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arXiv:2506.21299

Meson spectroscopy of exotic symmetries of Ising criticality in Rydberg atom arrays

Joseph Vovrosh, Julius de Hond, Sergi Julià-Farré, Johannes Knolle, Alexandre Dauphin

Published Jun 26, 2025• 0 citations
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Abstract

The Ising model serves as a canonical platform for exploring emergent symmetry in quantum critical systems. The critical point of the 1D Ising chain is described by a conformal Ising field theory, which remains integrable in the presence of a magnetic perturbation, leading to massive particles associated with the exceptional Lie algebra $E_8$. Weakly coupling two Ising chains into a ladder breaks this integrability and is predicted to confine the elementary excitations of each chain into a richer spectrum of bound states organized by a $\mathcal{D}^{(1)}_8$ symmetry. Experimental signatures of $E_8$ excitations have arguably been observed in scattering studies of the spin chain material CoNb$_2$O$_6$, but direct evidence of confinement driven by inter-chain coupling has remained elusive. Here, we probe these emergent symmetries in a Rydberg atom quantum processing unit, leveraging its tunable geometry to realize both chain and ladder configurations. We identify mass spectra consistent with $E_8$ at the single-chain critical point and, in the weakly coupled ladder, report the first signatures of confinement of Ising excitations into the bound-state spectrum predicted by $\mathcal{D}^{(1)}_8$ symmetry. Our results demonstrate the power of Rydberg platforms for investigating symmetry emergence in quantum many-body systems and provide a direct window into the interplay of confinement, geometry, and criticality.

DOI
10.48550/arxiv.2506.21299

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The Ising model serves as a canonical platform for exploring emergent symmetry in quantum critical systems. The critical point of the 1D Ising chain is described by a conformal Ising field theory, which remains integrable in the presence of a magnetic perturbation, leading to massive particles associated with the exceptional Lie algebra E8E_8E8​. Weakly coupling two Ising chains into a ladder breaks this integrability and is predicted to confine the elementary excitations of each chain into a richer spectrum of bound states organized by a D8(1)\mathcal{D}^{(1)}_8D8(1)​ symmetry. Experimental signatures of E8E_8E8​ excitations have arguably been observed in scattering studies of the spin chain material CoNb2_22​O6_66​, but direct evidence of confinement driven by inter-chain coupling has remained elusive. Here, we probe these emergent symmetries in a Rydberg atom quantum processing unit, leveraging its tunable geometry to realize both chain and ladder configurations. We identify mass spectra consistent with E8E_8E8​ at the single-chain critical point and, in the weakly coupled ladder, report the first signatures of confinement of Ising excitations into the bound-state spectrum predicted by D8(1)\mathcal{D}^{(1)}_8D8(1)​ symmetry. Our results demonstrate the power of Rydberg platforms for investigating symmetry emergence in quantum many-body systems and provide a direct window into the interplay of confinement, geometry, and criticality.
The Ising model serves as a canonical platform for exploring emergent symmetry in quantum critical systems. The critical point of the 1D Ising chain is described by a conformal Ising field theory, which remains integrable in the presence of a magnetic perturbation, leading to massive particles associated with the exceptional Lie algebra E 8 . Weakly coupling two Ising chains into a ladder breaks this integrability and is predicted to bind the elementary kink excitations of the decoupled chains, giving rise to a richer spectrum of excitations described by the D 8 ( 1 ) theory. Experimental signatures of E 8 excitations have arguably been observed in scattering studies of the spin chain material CoNb 2 O 6 , but direct observation of the bound-state spectrum emerging from interchain coupling has remained elusive. Here, we probe these emergent symmetries in a Rydberg atom quantum processing unit, leveraging its tunable geometry to realize both chain and ladder configurations. We identify mass spectra consistent with E 8 at the single-chain critical point and, in the weakly coupled ladder, report the first signatures of the reorganization of Ising excitations into the bound-state spectrum predicted by D 8 ( 1 ) symmetry. Our results demonstrate the power of Rydberg platforms for investigating symmetry emergence in quantum many-body systems and provide a direct window into the interplay of quasiparticle interactions, geometry, and criticality.
Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.
Quantum integrability emerging near a quantum critical point (QCP) is manifested by an exotic excitation spectrum that is organized by the associated algebraic structure. A well-known example is the emergent E8{E}_{8}E8​ integrability near the QCP of a transverse field Ising chain (TFIC), which was long predicted theoretically and initially proposed to be realized in the quasi-one-dimensional (q1D) quantum magnet CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. However, later measurements on the spin excitation spectrum of this material revealed a series of satellite peaks in low energies that go beyond the description of the E8{E}_{8}E8​ Lie algebra. Motivated by this experimental progress, we hereby revisit the spin excitations of CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​ by combining numerical calculation and analytical analysis. We show that, as effects of strong interchain fluctuations, the spectrum of the system near the 1D QCP is characterized by an emergent D8(1){D}_{8}^{(1)}D8(1)​ Lie algebra instead of the E8{E}_{8}E8​ one. We further show that the D8(1){D}_{8}^{(1)}D8(1)​ spectrum can be realized in a broad class of interacting quantum systems. Our results advance the exploration and manipulation of integrability in quantum critical systems.
The exceptional Lie algebra E₈ was predicted by Zamolodchikov (1989) to organize the mass spectrum of the 1D quantum Ising chain at its critical point under a small longitudinal magnetic field, with eight massive particles whose mass ratios are dictated by the geometry of the E₈ root system. The lowest ratio is the golden ratio, m₂/m₁ = 2cos(π/5) = ϕ. Sixteen years later, Coldea et al. (Science, 2010) reported the first experimental signatures of this prediction in the quasi-one-dimensional Ising-chain ferromagnet CoNb₂O₆ by inelastic neutron scattering. Since then, peer-reviewed work has confirmed E₈ signatures in a second material (BaCo₂V₂O₈, Zhang et al. 2020), demonstrated the limits of the framework in candidate materials where 3D ordering preempts the 1D quantum critical point (e.g. SrCo₂V₂O₈, Cui et al. 2019; Wang et al. 2015), and proposed a D₈⁽¹⁾ refinement of the spectrum motivated by satellite peaks that pure E₈ cannot describe (Xi et al., Phys. Rev. B 112 L241104, 2025). Most recently, a preprint by Vovrosh et al. (arXiv:2506.21299, v1 26 Jun 2025, v2 30 Apr 2026) reports mass spectra consistent with E₈ in a single-chain Rydberg-array geometry, together with what the authors identify as the first direct experimental signatures of D₈⁽¹⁾ confinement in a weakly coupled ladder geometry — claims that, if independently confirmed, would mark the first engineered-platform realization of this extended symmetry scenario. We compile these results, distinguish established findings from preprint-stage and open ones, articulate the four conditions for E₈ emergence that recur across this corpus, and identify the engineered-platform extension as the main qualitative novelty of the past two years. This note is a brief review of approximately twelve peer-reviewed and preprint sources; it makes no original theoretical or experimental contribution.
Exceptional E8_88​ symmetry in spin dynamics of quasi-one-dimensional antiferromagnet BaCo2_22​V2_22​O8_88​
Exotic excitations can emerge in the vicinity of a quantum phase transition. When the quantum critical point of the one-dimensional (1D) transverse field Ising model is perturbed by a longitudinal magnetic field, it was predicted that its massive excitations are precisely described by the exceptional E8_88​ Lie algebra. Here we show an unambiguous experimental realization of the massive E8_88​ phase in the quasi-1D antiferromagnetic material BaCo2_22​V2_22​O8_88​, via nuclear magnetic resonance and inelastic neutron scattering measurements, and detailed theoretical analysis. The large separation between the masked 1D and 3D quantum critical points of the system allows us to identify, for the first time, the full 8 single-particle E8_88​ excitations as well as various multi-particle states in the spin excitation spectrum. Our results open new experimental route for exploring the dynamics of quantum integrable systems and physics beyond integrability, and thus bridge key physics in condensed matter and statistical field theory.
In a recent experiment on CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​, R. Coldea et al. [Science 327, 177 (2010)] found experimental evidence of the exceptional Lie algebra E8{E}_{8}E8​. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​ incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-product-state method. The excitation spectra show bound states characteristic of the weakly broken E8{E}_{8}E8​ symmetry. We compare the observed bound-state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data.