# Homogeneous magnetic flux in Rydberg lattices

J Eix, R Bai, Thierry Lahaye, A Browaeys, H P Büchler, S Weber

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J Eix, R Bai, Thierry Lahaye, A Browaeys, H P Büchler, et al.. Homogeneous magnetic flux in Rydberg lattices. Physical Review Research, 2025, 7, pp.013062. 10.1103/physrevresearch.7.013062. hal-05102333

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Distributed under a Creative Commons Attribution 4.0 International License
Homogeneous magnetic flux in Rydberg lattices

J. Eix ◦ , 1,2,*,† R. Bai ◦ , 1,*,‡ T. Lahaye ◦ , 3 A. Browaeys ◦ , 3 H. P. Büchler ◦ , 1 and S. Weber ◦ 1,§

¹Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany

²Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

³Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau Cedex, France

<figure><img src="image_2.png" /></figure>

(Received 31 July 2024; accepted 10 December 2024; published 16 January 2025)

We present a method for generating homogeneous and tunable magnetic flux for bosonic particles in a lattice using Rydberg atoms. Our setup relies on Rydberg excitations hopping through the lattice by dipolar exchange interactions. The magnetic flux arises from complex hopping via ancilla atoms. Remarkably, the total flux within a magnetic unit cell directly depends on the ratio of the number of lattice sites to ancilla atoms, making it topologically protected to small changes in the positions of the atoms. This allows us to optimize the positions of the ancilla atoms to make the flux through the magnetic unit cell homogeneous. With this homogeneous flux, we get a topological band in the single-particle regime. In the many-body regime, we obtain indications of a bosonic fractional Chern insulator state at $v = 1/2$ filling.

DOI: 10.1103/PhysRevResearch.7.013062

# I. INTRODUCTION

Charged particles in the presence of magnetic fields exhibit many fascinating phenomena [1–7], including integer [8] and fractional [9] quantum Hall states, as well as topological fractional Chern-insulating states [10]. The observation of these many-body states requires strong interactions between the charged particles and the high magnetic fields. In this context, quantum simulators composed of ultracold atoms trapped in optical lattices have emerged as a pathway to achieve these conditions, owing to their many tunable experimental parameters [11–16]. However, ultracold atoms are charge neutral and do not experience the Lorentz force when exposed to external magnetic fields. Despite this obstacle, magnetic fields for neutral atoms can be implemented artificially [17–23]. There are several ways to generate effective magnetic fields, including rotating the optical lattice [24], using laser-assisted tunneling [17,22,23], implementing spatially inhomogeneous light fields [19–21], periodically perturbing the Hamiltonian [25,26], and lattice shaking [18,27]. In this work, we propose an alternative method to generate homogeneous magnetic flux in rectangular lattices of charge-neutral particles using only lattice geometry and Rydberg atoms.

Rydberg atoms trapped in optical tweezer arrays provide a convenient platform to study many-body physics for

charge-neutral particles in effective magnetic fields due to the experimental control of their long-range dipole-dipole interactions [28], the tunable geometry of the arranged atoms, and dipolar exchange interactions [29,30]. Effective magnetic fields arise from complex hopping, known as the Peierls phase [31], which can be realized solely from dipolar exchange interactions in Rydberg atoms without requiring inhomogeneous light fields or lattice shaking [32]. Rydberg atom arrays have been identified as a tool to explore different quantum states of matter such as symmetry-protected topological phases [33–35], fractional Chern-insulating states [36], and quantum spin liquid states [37]. However, it remains an open challenge to find setups where the effective magnetic field corresponds to a homogeneous magnetic flux.

Here, we provide a way to generate robust, homogeneous, and tunable magnetic flux in rectangular lattices of Rydberg atoms. Our method makes use of ancilla atoms in a background lattice which induce an effective magnetic flux arising from dipolar exchange interactions. Strikingly, the total flux per magnetic unit cell is determined by the ratio of ancilla atoms to Rydberg atoms. This ratio is a bulk characteristic of the system and is robust to perturbation; hence, the total flux through the magnetic unit cell is a topological property, but the flux can be tuned locally by varying the lattice geometry. Using this method, topological bands can be generated with arbitrary flux values. After establishing the robustness of the magnetic flux, we find indications for a bosonic fractional Chern insulator state at $v = 1/2$ filling in the many-body regime, in agreement with previous studies that theoretically predicted such a state in a bosonic Harper Hofstadter model [38–42].

This article is organized as follows. In Sec. II, we discuss our setup along with the Rydberg level scheme. In Sec. III, we examine obtainable magnetic flux patterns and show that the total flux through the magnetic unit cell is determined by the

<footnote>*These authors contributed equally to this work.</footnote>

<footnote>†Contact author: jeix@iastate.edu</footnote>

<footnote>‡Contact author: rukmani.bai@itp3.uni-stuttgart.de</footnote>

<footnote>§Contact author: sebastian.weber@itp3.uni-stuttgart.de</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 1. Setup with rectangular lattice A (blue sites) hosting $|−〉$ excitations which encode the hopping model. The background lattice B (red sites) of ancilla atoms hosts $|+〉$ excitations and generates complex hopping in lattice A. Level structures of lattice A and B sites are shown. Additionally depicted is an exemplary second-order process, which contributes a complex phase based on the angle $\phi_{isj}$ to the hopping $t_{ij}^−$, as described in Eq. (2). The magnetic unit cell is shaded in gray.</figcaption></figure>

ratio of sites between the background and rectangular lattices. Furthermore, we demonstrate how the magnetic flux can be made homogeneous by optimizing the lattice. In Sec. IV, we discuss the topological bands in the single-particle picture. Finally, in Sec. V, we investigate the many-body regime and obtain a topological state at $v = 1/2$ filling using the robust flux generated from our method.

## II. SETUP

We begin with the microscopic description of our setup. We focus on Rydberg atoms trapped individually by optical tweezers, where each atom can occupy one of two Rydberg states which are energetically isolated from all other Rydberg states. The atoms are arranged to form a rectangular lattice A (blue sites) and background lattice B of ancilla atoms (red sites), as depicted in Fig. 1.

We start by describing lattice A in detail. Lattice A has horizontal spacing $b$ and vertical spacing $a$. Rydberg atoms in lattice A have quantum number $n$ and an internal level structure consisting of two different Rydberg states, namely, a Rydberg $S$ state $|0⟩$ and a Rydberg $P$ state $|−⟩$, as shown in Fig. 1 [35]. The states can be isolated energetically from other states using, for example, a combination of electric and magnetic fields [32]. The state $|0⟩$ can be interpreted as the vacuum state, and the state $|−⟩$ can be interpreted as a bosonic excitation. Therefore, the model is described by hard-core bosons with bosonic operators satisfying $b_−^†|0⟩ = |−⟩$ [33]. Within the lattice, dipolar exchange interactions give rise to the hopping of $|−⟩$ excitations, which can in general be described by the Hamiltonian

$$ \mathcal{H} = \sum_{ij} t_{ij}^− b_{−,j}^† b_{−,i}, \quad (1) $$

where $t_{ij}^−$ is the hopping element between sites $i$ and $j$ of lattice A.

We use the ancilla lattice B to generate effective magnetic fields. We do so by allowing off-resonant hopping processes that occupy the ancilla atoms virtually, making the hopping elements $t_{ij}^−$ complex. To achieve these hopping processes, lattice B hosts Rydberg atoms with quantum number $n'$ and

an internal level structure consisting of a Rydberg $S$ state $|0⟩$ and a Rydberg $P$ state $|+⟩$, as shown in Fig. 1. Having two different quantum numbers, $n$ and $n'$, for the atoms forming the rectangular lattice A and the background lattice B allows us to choose level schemes where virtual hopping processes among atoms within the same lattice are negligible. Note that we specifically chose the $|−⟩$ and $|+⟩$ states to differ in the magnetic quantum number by 2. All lattice B sites are initialized in the ground state $|0⟩$. To ensure that the lattice B sites are only occupied virtually, we apply the energy shift $\mu = [E(|+⟩) - E(|0⟩') - E(|−⟩) + E(|0⟩)]$. This shift allows us to treat the hopping from a site $i$ to a site $j$ through an ancilla atom in second-order perturbation theory. We have illustrated such a second-order process in Fig. 1, where a $|−⟩$ excitation at site $i$ hops to site $j$ by first virtually occupying the ancilla atom $s$ with a $|+⟩$ excitation. The spin-flip hopping processes couple the $|+⟩$ state to the $|−⟩$ state, changing the internal angular momentum by 2, which yields a complex phase due to the conservation of angular momentum [32].

Taking into account these processes, the hopping element $t_{ij}^−$ becomes

$$ t_{ij}^− = |t_{ij}^−| e^{i\phi_{isj}} = -\frac{t^−}{|\mathbf{R}_{ij}|^3} - \sum_s \frac{w^2}{\mu |\mathbf{R}_{is}|^3 |\mathbf{R}_{sj}|^3} e^{2i\phi_{isj}}, \quad (2) $$

where the first term is the direct hopping from $i$ to $j$ with the hopping strength $t^−$, and the second term arises from second-order hopping processes. The phase factor of a second-order process is determined by the in-plane azimuthal angle $\phi_{isj}$ shown in Fig. 1. The sign of $\phi_{isj}$ is determined by $\operatorname{sgn} \mathbf{R}_{ij} \times \mathbf{R}_{is}$, where $\mathbf{R}_{ij}$ is the vector from $i$ to $j$. The amplitude of a second-order process is determined by the strength $w$ of the hopping between lattices A and B and the energy difference $\mu$ [33,35]. Because the value of $t^−/w$ is highly implementation dependent and the results of our paper stay qualitatively the same for different values of $t^−$, we consider the limit $w \gg t^−$ and set $t^− = 0$ for the rest of this work.

The ancilla lattice B is chosen such that it has the same periodicity in the $y$ direction as lattice A. The periodicity of lattice B in the $x$ direction is commensurable with the periodicity of lattice A. This allows us to specify a unit cell of the combined lattice with $p$ lattice B sites and $q$ lattice A sites, being invariant under translations by $qb\hat{x}$ and $a\hat{y}$. For an example of a unit cell with $q = 4$ and $p = 7$, see the gray shaded region in Fig. 1. This unit cell coincides with the magnetic unit cell of the lattice A. We align lattice B so that, in the $y$ direction, sites of lattice A are centered between sites of lattice B. This guarantees real hopping elements along $x$ since pairs of lattice B sites contribute to the second-order hopping with conjugate phases [43]. The integer-valued number of flux quanta through the magnetic unit cell is fully determined by the complex hoppings along $y$. Remarkably, as we show in the next section, the number of flux quanta is directly related to the numbers $q$ and $p$ of lattice sites in the magnetic unit cell.

## III. GENERATING HOMOGENEOUS MAGNETIC FIELDS

In this section, we show that the effective magnetic flux can be tuned to be homogeneous, and arbitrary rational numbers of flux quanta can be realized in the Rydberg lattice A. As
<figure><img src="image_4.png" /><figcaption>FIG. 2. (a) Two lattice A sites immersed in lattice B, which is held to be a rectangular lattice. (b) Upon smoothly translating the two sites by the horizontal lattice constant *c*, the phase angle *φ* of the complex hopping *t*(*x*) collects the phase 2*π*n, where *n* is the winding number. The winding number depends on the ratio *c*/*a*. The winding numbers are 0, 1, and 2 for the red, purple, and green curves, respectively.</figcaption></figure>

explained in the previous Sec. II, only the hopping along the
y direction is complex valued. To characterize this hopping,
we consider two sites of lattice A within an infinite lattice B,
as shown in Fig. 2(a). For our argument and without loss of
generality, lattice B is taken to be rectangular with horizontal
lattice spacing *c* and vertical spacing *a*. The two lattice A sites
are a pair of vertical nearest-neighbors separated by *a*.

While our discussion holds for arbitrary ratios of c/a, we
consider c = 4/7a for illustration, being consistent with the
setup depicted in Fig. 1 that realizes this value for q = 4, p =
7, and a = b. In the following, we move the pair of lattice
A sites along x while keeping track of the complex hopping
between them. The hopping is thus a function of horizontal
distance x and is denoted as t(x). The phase of t(x) is denoted
as φ(x).

The phase $\phi(x)$ is plotted in Fig. 2(b) as the pair of lattice A sites are translated along $x$ from 0 to $c$. This translation reveals a few general and key properties of $\phi(x)$. First, due to translational symmetry, $t(x)$ is invariant when $x$ changes by an integer multiple of $c$. Accordingly, $\phi(x)$ is invariant with $\phi(x+c) = \phi(x) \mod 2\pi$. Second, we observe that $\phi(mc+x) = -\phi(mc-x) \forall m \in \frac{1}{2}\mathbb{Z}$ because the corresponding geometrical arrangements are related by a reflection. The result here can be understood from the fact that $t_{ij} = t_{ji}^*$ from Eq. (2) and that a reflection effectively reverses the hopping direction. Third, the phase $\phi(x)$ is a continuous function in the interval $[0, c]$ as long as $t(x) \neq 0$. Importantly, these properties imply that $\phi(x)$ can only change by $2\pi n$ when $x$ increases from 0 to $c$. We call the integer value $n$ the winding number.

It turns out that the winding number is dependent on the
ratio c/a. For example, for c/a = 4/7, we observe that φ(x)
varies from π to -π as x varies from 0 to c, giving a winding
number of 1. In contrast, for c/a ≲ 0.41, the winding number
is found to be 0. For c/a ≳ 2.00, the winding number is 2.
In between, 0.41 ≲ c/a ≲ 2.00, the winding number is 1 [see
Fig. 2(b)]. The values c/a ≈ 0.41 and c/a ≈ 2.00 are critical
values which separate different regimes. These critical values

correspond to special cases in which the hopping *t*(*x*) vanishes
at the reflection symmetric point *x* = *c*/2; thus, the phase
*φ*(*c*/2) and the winding number are no longer defined.

This behavior of the phase φ(x) has implications for the
flux through a magnetic unit cell. Let us consider that the
lattice A has a rational horizontal spacing of b = l/q c. Then,
the magnetic unit cell will contain p lattice B sites and q lattice
A sites. Our analysis of the winding number showed that the
phase φ(x) of the hopping along y increments by 2πn with
n ∈ {0, 1, 2} between each lattice B site. Accordingly, the total
flux through the magnetic unit cell must be 2π pn because we
have p lattice B sites. The average flux through each lattice A
plaquette is thus 2π pn/q.

Notably, our argument did not rely on specific positions of
the lattice sites. Thus, rearranging the positions of the sites in
the magnetic unit cell does not change the total flux through it
as long as *t*(*x*) does not vanish. This allows us to optimize the
homogeneity of the flux and hopping strengths by adapting
the positions of the lattice sites.

While tuning the positions of each site is a high-
dimensional optimization problem that is challenging to solve
exactly, we use a heuristic that provides reasonably homoge-
neous magnetic fields and hoppings. Our heuristic takes place
in several stages. At the beginning, lattice A is initialized
with *a* = *b*, and the background lattice B is initialized with
vertical spacing *a* and horizontal lattice spacing *c* = (q/p)*a*
(we refer to this initial arrangement as the nonoptimized ar-
rangement). This fixes the average flux to 2π*p*/q, provided
that 0.41 ≲ *q*/*p* ≲ 2.00. As a first optimization step, the flux
is optimized for homogeneity by rearranging lattice B sites
within the magnetic unit cell along *x*. During this process,
we keep the magnetic unit cell symmetric under reflection.
As a second optimization step, we optimize for homogeneous
hopping strengths. To do so, we tune the ratio between *a* and
*b* and, in addition, allow lattice B sites to move out of the
plane in the *z* direction. The latter helps to tune the hopping
strengths not only because of the resulting distance changes
but also because of the anisotropy of the dipole-dipole inter-
action. Again, we only allow for rearrangements that keep
the reflection symmetry. Optimizing for homogeneous flux
and optimizing for hopping strength are not independent pro-
cesses; however, iterating the processes converges in that flux
variance and hopping strength variance may both be mini-
mized, as shown in Fig. 3. The optimized arrangement of sites
shows distances between lattice B sites ranging from 0.55 *b*
to 0.59 *b*, movements of lattice B sites along *z* of up to 0.2 *b*,
and *a*/*b* = 1.016, making the lattice A slightly rectangular. As
we see in the next section, having a homogeneous flux and
nearly homogeneous hoppings allows for obtaining gapped
topological bands in our model.

IV. TOPOLOGICAL BAND STRUCTURE

In this section, we present the topological band structure of
lattice A for the nonoptimized and optimized arrangements of
sites, which have been discussed in the previous section.

In Fig. 4, we show the resulting band structures and density
of states (D.O.S.) in the single-particle picture. The consid-
ered unit cell with *p* = 7 and *q* = 4 lattice sites provides four
bands as lattice A has *q* = 4 sites in its magnetic unit cell. For
<figure><img src="image_5.png" /><figcaption>FIG. 3. Optimization for homogeneous flux and hopping amplitude. Normalized variance $C[\chi] = \frac{\mathbb{E}[\chi^2]-\mathbb{E}[\chi]^2}{\mathbb{E}[\chi]^2}$ (where $\mathbb{E}[\chi]$ is the average over the whole lattice) of nearest-neighbor hopping amplitude $|t_{(ij)}|$ and plaquette flux $\Phi_p$ vs the linear interpolation parameter $\lambda$. $\lambda = 0$ corresponds to the initialized lattice, while $\lambda = 1$ corresponds to the fully optimized lattice. The mean flux is always $\pi/4$ while the mean of the hopping amplitude goes from $33.10 \frac{w^2}{\mu b^6}$ to $36.59 \frac{w^2}{\mu b^6}$. The inset shows the variance for both the hopping and the flux trends to 0.</figcaption></figure>

all analyzed band structures, the lowest band is topological with a Chern number of 1.

Figures 4(a) and 4(b) show results for nearest-neighbor hoppings. Figure 4(a) depicts the results if we do not optimize the positions of lattice sites. In the case of the optimized arrangement of lattice sites, the lowest band is flattened and exhibits a flatness ratio of 5.29, which is determined by the minimum direct band gap divided by the bandwidth. The density of states shows the opening of a band gap as shown in Fig. 4(b).

Because the dipole-dipole interaction decreases with $1/R^3$, where $R$ is the interatomic distance, we have to consider long-range hoppings for a more realistic analysis. In this case, the flux is no longer well-defined, and the term homogeneous flux refers to the homogeneous flux that is picked up by the dominant nearest-neighbor hoppings. As the long-range hoppings are significantly weaker than the nearest-neighbor hoppings, the lowest band remains gapped and topological with a flatness ratio of 1.28 [see Fig. 4(c)].

In summary, Fig. 4 shows that we have obtained a topological band for a system with $p/q = 7/4$. However, we want to point out that topological bands can be achieved for arbitrary flux values by adjusting the ratio $p/q$ and optimizing the magnetic unit cell through our proposed method. As demonstrated, the flatness of the topological bands can be increased by optimizing the positions of the lattice sites to make the flux that is picked up by the dominant nearest-neighbor hoppings homogeneous.

## V. MANY-BODY PHYSICS

In the single-particle picture, we have obtained nearly homogeneous flux and topological bands in the Rydberg lattices. Now, we extend our analysis to the many-body regime. To be specific, we aim to study a fractional Chern insulator (FCI) state, using the flux generated in the lattice through our method. The properties of our model in the single-particle

regime combined with the strong interaction between particles provided by the hard-core constraint, make our approach a promising tool for realizing FCI states using Rydberg atoms.

We use the exact diagonalization (ED) method to obtain the many-body ground state of our Hamiltonian provided by Eq. (1). In our implementation of ED, the Hilbert space dimension is reduced due to particle number conservation. To obtain the FCI state, we consider finite lattices with $p/q = 7/4$ and a filling fraction of $\nu = 1/2$. This corresponds to a lattice with 1/4 flux per plaquette and one particle for every two fluxes. We consider lattice sizes from 4 × 4 to 8 × 6 with periodic boundary conditions and obtain the many-body ground state. Due to periodic boundary conditions, the topology of the system is a torus. The observed ground state is degenerate, which is the signature of a FCI $\nu = 1/2$ state. To further support this interpretation of the ground state, we calculate the many-body Chern number for the ground state. For this, we use twisted periodic boundary conditions along the $x$ and $y$ directions with the twist angles $\theta_x$ and $\theta_y$:

$$ 
\begin{aligned}
T_x(L_x)\psi(x, y) &= e^{i\theta_x} \psi(x, y), \\
T_y(L_y)\psi(x, y) &= e^{i\theta_y} \psi(x, y),
\end{aligned} 
\qquad (3) $$

where $T_i(L_i)$ are operators which translate single-particle wave functions $\psi$ around the $i$th direction of the torus. The twist angles $\theta_x$ and $\theta_y$ correspond to additional fluxes which are threaded through the hole of the torus or through its interior. Thus, the many-body Hamiltonian and its eigenstates $\Psi(\theta_x, \theta_y)$ become functions of the twist angles. Within this parameter space, the non-Abelian Berry connection and the many-body Chern number are defined as

$$ A_j^{\alpha, \beta}(\theta_x, \theta_y) = i \langle \Psi^\alpha | \partial_j | \Psi^\beta \rangle, \qquad (4) $$

$$ C = \frac{1}{2\pi} \int_{0}^{2\pi} \int_{0}^{2\pi} \mathrm{Tr} \{ \partial_x A_y^{\alpha, \beta} - \partial_y A_x^{\alpha, \beta} \} d\theta_x d\theta_y, \qquad (5) $$

where $\partial_j$ is the partial derivative with respect to $\theta_j$, and $\alpha$ and $\beta$ correspond to degenerate eigenstates. We calculate the many-body Chern number in addition to the ground-state degeneracy to verify topological equivalence to Laughlin $\nu = 1/2$ states.

The lowest six eigenenergies for nearest-neighbor hopping as a function of twist angles are shown in Figs. 5(a) and 5(b) for lattices of sizes 4 × 4 and 8 × 6, respectively. Here, the results are computed across an 11 × 11 discretization of the twist-angle space. We observe that the ground state is degenerate and it is gapped from the first excited state for all twist angles. The doubly degenerate ground-state manifold has a Chern number $C = 1$. This corresponds with the FCI $\nu = 1/2$ state. The nearest-neighbor model also maps to the known models of FQH states considered in Ref. [38]. Therefore, we have established that our approach exhibits the FCI state if we consider hopping up to nearest-neighbor. To make a conclusive statement about the effect of long-range hoppings, larger systems would be required, which are not accessible via exact diagonalization. However, already the reduced flatness in the single-particle band structure [see Fig 4(c)] indicates that the topological state might be destabilized. On the other hand, it might be that more sophisticated and numerically challenging optimization strategies than those applied by us
<figure><img src="image_6.png" /></figure>

FIG. 4. Band structures and densities of states for magnetic unit cells comprised of $q = 4$ lattice A sites with a background lattice B of $p = 7$ sites. (a) Nearest-neighbor hoppings on the unoptimized lattice. (b) Nearest-neighbor hoppings on the optimized lattice. The band structure is clearly flattened, and the density of states shows the opening of a band gap that is on the order of the mean of the nearest-neighbor hopping amplitudes. (c) Long-range hoppings on the optimized lattice. Although longer-range hoppings roughen the bands, the bands remain gapped.

can find geometries that result in flatter bands. Thus, it is an open question whether, in the thermodynamic limit, the topological state survives in the presence of long-range hoppings.

## VI. CONCLUSION

We have proposed a method to generate arbitrary homogeneous flux and hopping in rectangular lattices of Rydberg atoms using dipolar exchange interactions and a background lattice of ancilla atoms. Note that while we focused on Rydberg atoms, our proposal can be easily adapted to polar molecules, where similar level structures and interactions can be obtained. We showed that within a magnetic unit cell, the flux is proportional to the ratio between the number of sites of the background lattice, $p$, and the number of sites of the rectangular lattice, $q$. We demonstrated that the flux is topologically robust and unaffected by changes in the atomic positions within the magnetic unit cell as long as hopping amplitudes do not vanish. To make the flux homogeneous,

we made use of the freedom to change the atomic positions. With this homogeneous flux, we obtained topological bands in the single-particle regime with an increased flatness. Further, in the many-body regime, we obtained a bosonic fractional Chern insulator state at $v = 1/2$ filling with nearest-neighbor hopping. This many-body state has the same topological properties as those of the Laughlin state $v = 1/2$ with the many-body Chern number $C = 1$. To fully understand the effect of long-range hoppings, larger system sizes must be explored. These were inaccessible by the exact diagonalization method applied by us and require numerical methods such as density matrix renormalization group. Alternatively, to suppress long-range hoppings altogether, one can implement our model using Rydberg $S$ and $D$ states that are only dipole-coupled in second order, giving rise to exchange interactions that scale with $1/R^6$, where $R$ is the interatomic distance. Future work can be done to explore the entire parameter space of our model.

<figure><img src="image_7.png" /></figure>

FIG. 5. Results of the exact diagonalization for optimized $p = 7$ and $q = 4$ lattices with half filling. In the left and middle plots, the lowest six eigenenergies are plotted against $\theta_x$ and $\theta_y$ while the other twist angle is held constant at $\pi$. In the right plots, the gap between the second and third lowest eigenenergies is shown across the whole twist-angle space. (a) Lattice size of $4 \times 4$ with nearest-neighbor hopping. Here the degeneracy of the lowest two eigenenergies and their many-body Chern number $C = 1$ correspond with the Laughlin $v = 1/2$ state. (b) Lattice size of $8 \times 6$ with nearest-neighbor hopping. The lowest two eigenstates remain degenerate and gapped to excitation, and the calculated many-body Chern number for the lowest two eigenstates remains $C = 1$.
ACKNOWLEDGMENTS

We gratefully acknowledge financial support by the Baden-
Württemberg Stiftung via Grant No. BWST ISF2019-017
under the program Internationale Spitzenforschung. This
project has also received funding from the French-German
collaboration for joint projects in NLE Sciences funded by
the Deutsche Forschungsgemeinschaft (DFG) and the Agence

Nationale de la Recherche (ANR, project RYBOTIN). A.B. and T.L. acknowledge funding by the Agence Nationale de la Recherche (Grant No. ANR-22-PETQ-0004 France 2030, project QuBitAF), Horizon Europe programme HORIZON-CL4-2022-QUANTUM-02-SGA via Project No. 101113690 (PASQuanS2.1), and the European Research Council (Advanced Grant No. 101018511-ATARAXIA).

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