Geometry-driven transitions in sparse long-range spin models with cold atoms

TL;DR

This work shows that geometry alone can drive quantum phase transitions in sparse long-range spin models. By tuning a power-of-two coupling graph (PWR2) and examining both classical and quantum regimes, the authors reveal four geometry-driven phases and identify critical points tied to the graph’s structure. They demonstrate that the same critical behavior arises in a tambourine-shaped Rydberg setup, and confirm 2D Ising universality with exponents $\nu\approx 1$, $z\approx1$, and central charge $c\approx\tfrac12$, using finite-size scaling and entanglement analyses. The results establish a concrete link between graph geometry, experimental realizations in Rydberg tweezers, and universal criticality, offering a practical route to study geometry-controlled phase transitions in near-term quantum devices.

Abstract

We explore the influence of geometry in the critical behavior of sparse long-range spin models. We examine a model with interactions that can be continuously tuned to induce distinct changes in the metric, topology, and dimensionality of the coupling graph. This underlying geometry acts as the driver of criticality, with structural changes in the graph coinciding with and dictating the phase boundaries. We further discuss how this framework connects naturally to realizations in tweezer arrays with Rydberg excitations. In certain cases, the effective geometry can be incorporated in the layout of atoms in tweezers to realize phase transitions that preserve universal features, simplifying their implementation in near-term experiments.

Figures (10)

• Figure 1: (a) 1D long-range Ising model on a power-of-two (PWR2) coupling graph, with transverse field $B$ in $x$-direction as in Eq. (\ref{['eq:H']}). A Monna map $\mathcal{M}$ transforms our 1D euclidean chain (left) into a hierarchal $2$-adic tree (right). (b) A slider schematic illustrating how continuous tuning of $s$ interpolates between different coupling geometries. (c) Schematic of quantum phase diagram for Eq. (\ref{['eq:H']}) as a function of geometry parameter $s$ and transverse field strength $B$. Dotted lines correspond to finite system sizes; solid lines indicate extrapolated thermodynamic behavior (upper right box). Shaded regions denote phases characterized by area-law (yellow), volume-law (blue), and intermediate (red) entanglement entropy scaling. Phases (i) antiferromagnetic, (ii) paramagnetic, and (iii) geometric are labeled. (d) Rydberg atom array in a ring geometry. Odd sites (red) are vertically displaced, as indicated by the arrow, to mimic the tuning of the coupling parameter $s$ in (b) and yields the "tambourine" geometry.
• Figure 2: Analytical energy gap $\Delta E$ as a function of coupling parameter $s$ for classical limit of PWR2 model $B/J=0$ in Eq. (\ref{['eq:H']}). Four distinct phases (labeled (1)-(4)) are present in the infinite system, separated by critical points (solid lines). At finite sizes an additional gap opens around $s=0$, giving rise to an extra region (dashed lines). Semi-transparent overlays (blue) depict the effective geometry associated with each region, illustrating how changes in connectivity coincide with the phase boundaries.
• Figure 3: Finite-size scaling analysis of the energy gap $\Delta E$ and half-chain entanglement entropy $S_{vN}$ for the PWR2 and tambourine Rydberg model showing consistency with 2D Ising class. (a) Scaling collapse of the rescaled gap $N^z \Delta E$ for the PWR2 model at system sizes $\times=(64,128)$, optimized at $\nu_{\times} \approx 0.993$ and $z_{\times}\approx1.026$ at critical point $s_{c\times} \approx -3.3939$ for $B/J=0.8$. Inset: Thermodynamic limit $z_{\infty}=1.000(2)$ and $\nu_{\infty}=0.99(1)$ from the best fit convergence of $z_{\times}$ and $\nu_{\times}$ with $M_{\times}=\sqrt{2}N$. (b) Extraction of the central charge from the entanglement entropy [Eq. \ref{['eq:c']}] at $N=128$, with best linear fit yielding $c=0.5000(3)$ at the thermodynamic value of $s_c$. (c) Scaling collapse for the tambourine Rydberg model at system sizes $\times=(128,256)$, optimized at $\nu_{\times} \approx 0.98$ and $z_{\times} \approx 1.01$ at $h_{c\times} \approx 0.9935$ for $B/J=0.8$. Inset: Thermodynamic limit $z_{\infty}=1.01(5)$ and $\nu_{\infty}=0.99(4)$ from the best fit convergence of $z_{\times}$ and $\nu_{\times}$. (d) Extraction of the central charge at $N=256$, with optimized linear fit when $c=0.5000(9)$.
• Figure S1: Local intrinsic dimensionality as a function of the coupling parameter $s$ for increasing system sizes, taking $k=\log_2(N)$. The data shows stable 1D behavior until $s=-2$, where a sharp peak emerges, followed by a growth in dimensionality up to $s=0$. Beyond $s=0$, the dimensionality decreases and settles into a plateau at $s>>0$. A finite-size dip appears at small positive $s$, but shifts back toward $s=0$ as the system grows, consistent with the classical phase diagram.
• Figure S2: Numerically computed energy gaps using the Monte Carlo Markov Chain method, which shows good agreement with the analytical results of Fig. \ref{['fig:2']}, except for numerical errors and finite-size effects that arise in the simulation in the gapless regimes.