R^{\prime }$ will enter $\mathcal{O}$ with a minus sign if $c^{\prime }$ is a cube labeled $-$ in the figure.