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arXiv:2606.01116

Ground-state phase diagram of Rydberg atoms in a triangular-prism array

Qing-Yuan Zuo, Shuo Geng, Shan-Wen Tsai, Jin Zhang

cond-mat.quant-gasphysics.atom-phquant-phSubmitted May 31, 2026• Announced Jun 2, 2026• 0 citations
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Abstract

We study the ground-state phase diagram of Rydberg atoms in a triangular-prism optical tweezer array using the density matrix renormalization group. By tuning the detuning-to-Rabi-frequency ratio and the Rydberg blockade radius, the system realizes several density-wave phases with spontaneous breaking of translational and leg-exchange symmetries. Unlike two-leg Rydberg ladders with $\mathbb{Z}_2$ leg-exchange symmetry, the triangular prism has $\mathbb{D}_3$ symmetry, leading to a richer set of ordered phases and transitions. For blockade radius moderately larger than the lattice spacing, a phase with alternating double and single Rydberg occupancy appears at large detuning. It breaks $\mathbb{Z}_2$ translational and $\mathbb{Z}_3$ rotational symmetry while preserving a rung reflection symmetry. Upon decreasing detuning, it melts through two Berezinskii-Kosterlitz-Thouless transitions with an intermediate critical phase described by a $\mathbb{Z}_6$ clock model. At larger blockade radius, a phase with one Rydberg excitation per triangle and broken $\mathbb{D}_3$ symmetry appears through a first-order transition. When double occupation of neighboring triangles is suppressed, rung-trimerized density waves develop as detuning increases from the disordered phase. Their melting follows the same structure as in Rydberg chains and two-leg ladders: the $\mathbb{Z}_2$ case has Ising critical lines, while the $\mathbb{Z}_3$ and $\mathbb{Z}_4$ cases have chiral critical lines, with Potts and Ashkin-Teller points only on the corresponding commensurate lines. Inside the $\mathbb{Z}_2$ rung-trimerized phase, an entanglement-entropy peak signals a crossover regime with enhanced period-2 density modulation before a first-order transition into a $\mathbb{Z}_2\times\mathbb{D}_3$ phase. Floating phases with incommensurate quasi-long-range order appear between trimerized states of different periods.

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We study the ground-state phase diagram of Rydberg atoms in a triangular-prism optical tweezer array using the density matrix renormalization group. By tuning the detuning-to-Rabi-frequency ratio and the Rydberg blockade radius, the system realizes several density-wave phases with spontaneous breaking of translational and leg-exchange symmetries. Unlike two-leg Rydberg ladders with Z2\mathbb{Z}_2Z2​ leg-exchange symmetry, the triangular prism has D3\mathbb{D}_3D3​ symmetry, leading to a richer set of ordered phases and transitions. For blockade radius moderately larger than the lattice spacing, a phase with alternating double and single Rydberg occupancy appears at large detuning. It breaks Z2\mathbb{Z}_2Z2​ translational and Z3\mathbb{Z}_3Z3​ rotational symmetry while preserving a rung reflection symmetry. Upon decreasing detuning, it melts through two Berezinskii-Kosterlitz-Thouless transitions with an intermediate critical phase described by a Z6\mathbb{Z}_6Z6​ clock model. At larger blockade radius, a phase with one Rydberg excitation per triangle and broken D3\mathbb{D}_3D3​ symmetry appears through a first-order transition. When double occupation of neighboring triangles is suppressed, rung-trimerized density waves develop as detuning increases from the disordered phase. Their melting follows the same structure as in Rydberg chains and two-leg ladders: the Z2\mathbb{Z}_2Z2​ case has Ising critical lines, while the Z3\mathbb{Z}_3Z3​ and Z4\mathbb{Z}_4Z4​ cases have chiral critical lines, with Potts and Ashkin-Teller points only on the corresponding commensurate lines. Inside the Z2\mathbb{Z}_2Z2​ rung-trimerized phase, an entanglement-entropy peak signals a crossover regime with enhanced period-2 density modulation before a first-order transition into a Z2×D3\mathbb{Z}_2\times\mathbb{D}_3Z2​×D3​ phase. Floating phases with incommensurate quasi-long-range order appear between trimerized states of different periods.
Using the density matrix renormalization group algorithm, we map the ground-state phase diagram of a two-leg Rydberg ladder array with lattice spacings ax=2ay{a}_{x}=2{a}_{y}ax​=2ay​. We identify various density wave phases that spontaneously break the translational symmetry or the top-bottom reflection symmetry within the ladder. By increasing the laser detuning from zero, where the system is in a disordered phase that preserves all symmetries, we observe density wave orders with spontaneous breaking of the translational Zp{\mathbb{Z}}_{p}Zp​ symmetries at intermediate detuning values, while the reflection symmetry is preserved. These orders exhibit nonzero bond orders with positive expectation values on every pthp\mathrm{th}pth rung, thus labeled as Zp+{\mathbb{Z}}_{p}^{+}Zp+​ phases. At larger detuning values, another spontaneous breaking of the reflection symmetry, which disrupted the bond orders on the rungs, occurs via an Ising phase transition. In these phases, either the top or the bottom site is occupied in a staggered way on every pthp\mathrm{th}pth rung, breaking the translational Z2p{\mathbb{Z}}_{2p}Z2p​ symmetry, thus labeled by Z2p{\mathbb{Z}}_{2p}Z2p​ phases. We locate and characterize the 3-state Potts point and Ashkin-Teller point along the commensurate lines, as well as the direct chiral phase transitions between the disordered phase and the Zp+{\mathbb{Z}}_{p}^{+}Zp+​ (p=3,4)(p=3,4)(p=3,4) phases. Critical exponents $\ensuremath{\nu}$ and zzz are calculated for both conformal and chiral phase transition points. We finally identify two types of floating phases in the phase diagram: one characterized by a quasi-long-range incommensurate bond-order wave, and the other by a quasi-long-range incommensurate wave of density differences in the rungs. Our work motivates further applications of Rydberg atom arrays in quantum simulation.
Dual-species Rydberg atom arrays extend single-species platforms by introducing competing interaction scales and enhanced quantum fluctuations, enabling phenomena beyond homogeneous settings. In this work, we study the ground-state phase diagram of a one-dimensional dual-species Rydberg atom ladder using large-scale density-matrix renormalization group calculations. We identify disordered phases, multiple ordered phases with Z2\mathbb{Z}_2Z2​, Z3\mathbb{Z}_3Z3​, and Z4\mathbb{Z}_4Z4​ symmetry, as well as floating phases characterized by incommensurate wave vectors and algebraically decaying correlations. Importantly, we observe a smooth crossover between distinct Z2\mathbb{Z}_2Z2​-ordered regimes, reflecting a reorganization of low-energy degrees of freedom rather than a true phase transition, which is absent in single-species Rydberg arrays. We further uncover a multi-critical point at the boundary between the Z2⊗Z2\mathbb{Z}_2 \otimes \mathbb{Z}_2Z2​⊗Z2​ and Z3⊗Z3\mathbb{Z}_3 \otimes \mathbb{Z}_3Z3​⊗Z3​ ordered phases, where Ising, chiral, and first-order transition lines intersect. Our results demonstrate that dual-species Rydberg atom arrays provide a unique platform for realizing crossover physics and multi-critical behavior inaccessible in single-species architectures.
Arrays of Rydberg atoms in the blockade regime realize a wealth of strongly correlated quantum physics, but theoretical analysis beyond the chain is rather difficult. Here we study a tractable model of Rydberg-blockade atoms on the square ladder with a ${\mathbb{Z}}_{2}\phantom{\rule{0.16em}{0ex}}\ifmmode\times\else\texttimes\fi{}\phantom{\rule{0.16em}{0ex}}{\mathbb{Z}}_{2}$ symmetry and at most one excited atom per square. We find D4{D}_{4}D4​, Z2{\mathbb{Z}}_{2}Z2​, and Z3{\mathbb{Z}}_{3}Z3​ density-wave phases separated by critical and first-order quantum phase transitions. A noninvertible remnant of U(1)U(1)U(1) symmetry applies to our full three-parameter space of couplings, and its presence results in a larger critical region as well as two distinct Z3{\mathbb{Z}}_{3}Z3​-broken phases. Along an integrable line of couplings, the model exhibits a self-duality that is spontaneously broken along a first-order transition. Aided by numerical results, perturbation theory, and conformal field theory, we also find critical Ising2{\mathrm{Ising}}^{2}Ising2 and three-state Potts transitions, and provide good evidence that the latter can be chiral.
Experiments on chains of Rydberg atoms appear as a playground to study quantum phase transitions in 1D. As a natural extension, we report a quantitative ground-state phase diagram of Rydberg atoms arranged in a two-leg ladder interacting via van der Waals potential. We address this problem numerically, using the density matrix renormalization group algorithm. Our results suggest that, quite remarkably, Z k crystalline phases, with the exception of the checkerboard phase, appear in pairs characterized by the same pattern of occupied rungs but distinguishable by a spontaneously broken Z ̃ 2 symmetry between the two legs of the ladder. Within each pair, the two phases are separated by a continuous transition in the Ising universality class, which eventually fuses with the Z k transition, whose nature depends on k . According to our results, the transition into the Z 2 ⊗ Z ̃ 2 phase changes its nature multiple times, including an Ashkin-Teller transition that is surprisingly stable over an extended interval, followed by the Z 4 -chiral transition, and finally in a two step-process mediated melting via the floating phase. The transition into the Z 3 phase with resonant states on the rungs belongs to the three-state Potts universality class at the commensurate point, to the Z 3 -chiral Huse-Fisher universality class away from it, and eventually it is through an intermediate floating phase. The Ising transition between Z 3 and Z 3 ⊗ Z ̃ 2 phases, entering the floating phase, opens the possibility to realize lattice supersymmetry in Rydberg quantum simulators.
The neutral atoms coupled to a highly excited Rydberg state on a two-dimensional triangular lattice are investigated by employing the density matrix renormalization group technique in the matrix product state form. The full ground-state phase diagram as a function of blockade radius and the detuning of the exciting laser is determined by the behavior of entanglement entropy. We find several quantum phases including stripe-ordered and symmetry-breaking density-wave-ordered phases featured with regular excitation patterns of different excitation densities ρ = 1/3, 1/4, and 1/7. In addition, a ρ = 2/3 ordered phase and an interesting "order-by-disorder" phase, which has been prepared experimentally, are also observed in this work. Our work provides an exploration of the possible quantum phases that can occur in a triangularly arrayed Rydberg system, and thus could be a faithful theoretical guide for further experimental research.
We consider cold Rydberg atoms in a one-dimensional optical lattice in the Mott regime with a single atom per site at zero temperature. An external laser drive with Rabi frequency $\ensuremath{\Omega}$ and laser detuning $\ensuremath{\Delta}$ creates Rydberg excitations whose dynamics is governed by an effective spin-chain model with (quasi) long-range interactions. This system possesses intrinsically a large degree of frustration resulting in a ground-state phase diagram in the $(\ensuremath{\Delta},\ensuremath{\Omega})$ plane with a rich topology. As a function of $\ensuremath{\Delta}$, the Rydberg blockade effect gives rise to a series of crystalline phases commensurate with the optical lattice that form a so-called devil's staircase. The Rabi frequency $\ensuremath{\Omega}$, on the other hand, creates quantum fluctuations that eventually lead to a quantum melting of the crystalline states. Upon increasing $\ensuremath{\Omega}$, we find that generically a commensurate-incommensurate transition to a floating Rydberg crystal that supports gapless phonon excitations occurs first. For even larger $\ensuremath{\Omega}$, dislocations within the floating Rydberg crystal start to proliferate and a second, Kosterlitz-Thouless-Nelson-Halperin-Young dislocation-mediated melting transition finally destroys the crystalline arrangement of Rydberg excitations. This latter melting transition is generic for one-dimensional Rydberg crystals and persists even in the absence of an optical lattice. The floating phase and the concomitant transitions can, in principle, be detected by Bragg scattering of light.