# Construction of Fractal Order and Phase Transition with Rydberg Atoms

Nayan E. Myerson-Jain  Stephen Yan  David Weld  Cenke Xu  Department of
Physics, University of California, Santa Barbara, CA 93106, USA

###### Abstract

We propose the construction of a many-body phase of matter with fractal
structure using arrays of Rydberg atoms. The degenerate low energy excited
states of this phase form a self-similar fractal structure. This phase is
analogous to the so-called “type-II fracton topological states”. The main
challenge in realizing fracton-like models in standard condensed matter
platforms is the creation of multi-spin interactions, since realistic systems
are typically dominated by two-body interactions. In this work, we demonstrate
that the Van der Waals interaction and experimental tunability of Rydberg-
based platforms enable the simulation of exotic phases of matter with fractal
structures, and the study of a quantum phase transition involving a fractal
ordered phase.

In recent years, tremendous progress has been made in simulating quantum many-
body systems with highly tunable arrays of Rydberg atoms Browaeys and Lahaye
(2020); Gross and Bloch (2017); Saffman et al. (2010). In many such
experiments, the ground state and a high-lying excited state of the atom
constitute a qubit, the fundamental element of numerous exotic quantum many-
body states Wen (2017); Sachdev (2018). Recently, the construction of
unconventional many-body states like $Z_{2}$ quantum spin liquids has been
explored Verresen et al. (2021); Samajdar et al. (2021); Semeghini et al.
(2021), demonstrating the highly promising potential of the Rydberg-based
platforms. The possibility to extend these platforms to realize quantum many-
body systems beyond the currently well-understood theoretical paradigm such as
those exhibiting $Z_{2}$ topological order Read and Sachdev (1991); Wen
(1991); Kitaev (2003) would be extremely exciting.

“Fracton” phases of matter provide a natural playground for exotic physics.
These phases host excitations with restricted dynamics, and a ground-state
degeneracy that scales with the system size Haah (2011); Chamon (2005); Bravyi
et al. (2011); Yoshida (2013); Vijay et al. (2015, 2016); Nandkishore and
Hermele (2019); Pretko et al. (2020). Fracton related models are loosely
classified by their qualitative features: “type-I” models have excitations
whose dynamics are restricted to standard submanifolds, e.g lines and planes
in space, while excitations of the more exotic “type-II” models are created at
the end of a fractal subset of the lattice and are completely immobile Vijay
et al. (2016). The prototypical example of the type-II model is Haah’s code
Haah (2011), which is defined on the cubic lattice and has a ground-state
degeneracy which scales subextensively with the system size. While fracton
related phases are of great theoretical interest, as they represent a world
beyond well-studied topological quantum field theories, much less progress has
been made in realizing these models experimentally.

In this work, we propose an experimental realization of a two-dimensional
analogue of a type-II fracton phase, as well as a quantum phase transition
between a phase with “fractal order” and a trivial phase. The fractal order
spontaneously breaks a fractal subsystem symmetry, and its low energy
excitations form a Sierpinski triangle on the lattice, which is a fractal
shape with Hausdorff dimension $d_{H}=\ln 3/\ln 2$, and only costs energy at
the corners of the Sierpinski triangle. The corners of the triangle are also
completely immobile, as their motion inevitably creates higher energy
excitations. We stress that the fractal order we consider is defined as
spontaneous breaking of a fractal subsystem symmetry; this phase does not have
topological order as in Haah’s code Haah (2011).

— Sierpinski Triangle Model

Figure 1: (Left): Three-body spin interaction on each downward facing triangle
in Eq. 1. (Right): one of the low energy excitations of Eq. 1; starting with
the obvious ground state with $\sigma^{z}=+1$, the spins are flipped to
$\sigma^{z}=-1$ on a $\ell=3$ Sierpinski triangle (labelled blue), this
configuration only costs energy on the unit triangles at the three corners
(labelled red).

The Sierpinski triangle model Newman and Moore (1999); Yoshida (2013) is the
paradigmatic model with fractal order. It is a classical statistical
mechanical model for an Ising system on a triangular lattice whose Hamiltonian
takes a simple form: one sums over all three-body interactions on downward
facing unit triangles $\bigtriangledown$ as opposed to upward facing triangles
$\triangle$ (Fig. 1):

$H_{\mathrm{ST}}=\sum_{\bigtriangledown}-K\sigma^{z}_{1}\sigma^{z}_{2}\sigma^{z}_{3},$
(1)

where the $\sigma^{z}=\pm 1$ are the Ising degrees of freedom placed at the
vertices of each downward unit triangle. The low energy excited states of this
model have a fractal structure: starting with the obvious ground state with
uniform $\sigma^{z}_{j}=+1$, the low energy excited states correspond to
flipping spins in the shape of a Sierpinski triangle, which does not cost
energy anywhere except for the three corners of the Sierpinski triangle. If we
view the corners of the Sierpinski triangle as point particles with finite
energy, these particles cannot move along the lattice without creating more
excitations that cost higher energy. Hence the mobility of these particles is
highly restricted, and it is in this sense that they are fractons.

A quantum version of the Sierpinski triangle model was discussed in Ref.
Vasiloiu et al. (2020); Zhou et al. (2021). Just like the classic quantum
Ising model, the quantum Sierpinski triangle model has an extra transverse
field

$\displaystyle
H_{q\mathrm{ST}}=\sum_{\bigtriangledown}-K\sigma^{z}_{1}\sigma^{z}_{2}\sigma^{z}_{3}-\sum_{j}h\sigma^{x}_{j}.$
(2)

It was shown that the quantum Sierpinski triangle model has two highly
desirable features:

(1) It is “self-dual”, meaning that under a duality transformation the $K$ and
$h$ term will exchange. This self-duality is analogous to that of the Kramers-
Wannier duality of the $1d$ quantum Ising model Kramers and Wannier (1941);
Kogut (1979), and the self-duality of the quantum plaquette model on the
square lattice Xu and Moore (2004). The self-duality implies that, if there
were a quantum phase transition reached by tuning $h/K$ in Eq. 2, it must
happen at $h=K$.

(2) Numerical simulation of the quantum Sierpinski triangle model suggests
that the system may have a second order quantum phase transition at the self-
dual point $h=K$ Zhou et al. (2021) (though earlier numerics suggested a first
order transition Vasiloiu et al. (2020)); at the quantum phase transition, the
energy density has a fractal dimension $d_{H}=\ln 3/\ln 2$ rather than scaling
dimension 2 as in ordinary quantum phase transitions in $2d$ Zhou et al.
(2021). This quantum phase transition is associated with the spontaneous
breaking of a “fractal symmetry”; the phase with $h<K$ is identified as a
“fractal order”, while the phase with $h>K$ is a disordered phase of the
fractal symmetry (see the appendix for more discussion).

The nature of the quantum phase transition at $h=K$ in Eq. 2 is far from being
understood, and the ordinary Landau-Ginzburg paradigm is no longer applicable.
Numerics suggest that this transition is likely continuous, but many questions
await answers. For example: is the continuous quantum phase transition stable
against perturbations? For ordinary quantum phase transitions, this question
is answered through the well-established renormalization group (RG) method
Wilson (1971); Wilson and Fisher (1972); Wilson (1975), by evaluating the
relevance or irrelevance of certain perturbations. But for the quantum phase
transition under discussion, no reliable RG procedure has been established.
Hence, key aspects of the transition must be explored experimentally. A
tunable experimental realization of the classical and quantum Sierpinski
triangle model would be extremely useful in understanding quantum phase
transitions involving fractal geometry.

— Realizing the Sierpinski Triangle model with Rydberg atoms

The goal of this work is to describe a construction of both the classical and
quantum Sierpinski triangle models from arrays of Rydberg atoms. To begin, let
us consider a single atom for which we couple the ground state $|g\rangle$ and
an excited Rydberg state $|r\rangle$ via a laser detuned from resonance. The
two states coupled by the laser are the atom-field product states labelled
$|g,N_{\gamma}+1\rangle$ and $|r,N_{\gamma}\rangle$, where $N_{\gamma}$ is the
photon number of the laser so that $|g,N_{\gamma}+1\rangle$ has one extra
photon compared to $|r,N_{\gamma}\rangle$. In the effective two-state problem,
the Rabi frequency enters as a term coupling these two states. The simplest
manifestation of the Rabi oscillations is as a term in the Hamiltonian
$\Omega\sigma^{x}$ where $\sigma^{x}=|g,N_{\gamma}+1\rangle\langle
r,N_{\gamma}|+|r,N_{\gamma}\rangle\langle g,N_{\gamma}+1|$.

If we blue-detune the laser from resonance, the energy gained by the atom
being in the excited state $|r,N_{\gamma}\rangle$ relative to being in the
state $|g,N_{\gamma}+1\rangle$ is $-\delta$, where $\delta$ is the detuning of
the laser. This detuning then contributes a diagonal term to the effective
Hamiltonian $-\delta\hat{n}$ where $\hat{n}$ is $0$ if the atom is in the
state $|g,N_{\gamma}+1\rangle$ and $1$ if the atom is in the state
$|r,N_{\gamma}\rangle$. This allows us to write down an effective two-state
Hamiltonian in the basis of atom-field product states for the single atom

$H_{1\text{ atom}}=\Omega\sigma^{x}-\delta\hat{n}.$ (3)

Two atoms in $s-$orbital Rydberg states interact through a force that can be
modelled by a Van der Waals potential $V(r)=C/r^{6}$ when the inter-atomic
separation $r$ is large, where $C$ is a constant that scales very strongly
with the principal quantum numbers of the two Rydberg states. For two
identical Rydberg atoms with principal quantum number $n$ (not to be confused
with the number operator $\hat{n}$), the coefficient $C$ of the Van der Waals
interaction roughly scales with $\sim n^{11}$. In the remaining of the paper
we will use the more detailed evaluation of the Van der Waals interaction
given in Ref. Šibalić et al. (2017). As such, the total effective many-body
Hamiltonian that describes a lattice of these atoms is

$\displaystyle H=\sum_{i}\Omega_{i}\sigma^{x}_{i}+H_{0},\ \
H_{0}=-\sum_{i}\delta_{i}\hat{n}_{i}+\sum_{ij}V_{ij}\hat{n}_{i}\hat{n}_{j},$
(4)

where $V_{ij}=C_{ij}/|i-j|^{6}$ and $i,j$ label the lattice sites.

We will start with the small Rabi frequency (relative to the detuning) limit
of this model so that we may first ignore the $\sigma^{x}$ terms and focus on
the classical part of the Hamiltonian $H_{0}$. To realize the classical
Sierpinski triangle model of Eq. 1, we need to select the parameters in
$H_{0}$ which yield low energy states that can be mapped to those of the
Sierpinski triangle model. We consider the honeycomb lattice with two
sublattices $\mathcal{A}$ and $\mathcal{B}$. We trap an “auxiliary” atom at
each site in $\mathcal{A}$ and a “target” atom at each site in $\mathcal{B}$.
An equivalent picture is that we take the triangular lattice and decorate each
vertex with a target atom and the center of each downward facing triangle with
an auxiliary atom (Fig. 2). We aim to reproduce the states of Eq. 1 only on
the $\mathcal{B}$ sublattice with target atoms. The auxiliary atoms enlarge
the Hilbert space and hence the states of model Eq. 1 with multi-spin
interactions can be reproduced through two-body interactions only in the low
energy subspace of the atomic system.

We assign different principal quantum numbers $n_{\mathcal{A}}$ and
$n_{\mathcal{B}}$ for the Rydberg states of the auxiliary atoms located on
sublattice $\mathcal{A}$ and target atoms on sublattice $\mathcal{B}$.
Crucially, with a proper choice of $n_{\mathcal{A}}$, $n_{\mathcal{B}}$, as
well as the detuning, the Hamiltonian $H_{0}$ can be organized as
$H_{0}=\sum_{a\in\mathcal{A}}H_{0,a}$:

$\displaystyle H_{0,a}$ $\displaystyle=$ $\displaystyle
V\left(2\hat{n}_{a}+\hat{n}_{t,1}+\hat{n}_{t,2}+\hat{n}_{t,3}-2\right)^{2}+\sum_{i=1}^{3}v\hat{n}_{a}\hat{n}_{t,i}$
(5) $\displaystyle\sim$ $\displaystyle
V\left(\sum_{i=1}^{3}4\hat{n}_{a}\hat{n}_{t,i}+\sum_{i<j}2\hat{n}_{t,i}\hat{n}_{t,j}-4\hat{n}_{a}-\sum_{i=1}^{3}3\hat{n}_{t,i}\right)$
(7) $\displaystyle+$
$\displaystyle\sum_{i=1}^{3}v\hat{n}_{a}\hat{n}_{t,i}\cdots.$ (9)

The sum $H_{0}=\sum_{a\in\mathcal{A}}H_{0,a}$ is over all sublattice sites
$\mathcal{A}$, and each term in the sum involves an auxiliary atom
($\hat{n}_{a}=0,1$), as well as its three neighboring target atoms
($\hat{n}_{t,i}=0,1$) which form a downwards facing triangle on the honeycomb
lattice. The second line of Eq. 9 uses the fact that
$\hat{n}_{a}^{2}=\hat{n}_{a}$, $\hat{n}_{t}^{2}=\hat{n}_{t}$ for
$\hat{n}=0,1$. $H_{0}$ contains a two-body repulsive interaction
$V_{\mathcal{A}\mathcal{B}}=4V+v$ between the auxiliary atom and its
neighboring target atoms, as well as a repulsive interaction
$V_{\mathcal{B}\mathcal{B}}=2V$ between two nearest neighbor target atoms
(Fig. 2).

When $v>0$, there are two classes of configurations of $(\hat{n}_{t,i};\
\hat{n}_{a})$ on each downward triangle, both of which are the ground states
of $H_{0}$:

$\displaystyle(1)\ \ \hat{n}_{a}=1,\ \hat{n}_{t,i}=0;$ (10)
$\displaystyle\mathrm{or}$ $\displaystyle(2)\ \ \hat{n}_{a}=0,\ \text{two of
}\hat{n}_{t,i}=1.$ (12)

As a comparison, the ground states of the classical Sierpinski triangle model
$H_{\mathrm{ST}}$ also have two types of ground states on each downwards
facing triangle:

$\displaystyle(1)\ \ \sigma^{z}_{1}=\sigma^{z}_{2}=\sigma^{z}_{3}=+1;$ (13)
$\displaystyle\mathrm{or}$ $\displaystyle(2)\ \ \text{Two of
}\sigma^{z}_{i}=-1.$ (15)

Now the ground states of $H_{0}$ can be one-to-one mapped to the ground states
of the classical Sierpinski triangle model, as long as we identify
$\sigma^{z}_{i}=1-2\hat{n}_{t,i}$. In fact in the appendix we will show that,
all the states of Eq. 1 (ground and excited states) can be mapped one-to-one
to the low energy subspace of $H_{0}$ when $v>0$.

Figure 2: We trap atoms on both the vertices and the center of each downward
triangle of a triangular lattice, which together form a honeycomb lattice
(three hexagons of the honeycomb lattice are shown in the figure). The atoms
on the vertices and the triangle centers are the “target” and “auxiliary
atoms” respectively. The Hamiltonian Eq. 9 reduces to
$V_{\mathcal{A}\mathcal{B}}=4V+v$ between a auxiliary atom and its neighboring
target atoms, and interaction $V_{\mathcal{B}\mathcal{B}}=2V$ between two
neighboring target atoms. The ground state of Eq. 9 demands that, on each
downward facing triangle, either $\hat{n}_{a}=1$ (labelled orange) and all
$\hat{n}_{t}=0$; or $\hat{n}_{a}=0$ and two out of three $\hat{n}_{t}=1$
(labelled blue).

The relation between $V_{\mathcal{A}\mathcal{B}}$ and
$V_{\mathcal{B}\mathcal{B}}$ can be tuned by choosing the principal quantum
numbers $n_{\mathcal{A}}$ and $n_{\mathcal{B}}$ properly. For example, for
potassium atoms, if we choose $n_{\mathcal{A}}=76$ and $n_{\mathcal{B}}=113$,
then using the techniques in Ref. Šibalić et al. (2017) and the fact that the
interatomic distances are related by
$r_{\mathcal{B}\mathcal{B}}=\sqrt{3}r_{\mathcal{A}\mathcal{B}}$, we found that
both interactions are repulsive and satisfy
$V_{\mathcal{A}\mathcal{B}}/V_{\mathcal{B}\mathcal{B}}\sim 2.628$ ($v\sim
1.26V$). Note that different choices are possible; our results apply more
broadly than just to this specific choice of atom and principal quantum
numbers.

In the real system, there are perturbations to Eq. 9. These include other
terms induced by the Van der Waals interaction, for example the repulsion
between Rydberg states on two neighboring auxiliary atoms, whose strength
$V_{\mathcal{A}\mathcal{A}}$ compared with $V_{\mathcal{B}\mathcal{B}}$ is
$V_{\mathcal{A}\mathcal{A}}/V_{\mathcal{B}\mathcal{B}}\sim 0.011$ using the
example principal quantum numbers we chose above. The repulsion
$V_{\mathcal{B}\mathcal{B}}^{\prime}$ between the target atoms on two second
neighbor $\mathcal{B}$ sites is also much weaker than
$V_{\mathcal{B}\mathcal{B}}$ due to the rapid decay of the Van der Waals
interaction with distance. Another notable perturbation is the interaction
$V^{\prime}_{\mathcal{A}\mathcal{B}}$ between an auxiliary atom and its next-
neighbor target atom. Compared with $V_{\mathcal{B}\mathcal{B}}$, the two
perturbations $V_{\mathcal{B}\mathcal{B}}^{\prime}$ and
$V^{\prime}_{\mathcal{A}\mathcal{B}}$ are

$\displaystyle\frac{V^{\prime}_{\mathcal{B}\mathcal{B}}}{V_{\mathcal{B}\mathcal{B}}}=\frac{1}{(\sqrt{3})^{6}}\sim
0.037,\ \
\frac{V^{\prime}_{\mathcal{A}\mathcal{B}}}{V_{\mathcal{B}\mathcal{B}}}=\frac{1}{2^{6}}\frac{V_{\mathcal{A}\mathcal{B}}}{V_{\mathcal{B}\mathcal{B}}}\sim
0.041.$ (16)

These perturbations shift the energy of the excited state of the Sierpinski
triangle shape. Let us consider an excited state with flipped “spins”
($\hat{n}_{t}=1$) on a Sierpinski triangle with side length $L=2^{\ell}$. In
the ideal model of Eq. 1, the energy of this excited state does not depend on
$L$ or $\ell$: all the energy cost arises from the corners of the Sierpinski
triangle and the energy compared with the ground state is
$E_{\text{ex}}=E-E_{g}=6K=3V=3/2\times V_{\mathcal{B}\mathcal{B}}$. However,
in the real system the leading order perturbations
$V_{\mathcal{A}\mathcal{B}}^{\prime},V_{\mathcal{A}\mathcal{A}}$ and
$V_{\mathcal{B}\mathcal{B}}^{\prime}$ cause the energy cost of creating a
Sierpinski triangle to scale with its size. In particular, the energy of the
excitation relative to the ground state with uniform $\hat{n}_{a}=1$ and
$\hat{n}_{t}=0$ is estimated to be (details are given in the appendix)

$E_{\text{ex}}^{\text{Ry}}=(3/2-0.1+0.47\times
3^{\ell-3})V_{\mathcal{B}\mathcal{B}}$ (17)

for Sierpinski triangles of $\ell\geq 2$. Since the actual energy cost of
creating a Sierpinski triangle increases with the size of the triangle, the
perturbations can no longer be ignored for large enough Sierpinski triangles.
Finite-size fractal excitations, however, will still be observable.

— Experimental Proposals

Figure 3: With Hamiltonian Eq. 9, if three target atoms at the corners of a
triangle with side length $L=8$ are excited from the ground state to the
Rydberg state, it costs energy $3V+3v$ at each corner. If we apply an extra
detuning $-\bar{\delta}_{\mathrm{C}}\hat{n}_{t}$ on the three corners of the
triangle, for sufficiently large $\bar{\delta}_{\mathrm{C}}$ the ground state
of the system is given by the fractal configuration in Fig. 1.

Next, we outline a procedure to enable experimental observation of a
spontaneously generated fractal-shaped excitation. This can be done by
adiabatically evolving a prepared ground state of Eq. 9 to the Sierpinski
triangle excitation, which is the ground-state of a new Hamiltonian achieved
by slowly and carefully varying the detuning and Rabi frequency. Note that
this procedure requires a level of local control beyond that present in most
current Rydberg experimental platforms, but the elements of which have been
demonstrated experimentally Labuhn et al. (2014). We start with the
Hamiltonian of Eq. 9 with a small and finite Rabi frequency. To ensure a
unique ground state (to enable an adiabatic evolution), we first deform Eq. 9
with a small extra detuning $\bar{\delta}\hat{n}_{t}$ on all target atoms
outside of a triangle with side $L=2^{\ell}$. We then prepare an initial state
with all target atoms $\hat{n}_{t}=0$ and $\hat{n}_{a}=1$, which is equivalent
to $\sigma^{z}=1$ uniformly in Eq. 1 and represents the unique ground state of
the Hamiltonian prepared above. We slowly deform the Hamiltonian with time-
dependent Rabi frequency $\Omega(t)$ and detuning $\delta_{i}(t)$ within the
triangle, reaching the final Hamiltonian with extra detuning
$-\bar{\delta}_{\mathrm{C}}\hat{n}_{t}$ localized to three target atoms at the
corners of the triangle (Fig. 3). With sufficiently large
$\bar{\delta}_{\mathrm{C}}$, the unique ground state of the final Hamiltonian
contains a Sierpinski triangle arrangement of all the atoms inside the
triangle, as shown in Fig. 1 and Fig. 2, despite the fact that the extra
detuning $-\bar{\delta}_{\mathrm{C}}\hat{n}_{t}$ is only applied locally at
the corners rather than throughout the entire interior. Based on our estimate
of energy in Eq. 17 which includes further leading neighbor interactions
arising from the Van der Waals interaction, this phenomenon can hold up to
Sierpinski triangles with $\ell=4$ (side length $L=16$, containing 81 atoms),
as a single Sierpinski triangle configuration still has lower energy than
fragmented configurations.

We have specified the initial and final Hamiltonian for the desired evolution;
an adiabatic path of the detuning and Rabi frequency for observing Ising-like
crystallization of Rydberg atoms has already been demonstrated Schauß et al.
(2015); browaeys2daf; lukinphases. Leveraging the aforementioned power of
local control, we could also create arbitrary configurations of the Rydberg
array, including the Sierpinski triangle. Spectroscopic measurement of the
energy of the Sierpinski configuration compared with that of fragmented
configurations could then confirm that it represents a low-lying excited
state. This technique would no longer rely on adiabatic evolution.

This experimental platform also gives us the potential to probe quantum phase
transitions by controlling the Rabi frequency. When the Rabi frequency
$\Omega$ is increased uniformly on all the target atoms, around $\Omega\sim V$
there is expected to be a quantum phase transition similar to the one recently
studied numerically Vasiloiu et al. (2020); Zhou et al. (2021). In the large
$\Omega$ phase, one is not supposed to observe the fractal configuration in
the proposed experiment above, i.e. the configuration of Fig. 3 will not
evolve to the one in Fig. 1 and Fig. 2. As we pointed out before, the nature
of this transition is far from being understood. Hence the realization of the
quantum phase transition in experimental systems with high tunability is
crucial to understand these exotic quantum phase transitions, and to test
possible theoretical paradigms developed in the future.

The fractal structure of the fractal orders also manifests at the level of the
correlation functions. In both the classical and quantum Sierpinski triangle
models, the three-point correlation function
$\mathcal{C}_{3}=\langle\sigma^{z}_{j}\sigma^{z}_{j+L\hat{x}}\sigma^{z}_{j+L(\hat{x}-\hat{y})}\rangle$
is a characteristic quantity which plays the role of the correlation function
of ordinary quantum many-body systems. The three-point correlation decays
hyper-exponentially with the Hausdorff dimension at finite temperature Yoshida
(2013); Newman and Moore (1999) and its scaling at the quantum critical point
$h=K$ was computed in Ref. Zhou et al. (2021). In the experimental realization
of Eq. 9, the three-point correlation function $\mathcal{C}_{3}$ can be
reconstructed by averaging over multiple single-site resolution snapshots of
the configuration of the target atoms taken in separate experimental
realizations. Similar techniques have been used previously in cold-atom
experiments to reconstruct quantities such as the spin correlation functions
in Fermi-Hubbard systems Parsons et al. (2016); Endres et al. (2011).
Experimental measurement of $\mathcal{C}_{3}$, whose scaling with distance
diagnoses the fractal physics of the model, will be crucial in understanding
the nature of this quantum phase transition.

— Discussion

Previous proposals for realizing fracton related states mostly focused on
states that are analogous to type-I fractons Xu and Fisher (2007); Pretko
(2019); Sous and Pretko (2020); Giergiel et al. (2021). For example,
theoretical efforts on constructing fracton related states were made based on
localized Majorana zero modes for both type-I and type-II states Xu and Fu
(2010); You and von Oppen (2019). Compared with previous proposals, the
platform of Rydberg atoms discussed in the current work is highly tunable with
precision at the level of a single atom. Another advantage of the platform of
Rydberg atoms, and in general cold atom systems, is fast manipulation of
parameters in the Hamiltonian, which can either periodically drive the system,
or cause a quantum quench Polkovnikov et al. (2011); Wintersperger et al.
(2020); Potirniche et al. (2017); Singh et al. (2019); Chen et al. (2019);
Bluvstein et al. (2021); Keesling et al. (2019); Bernien et al. (2017). Many
exotic features are expected in the quantum dynamics of fracton related models
due to the restricted motion of the fracton excitations Pai and Pretko (2019);
Prem et al. (2017); Feldmeier et al. (2020); Kim and Haah (2016). By quickly
tuning parameters such as the Rabi frequency in Eq. 4, one can compare the
quantum dynamics of the fractal order simulated with Rydberg atoms with future
analytical and numerical analysis. Furthermore, our construction of Eq. 9 used
to reproduce multi-spin interactions in Rydberg Hamiltonians with only two-
body interactions but an enlarged Hilbert space, can be extended to fracton
related models richer than the Sierpinski triangle model. One example of such
extension is given in the appendix.

The authors thank Chao-Ming Jian and Hannes Bernien for very helpful
discussions. C.X. is supported by NSF Grant No. DMR-1920434, and the Simons
Investigator program. D.W. acknowledges support from the Army Research Office
(MURI W911NF1710323), and the University of California Multicampus Research
Programs and Initiatives (MRP19-601445). We gratefully acknowledge support via
the UC Santa Barbara NSF Quantum Foundry funded via the Q-AMASE-i program
under award DMR-1906325. This material is based upon work supported by the
U.S. Department of Energy, Office of Science, National Quantum Information
Science Research Centers, Quantum Science Center.

## References

  * Browaeys and Lahaye (2020) A. Browaeys and T. Lahaye, Nature Physics 16, 132 (2020), URL https://doi.org/10.1038/s41567-019-0733-z.
  * Gross and Bloch (2017) C. Gross and I. Bloch, Science 357, 995 (2017).
  * Saffman et al. (2010) M. Saffman, T. G. Walker, and K. Mølmer, Reviews of modern physics 82, 2313 (2010).
  * Wen (2017) X.-G. Wen, Rev. Mod. Phys. 89, 041004 (2017), URL https://link.aps.org/doi/10.1103/RevModPhys.89.041004.
  * Sachdev (2018) S. Sachdev, Reports on Progress in Physics 82, 014001 (2018), URL https://doi.org/10.1088/1361-6633/aae110.
  * Verresen et al. (2021) R. Verresen, M. D. Lukin, and A. Vishwanath, Phys. Rev. X 11, 031005 (2021), URL https://link.aps.org/doi/10.1103/PhysRevX.11.031005.
  * Samajdar et al. (2021) R. Samajdar, W. W. Ho, H. Pichler, M. D. Lukin, and S. Sachdev, Proceedings of the National Academy of Sciences 118 (2021), ISSN 0027-8424, eprint https://www.pnas.org/content/118/4/e2015785118.full.pdf, URL https://www.pnas.org/content/118/4/e2015785118.
  * Semeghini et al. (2021) G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, et al. (2021), eprint 2104.04119.
  * Read and Sachdev (1991) N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991), URL https://link.aps.org/doi/10.1103/PhysRevLett.66.1773.
  * Wen (1991) X. G. Wen, Phys. Rev. B 44, 2664 (1991), URL https://link.aps.org/doi/10.1103/PhysRevB.44.2664.
  * Kitaev (2003) A. Kitaev, Annals of Physics 303, 2 (2003), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/pii/S0003491602%000180.
  * Haah (2011) J. Haah, Phys. Rev. A 83, 042330 (2011), URL https://link.aps.org/doi/10.1103/PhysRevA.83.042330.
  * Chamon (2005) C. Chamon, Phys. Rev. Lett. 94, 040402 (2005), URL https://link.aps.org/doi/10.1103/PhysRevLett.94.040402.
  * Bravyi et al. (2011) S. Bravyi, B. Leemhuis, and B. M. Terhal, Annals of Physics 326, 839 (2011), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/pii/S0003491610%001910.
  * Yoshida (2013) B. Yoshida, Phys. Rev. B 88, 125122 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.88.125122.
  * Vijay et al. (2015) S. Vijay, J. Haah, and L. Fu, Phys. Rev. B 92, 235136 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.235136.
  * Vijay et al. (2016) S. Vijay, J. Haah, and L. Fu, Phys. Rev. B 94, 235157 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.94.235157.
  * Nandkishore and Hermele (2019) R. M. Nandkishore and M. Hermele, Annual Review of Condensed Matter Physics 10, 295 (2019), eprint https://doi.org/10.1146/annurev-conmatphys-031218-013604, URL https://doi.org/10.1146/annurev-conmatphys-031218-013604.
  * Pretko et al. (2020) M. Pretko, X. Chen, and Y. You, International Journal of Modern Physics A 35, 2030003 (2020), eprint https://doi.org/10.1142/S0217751X20300033, URL https://doi.org/10.1142/S0217751X20300033.
  * Newman and Moore (1999) M. E. J. Newman and C. Moore, Phys. Rev. E 60, 5068 (1999), URL https://link.aps.org/doi/10.1103/PhysRevE.60.5068.
  * Zhou et al. (2021) Z. Zhou, X.-F. Zhang, F. Pollmann, and Y. You (2021), eprint 2105.05851.
  * Vasiloiu et al. (2020) L. M. Vasiloiu, T. H. E. Oakes, F. Carollo, and J. P. Garrahan, Phys. Rev. E 101, 042115 (2020), URL https://link.aps.org/doi/10.1103/PhysRevE.101.042115.
  * Kramers and Wannier (1941) H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252 (1941), URL https://link.aps.org/doi/10.1103/PhysRev.60.252.
  * Kogut (1979) J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979), URL https://link.aps.org/doi/10.1103/RevModPhys.51.659.
  * Xu and Moore (2004) C. Xu and J. E. Moore, Phys. Rev. Lett. 93, 047003 (2004), URL https://link.aps.org/doi/10.1103/PhysRevLett.93.047003.
  * Wilson (1971) K. G. Wilson, Phys. Rev. B 4, 3174 (1971), URL https://link.aps.org/doi/10.1103/PhysRevB.4.3174.
  * Wilson and Fisher (1972) K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972), URL https://link.aps.org/doi/10.1103/PhysRevLett.28.240.
  * Wilson (1975) K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975), URL https://link.aps.org/doi/10.1103/RevModPhys.47.773.
  * Šibalić et al. (2017) N. Šibalić, J. Pritchard, C. Adams, and K. Weatherill, Computer Physics Communications 220, 319 (2017), ISSN 0010-4655, URL http://dx.doi.org/10.1016/j.cpc.2017.06.015.
  * Labuhn et al. (2014) H. Labuhn, S. Ravets, D. Barredo, L. Béguin, F. Nogrette, T. Lahaye, and A. Browaeys, Phys. Rev. A 90, 023415 (2014), URL https://link.aps.org/doi/10.1103/PhysRevA.90.023415.
  * Schauß et al. (2015) P. Schauß, J. Zeiher, T. Fukuhara, S. Hild, M. Cheneau, T. Macrì, T. Pohl, I. Bloch, and C. Gross, Science 347, 1455 (2015), ISSN 0036-8075, eprint https://science.sciencemag.org/content/347/6229/1455.full.pdf, URL https://science.sciencemag.org/content/347/6229/1455.
  * Parsons et al. (2016) M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, Science 353, 1253 (2016), ISSN 0036-8075, eprint https://science.sciencemag.org/content/353/6305/1253.full.pdf, URL https://science.sciencemag.org/content/353/6305/1253.
  * Endres et al. (2011) M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M. C. Bañuls, L. Pollet, I. Bloch, et al., Science 334, 200 (2011), ISSN 0036-8075, eprint https://science.sciencemag.org/content/334/6053/200.full.pdf, URL https://science.sciencemag.org/content/334/6053/200.
  * Xu and Fisher (2007) C. Xu and M. P. A. Fisher, Phys. Rev. B 75, 104428 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.104428.
  * Pretko (2019) M. Pretko, Phys. Rev. B 100, 245103 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.100.245103.
  * Sous and Pretko (2020) J. Sous and M. Pretko, Phys. Rev. B 102, 214437 (2020), URL https://link.aps.org/doi/10.1103/PhysRevB.102.214437.
  * Giergiel et al. (2021) K. Giergiel, R. Lier, P. Surówka, and A. Kosior, _Bose-hubbard realization of fracton defects_ (2021), eprint 2107.06786.
  * Xu and Fu (2010) C. Xu and L. Fu, Phys. Rev. B 81, 134435 (2010), URL https://link.aps.org/doi/10.1103/PhysRevB.81.134435.
  * You and von Oppen (2019) Y. You and F. von Oppen, Phys. Rev. Research 1, 013011 (2019), URL https://link.aps.org/doi/10.1103/PhysRevResearch.1.013011.
  * Polkovnikov et al. (2011) A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011), URL http://link.aps.org/doi/10.1103/RevModPhys.83.863.
  * Wintersperger et al. (2020) K. Wintersperger, C. Braun, F. N. Ünal, A. Eckardt, M. D. Liberto, N. Goldman, I. Bloch, and M. Aidelsburger, Nature Physics 16, 1058 (2020), ISSN 1745-2481, URL http://dx.doi.org/10.1038/s41567-020-0949-y.
  * Potirniche et al. (2017) I.-D. Potirniche, A. C. Potter, M. Schleier-Smith, A. Vishwanath, and N. Y. Yao, Phys. Rev. Lett. 119, 123601 (2017), URL https://link.aps.org/doi/10.1103/PhysRevLett.119.123601.
  * Singh et al. (2019) K. Singh, C. J. Fujiwara, Z. A. Geiger, E. Q. Simmons, M. Lipatov, A. Cao, P. Dotti, S. V. Rajagopal, R. Senaratne, T. Shimasaki, et al., Phys. Rev. X 9, 041021 (2019), URL https://link.aps.org/doi/10.1103/PhysRevX.9.041021.
  * Chen et al. (2019) Z. Chen, T. Tang, J. Austin, Z. Shaw, L. Zhao, and Y. Liu, Phys. Rev. Lett. 123, 113002 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.113002.
  * Bluvstein et al. (2021) D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Semeghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, et al., Science 371, 1355 (2021), ISSN 0036-8075, eprint https://science.sciencemag.org/content/371/6536/1355.full.pdf, URL https://science.sciencemag.org/content/371/6536/1355.
  * Keesling et al. (2019) A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, et al., Nature 568, 207 (2019), URL https://doi.org/10.1038/s41586-019-1070-1.
  * Bernien et al. (2017) H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Nature 551, 579 (2017), URL https://doi.org/10.1038/nature24622.
  * Pai and Pretko (2019) S. Pai and M. Pretko, Phys. Rev. Lett. 123, 136401 (2019), URL https://link.aps.org/doi/10.1103/PhysRevLett.123.136401.
  * Prem et al. (2017) A. Prem, J. Haah, and R. Nandkishore, Phys. Rev. B 95, 155133 (2017), URL https://link.aps.org/doi/10.1103/PhysRevB.95.155133.
  * Feldmeier et al. (2020) J. Feldmeier, P. Sala, G. De Tomasi, F. Pollmann, and M. Knap, Phys. Rev. Lett. 125, 245303 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.245303.
  * Kim and Haah (2016) I. H. Kim and J. Haah, Phys. Rev. Lett. 116, 027202 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.116.027202.
  * Devakul et al. (2019) T. Devakul, Y. You, F. J. Burnell, and S. L. Sondhi, SciPost Phys. 6, 7 (2019), URL https://scipost.org/10.21468/SciPostPhys.6.1.007.
  * Garrahan (2002) J. P. Garrahan, Journal of Physics: Condensed Matter 14, 1571 (2002), URL https://doi.org/10.1088/0953-8984/14/7/314.

##  Appendix A Estimate of energy of Sierpinski triangle excitation in real
system

With the inclusion of the perturbations of
$V_{\mathcal{AA}},V_{\mathcal{BB}}^{\prime}$ and $V_{\mathcal{AB}}^{\prime}$,
we can calculate the relative cost of creating a Sierpinski triangle excited
state by exciting target atoms and de-exciting auxiliary atoms in the
perturbed ground state with $\hat{n}_{a}=1$ and $\hat{n}_{t}=0$ uniformly. For
a Sierpinski triangle of length $L=2^{\ell}$, the energy cost
$E_{\text{ex}}^{\text{Ry}}$ is

$\displaystyle
E_{\text{ex}}^{\text{Ry}}=3V+N^{\prime}_{\mathcal{BB}}V_{\mathcal{BB}}^{\prime}+N^{\prime}_{\mathcal{AB}}V_{\mathcal{AB}}^{\prime}-N_{\mathcal{AA}}V_{\mathcal{AA}},$
(18)

where $N^{\prime}_{\mathcal{BB}}$, $N^{\prime}_{\mathcal{AB}}$ and
$N_{\mathcal{AA}}$ are the number of interactions $V_{\mathcal{BB}}^{\prime}$,
$V_{\mathcal{AB}}^{\prime}$ we have turned on, and the number of interactions
$V_{\mathcal{AA}}$ we have turned off, respectively, in order to create the
Sierpinski triangle.

Figure 4: The next neighbor Van der Waals interaction between target atoms.
Figure 5: The next neighbor interaction between auxiliary atoms and target
atoms. Figure 6: The nearest neighbor interaction between auxiliary atoms that
are “lost” while creating the Sierpinski triangle excitation.

Here $N_{\mathcal{BB}}^{\prime}$ as a function of $\ell$ is

$\displaystyle N^{\prime}_{\mathcal{BB}}(\ell)=(3^{\ell}-3)\text{ for
}\ell\geq 2.$ (19)

This result can be seen from the following: each Sierpinski triangle of length
$2^{\ell-1}$ has a corresponding number of $V_{\mathcal{BB}}^{\prime}$
interactions $N^{\prime}_{\mathcal{BB}}(\ell-1)$. The Sierpinski triangle of
length $2^{\ell}$ is constructed by gluing three copies of of the smaller
Sierpinski triangle of length $2^{\ell-1}$ together at the corners so that it
has $3N^{\prime}_{\mathcal{BB}}(\ell-1)$ many interactions, as well as the
added interactions between target atoms belonging to different smaller
Sierpinski triangles (Fig. 4). Including these added interactions, of which
there are $6$, we recover a recurrence relation for
$N^{\prime}_{\mathcal{BB}}(\ell)$:

$N^{\prime}_{\mathcal{BB}}(\ell)=3N^{\prime}_{\mathcal{BB}}(\ell-1)+6$

with the number of interactions for the $\ell=2$ Sierpinski triangle as the
initial condition, $N^{\prime}_{\mathcal{BB}}(2)=6$. This recurrence relation
can be solved for

$N^{\prime}_{\mathcal{BB}}(\ell)=(3^{\ell}-3)\text{ for }\ell\geq 2,$

which is the stated result.

We can compute the number of $V_{\mathcal{AB}}^{\prime}$ interactions that we
have turned on in a similar manner. The total number of these interactions is

$N^{\prime}_{\mathcal{AB}}(\ell)=(3^{\ell}+3)\text{ for }\ell\geq 2.$

Since these are interactions between auxiliary atoms and target atoms, these
occur between the target atoms on the edges of the Sierpinski triangle and
surrounding auxiliary atoms, as well as between target atoms and the auxiliary
atoms in the interior of the Sierpinski triangle. For a Sierpinski triangle of
length $L=2^{\ell}$, the total number of interactions coming from the target
atoms at the edges is $3L=3\times 2^{\ell}$. Counting the interactions coming
from the interior of the Sierpinski triangle
$N_{\mathcal{AB}}^{\text{int}}(\ell)$ is more complicated as it depends on the
length of the previous iteration of the Sierpinski triangle. These can be
counted as the contribution from the largest missing triangle
$3\times(2^{\ell-1}-2)$, plus three times the number of interactions from the
last iteration $3N_{\mathcal{AB}}^{\text{int}}(\ell-1)$. As a result, we are
yet again left with a recurrence relation

$N_{\mathcal{AB}}^{\text{int}}(\ell)=3N_{\mathcal{AB}}^{\text{int}}(\ell-1)+3\times(2^{\ell-1}-2)$

with initial condition $N_{\mathcal{AB}}^{\text{int}}(2)=0$. This can be
solved for

$\displaystyle N_{\mathcal{AB}}^{\text{int}}=3^{\ell}-3\times 2^{\ell}+3.$

Therefore the total number of $V_{\mathcal{AB}}^{\prime}$ interactions we have
turned on is

$\displaystyle N_{\mathcal{AB}}$
$\displaystyle=N_{\mathcal{AB}}^{\text{int}}+3\times 2^{\ell},$
$\displaystyle=(3^{\ell}+3)\text{ for }\ell\geq 2.$

The calculation of the number of $V_{\mathcal{AA}}$ interactions lost when
creating the Sierpinski triangle excitation is nearly identical to the
previous calculation. The total number of lost interactions can be calculated
as

$\displaystyle N_{\mathcal{AA}}=\frac{1}{2}(11\times 3^{\ell}+21)\text{ for
}\ell\geq 2.$ (20)

There are three sources of the loss of $V_{\mathcal{AA}}$ interactions: 1) the
loss of interaction between two de-excited auxiliary atoms in downwards
triangles in the Sierpinski triangle, 2) the loss of interaction between a de-
excited auxiliary atom on the edges and corners of the Sierpinski triangle and
an excited auxiliary atom surrounding the Sierpinski triangle and 3) the loss
of interaction between a de-excited auxiliary atom in the Sierpinski triangle
and an excited auxiliary atom in the interior of the Sierpinski triangle.

The contribution to the total number of lost $V_{\mathcal{AA}}$ interactions
from the first two cases 1) and 2) can be found to be $\frac{1}{2}(7\times
3^{\ell}-3)+6(2^{\ell}+1)$. There are an additional $12$ lost interactions
that come from de-excited auxiliary atoms at the corners of the Sierpinski
triangle and do not scale with $\ell$. For the $\ell=2$ Sierpinski triangle
there are $N_{\mathcal{AA}}^{1)}(2)=30$ lost interactions not including the
ones from the edges and corners. Gluing these triangles to generate a $\ell=3$
Sierpinski triangle gives us $3$ times as many interactions as well as an
additional $3$ lost interactions between auxiliary atoms belonging to
different smaller Sierpinski triangles. Hence we are left with a recursion
relation

$N^{1)}_{\mathcal{AA}}(\ell)=3N^{1)}_{\mathcal{AA}}(\ell-1)+3.$

Using $N_{\mathcal{AA}}^{1)}(2)=30$ as the initial condition, we have a
solution of

$N_{\mathcal{AA}}^{1)}(\ell)=\frac{1}{2}(7\times 3^{\ell}-3)\text{ for
}\ell\geq 2.$

$N_{\mathcal{AA}}^{2)}$ is straightforward:
$N_{\mathcal{AA}}^{2)}=6(2^{\ell}-1)+12=6(2^{\ell}+1)$ including the
interactions at the corners. Hence, restoring the additional $12$
interactions, $N_{\mathcal{AA}}^{1)}+N_{\mathcal{AA}}^{2)}=\frac{1}{2}(7\times
3^{\ell}-3)+6(2^{\ell}+1)$.

We recover another recurrence relation for the number of lost interactions in
the interior for case 3):

$N_{\mathcal{AA}}^{3)}(\ell)=3N_{\mathcal{AA}}^{3)}(\ell-1)+6\times(2^{\ell-1}-2)$

with initial condition $N_{\mathcal{AA}}^{3)}(2)=0$. This can be solved for

$N_{\mathcal{AA}}^{3)}(\ell)=2\times 3^{\ell}-3\times 2^{\ell+1}+6.$

The total number of lost $V_{\mathcal{AA}}$ interactions is then

$\displaystyle N_{\mathcal{AA}}$
$\displaystyle=N_{\mathcal{AA}}^{3)}(\ell)+\frac{1}{2}(7\times
3^{\ell}-3)+6(2^{\ell}+1),$ $\displaystyle=\frac{1}{2}(11\times
3^{\ell}+21)\text{ for }\ell\geq 2.$

Combining all the perturbations evaluated above, $E_{\text{ex}}^{\text{Ry}}$
is given by

$\displaystyle E_{\text{ex}}^{\text{Ry}}$
$\displaystyle=3V+N^{\prime}_{\mathcal{BB}}V_{\mathcal{BB}}^{\prime}+N^{\prime}_{\mathcal{AB}}V_{\mathcal{AB}}^{\prime}-N_{\mathcal{AA}}V_{\mathcal{AA}},$
(21) $\displaystyle\approx(3/2-0.1+0.47\times 3^{\ell-3})V_{\mathcal{BB}}.$
(23)

##  Appendix B Mapping between states

On a unit downward facing triangle, the mapping between states in Eq. 1 and
the states in Eq. 9 is:

$\displaystyle\sigma^{z}=(+1,+1,+1),\ \ \mathrm{Energy}=-K$ (24)
$\displaystyle\rightarrow$
$\displaystyle(\hat{n}_{t,i};\hat{n}_{a})=(0,0,0;1),\ \ \mathrm{Energy}=0;$
(28) $\displaystyle\sigma^{z}=(-1,-1,+1),\ \ \mathrm{Energy}=-K,$
$\displaystyle\rightarrow$
$\displaystyle(\hat{n}_{t,i};\hat{n}_{a})=(1,1,0;0),\ \ \mathrm{Energy}=0;$
(32) $\displaystyle\sigma^{z}=(-1,+1,+1),\ \ \mathrm{Energy}=+K,$
$\displaystyle\rightarrow$
$\displaystyle(\hat{n}_{t,i};\hat{n}_{a})=(1,0,0;0),\ \ \mathrm{Energy}=V;$
(36) $\displaystyle\sigma^{z}=(-1,-1,-1),\ \ \mathrm{Energy}=+K$
$\displaystyle\rightarrow$
$\displaystyle(\hat{n}_{t,i};\hat{n}_{a})=(1,1,1;0),\ \ \mathrm{Energy}=V.$
(38)

The Hilbert space of the atoms is bigger than the one of the spins; hence
there are some extra states with even higher energy. The
$v\hat{n}_{a}\hat{n}_{t,i}$ term in $H_{0}$ does not affect the energy of the
states list above, but they will separate the other states in the atomic
Hilbert space with extra energy cost:

$\displaystyle(\hat{n}_{t,i};\hat{n}_{a})=(1,0,0;1),\ \ \mathrm{Energy}=V+v;$
(39) (40) $\displaystyle(\hat{n}_{t,i};\ \hat{n}_{a})=(1,1,0;1),\ \
\mathrm{Energy}=4V+2v;$ (41) (42) $\displaystyle(\hat{n}_{t,i};\
\hat{n}_{a})=(1,1,1;1),\ \ \mathrm{Energy}=9V+3v;$ (43) (44)
$\displaystyle(\hat{n}_{t,i};\ \hat{n}_{a})=(0,0,0;0),\ \ \mathrm{Energy}=4V.$
(45)

Hence with positive $v$, the low energy states of the Rydberg atom systems can
be exactly mapped to the states of the Ising spins in Eq. 9, with $V=2K$.

##  Appendix C Fractal symmetry in the quantum Sierpinski triangle model

Figure 7: (Left): when the system is defined on a triangular lattice with
periodic boundary conditions, and $L_{x}=L_{y}=2^{k}-1$ ($k=3$ in the figure),
there exist symmetry generation operators
$\hat{G}(\mathcal{L})=\prod_{j\in\mathcal{L}}\sigma^{x}_{j}$ where
$\mathcal{L}$ are all the sites labeled in blue. (Right): we duplicate the
bottom row on the top, and the left column on the right of the lattice to
demonstrate the periodic boundary conditions.

For a quantum system to have a symmetry, the Hamiltonian of the system needs
to commute with a symmetry generation operator $\hat{G}$. For example, the
standard quantum Ising model $H=\sum_{\langle
i,j\rangle}-K\sigma^{z}_{i}\sigma^{z}_{j}-\sum_{j}h\sigma^{x}_{j}$ has a
$Z_{2}$ spin symmetry that is generated by the operator
$\hat{G}=\prod_{j}\sigma^{x}_{j}$ which commutes with the entire Hamiltonian.
Although the Hamiltonian has $Z_{2}$ spin symmetry, in the Ising ordered phase
with $K\gg h$, (at least in a certain basis) the ground state wave function of
the system in the thermodynamics limit is not invariant under the operation of
$\hat{G}$. The ground states are degenerate in the ordered phase, and
$\hat{G}$ would take one ground state to another. In this sense, the $Z_{2}$
symmetry is spontaneously broken in the Ising ordered phase. On the other
hand, in the Ising disordered phase $h\gg K$ the $Z_{2}$ symmetry is
preserved,i.e. the ground state wave function is unique and invariant under
operation of $\hat{G}$.

In the quantum Sierpinski triangle model Eq. 2, let us consider periodic
boundary conditions for convenience. The Hamiltonian does not have exact
subsystem symmetry for most system sizes; however, when the system is a
parallelogram with side length $L_{x}=L_{y}=2^{k}-1$ and $k$ being a positive
integer, there exists a series of symmetry generation operators
$\hat{G}(\mathcal{L})$ which are a product of $\sigma^{x}_{j}$ on a subset of
the lattice that belong to a Sierpinski triangle:

$\hat{G}(\mathcal{L})=\prod_{j\in\mathcal{L}}\sigma_{j}^{x}.$ (46)

The shape of $\mathcal{L}$ is illustrated in Fig. 7. Notice that due to the
subtlety of the periodic boundary conditions, $\hat{G}(\mathcal{L})$ commutes
with the entire Hamiltonian when and only when the system size
$L_{x}=L_{y}=2^{k}-1$. And when $K>h$, in the thermodynamics limit
$(k\rightarrow\infty)$ the system has many degenerate ground states which are
not invariant under the operation of $\hat{G}(\mathcal{L})$, hence the phase
$K>h$ spontaneously breaks the fractal symmetry generated by
$\hat{G}(\mathcal{L})$. While in the phase $h>K$ the ground state is unique
and invariant under operation of $\hat{G}(\mathcal{L})$.

##  Appendix D The Sierpinski tetrahedron model

Figure 8: The diamond lattice constructed out of tetrahedra with target atoms
$\hat{n}_{t,1},\dots,\hat{n}_{t,4}$ decorating the vertices and auxiliary
atoms supporting two different quantum numbers $\hat{n}_{a}^{\alpha,\beta}$
decorating the centers.

One of the main obstacles in constructing many exotic states of matter
considered theoretically is the multi-spin interaction in the Hamiltonian. Our
platform constructed with Rydberg atoms can simulate much of the low energy
physics of the Sierpinski triangle model, which originally involves three-body
interactions. In fact, our construction can be expanded to other fractal
models with more general multi-spin interactions as well. An example of such a
model that we might hope to realize is the natural extension of the Sierpinski
triangle model to three-dimensions: the Sierpinski tetrahedron model Garrahan
(2002); Devakul et al. (2019). To construct this model, we first glue
tetrahedra with effective spin-$1/2$ degrees of freedom $(\sigma^{z}=\pm 1)$
at the vertices together at the corners to form a lattice. As each tetrahedron
has four vertices, one may sum over all four-body interactions on each
tetrahedron,

$H_{\text{tetra}}=\sum_{\text{tetra}}-K\sigma_{1}^{z}\sigma_{2}^{z}\sigma_{3}^{z}\sigma_{4}^{z}.$
(47)

Many of the technical features of this model are similar to those of the
Sierpinski triangle model of Eq. 1:

(1) The low energy excitations above the ground state are created at the
corners of a Sierpinski tetrahedron (a self-similar fractal with Hausdorff
dimension $d_{H}=2$) of flipped spins and are fractons.

(2) The quantum model with an added transverse field
$\sum_{j}-h\sigma^{x}_{h}$ is self-dual and there is a fractal symmetry that
is spontaneously broken in the phase $K>h$. A quantum phase transition is
expected at $h=K$, although the nature of the transition needs further study.

Fractal-ordered phases such as the Sierpinski triangle and Sierpinski
tetrahedron models tend to have their fractal structure manifest at the level
of the correlation functions. In the Sierpinski triangle model, the three-
point correlation function
$\mathcal{C}_{3}=\langle\sigma^{z}_{j}\sigma^{z}_{j+L\hat{x}}\sigma^{z}_{j+L(\hat{x}-\hat{y})}\rangle$
is known to be non-vanishing only when $L=2^{\ell}$ and decays hyper-
exponentially with the Hausdorff dimension at finite temperature Yoshida
(2013); Newman and Moore (1999).

In the Sierpinski tetrahedron model, the four-point correlation function
$\mathcal{C}_{4}=\langle\sigma^{z}_{j}\sigma^{z}_{j+L\hat{x}}\sigma^{z}_{j+L\hat{y}}\sigma^{z}_{j+L\hat{z}}\rangle$
displays the same smoking-gun signatures of fractal order as $\mathcal{C}_{3}$
in the Sierpinski triangle model. This correlation function may be computed in
a similar fashion as one might compute the two-point correlation function of
the $1d$ Ising model, $\langle\sigma^{z}_{j}\sigma^{z}_{j+L}\rangle$. Since
there is a one-to-one correspondence between tetrahedra and vertices, we
define new spin-1/2 variables at the center of each tetrahedron
$\tau^{z}_{T}=(\sigma^{z}_{1}\sigma^{z}_{2}\sigma^{z}_{3}\sigma^{z}_{4})$. The
Hamiltonian now describes a collection of statistically independent spin
variables $\tau_{T}$,

$H_{\text{tetra,d}}=\sum_{T\in\text{tetra}}-K\tau_{T}.$ (48)

Since the $\tau$ variables are independent from one another, the correlations
of products of $\tau$’s factor nicely such that

$\displaystyle\bigg{\langle}\prod_{j}\tau_{j}\bigg{\rangle}$ $\displaystyle=$
$\displaystyle\prod_{j}\langle\tau_{j}\rangle,$ (49) $\displaystyle=$
$\displaystyle\prod_{j}\frac{\sum_{\\{\tau\\}}\tau_{j}e^{\beta
K\sum_{T}\tau_{T}}}{\sum_{\\{\tau\\}}e^{\beta K\sum_{T}\tau_{T}}},$ (51)
$\displaystyle=$ $\displaystyle\prod_{j}\bigg{(}\frac{e^{\beta K}-e^{-\beta
K}}{e^{\beta K}+e^{-\beta K}}\bigg{)},$ (53) $\displaystyle=$
$\displaystyle\tanh(\beta K)^{\\#\text{ of }\tau\text{'s}}.$ (55)

Taking products of spins over a Sierpinski tetrahedron of length $L=2^{\ell}$
for integer $\ell$ allows us to write the four-point correlation as

$\displaystyle\mathcal{C}_{4}$ $\displaystyle=$
$\displaystyle\bigg{\langle}\prod_{j\in\text{ST}(L)}\tau_{j}\bigg{\rangle},$
(56) $\displaystyle=$ $\displaystyle\tanh(\beta K)^{\\#\text{ of
tetrahedra}},$ (58) $\displaystyle=$ $\displaystyle\tanh(\beta K)^{L^{2}}.$
(60)

The four point correlations for $L\neq 2^{\ell}$ must vanish.

The approach we have taken in realizing the Sierpinski triangle model in Eq. 9
can be extended to this Sierpinski tetrahedron model with a four-body
interaction. Starting from the diamond lattice, we decorate each vertex and
center of every tetrahedron with target atoms and an auxiliary atom
respectively as shown in Fig. 8. We couple the auxiliary atom to two different
Rydberg states using lasers with different frequencies, and the two Rydberg
states are labelled by their principal quantum numbers $n^{\alpha}$ and
$n^{\beta}$ ($n^{\alpha}>n^{\beta}$ without loss of generality). Then we
propose the following Hamiltonian:

$\displaystyle
H_{0}\sim\sum_{a}V\bigg{(}4\hat{n}_{a}^{\alpha}+2\hat{n}_{a}^{\beta}+\sum_{v\in\text{tetra}}\hat{n}_{t,v}-4\bigg{)}^{2}+\dots$
(61)

where the ellipsis include all the longer-range couplings of auxiliary and
target atoms and the second sum is over all vertices of the tetrahedron with
$\hat{n}_{a}^{\alpha,\beta}$ at its center. Again, as we did for the
Sierpinski triangle, we may select principal quantum numbers $n^{\alpha}_{a}$,
$n^{\beta}_{a}$, and $n_{t,i}$ such that the ratio of Van der Waals
interaction strengths yields our desired Hamiltonian Eq. 61. In Eq. 61, the
expansion will formally include terms such as $(\hat{n}_{a}^{\alpha})^{2}$,
$(\hat{n}_{a}^{\beta})^{2}$ and $\hat{n}_{t,v}^{2}$, but these terms can be
absorbed by shifting the detuning as $\hat{n}^{2}=\hat{n}$ for $\hat{n}=0,1$.
The expansion of Eq. 61 also contains an interaction between
$\hat{n}_{a}^{\alpha}$ and $\hat{n}_{a}^{\beta}$ which seems unphysical.
However this terms does not change the energy of any physical states because
$\hat{n}^{\alpha}$ and $\hat{n}^{\beta}$ cannot be both nonzero at the same
site.

The ground states of Eq. 61 are found when each term in the sum vanishes. The
possible configurations of $\hat{n}$ on each tetrahedra when this occurs are

$\displaystyle(1)\ \ \hat{n}_{a}^{\alpha}=1,\ \hat{n}_{a}^{\beta}=0,\
\hat{n}_{t,i}=0;$ (62) $\displaystyle\mathrm{or}$ $\displaystyle(2)\ \
\hat{n}_{a}^{\alpha}=0,\ \hat{n}_{a}^{\beta}=1,\ \text{two of
}\hat{n}_{t,i}=0;$ (64) $\displaystyle\mathrm{or}$ $\displaystyle(3)\ \
\hat{n}_{a}^{\alpha}=0;\ \ \hat{n}_{a}^{\beta}=0;\ \ \hat{n}_{t,i}=1.$ (66)

These ground states of each tetrahedron of Eq. 61 can be one-to-one mapped to
the ground states of Eq. 47: (1)
$\sigma^{z}_{1}=\sigma^{z}_{2}=\sigma^{z}_{3}=\sigma^{z}_{4}=1$, or (2) two of
$\sigma^{z}_{i}=-1$ or (3)
$\sigma^{z}_{1}=\sigma^{z}_{2}=\sigma^{z}_{3}=\sigma^{z}_{4}=-1$, under the
correspondence $\sigma^{z}_{i}=1-2\hat{n}_{t,i}$. Therefore our platform
provides a natural extension to the modeling of classical fracton models and
multi-spin interactions beyond the Sierpinski triangle model.


A Beluga Formalization of the
Harmony Lemma in the π-Calculus
Gabriele Cecilia
Dipartimento di Matematica,

Università degli Studi di Milano, Italy
Alberto Momigliano
Dipartimento di Informatica,

Università degli Studi di Milano, Italy
Cecilia & Momigliano
Formalizing the Harmony Lemma in Beluga

The “Harmony Lemma”, as formulated by Sangiorgi & Walker,
establishes the equivalence between the labelled transition
semantics and the reduction semantics in the $\pi$-calculus. Despite
being a widely known and accepted result for the standard
$\pi$-calculus, this assertion has never been rigorously proven,
formally or informally. Hence, its validity may not be immediately
apparent when considering extensions of the
$\pi$-calculus. Contributing to the second challenge of the
Concurrent Calculi Formalization Benchmark — a set of challenges
tackling the main issues related to the mechanization of concurrent
systems — we present a

formalization of this result for the fragment of the $\pi$-calculus
examined in the Benchmark. Our formalization is implemented in
Beluga and draws inspiration from the HOAS formalization of the LTS
semantics popularized by Honsell et al. In passing, we
introduce a couple of useful encoding techniques for handling
telescopes and lexicographic induction.

[1]
authorAndreas Abel,
authorGuillaume Allais,
authorAliya Hameer,
authorBrigitte Pientka,
authorAlberto Momigliano,
authorSteven Schäfer &
authorKathrin Stark
(year2019): titlePOPLMark Reloaded:
Mechanizing Proofs by Logical Relations.
journalJ. Funct. Program.
volume29:eid19, 10.1017/S0956796819000170.

[2]
authorBrian E. Aydemir,
authorAaron Bohannon,
authorMatthew Fairbairn,
authorJ. Nathan Foster,
authorBenjamin C. Pierce,
authorPeter Sewell,
authorDimitrios Vytiniotis,
authorGeoffrey Washburn,
authorStephanie Weirich &
authorSteve Zdancewic
(year2005): titleMechanized Metatheory for the
Masses: The POPLMark Challenge.
In editorJoe Hurd &
editorTom Melham, editors: booktitleTheorem Proving in Higher Order Logics,
publisherSpringer, addressBerlin & Heidelberg, pp.
pages50–65, 10.1007/11541868_4.

[3]
authorJesper Bengtson &
authorJoachim Parrow
(year2009): titleFormalising the
\(\pi\)-Calculus using Nominal Logic.
journalLog. Methods Comput. Sci.
volume5, 10.2168/LMCS-5(2:16)2009.

[4]
authorMarco Carbone,
authorDavid Castro-Perez,
authorFrancisco Ferreira,
authorLorenzo Gheri,
authorFrederik Krogsdal Jacobsen,
authorAlberto Momigliano,
authorLuca Padovani,
authorAlceste Scalas,
authorDawit Tirore,
authorMartin Vassor,
authorNobuko Yoshida &
authorDaniel Zackon
(year2024): titleThe Concurrent Calculi
Formalisation Benchmark.
In editorIlaria Castellani &
editorFrancesco Tiezzi, editors:
booktitleCoordination Models and Languages,
publisherSpringer Nature Switzerland, addressCham,
pp. pages149–158, 10.1007/978-3-031-62697-5_9.

[5]
authorGabriele Cecilia
(year2024): titleFormalizing the Operational
Semantics of the $\pi$-Calculus.
Master's thesis, schoolUniversità degli Studi di Milano.

[6]
authorJoëlle Despeyroux
(year2000): titleA Higher-Order Specification of
the \(\pi\)-Calculus.
In editorJan van Leeuwen,
editorOsamu Watanabe,
editorMasami Hagiya,
editorPeter D. Mosses &
editorTakayasu Ito, editors: booktitleTheoretical Computer Science, Exploring New Frontiers of
Theoretical Informatics, International Conference IFIP TCS 2000, Sendai,
Japan, August 17-19, 2000, Proceedings, seriesLecture
Notes in Computer Science volume1872,
publisherSpringer, pp. pages425–439,

[7]
authorAmy P. Felty,
authorAlberto Momigliano &
authorBrigitte Pientka
(year2018): titleBenchmarks for Reasoning with
Syntax Trees Containing Binders and Contexts of Assumptions.
journalMath. Struct. Comput. Sci.
volume28(number9), pp. pages1507–1540,

[8]
authorDaniel Hirschkoff
(year1997): titleA Full Formalisation of
\(\pi\)-Calculus Theory in the Calculus of Constructions.
In editorElsa L. Gunter &
editorAmy P. Felty, editors: booktitleTheorem Proving in Higher Order Logics, 10th
International Conference, TPHOLs'97, Murray Hill, NJ, USA, August 19-22,
1997, Proceedings, seriesLecture Notes in Computer
Science volume1275, publisherSpringer, pp.
pages153–169, 10.1007/BFB0028392.

[9]
authorFurio Honsell,
authorMarina Lenisa,
authorUgo Montanari &
authorMarco Pistore
(year1998): titleFinal Semantics for the
\(\pi\)-Calculus.
In editorDavid Gries &
editorWillem P. de Roever, editors:
booktitleProgramming Concepts and Methods, IFIP
TC2/WG2.2,2.3 International Conference on Programming Concepts and Methods
(PROCOMET '98) 8-12 June 1998, Shelter Island, New York, USA, seriesIFIP Conference Proceedings volume125,
publisherChapman & Hall, pp. pages225–243,

[10]
authorFurio Honsell,
authorMarino Miculan &
authorIvan Scagnetto
(year2001): title\(\pi\)-Calculus in
(Co)Inductive Type Theory.
journalTheor. Comput. Sci.
volume253(number2), pp. pages239–285,

[11]
authorT. F. Melham (year1994):
titleA Mechanized Theory of the $\pi$-Calculus in HOL.
journalNordic J. of Computing
volume1(number1), p. pages50–76,

[12]
authorDale Miller (year1994):
titleSpecification of the \(\pi\)-Calculus.

[13]
authorDale Miller &
authorAlwen Tiu (year2005):
titleA Proof Theory for Generic Judgments.
journalACM Trans. Comput. Log.
volume6(number4), pp. pages749–783,

[14]
authorAlberto Momigliano,
authorBrigitte Pientka &
authorDavid Thibodeau
(year2019): titleA Case-Study in Programming
Coinductive Proofs: Howe's Method.
journalMath. Struct. Comput. Sci.
volume29(number8), pp. pages1309–1343,

[15]
authorJoachim Parrow
(year2001): titleAn Introduction to the
\(\pi\)-Calculus.
In editorJan A. Bergstra,
editorAlban Ponse &
editorScott A. Smolka, editors:
booktitleHandbook of Process Algebra,
publisherNorth-Holland / Elsevier, pp. pages479–543,

[16]
authorFrank Pfenning
(year2000): titleStructural Cut Elimination: I.
Intuitionistic and Classical Logic.
journalInf. Comput.
volume157(number1-2), pp. pages84–141,

[17]
authorBrigitte Pientka &
authorAndrew Cave
(year2015): titleInductive Beluga: Programming
In editorAmy P. Felty &
editorAart Middeldorp, editors:
booktitleAutomated Deduction - CADE-25 - 25th
International Conference on Automated Deduction, Berlin, Germany, August 1-7,
2015, Proceedings, seriesLecture Notes in Computer
Science volume9195, publisherSpringer, pp.
pages272–281, 10.1007/978-3-319-21401-6_18.

[18]
authorBrigitte Pientka &
authorJana Dunfield
(year2010): titleBeluga: A Framework for
Programming and Reasoning with Deductive Systems (System Description).
In editorJürgen Giesl &
editorReiner Hähnle, editors:
booktitleAutomated Reasoning, 5th International Joint
Conference, IJCAR 2010, Edinburgh, UK, July 16-19, 2010. Proceedings,
seriesLecture Notes in Computer Science
volume6173, publisherSpringer, pp.
pages15–21, 10.1007/978-3-642-14203-1_2.

[19]
authorDavide Sangiorgi &
authorDavid Walker
(year2001): titleThe \(\pi\)-Calculus - a
Theory of Mobile Processes.
publisherCambridge University Press,

[20]
authorChuta Sano, authorRyan
Kavanagh & authorBrigitte Pientka (year2023): titleMechanizing
Session-Types Using a Structural View: Enforcing Linearity without
journalProc. ACM Program. Lang.
volume7(numberOOPSLA), pp.
pages235:374–235:399, 10.1145/3622810.

[21]
authorAlwen Tiu & authorDale
Miller (year2010):
titleProof Search Specifications of Bisimulation and Modal
Logics for the \(\pi\)-Calculus.
journalACM Trans. Comput. Log.
volume11(number2), pp. pages13:1–13:35,