#  $p$-adic Holography from the Hyperbolic Fracton Model

Han Yan (闫寒)  Department of Physics and Astronomy, Rice University, Houston,
TX 77005, USA  Smalley-Curl Institute, Rice University, Houston, TX 77005, USA
Christian B. Jepsen  Simons Center for Geometry and Physics, SUNY, Stony
Brook, NY 11794, USA  Yaron Oz  Raymond and Beverly Sackler School of Physics
and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel

###### Abstract

We reveal a low-temperature duality between the hyperbolic lattice model
featuring fractons and infinite decoupled copies of Zabrodin’s $p$-adic model
of AdS/CFT. The core of the duality is the subsystem symmetries of the
hyperbolic fracton model, which always act on both the boundary and the bulk.
These subsystem symmetries are associated with fractal trees embedded in the
hyperbolic lattice, which have the same geometry as Zabrodin’s model. The
fracton model, rewritten as electrostatics theory on these trees, matches the
equation of motion of Zabrodin’s model. The duality extends from the action to
lattice defects as $p$-adic black holes.

Introduction. –  The contemporary landscape of theoretical physics is marked
by an exhilarating fusion of quantum many-body systems, quantum gravity, and
quantum information, guided by the holographic principle and the anti-de
Sitter/conformal field theory (AdS/CFT) correspondence [1, 2]. This duality
not only provides a profound insight into quantum gravity but also offers a
robust tool for addressing condensed matter problems [3, 4]. Additionally, the
development of numerous tensor-network holographic toy models reveals the
quantum-informational correcting feature of holographic entanglement [5, 6,
7].

Fracton states of matter [8, 9, 10, 11, 12, 13, 14, 15, 16], closely tied with
Lifshitz gravity [17, 18], have recently been incorporated into the AdS/CFT
landscape. Prior investigations [19, 20, 21] have indeed revealed that various
holographic information properties of fracton models in hyperbolic space are
consistent, akin to holographic states built via tensor networks [6, 7, 22].
However, despite these advancements, essential questions regarding the
effective AdS bulk theory and corresponding boundary CFT for these states
remain a mystery.

In this work, we help answer these questions by establishing a connection
between the $p$-adic Zabrodin model [23], which is arguably the simplest toy
model of AdS/CFT, and hyperbolic lattice models featuring fractons. Notably,
the lattice model’s subsystem symmetry is described by fractal trees identical
to the $(p+1)$-fractal tree in the Zabrodin model, forming the basis for the
connection. The connection between the two models can be established from the
action and information properties to $p$-adic black holes (BHs) as lattice
defects. This discovery builds a valuable framework for understanding physics
on both sides of the correspondence. Furthermore, it enriches the
AdS/condensed matter theory (CMT) program [3, 4, 24] by presenting an example
of condensed matter theory within the bulk, rather than conventionally on the
boundary.

Zabrodin’s model. –  Zabrodin’s $p$-adic model [23], a simple AdS/CFT toy
model, uses Gaussian fields on the vertices of an infinite $(p+1)$-regular
tree $\mathcal{T}_{p}$ to discretize the AdS bulk. This tree is also known as
the Bruhat-Tits tree or $(p+1)$-degree fractal tree, as shown in Fig. 3a for a
$3$-regular tree. Its action is given by

$\displaystyle
S_{\text{bulk}}=\frac{\gamma}{2}\sum_{v\in\mathcal{T}_{p}}\sum_{v^{\prime}\sim
v}(\phi_{v}-\phi_{v^{\prime}})^{2}\,,$ (1)

where $v^{\prime}$ denotes neighbor vertices of $v$. The model’s equation of
motion is given in terms of a graph Laplacian:

$\mathop{}\\!\mathbin{\bigtriangleup}\phi_{v}\equiv(p+1)\phi_{v}-\sum_{v^{\prime}\sim
v}\phi_{v^{\prime}}=0.$ (2)

Focusing on prime $p$, Zabrodin established a duality between this discrete
theory and a continuous boundary theory,

$\displaystyle
S_{\text{boundary}}=\frac{\gamma}{4}\frac{p(p-1)}{p+1}\int_{\mathbb{Q}_{p}}dx\,dy\,\frac{(\phi(x)-\phi(y))^{2}}{|x-y|_{p}^{2}}\,,$
(3)

where $\mathbb{Q}_{p}$ is the field of $p$-adic numbers and $|\cdot|_{p}$ is
the $p$-adic norm [25, 26]. In the limit, where source terms in (1) approach
the tree boundary, the partition functions of (1) and (3) become identical.
The theory (3) had previously been introduced as an effective theory for the
endpoints of $p$-adic strings, from which the $p$-adic Veneziano amplitude
[27] can be derived [28, 29]. With his model (3), Zabrodin had discovered the
worldsheet theory of $p$-adic strings. With the advent of $p$-adic AdS/CFT
[30, 31], Zabrodin’s duality was reinterpreted as the first example of
$p$-adic holography. From this new perspective, the tree $\mathcal{T}_{p}$ is
instead viewed as the $p$-adic version of AdS bulk, and the $p$-adic numbers
is the boundary space in which $p$-adic CFTs live.

Hyperbolic fracton model with Ising perturbations. –  In this work, we discuss
the Hyperbolic Fracton Model (HFM) with ferromagnetic Ising interactions
(HFM+Ising). These models are placed on hyperbolic lattices, tessellations of
a 2D plane with constant negative curvature by polygons. The lattices are
characterized by Schläfli symbols, pairs of integers $(m,n)$, which represent
tessellations of $m$-sided regular polygons with $n$ polygons meeting at each
vertex. This configuration necessitates the condition $m^{-1}+{n}^{-1}<1/2$,
permitting infinitely many different hyperbolic tessellations. An Example with
Schläfli symbol $(5,6)$ are shown in Fig. 1. Our study is primarily concerned
with tessellations having an even $n$, and uses the $(5,6)$ lattice as the
concrete example.

Figure 1:  $(5,6)$ tessellation of the hyperbolic plane on the Poincaré disk:
all polygons on the disk are identical, but look smaller when drawn farther
from the center. The thick blue and red lines are fractal trees, and the
shaded regions are associated with different subsystem symmetries.

In the HFM, the scalar degrees of freedom $Z_{p_{i}}$ are located at the
centers of polygons labeled by $p_{i}$’s. The total Hamiltonian is given by

$\mathcal{H}_{\text{HFM-
Ising}}=\mathcal{H}_{\text{HFM}}+\mathcal{H}_{\text{Ising}}.$ (4)

It includes the HFM part $\mathcal{H}_{\text{HFM}}$ and the Ising part
$\mathcal{H}_{\text{Ising}}$. The Ising part is the familiar Ising model with
ferromagnetic nearest-neighbor interactions,

$\mathcal{H}_{\text{Ising}}=\alpha\sum_{\langle
i,j\rangle}(Z_{p_{i}}-Z_{p_{j}})^{2}.$ (5)

We now elaborate on the HFM component, $\mathcal{H}_{\text{HFM}}$. The HFM
extends the model initially presented in Refs. [19, 20, 21], with excitations
analogous to fractons in gapped fracton orders. Its Hamiltonian is:

$\mathcal{H}_{\text{HFM}}=U\sum_{v\in\text{vertices}}\left(Z_{p_{1}}-Z_{p_{2}}+\dots-
Z_{p_{n}}\right)^{2},$ (6)

where $p_{1},\dots,p_{n}$ indicate the $n$ sites encircling vertex $v$ in a
clockwise sequence (refer to Fig. 1 for an example).

Alternatively, considering $Z_{p_{i}}$ as $\mathbb{Z}_{N}$ numbers, one can
replace $(Z_{p_{1}}-Z_{p_{2}}+\dots-Z_{p_{n}})^{2}$ with
$(\frac{N}{\pi})^{2}\sin^{2}\left[\frac{\pi}{N}(Z_{p_{1}}-Z_{p_{2}}+\dots-
Z_{p_{n}})\right]$. In the case where $Z_{p_{i}}$ adopts $\mathbb{Z}_{2}$
values, the model has been examined for the $(5,4)$ tessellation in Refs. [19,
20], representing a simplified extreme of the general HFM.

A fracton is an excitation of the vertex term. Note that by changing the value
of a single $Z_{p_{i}}$, we create an $m$-multipole of fracton instead of
dipole, which leads to the immobility of a single fracton in the system. This
study concentrates on the low-energy sector in the limit $U\gg\alpha$ and
$U\gg T$ (the sector devoid of fracton excitations). We illustrate that the
HFM+Ising model is congruent to an infinite set of $p$-adic Zabrodin models,
further exploring its holographic information properties.

Emergent Zabrodin model from HFM. –  We first examine
$\mathcal{H}_{\text{HFM}}$ without Ising interactions, which will be shown to
match the tensionless-string limit of the Zabrodin’s model. The primary result
is the model’s duality to multiple copies of the electrostatics theory on
distinct $n/2-$degree fractal trees ($n/2$-regular trees). In the dual model,
ground states on each tree satisfy the EOM of the Zabrodin model (Eq. (2)).
Additionally, we can verify the holographic information characteristics of the
model within this limit.

The HFM, described by Eq. (6), exhibits a large ground state degeneracy, a
feature dictated by its subsystem symmetries that commute with the
Hamiltonian. The key to finding them is to notice that changing an even number
of neighboring variables $Z_{p_{i}}$ at a given vertex by the same constant
$C$ leaves the vertex term invariant. This results in subsystem symmetries
defined by the boundaries of $n/2$-degree fractal trees, pivotal to the
underlying physics. Two examples, represented by blue and red trees in thick
lines, are depicted in Fig. 1.

To construct these trees more formally, begin at an arbitrarily chosen vertex
and select $n/2$ nonadjacent edges from the total of $n$ edges, keeping in
mind that we’re focusing on even-$n$ tessellations. Extend the chosen edges to
the neighboring vertices and repeat the selection process, making sure to
include the previously chosen edges, thus making the choice unique. Continuing
this process, we construct a degree-$n/2$ fractal tree embedded within the
hyperbolic lattice, extending the selected edges indefinitely. This method
yields an infinitely many trees on the lattice in the thermodynamic limit.

We then define the fractal-tree wedge as the part of the lattice bounded by
the edges of a tree, a pattern also seen in hyperbolic fracton orders [32]. In
Fig. 1, the blue and red shaded regions serve as examples. Given a classical
state, denoted as $\prod_{p_{i}}\ket{Z_{p_{i}}}$, the associated subsystem
symmetry for a particular wedge $w$ can be defined via the following
operation:

$X^{\dagger}_{\text{wedge
}w}(C)\ket{Z_{p_{i}}}=\ket{Z_{p_{i}}+C},\quad\forall{p_{i}}\in w,$ (7)

while $X^{\dagger}_{\text{wedge }w}(C)$ acts as the identity operator on
$\ket{Z_{p_{j}}}$ for all plaquettes $p_{j}$ outside $w$.

(a)

(b)

Figure 2:  Degree of freedom (DOF) $Z$, $\bm{E}$, and $\phi$ defined on the
plaquette centers, edges, and vertices.

We’ll now formulate the dual model to the HFM, aiding our understanding of
subsystem symmetries and illuminating its connection to the Zabrodin $p$-adic
model.

We start by defining 1$d$ vector DOFs, denoted as $\bm{E}_{e_{i}}$, which
reside on the lattice’s edges, $e_{i}$, and point along them. Each edge, $e$,
can be described in two ways: it’s sandwiched by the plaquettes $p_{i},\
p_{j}$, denoted as $e=p_{i,j}$, and it has vertices $v_{k},\ v_{l}$ at its
endpoints, denoted as $e=v_{k,l}$. For an edge as depicted in Fig. 2a, we
define:

$\bm{E}_{v_{1,2}}\equiv(Z_{p_{1}}-Z_{p_{2}})\hat{\bm{r}}_{v_{1,2}}=(-Z_{p_{1}}+Z_{p_{2}})\hat{\bm{r}}_{v_{2,1}}=\bm{E}_{v_{2,1}}.$
(8)

Note that the order $v_{1}\rightarrow p_{1}\rightarrow v_{2}\rightarrow p_{2}$
around the edge is in a clockwise direction (see Appendix for the counting of
DOFs).

At a vertex, the term $(Z_{p_{1}}-Z_{p_{2}}+\dots-Z_{p_{q}})^{2}$ can be
recast in terms of $\bm{E}$ as illustrated in Fig. 2b for the special case
$q=6$,

$\begin{split}&(Z_{p_{1}}-Z_{p_{2}}+\dots-Z_{p_{6}})^{2}\\\
=&(\bm{E}_{v_{0,1}}\cdot\hat{\bm{r}}_{v_{0,1}}+\bm{E}_{v_{0,3}}\cdot\hat{\bm{r}}_{v_{0,3}}+\bm{E}_{v_{0,5}}\cdot\hat{\bm{r}}_{v_{0,5}})^{2}\\\
=&(\nabla\cdot\bm{E})^{2}\text{ on the three blue edges}\end{split}$ (9)

Notably, the $\bm{E}$ fields associated with an $n/2$-degree fractal tree
correlate, while remaining uncoupled from DOFs on other tree, except for an
exact inter-tree constraint at each vertex,

$(\nabla\cdot\bm{E})^{2}\text{ on blue edges}=(\nabla\cdot\bm{E})^{2}\text{ on
red edges}.$ (10)

But in the low-energy sector where fracton excitations are absent, these
constraints no longer couple the fractal trees to each other.

Let’s delve into the physics on a single fractal tree. The Hamiltonian on the
tree of interest is

$\mathcal{H}_{\text{tree}}=U\sum_{v_{i}\text{ on
}\mathcal{T}}(\nabla\cdot\bm{E})^{2}.$ (11)

This Hamiltonian unveils that the model equates to an electrostatics theory on
the tree, involving solely the electric field sector of electrodynamics. It
imposes an energy cost on charge excitations, thus the low-energy sector
corresponds to charge-free electric field configurations. The subsystem
symmetry in the dual model is expressed as:

$\begin{split}&X^{\dagger}_{\text{wedge
}w}(C)\ket{\bm{E}_{e_{i}}}=\ket{\bm{E}_{e_{i}}+C\hat{\bm{r}}_{e_{i}}},\\\
&\quad\text{ for edges $e_{i}$ on the boundary of wedge $w$.}\end{split}$ (12)

In the context of electrostatics, this has a clear interpretation: the
injection of an electric field string from one end on the boundary to the
other. Given the absence of loops on a fractal tree, this is the sole method
of flux injection without charge creation, as depicted in Fig. 3a. Thus we
identify a set of multiple copies of electrostatics on $n/2$-degree fractal
trees as a dual model to the HFM.

This identification aids in elucidating certain holographic properties such as
the AdS-Rindler reconstruction. Using Fig. 3b as an example, an observer
measuring boundary electric fields $E_{1}$ to $E_{4}$ can reconstruct bulk
DOFs up to $E_{7}$ (minimal covering surface) using the charge-free condition
at every vertex, but not beyond (see Appendix for details).

(a)

(b)

Figure 3:  (a) Subsystem symmetry in the dual model as injecting a flux of
electric field. (b) AdS-Rindler reconstruction: knowing boundary electric
fields $E_{1}$ to $E_{4}$ allows reconstruction up to $E_{7}$ (minimal
covering surface), but not beyond.

Furthermore, we can recast the fractal-tree electrostatics using the electric
field potential, whose low energy sector explicitly obeys the EOM of the
Zabrodin $p$-adic model. We assign electric potential $\phi_{v_{i}}$’s to the
vertices. The electric field between vertices $v_{1},\ v_{2}$ is
$\bm{E}_{v_{1,2}}=(\phi_{v_{1}}-\phi_{v_{2}})\hat{\bm{r}}_{v_{1,2}}$. The
Hamiltonian (11) becomes

$\begin{split}\mathcal{H}_{\text{tree}}&=U\sum_{v_{i}\text{ on
}\mathcal{T}}\left((\frac{n}{2})\phi_{v_{i}}-\phi_{v_{i,1}}-\dots-\phi_{v_{i,n/2}}\right)^{2}\\\
&\equiv U\sum_{v_{i}\text{ on
}\mathcal{T}}\left(\mathop{}\\!\mathbin{\bigtriangleup}\phi\right)^{2}.\end{split}$
(13)

Here, $v_{i,j}$ denote the $n/2$ vertices next to $v_{i}$, and
$\mathop{}\\!\mathbin{\bigtriangleup}$ denotes the Laplacian operator on the
tree. Hence, the ground states of the dual model to HFM are precisely the
solutions to the EOM of the Zabrodin model (Eq. (2)), with the prime $p$ of
Zabrodin’s model being related to the second integer $n$ of the Schläfli
symbol via the relation $n=2p+2$. In the case when $(n-2)/2$ is not a prime,
the bulk HFM does not change behaviour substantially, but the dual tree model
no longer admits a convenient holographic description in terms of $p$-adic
numbers; what is lost is the multiplicative property of the norm:
$|ab|_{p}=|a|_{p}|b|_{p}$.

Note that the degenerate ground states of $\mathcal{H}_{\text{HFM}}$
contribute unequally to the action of the Zabrodin model, (Eq. (1))

$S_{\text{bulk}}=\frac{\gamma}{2}\sum_{v\in\mathcal{T}_{n/2}}\sum_{v^{\prime}\sim
v}(\phi_{v}-\phi_{v^{\prime}})^{2}=\gamma\sum_{e\in\mathcal{T}_{n/2}}{\bm{E}_{e}}^{2},$
(14)

which differentiates the charge-free electric field configurations by the
familiar $\gamma{\bm{E}}^{2}$ term, instilling tension to the strings on the
fractal tree. Consequently, the pure HFM model realizes the Zabrodin model’s
tensionless-string limit.

To account for the omitted string tension contribution, we should enable the
Ising sector $\mathcal{H}_{\text{Ising}}$ (Eq. (5)) and match $\alpha=\gamma$,
so that

$\alpha(Z_{p_{i}}-Z_{p_{j}})^{2}=\gamma{\bm{E}}_{e={p_{i,j}}}^{2}.$ (15)

Once the Ising Hamiltonian is turned on, the Hamiltonian associated to each of
the trees exactly matches that of Zabrodin’s model. An exact duality is not
present generically, however, owing to the fact that two trees intersect at
each vertex on the lattice, and the edge degrees of freedom of these trees are
constrained to produce identical vertex terms. Only in the limit when $U$ is
much greater than $kT=\beta^{-1}$, do the trees decouple, as the vertex terms
are all required to assume their lowest possible value. This is a low-
temperature limit according to the scale set by $U$, but not according to the
scale set by $\alpha$: we do not assume $\alpha>>kT$.

We see here that the large $U$ limit of the HFM+Ising in its action on a
single fractal tree, as an electrostatics theory, is equivalent to the
Zabrodin model. Consequently, the HFM+Ising model on the full hyperbolic
lattice, as infinitely many fractal trees defining the subsystem symmetries,
is a model hosting infinitely many holographic $p$-adic Zabrodin models.

Lattice defects as $p$-adic black holes. – Finally, we explain how lattice
defects on HFM corresponds to BTZ black holes in $p$-adic AdS/CFT. We set
$\alpha=0$ and turn off the Ising interaction, because in this case the
electrostatics picture of the BH offers a particularly simple intuitive
understanding of the microstates and entropy of the BH.

Defined on a rigid lattice, the $p$-adic model has no notion of dynamical
gravity. But Refs. [33, 31] proposed the notion of $p$-adic BHs as topological
objects, constructed in analogy with the arithmetic interpretation of the BTZ
BH [34] as a quotient of the isometry group of AdS by a discrete subgroup
[35]. The $p$-adic BTZ BH boils down to the same action defined on a graph
with a loop of $L$ edges and $L$ vertices, attached to $L$ fractal trees of
coordination number $(p+1)$, see Fig. 4. Subsequent papers [36, 37, 38, 39,
40] have corroborated the identification of this graph with a type of BTZ BH,
with $L$ as the size of the BH perimeter.

Figure 4: Quotient construction of $p$-adic BTZ black hole. Top: Picking any
doubly-infinite path in the Bruhat-Tits tree, the shifting of all vertices by
$L$ edges along this path is an isometry for any $L\in\mathbb{N}$. Bottom:
Identifying vertices related by such an isometry results in the $p$-adic BTZ
black hole. In this case $L=4$ and $p=2$.

We now explain how a defect in the HFM can lead to the emergence of a $p$-adic
BH. The defect sits at a vertex, as shown by the green disk in Fig. 5, and
takes the form of a new plaquette with DOF $Z_{p_{0}}$ there.

Figure 5:  A defect on the hyperbolic lattice and the merger of two fractal
trees that it engenders.

Initially, the vertex has two trees, $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$,
whose respective wedges determine the subsystem symmetries of the HFM model.
However, the inclusion of the defect merges $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ into a single tree $\mathcal{T}^{\prime}$, with a loop at
its center, as shown in Fig. 5. Every wedge bounded by the new tree
$\mathcal{T}^{\prime}$ defines a subsystem symmetry. The low-energy domain of
the electrostatics theory living on $\mathcal{T}^{\prime}$ abides by the EOM
for the $p$-adic theory with the BTZ BH. With the Ising term included, it also
matches the action.

The introduction of a defect and suppression of the vertex term leads to an
increase in the ground state degeneracy, corresponding to the microstates of
the $p$-adic BH and its resultant entropy. Consider the simplest BH from a
single defect, depicted in Fig. 5. Initially, the electric fields on trees
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ comply with individual charge-free
conditions, $\rho_{1}=0$ and $\rho_{2}=0$ if one measured the electric fields
on the boundary of a region containing the BH. After merging these into a
single tree $\mathcal{T}^{\prime}$ (or equivalently
$\mathcal{T}_{1}\cup\mathcal{T}_{2}$ for observers outside), only a single
charge-free condition applies to the whole tree:
$\rho^{\prime}=\rho_{1}+\rho_{2}=0$. Consequently, $\rho_{1}(=-\rho_{2})$ can
take non-zero values. These are the new states affiliated with the BH.
Meanwhile, boundary observers measuring $\bm{E}$ there will struggle to
extract any BH information ($\rho_{1}$) unless they simultaneously measure a
large boundary region covering all ends on either original tree,
$\mathcal{T}_{1}$ or $\mathcal{T}_{2}$. This concept aligns with physics
discussed in simpler cases where trees represent geodesics, as mentioned in
Refs. [19, 20].

A defect encompassing a larger region merges $N$ trees, with a longer loop
connecting all branches outside the loop. Here, the original $N$ conditions
$\rho_{i}=0\ \forall\ i=1,\dots,N$ on the ground state are reduced into a
single one, $\sum_{i}\rho_{i}=0$. So the BH DOF is then labeled by a series of
numbers $\rho_{1},\ \rho_{2},\ \dots,\ \rho_{N-1}$. The count of $N$
corresponds to the count of trees partially overlapping with the defect
region. In the Appendix, we show that for a region of large radius (number of
layers) $R$, the number of BH DOF is

$N_{\text{BH}}\sim e^{R\log\lambda_{+}}\sim\sinh{(R\log\lambda_{+})},$ (16)

where $\lambda_{+}$ is a constant depending on the lattice tessellation
$(m,n)$. The number of DOF $N_{\text{BH}}$, which measures the entropy of the
BH, is proportional to its horizon size.

Discussion. –  We’ve demonstrated that, at low temperatures, the simplest HFM
coupled with weak Ising interactions is equivalent to the $p$-adic Zabrodin
model – a simple toy model of AdS/CFT. This equivalence enriches our
understanding of both models. In particular, the dual electrostatics model is
instrumental in illustrating the holographic information properties inherent
to the Zabrodin model, and the Zabrodin model’s effective boundary action (Eq.
(3)) can be leveraged to compute the 3- and 4-point correlation functions of
the HFM. A challenge is presented, however, by the fact that the Ising-fracton
hyperbolic model is dual to not one but a large set of copies of the Zabrodin
model, and the boundary duals of these theories are all scrambled together.

Our findings invite future quantitative exploration of emerging new concepts.
Going beyond the low-energy sector, the series of Zabrodin models that make up
the dual of the HFM become interlinked. Each tree is described using $p$-adic
geometry, yet when they’re combined, conventional hyperbolic geometry is
restored. This raises intriguing questions around whether this transition of
geometry carries a deeper meaning.

Furthermore, when the fractal trees reduce to geodesics (the equivalent of
“straight lines” in hyperbolic space) in the limit of $2$-adic geometry [19,
20, 21], the associated electrostatics theory represents a classical analogue
of a “bit thread” [41, 42, 43]. This notion of a “bit thread” is a string in
the bulk with its endpoints entangled on the boundary. Our work proposes its
extension as a “bit tree” in the bulk with multiple entangled endpoints. This
opens up fascinating possibilities; for instance, could these “bit trees”
offer fresh insights into the structure of holographic entanglement entropy
and holographic code based on tensor networks? We hope that the equivalence
demonstrated in our work can serve as a key node, concretely linking various
facets of AdS/CFT.

## Acknowledgment

H.Y. thanks Hao-Yu Sun and Andriy H. Nevidomskyy for helpful discussions. H.Y.
was supported by the U.S. National Science Foundation Division of Materials
Research under the Award DMR-1917511. Y.O. is supported by the ISF center of
excellence.

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Appendix for “$p$-adic Holography from Hyperbolic Fracton Model”

##  I Counting the DOF

Let us discuss the counting of DOFs when we switch from the original HFM to
the dual one living on the edges/vertices. The edge terms in
$\mathcal{H}_{\text{Ising}}$ can be thought of as living on all the fractal
trees of coordination number $n/2=(p+1)$ that span the hyperbolic lattice,
with two trees intersecting at each vertex of the lattice. The vertex terms in
$\mathcal{H}_{\text{HFM}}$ couple the trees together, but in the limit where
$U$ tends to infinity in units of inverse temperature, the trees decouple, and
the model $\mathcal{H}_{\text{HFM-Ising}}$ becomes equivalent to a decoupled
set of Zabrodin-models. To see how the degrees of freedom align, consider a
finite, cut-off version of the hyperbolic lattice with $N_{e}$ edges, $N_{v}$
vertices, and $N_{p}$ plaquettes. For each plaquette, we have one variable
$Z_{p_{i}}$, but in the $U\rightarrow\infty$ limit, we also have one rigid
constraint for each vertex. Additionally, $\mathcal{H}_{\text{HFM-Ising}}$ is
invariant if we send every plaquette variable to its negative:
$Z_{p_{i}}\rightarrow-Z_{p_{i}}$. In total we are left with $N_{p}-N_{v}-1$
degrees of freedom. In the tree-description, we have one variable ${\bf
E}_{e_{i}}$ for each edge. There are more edges than vertices on the lattice,
but in the tree-description, each vertex term imposes a rigid constraint on
two intersecting trees, leaving us with $N_{e}-2N_{v}$ degrees of freedom.
Including also the external face in counting the total number of faces
$N_{f}=N_{p}+1$, the equality between the degrees of freedom in the two
descriptions amounts to Euler’s formula for connected plane graphs:
$N_{v}-N_{e}+N_{f}=2$.

##  II Rindler reconstruction

The fractal-tree electrostatics model, dual to an HFM subsector, helps the
demonstration of holographic properties. A notable example is the Rindler
reconstruction, asserting that part of the boundary data can reconstruct the
bulk within the wedge of the minimal surface, as mentioned in Fig. 3b.

Let’s explain in more detail the reconstruction using Fig. S1. Here, the known
boundary electric fields are denoted as $\bm{E}_{e_{1,2,3,4}}$. The aim of the
Rindler reconstruction is to determine which bulk electric fields can be
reconstructed.

Figure S1:  AdS-Rindler reconstruction of the electrostatics theory on the
fractal tree.

In the low-energy limit, the charge-free condition imposed on each vertex
leads to:

$\sum_{i=1,2,5}\bm{E}_{e_{i}}\cdot\hat{\bm{r}}_{e_{i}}=0,\quad\sum_{i=3,4,6}\bm{E}_{e_{i}}\cdot\hat{\bm{r}}_{e_{i}}=0,$
(S1)

Using these equations, we can reconstruct the values of $\bm{E}_{e_{5}}$ and
$\bm{E}_{e_{6}}$ from $\bm{E}_{e_{1,2,3,4}}$, which further allows us to
determine $\bm{E}_{e_{7}}$. However, the reconstruction process stops beyond
$\bm{E}_{e_{7}}$. Beyond this point, we encounter two edges with unknown
electric fields, halting further reconstruction. It’s important to note that
$\bm{E}_{e_{7}}$ represents the minimal “surface” in the bulk that separates
the known boundary segment from the unknown. This demonstrates the Rindler
reconstruction on a fractal tree.

##  III Scaling of the lattice

Let us explain the derivation of Eq. 16, a key result in our work.

Figure S2:  Inflation rule for expanding the hyperbolic lattice with Schläfli
symbol (5,4).

To investigate how the number of degrees of freedom and subsystem symmetries
scale with system size, we can count the number of vertices and polygons in
finite versions of the tessellations with Schläfli symbols $(m,n)$. This is a
straightforward exercise, using the technology of Ref. [44]. The basic idea is
to imagine that we generate a tessellation by iteratively performing inflation
steps. At each step we add another of layer of polygons as in figure S2. In
order to perform the counting of vertices and polygons, it is useful to
distinguish between two types of vertices, $a$ and $b$, coloured respectively
in blue and red in figure S2. A red vertex sits at the boundary between two
polygons of the previous layer, while a blue vertex touches only one polygon
in the previous generation. If we introduce two-component vectors given by

$\displaystyle\vec{v}(R)=\left(\begin{matrix}\\#\text{ of blue vertices in
layer $R$}\\\ \\#\text{ of red vertices in layer $R$}\end{matrix}\right)\,,$
(S2)

then these vectors are given by

$\displaystyle\vec{v}(1)=\left(\begin{matrix}m\\\
0\end{matrix}\right)\,,\hskip 56.9055pt\vec{v}(R+1)=M\vec{v}(R)\,,$ (S3)

where the matrix relating the number of vertex types in neighbouring layers,
assuming $m>3$, is given by

$\displaystyle M=\left(\begin{matrix}m-4+(m-3)(n-3)&&&m-4+(m-3)(n-4)\\\
n-2&&&n-3\end{matrix}\right)\,.$ (S4)

This matrix has the eigenvalues

$\displaystyle\lambda_{\pm}=\frac{PQ-2}{2}\pm\frac{1}{2}\sqrt{PQ(PQ-4)}\,,$
(S5)

where we have introduced the definitions $P\equiv m-2$ and $Q\equiv n-2$ to
make the expressions a bit more compact. By decomposing $\vec{v}(1)$ into
eigenvectors of $M$, it becomes easy to compute $\vec{v}(R)$ for arbitrary
$R$. By summing over the number of vertices in each layer, we find that the
total number of internal vertices $N_{v}(R)$ in a finite tessellation
generated by $R$ inflation steps is given by

$\displaystyle N_{v}(R)=\,$
$\displaystyle\sum_{i=1}^{R}\Big{(}v_{1}(i)+v_{2}(i)\Big{)}$ (S6)
$\displaystyle=\,$
$\displaystyle\frac{P+2}{2}\Bigg{(}\frac{\lambda_{-}^{R}-1}{\lambda_{-}-1}\bigg{(}1-\sqrt{\frac{PQ}{PQ-4}}\bigg{)}+\frac{\lambda_{+}^{R}-1}{\lambda_{+}-1}\bigg{(}1+\sqrt{\frac{PQ}{PQ-4}}\bigg{)}\Bigg{)}\,.$
(S7)

Using the fact that each blue vertex gives rise to $n-2$ polygons in the next
layer, and each red vertex gives rise to $n-3$ polygons in the next layer, we
can add together the polygons in each layer to determine the number of
polygons $N_{p}(R)$ in a finite tessellation generated by $R$ inflation steps
to equal

$\displaystyle N_{p}(R)=\,$ $\displaystyle
1+\sum_{i=1}^{R}\bigg{(}(n-2)v_{1}(i)+(n-3)v_{2}(i)\bigg{)}$ (S8)
$\displaystyle=\,$ $\displaystyle
1+\frac{P+2}{2}\Bigg{(}\frac{\lambda_{-}^{R}-1}{\lambda_{-}-1}\bigg{(}Q-(PQ-2)\sqrt{\frac{Q}{P(PQ-4)}}\bigg{)}+\frac{\lambda_{+}^{R}-1}{\lambda_{+}-1}\bigg{(}Q+(PQ-2)\sqrt{\frac{Q}{P(PQ-4)}}\bigg{)}\Bigg{)}\,.$
(S9)

In the limit as the number of inflation steps tends to infinity, the ratio of
polygons to vertices tends to a finite number:

$\displaystyle\lim_{R\rightarrow\infty}\frac{N_{p}(R)}{N_{v}(R)}=\frac{1}{2}\left(Q+\sqrt{\frac{Q(PQ-4)}{P}}\right)\,.$
(S10)

We can also use the eigendecomposition of $M$ to count the number of trees in
a finite hyperbolic graph. For it can be checked that the total number
$N_{t}(n)$ of trees equals the total number of blue vertices, wherefore
$N_{t}(R)$ is given by

$\displaystyle
N_{t}(R)=\sum_{i=1}^{R}v_{1}(i)=\frac{P+2}{2}\Bigg{(}\frac{\lambda_{-}^{R}-1}{\lambda_{-}-1}\bigg{(}1-(P-2)\sqrt{\frac{Q}{P(PQ-4)}}\bigg{)}+\frac{\lambda_{+}^{R}-1}{\lambda_{+}-1}\bigg{(}1+(P-2)\sqrt{\frac{Q}{P(PQ-4)}}\bigg{)}\Bigg{)}\,.$
(S11)

This condenses to the result of Eq. (16).