# Y-cube model and fractal structure of subdimensional particles on hyperbolic
lattices

Han Yan (闫寒<EMAIL_ADDRESS>Department of Physics & Astronomy, Rice
University, Houston, Texas 77005, USA  Kevin Slagle  Department of Electrical
and Computer Engineering, Rice University, Houston, Texas 77005 USA
Department of Physics, California Institute of Technology, Pasadena,
California 91125, USA  Institute for Quantum Information and Matter and Walter
Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, California 91125, USA  Andriy H. Nevidomskyy  Department of Physics
& Astronomy, Rice University, Houston, Texas 77005, USA

###### Abstract

Unlike ordinary topological quantum phases, fracton orders are intimately
dependent on the underlying lattice geometry. In this work, we study a
generalization of the X-cube model, dubbed the Y-cube model, on lattices
embedded in $H_{2}\times S^{1}$ space, i.e., a stack of hyperbolic planes. The
name ‘Y-cube’ comes from the Y-shape of the analog of the X-cube’s X-shaped
vertex operator. We demonstrate that for certain hyperbolic lattice
tesselations, the Y-cube model hosts a new kind of subdimensional particle,
treeons, which can only move on a fractal-shaped subset of the lattice. Such
an excitation only appears on hyperbolic geometries; on flat spaces treeons
becomes either a lineon or a planeon. Intriguingly, we find that for certain
hyperbolic tesselations, a fracton can be created by a membrane operator (as
in the X-cube model) _or_ by a fractal-shaped operator within the hyperbolic
plane.

Introduction:  Fracton orders Pretko  _et al._ (2020); Nandkishore and Hermele
(2019); Bravyi  _et al._ (2011); Haah (2011); Vijay  _et al._ (2015, 2016);
Chamon (2005) are examples of highly entangled gapped phases of matter that
lie beyond the Landau–Ginzburg paradigm in that no symmetry is broken, yet
they are also distinct from the more familiar topological phases of matter in
that they do not possess a universal long-wavelength description in terms of a
topological quantum field theory (TQFT) Seiberg and Shao (2021); Gorantla  _et
al._ (2021); Slagle (2021); Ohmori and Shimamura (2022); Fontana  _et al._
(2022); Slagle and Kim (2017); Aasen  _et al._ (2020); Wen (2020); Wang
(2020); Slagle  _et al._ (2019); Qi  _et al._ (2021). Rather, fracton orders
have a nontrivial ground state degeneracy that depends not only on the
topology but also on the system size Bravyi  _et al._ (2011); Haah (2011);
Vijay  _et al._ (2015, 2016) and lattice geometry Shirley  _et al._ (2018);
Slagle and Kim (2017). Furthermore, fracton orders support excitations whose
mobility is restricted when we do not allow any additional excitations to be
created: fractons are immobile; lineons can only move along a one-dimensional
line, and planeons can only move within a two-dimensional plane Pai and
Hermele (2019); Li and Ye (2020).

Much attention has been devoted to exactly solvable models that host fracton
order in flat space, such as the X-cube model formulated on the cubic lattice
Vijay  _et al._ (2016). By comparison, relatively little is known about the
behavior of such models in curved spaces Yan (2019, 2019, 2020); Gorantla  _et
al._ (2022a, b); Radicevic (2019) (see Refs. Slagle  _et al._ (2019); Bidussi
_et al._ (2022); Jain and Jensen (2022); Gromov (2019) for works that study
gapless fracton models Pretko (2017); Seiberg and Shao (2020); Pretko and
Radzihovsky (2018); Chen  _et al._ (2022) on curved spaces ). The fundamental
motivation for introducing curvature is to investigate how it affects the
properties of the fracton order, in much the same way as placing a TQFT on a
manifold with a different genus teaches us about the topological nature of the
ground state degeneracy. For example, it has been previously noted that
curvature can lead to a robust ground state degeneracy of X-cube model even on
manifolds that are topologically trivial Slagle and Kim (2017); and curvature
can grant the subdimensional particles additional mobility Slagle and Kim
(2018); Slagle  _et al._ (2019). Another practical motivation for introducing
curvature is to search for better error correcting fracton codes. In
particular, codes on hyperbolic spaces can have favorable quantum error
correcting properties Breuckmann and Eberhardt (2021). It is thus interesting
to examine this aspect for fracton order Tian  _et al._ (2020); Tian and Wang
(2019).

In this Letter, we investigate a generalization of the X-cube model to the
hyperbolic space $H_{2}\times S^{1}$, which can be visualized as a stack of
hyperbolic planes ($H^{2}$) with the top and bottom layers identified. Unlike
the flat space, which only permits a small number of different lattices viewed
as tessellations by regular polygons/polyhedra (i.e. the familiar square,
triangular and hexagonal lattices in two dimensions), the number of distinct
tessellations is infinite in hyperbolic spaces. Regular two-dimensional
hyperbolic plane tessellations are enumerated by a pair of integers $(p,q)$
satisfying $\frac{1}{p}+\frac{1}{q}<1/2$, which is called the Schläfli symbol.
These tessellations consist of of $p$-gonal regular polygons, with $q$
polygons meeting at each vertex. Two examples of such tessellations are shown
in Fig. 1.

We find that the generalized X-cube model on this hyperbolic geometry depends
sensitively on the tessellation. The simplest is the case of $q\\!=\\!4$,
where each vertex is locally isomorphic to that of a cubic lattice, allowing
for the standard definition of the vertex operators as products of four Pauli
$X$ in each of the three locally orthogonal intersecting planes. The resulting
$(p,4)$ model has one-dimensional (1D) particles, lineons, which propagate
along the geodesics of the $H_{2}$ plane (as opposed to straight lines in the
flat space), but otherwise are very similar to the X-cube lineons. There are
nuances with the operators necessary to create individual fractons however, as
we shall see below.

The most intriguing findings are for tessellations of $q>4$. For even $q>4$,
we find new models, which we dub Y-cube (because of the Y shape of the in-
plane part of two of the vertex operators in the simplest $q=6$ case
illustrated in Fig. 1b). Unlike the X-cube model, the Y-cube model with $q>4$
does not possess lineons; instead the lineons are replaced with a new kind of
quasiparticles, treeons, that can only propagate on a fractal tree as shown in
Fig. 5b. Moreover, a pair or “dipole” of neighboring fractons remains immobile
within the $H_{2}$ plane, in contrast to X-cube model where fracton dipoles
forms a planeon.

(a) (5,4) tessellation

(b) (4,6) tessellation

Figure 1: Examples of tessellations of the $H_{2}\times S^{1}$manifold using
the Poincaré disk representation: all polygons on the disk have identical
area, but look smaller when drawn farther from the center. (a) Hyperbolic
tessellation with $(p,q)=(5,4)$ (left) and Hamiltonian terms (right). (b)
Hyperbolic tessellation with $(p,q)=(4,6)$ (left) and Hamiltonian terms
(right).

X-cube and Y-cube models in $H_{2}\times S^{1}$:  The generalized X-cube
models are constructed as shown in Fig. 1. The Hamiltonian consists of two
types of terms: the vertex and the prism (generalization of the cube) terms.
For $(p,q=4)$ tessellations, the model is the natural generalization of the
X-cube model Shirley  _et al._ (2018); Slagle and Kim (2018): the vertex terms
are identical to those in the X-cube model, while the “cube” terms become
products of $Z$ operators over the edges of the $p$-gonal prisms, as shown in
Fig. 1a.

In the general case of $(p,q)$ tessellations (with even $q$), we keep the
$p$-gonal prism $Z$-operators in the Hamiltonian. There are in general two
kinds of vertex terms: (1) a product of $X$ operators on the two neighboring
out-of-plane edges and $q/2$ nonadjacent in-plane edges neighboring the
vertex; and (2) a product of $X$ operators on the $q$ in-plane edges
neighboring the vertex. See Fig. 1b for a $q=6$ example. Each vertex term has
an even number of X operators that overlap with a prism Z operators, making
the model a stabilizer code Hamiltonian. We name this model the Y-cube model,
alluding to the “Y” shape of the in-plane vertex terms when $q=6$ [Fig. 1b].

Before we move on to describe the new features of the hyperbolic X/Y-cube
models, we briefly note that all of them share some common properties due to
the flat $S^{1}$ dimension. Acting with an $X$ operator on an in-plane edge
will create four fracton excitations in the four prisms neighboring the edge.
A pair of fractons displaced out of the plane is mobile within the hyperbolic
plane (via $X$ operators acting on the in-plane edges). A pair of fractons
displaced in-plane (see e.g. Fig. 2b) is a lineon that can move in the out of
plane direction. A pair of fractons displaced out of the plane is mobile
within the hyperbolic plane (via $X$ operators acting on the in-plane edges).
Acting with a $Z$ operator on an out-of-plane edge will create two lineons on
the two vertices at the ends of the edge. These lineons are free to move in
the out-of-plane direction. These excitations are similar to excitations in
the cubic lattice X-cube model. However the excitations (fractons, lineons,
and the new treeons) created otherwise—via $X$ operators acting on a out-of-
plane edges or $Z$ operators on in-plane edges—have new physics that depends
on the tessellation, as we discuss in detail below.

(a)

(b)

(c)

Figure 2:  Fracton operators for the $(5,4)$ tessellation: (a) An $X$ operator
(red dot) on an out-of-plane edge (perpendicular to the shown hyperbolic
plane) creates four fractons (yellow stars). (b) A truncated geodesic of $X$
operators creates two fractons, which can move along the geodesic. (c) $X$
operators on a stack of truncated geodesics create a single fracton.

Fractons in $(5,4)$ X-cube model:  Let us first consider the model on the
hyperbolic lattice with $(p,q)=(5,4)$ [Fig. 1a], whose physics generalizes
straight-forwardly to all $(\text{odd }p\geq 5,q=4)$ tessellations.

We first examine the effects of an $X$ operator acting on an out-of-plane
edge. It creates four fractons (a quadrupole) on the four neighboring prisms
[Fig. 2a]. By consecutively applying $X$ operator on the out-of-plane edges
attached to the same geodesic of the $H_{2}$ plane, a pair (or dipole) of the
fractons can be moved away. Extending one side of the string to the infinite
boundary of the hyperbolic plane will leave a single pair of fractons in the
bulk. Equivalently, a truncated geodesic of $X$ operators creates a pair of
fractons at its end [Fig. 2b].

Unlike in the X-cube model, a single fracton cannot be created in the bulk at
the corner of a membrane operator. This is because a membrane operator creates
fractons inside the membrane since each pentagon prism is surrounded by an odd
number ($p=5$) of out-of-plane edges. Instead, a single fracton can be created
using a series of truncated geodesic strings of $X$ operators, as illustrated
in Fig. 2c. Each truncated geodesic of out-of-plane $X$ operators creates a
pair of fractons. Thus, the first truncated geodesic operator creates a pair
of fractons, and the others moves one fracton in the pair away to infinity.

(a)

(b)

(c)

Figure 3:  Lineon operators for the $(5,4)$ tessellation: (a) A $Z$ operator
(teal) on an in-plane edge creates two lineons (blue diamonds). The inset
shows the two excited terms in the Hamiltonian for a single lineon. (b)
Applying a string of $Z$ operators along a geodesic of in-plane edges creates
a single lineon, which can move along the geodesic. (c) The logical operator
constructed by $Z$ operators, which can also be viewed as moving a lineon from
one boundary to the other.

Lineons in $(5,4)$ X-cube model:  Next we examine the action of $Z$ operators
on in-plane edges. When $q=4$, each vertex is locally identical to a vertex of
the cubic lattice. Hence, a $Z$ operator on an in-plane link creates two
lineons [Fig. 3a] that behave similarly to lineons in the X-cube model on a
cubic lattice. Each lineon is an excited state of two vertex operators, shown
in the inset of Fig. 3a. Lineons are restricted to move on a $H_{2}$ geodesic,
as shown in Fig. 3b. Under rough boundary conditions (which condense lineons)
of the hyperbolic planes, the product of $Z$ operators along a geodesic [Fig.
3c] becomes a logical operator that does not create any excitations.

(a)

(b)

(c)

(d)

(e)

Figure 4:  Fracton operators for the $(4,5)$ tessellation: (a) A bulk logical
$X$ operator on a fractal tree (thick black), which is a product of $X$
operators (red dots) on out-of-plane edges neighboring one (out-of-plane) side
of the fractal tree. (b) A pruned fractal-tree of $X$ operators creates a pair
of fractons, which is a lineon with mobility only in the out-of-plane
direction. (c) An infinite series of the pruned fractal-trees creates a single
fracton. (d) A bulk logical $X$ operator on a fractal tree wedge (colored
red). It is the product of all $X$ operators (red dots) on the out-of-plane
edges neighboring one side of the wedge. The fractal tree is drawn in thick,
dashed line. (e) A membrane of $X$ operators supported on the intersection of
two wedges (red and teal) also creates a single fracton.

Fractons in $(4,6)$ Y-cube model:  Next we consider the Y-cube model on
$(\text{even }p\geq 4,q\geq 6)$ tessellations. We find that $q\geq 6$ results
in novel physics with a new kind of restricted particle mobility. We focus on
the representative example of $(4,6)$, with the Hamiltonian shown in Fig. 1b.
We first discuss the properties of fractons, then we consider the treeons
(which are analogs of lineons).

On the $(4,6)$ tessellation, there are two ways to create fractons,
illustrated in Fig. 4. To understand the first way, it is helpful to first
construct logical out-of-plane $X$ operators that do not create any fractons
in the bulk, shown in Fig. 4a. The logical operator is a product of $X$
operators on a fractal tree (also known as a Bruahat-Tits tree). The fractal
tree is constructed by choosing $q/2$ non-adjacent edges at a vertex and
repeating this procedure at every vertex it extends to. We shall refer to
these logical operators as $T_{X}$. The $T_{X}$ operators anti-commute with
strings of $Z$ operators in the out-of-plane direction. Here, we assume either
an infinite hyperbolic plane or a finite plane with a boundary that condenses
fracton dipoles Bulmash and Iadecola (2019); Luo  _et al._ (2022).

Now we can see how single fracton and fracton dipoles are created by the first
type of fractal tree operator: an in-plane fracton dipole is created by
pruning the fractal tree operator $T_{X}$ at a vertex in the bulk, as shown in
Fig. 4b. Unlike the X-cube model on a cubic lattice or a $(p,q=4)$
tessellation, for even $q\geq 6$ the in-plane fracton dipoles are lineons that
can only move in the out-of-plane direction. A single fracton can be created
by aligning many pruned fractal trees along a series of adjacent vertices, as
shown in Fig. 4c. Each pruned tree, creating a dipole, serves the purpose of
moving a fracton closer to the boundary.

Now we turn to a second type of fracton-creation operator. Recall that when
$p$ is odd (such as $p=5$ in the earlier example in Fig. 2), a membrane of $X$
operators creates an extensive number of excitations because each prism term
overlaps with the membrane operate by an odd number ($p$) of out-of-plane
edges. However when $p$ is even, $X$ membrane operators commute with the prism
term in the bulk.

The key to constructing such membrane operators is that its boundary should
overlap with the neighbouring prisms by two out-of-plane edges. Following this
principle, the membrane boundary should follow this pattern: we first find a
$q/2$-degree fractal tree we constructed earlier. Then, we select a region
bounded by the branches of the tree that contains the entire tree [shaded
region in Fig. 4d]. The membrane operator is the product of all the $X$
operators on the out-of-plane edges attached to this region (on the top or
bottom side of the $H^{2}$ plane). We name this geometric shape the fractal-
tree wedge.

To construct a membrane operator that creates a single fracton, we can select
two partially overlapping fractal-tree wedges. The membrane operator supported
on the overlap creates a single fracton near the intersection of wedge
boundaries, as shown in Fig. 4e.

(a)

(b)

(c)

Figure 5:  Treeon operators for the $(6,4)$ tessellation: (a) A $Z$ operator (teal) on an in-plane edge creates two treeons (blue stars). The inset shows the two excited terms in the Hamiltonian for a single treeon. (b) Applying a string of $Z$ operators (colored teal) within the fractal tree (colored light teal) creates a single treeon. The treeon can only move within the fractal tree. (c) An infinite string of $Z$ operators within the fractal tree is a logical operator.  Table 1:  Properties of hyperbolic X/Y-cube models on different tessellations of $H_{2}\times S^{1}$ lattice and model |  | in-plane
---
lineon/treeon
| in-plane
---
fracton dipole
| in-plane
---
$X$ logical op.
| in-plane
---
fracton creation op.
$(p\text{ odd},q=4)$ X-cube | lineon, Fig. 3b | 1D mobility, Fig. 2b | geodesic | truncated geodesics, Fig. 2b
$(p\text{ even},q=4)$ X-cube | lineon | 1D mobility |  | geodesic,
---
geodesic wedge, Fig. 6a
| truncated geodesics,
---
wedge corner, Fig. 6b
$(p\text{ odd},q\geq 6)$ Y-cube | treeon | no mobility | fractal tree | pruned fractal trees
$(p\text{ even},q\geq 6)$ Y-cube | treeon, Fig. 5b | no mobility, Fig. 4b |  | fractal tree, Fig. 4a,
---
fractal-tree wedge, Fig. 4d
| pruned fractal trees, Fig. 4c,
---
wedge corner, Fig. 4e

Treeons in $(4,6)$ Y-cube model:  Let us now discuss the excitations that
result from $Z$ operators acting on the in-plane edges in the $(4,6)$ Y-cube
model. A single in-plane $Z$ operator creates two composite excitations, each
consisting of two excited vertex operators, as shown in Fig. 5a. The two
excited vertex operators (inset of Fig. 5a) share three in-plane edges. Acting
with a $Z$ operator on one of these shared edges will move the composite
excitation to the adjacent vertex on the other side of the edge. The
excitation cannot move along other edges without creating additional
excitations.

Repeating this procedure, we find that this composite excitation can move
anywhere on the fractal tree shown in Fig. 5b. The construction of such a tree
is the same as $T_{X}$ in Fig. 4a. This composite excitation is similar to a
lineon, except at each vertex it can choose between multiple fractal paths. We
call this new kind of mobility-restricted excitation a _treeon_ since its
mobility is restricted to a fractal tree.

One non-trivial consequence of the treeons is how they form logical operators.
A treeon can travel from any one of the many branches of the tree to any other
one. The product of the $Z$ operators along any such path is then a logical
operator (assuming rough boundary conditions that condense treeons). One
example is shown in Fig. 5c.

Generalization to all tessellations:  Let us now summarize some properties of
the hyperbolic X/Y-cube models on all $(p,q)$ tessellations with even $q$.
When $q\geq 6$, X-cube lineons are replaced by a new type of excitations,
treeons, that move on the fractal tree. The $q=4$ case can be viewed as a
special limit of a tree with only two branches at each vertex, which becomes a
geodesic. In this limit the treeons become lineons, and fracton dipoles gain
mobility along the geodesics. The properties of different hyperbolic lattices
are summarized in Table. 1.

When $p$ is odd, a fracton can be created at the end of a series of truncated
geodesics or fractal trees [Figs. 4c,2c]. When $p$ is even, logical $X$
membrane operators are allowed in the shape of fractal-tree wedges for $q\geq
6$ [Fig. 4e] or geodesic wedges for $q=4$ [Fig. 6a]. The intersection of two
of these logical operators creates a single fractons at the corner [Figs. 4e
and 6b].

Finally, tessellations of $1/p+1/q=1/2$ are special limits of the embedding
space becoming flat rather than hyperbolic. In the case of $p=q=4$ (square
lattice) we recover the 3D X-cube model on a cubic lattice. When $(p=3,q=6)$,
the 2D tessellation forms a triangular lattice, and we can define the Y-cube
model on a stack of these triangular lattices. In this case, the Y-cube model
treeons we encountered for $(p>3,q=6)$ become planeons that move on a
honeycomb network embedded in the flat triangular layer. This results because
when the hyperbolic geometry is made flat, the fractal tree that the treeon
can traverse collapses onto itself and reduces to a 2D honeycomb, see the
Supplementary Materials for more detail (although we leave the in-depth study
of this triangular model to future work).

(a)

(b)

Figure 6:  (a) A logical membrane $X$ operator as the geodesic wedge for the
$(6,4)$ model. (b) A membrane $X$ operator with a corner that creates a
fracton.

Summary and outlook:  We introduced the Y-cube model on stacked tessellations
of the hyperbolic plane. We discovered that the Y-cube model features a new
kind of particle with restricted mobility: a treeon, which is constrained to
move along a fractal tree. We also find that in the hyperbolic X-cube and
Y-cube models with even $p$, fractons can be created by either in-plane
membrane or fractal operators (Figs. 4c and 4). We are not aware of any
previously studied models with this property.

These models also serve as a concrete example of how the lattice geometry
(tessellation) of fracton orders can determine their fundamental properties,
even when the embedding space (in this case $H_{2}\times S^{1}$) is the same.
Our discovery suggests that there are still many fracton orders with new and
exotic features not seen before, especially when their underlying
graphs/lattices are beyond the flat space ones, i.e., lattices with no
translational symmetry. Like Type-II fracton orders, the hyperbolic Y-cube
models are not foliated fracton orders Shirley  _et al._ (2018, 2019),
challenging us for new insight of classification schemes of fracton order Dua
_et al._ (2020).

This work provides one of the simplest examples of fracton order beyond the
flat space, but there is much room left for future exploration. One future
topic is to impose boundary conditions on the hyperbolic plane and study the
ground state degeneracy, logical operators, and quantum information encoding.
It is also useful to ask if the new physics from the hyperbolic structure
provides benefits in quantum memory storage. Another direction is to
investigate fracton models on the 3D hyperbolic space $H_{3}$, or general
graphs without translational symmetry Gorantla  _et al._ (2022a).

H.Y. and A.H.N. were supported by the National Science Foundation Division of
Materials Research under the Award DMR-1917511. K.S. was partially supported
by the Walter Burke Institute for Theoretical Physics at Caltech; and the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Quantum Science Center.

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Supplementary Materials for “Y-cube model and fractal structure of
subdimensional particles on hyperbolic lattices”

##  I The case of $(3,6)$ tessellation

Here, we discuss some properties of the Y-cube model on the $(3,6)$
tessellation (triangular lattice) $\times S^{1}$. This is a special limit of
the $(p,q)$ tessellations that is geometrically flat instead of hyperbolic,
which drastically affects the mobility properties of the excitations. We leave
a more complete study of this model to future work.

The Hamiltonian consists of three types of terms, shown in Fig. S1. The first
two types are the product of $Z$’s on the edges of a triangle prism, and
certain products of $X$’s around vertices. On this flat tessellation, an
additional term can be added to the Hamiltonian, which is not analogous to any
term in hyperbolic tessellations: namely the product of $Z$’s around a
hexagon. The existence of this third term only on flat space is related to the
fact (explained below) that $Z$ operators on in-plane edges create planons,
rather than treeons as in the hyperbolic Y-cube model.

Figure S1:  The triangular$\times S^{1}$ lattice and its Hamiltonian terms.

We first examine the excited states of the vertex terms. Recall from the main
text that when $p>3$ and $q=6$, $Z$ operators acting on in-plane edges create
treeon excitations that are restricted to move on a fractal tree. However on
the $(3,6)$ tessellation, we find that these vertex excitations are instead
planeons. This can be seen as follows. Locally, a $Z$ operator on an in-plane
edge creates two such planeons [Fig. S2a]. Each planeon is in the excited
state of two vertex operators [inset of Fig. S2a]. Similar to a treeon, these
planeons can move along any one of three in-plane edges connected to its
vertex. However since the lattice is no longer hyperbolic, this mobility
results in the mobility of a planeon, which can move within a hexgonal
sublattice, as shown in Fig. S2b. There are three flavors of this planeon, one
flavor for each hexagonal sublattice. The flavor of the planeon can be changed
at the expense of creating the fully mobile excitation described in the
following paragraph.

(a)

(b)

Figure S2:  (a) A $Z$ operator (teal) acting on an in-plane edge creates two
planeons (blue stars). The inset shows the two excited terms in the
Hamiltonian for a single planeon. (b) A single planeon can travel on the
hexagonal sublattice of the triangular lattice, colored in light blue.

In hyperbolic geometry, the composite excited state of two vertex operators
created by an out-of-plane $Z$ operator is a lineon that can move in the
vertical direction only. On the $(3,6)$ tessellation, the analogous excitation
is instead free to move in 3D. Three $Z$ operators on the edges of an in-plane
triangle creates three such excitations [Fig. S3a]. Each composite excitation
is in the excited state of two vertex operators [inset of Fig. S3a]. Unlike on
the hyperbolic plane, the flat geometry allows these composite excitations to
move along six in-plane directions, as shown in Fig. S3b. Repeated in-plane
movement can span a triangular sub-lattice on the original lattice. These
composite excitations can also move along the vertical direction as shown in
Fig. S3c, giving them 3D mobility.

(a)

(b)

(c)

Figure S3:  (a) Three $Z$ operators (teal) acting on an in-plane triangle
creates three excitations (green hexagons). The inset shows the two excited
terms in the Hamiltonian for a single excitation. (b) A single excitation can
travel on a sub-triangular lattice of the original triangular lattice. This
movement is done using four $Z$ operators shown on the figure. (c) A single
excitation can also travel vertically. This movement is done using a $Z$
operator on the vertical link shown on the figure.  Figure S4:  A flux
operator consisting of a product of $X$ operators (red) within a 2D membrane.
The flux operator anticommutes with the string operator (Fig. S3b) of the
mobile charge excitation.

These composite vertex excitations are similar to 3D toric code charges.
Naively, there are three flavors: one on each of the 3 sublattices. But by
acting with a triangle of $Z$ operators [Fig. S3a], the composite of three
flavors annihilate. Thus, there are actually only two independent flavors. The
corresponding flux operator (similar to a 3D toric code flux operator) is the
membrane operator consisting of an out-of-plane stack of products of $X$
operators on the red links in Fig. S4. This 2D membrane operator creates a
loop excitation around its boundary. Presumably, there are two flavors of this
membrane operator.

The action of in-plane $X$ operators creates planeon excitations with in-plane
mobility. The planeon is a composite excitations of the $Z$ terms shown in
Fig. S5. This planeon anticommutes with the planeon shown in Fig. S2.

Figure S5:  The action of an in-plane $X$ operator moves a planeon (orange
triangle in the top panel). The planeon is a composite excitations of the five
$Z$ terms (teal) shown in the lower panel.

Finally, we note that for an $L\times L\times L$ periodic lattice (with
lattice constants $\bm{\hat{a}}_{1,2,3}$ shown in Fig. S1), the ground state
degeneracy is GSD = $2^{6+2L}$ when $L$ is multiple of $3$ (for which
different flavors of particles do not turn into each other due to boundary
conditions).

Therefore, this model supports a pair of anticommuting planeons on each layer,
similar to stacks of toric code. The model also supports two sets of fully
mobile charges along with flux-line excitations, similar to two copies of 3D
toric code. Thus, it seems plausible that the model is local-unitary
equivalent Chen  _et al._ (2010) to the following hybrid fracton order
Tantivasadakarn  _et al._ (2021): two copies of 3D toric code and decoupled
stacks of 2D toric codes. Indeed, the ground state degeneracy is also
consistent with this possibility. We leave the resolution of this possibility
to future work.


in part by the Australian Research Council and the Australian Government
Research Training Program Scholarship initiative.

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