Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.20221

Universal meson spectra near $(1+1)$-dimensional Ising criticality

Xiao Wang, Jianda Wu

Published Aug 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Near $(1+1)$-dimensional [$(1+1)$D] Ising criticality, a magnetic perturbation induces confinement and produces a cascade of bound-state excitations known as mesons. Here we show these mesons share a universal mass scaling after independently rescaling the model-dependent microscopic couplings. The number of stable mesons is controlled by the lightest two-meson threshold, while the lightest-meson mass follows a continuous trajectory characterized by a single scaling parameter. Using Hamiltonian truncation method, we obtain the trajectory numerically in both Ising field theory and the near-critical mixed-field Ising chain (MFIC). Under the rescaling, the trajectory and stable-meson-count crossover windows of MFIC both collapse onto the field-theory results. To further demonstrate the above universal organization of the meson spectra, we consider a class of four-periodic spin-$1/2$ Heisenberg-Ising chains under transverse fields, whose parameter space contains a family of quantum Ising critical points. The Hamiltonian family includes effective spin models for the quasi-one-dimensional antiferromagnets Ba(Sr)Co$_2$V$_2$O$_8$. Using tensor-network calculations, we obtain the corresponding lightest-meson mass trajectory for BaCo$_2$V$_2$O$_8$ and find that it also collapse onto the same universal curve given by field-theory result. Our results suggest that the universal scaling structure of quantum Ising criticality extends into the nearby confining regime, governing the organization of the meson spectrum. They thereby provide a practical criterion for interpreting excitations of quasi-1D Ising-like magnets in mixed fields beyond $E_8$ integrability.

DOI
10.48550/arxiv.2608.20221

Related papers

Mesons in a quantum Ising ladder

Yunjing Gao, Yunfeng Jiang, Jianda Wu

A bstract When two transverse-field Ising chains (TFICs) with magnetic order are coupled, the original free excitations become confined, giving rise to meson-like bound states. In this work, we study such bound states systematically. The mesons are characterized by their fermion number parity and chain-exchanging properties, which lead to distinct sets of mesonic states. The meson masses are determined by solving the Bethe-Salpter equation. An interesting observation is the additional degeneracy in the chain-exchanging odd sectors. Beyond the two particle approximation, we exploit the truncated free fermionic space approach to calculate the spectrum numerically. Corrections to the meson masses are obtained, and the degeneracy is further confirmed. The characterization and degeneracy can be connected to the situation when each chain is tuned to be quantum critical, where the system is described by the $$ {\textrm{Ising}}_h^2 $$ Ising h 2 integrable model, a sine-Gordon theory with ℤ 2 orbifold. Here we establish a clear correspondence between the particles in the bosonized form and their fermionic counterparts. Near this point, the stability of these particles is analyzed using the form factor perturbation scheme, where four particles are always present. Additionally, we calculate the evolution of the dominant dynamical structure factor for local spin operators, providing further insight into the low-energy excitations and their role in the system’s behavior. The two-particle confinement framework as well as the parity classifications may inspire the study for other coupled bi-partite systems.

arXiv:2502.15463•Journal of High Energy Physics•Jul 4, 2025•3 citations

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

Joseph Vovrosh, Julius de Hond, Sergi Julià-Farré +2 more

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.

arXiv:2506.21299•Jun 26, 2025•0 citations

Meson Spectroscopy of Exotic Symmetries of Ising Criticality in Rydberg Atom Arrays

Joseph Vovrosh, Julius de Hond, Sergi Julià-Farré +2 more

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.

PRX Quantum•Sep 8, 2026•0 citations

Universality of Heisenberg–Ising chain in external fields

Haiyuan Zou, Rong Yu, Jianda Wu

Abstract Motivated by the recent surge of transverse-field experiments on quasi-one-dimensional (1D) antiferromagnets Sr(Ba)Co 2 V 2 O 8 , we investigate the quantum phase transition (QPT) in a Heisenberg–Ising chain under a combination of two in-plane inter-perpendicular transverse fields and a four-period longitudinal field, where the in-plane transverse field is either uniform or staggered. We show that the model can be unitary mapped to the 1D transverse-field Ising model (1DTFIM) when the x and y components of the spin interaction and the four-period field are absent. When these two terms are present, following both analytical and numerical efforts, we demonstrate that the system undergoes a second-order QPT with increasing transverse fields, where the critical exponents as well as the central charge fall into the universality of 1DTFIM. Our results naturally identify the 1DTFIM universality of 1D quantum phase transitions observed in the existed experiments in Sr(Ba)Co 2 V 2 O 8 with transverse field applied along either [ ] or [ ] direction. Upon varying the tuning parameters, a critical surface with 1DTFIM universality is determined and silhouetted to exhibit the general presence of the universality in a much wider scope of models than conventional understanding. Our results provide a broad guiding framework to facilitate the experimental realization of 1DTFIM universality in real materials.

arXiv:1904.11398•Journal of Physics Condensed Matter•Oct 9, 2019•8 citations

Discrete Scale Invariance of Ising Mesons

Denis V. Vasilyev, Abbas Ali Saberi

Discrete scale invariance often arises from resonant few-body or singular scale-invariant dynamics. We show that it can instead emerge in the internal motion of a composite excitation in an ordered quantum magnet. In the marginal $1/r^4$ transverse-field Ising chain, long-range bonds generate an asymptotic inverse-square attraction between dressed domain walls, with strength fixed by the spontaneous magnetization. The dressed two-kink threshold curvature sets the relative kinetic scale. Together, these independently accessible quantities determine the scale-anomaly exponent and hence the meson hierarchy. In the supercritical regime, the same exponent governs geometric binding-energy and size ratios, logarithmic level accumulation, and log-periodic threshold scattering. Full-spin exact diagonalization and a fourth-order weak-field Hamiltonian support the coupled energy--size scaling in the finite window between core and ring effects. This provides a microscopic many-body realization of quantum limit-cycle physics, with its universal scaling fixed by magnetic order and kink mobility.

cond-mat.dis-nnCross-listedarXiv:2609.35940•Sep 29, 2026•0 citations

E 8 Spectra of Quasi-One-Dimensional Antiferromagnet BaCo 2 V 2 O 8 under Transverse Field

Haiyuan Zou, Yi Cui, Xiao Wang +16 more

We report ^{51}V NMR and inelastic neutron scattering (INS) measurements on a quasi-1D antiferromagnet BaCo_{2}V_{2}O_{8} under transverse field along the [010] direction. The scaling behavior of the spin-lattice relaxation rate above the Néel temperatures unveils a 1D quantum critical point (QCP) at H_{c}^{1D}≈4.7 T, which is masked by the 3D magnetic order. With the aid of accurate analytical analysis and numerical calculations, we show that the zone center INS spectrum at H_{c}^{1D} is precisely described by the pattern of the 1D quantum Ising model in a magnetic field, a class of universality described in terms of the exceptional E_{8} Lie algebra. These excitations are nondiffusive over a certain field range when the system is away from the 1D QCP. Our results provide an unambiguous experimental realization of the massive E_{8} phase in the compound, and open a new experimental route for exploring the dynamics of quantum integrable systems as well as physics beyond integrability.

arXiv:2005.13302•Physical Review Letters•Aug 9, 2021•57 citations
Universal meson spectra near (1+1)(1+1)(1+1)-dimensional Ising criticality
Near (1+1)(1+1)(1+1)-dimensional [(1+1)(1+1)(1+1)D] Ising criticality, a magnetic perturbation induces confinement and produces a cascade of bound-state excitations known as mesons. Here we show these mesons share a universal mass scaling after independently rescaling the model-dependent microscopic couplings. The number of stable mesons is controlled by the lightest two-meson threshold, while the lightest-meson mass follows a continuous trajectory characterized by a single scaling parameter. Using Hamiltonian truncation method, we obtain the trajectory numerically in both Ising field theory and the near-critical mixed-field Ising chain (MFIC). Under the rescaling, the trajectory and stable-meson-count crossover windows of MFIC both collapse onto the field-theory results. To further demonstrate the above universal organization of the meson spectra, we consider a class of four-periodic spin-1/21/21/2 Heisenberg-Ising chains under transverse fields, whose parameter space contains a family of quantum Ising critical points. The Hamiltonian family includes effective spin models for the quasi-one-dimensional antiferromagnets Ba(Sr)Co2_22​V2_22​O8_88​. Using tensor-network calculations, we obtain the corresponding lightest-meson mass trajectory for BaCo2_22​V2_22​O8_88​ and find that it also collapse onto the same universal curve given by field-theory result. Our results suggest that the universal scaling structure of quantum Ising criticality extends into the nearby confining regime, governing the organization of the meson spectrum. They thereby provide a practical criterion for interpreting excitations of quasi-1D Ising-like magnets in mixed fields beyond E8E_8E8​ integrability.
A bstract When two transverse-field Ising chains (TFICs) with magnetic order are coupled, the original free excitations become confined, giving rise to meson-like bound states. In this work, we study such bound states systematically. The mesons are characterized by their fermion number parity and chain-exchanging properties, which lead to distinct sets of mesonic states. The meson masses are determined by solving the Bethe-Salpter equation. An interesting observation is the additional degeneracy in the chain-exchanging odd sectors. Beyond the two particle approximation, we exploit the truncated free fermionic space approach to calculate the spectrum numerically. Corrections to the meson masses are obtained, and the degeneracy is further confirmed. The characterization and degeneracy can be connected to the situation when each chain is tuned to be quantum critical, where the system is described by the Isingh2 {\textrm{Ising}}_h^2 Isingh2​ Ising h 2 integrable model, a sine-Gordon theory with ℤ 2 orbifold. Here we establish a clear correspondence between the particles in the bosonized form and their fermionic counterparts. Near this point, the stability of these particles is analyzed using the form factor perturbation scheme, where four particles are always present. Additionally, we calculate the evolution of the dominant dynamical structure factor for local spin operators, providing further insight into the low-energy excitations and their role in the system’s behavior. The two-particle confinement framework as well as the parity classifications may inspire the study for other coupled bi-partite systems.
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.
Abstract Motivated by the recent surge of transverse-field experiments on quasi-one-dimensional (1D) antiferromagnets Sr(Ba)Co 2 V 2 O 8 , we investigate the quantum phase transition (QPT) in a Heisenberg–Ising chain under a combination of two in-plane inter-perpendicular transverse fields and a four-period longitudinal field, where the in-plane transverse field is either uniform or staggered. We show that the model can be unitary mapped to the 1D transverse-field Ising model (1DTFIM) when the x and y components of the spin interaction and the four-period field are absent. When these two terms are present, following both analytical and numerical efforts, we demonstrate that the system undergoes a second-order QPT with increasing transverse fields, where the critical exponents as well as the central charge fall into the universality of 1DTFIM. Our results naturally identify the 1DTFIM universality of 1D quantum phase transitions observed in the existed experiments in Sr(Ba)Co 2 V 2 O 8 with transverse field applied along either [ ] or [ ] direction. Upon varying the tuning parameters, a critical surface with 1DTFIM universality is determined and silhouetted to exhibit the general presence of the universality in a much wider scope of models than conventional understanding. Our results provide a broad guiding framework to facilitate the experimental realization of 1DTFIM universality in real materials.
Discrete scale invariance often arises from resonant few-body or singular scale-invariant dynamics. We show that it can instead emerge in the internal motion of a composite excitation in an ordered quantum magnet. In the marginal 1/r41/r^41/r4 transverse-field Ising chain, long-range bonds generate an asymptotic inverse-square attraction between dressed domain walls, with strength fixed by the spontaneous magnetization. The dressed two-kink threshold curvature sets the relative kinetic scale. Together, these independently accessible quantities determine the scale-anomaly exponent and hence the meson hierarchy. In the supercritical regime, the same exponent governs geometric binding-energy and size ratios, logarithmic level accumulation, and log-periodic threshold scattering. Full-spin exact diagonalization and a fourth-order weak-field Hamiltonian support the coupled energy--size scaling in the finite window between core and ring effects. This provides a microscopic many-body realization of quantum limit-cycle physics, with its universal scaling fixed by magnetic order and kink mobility.
We report ^{51}V NMR and inelastic neutron scattering (INS) measurements on a quasi-1D antiferromagnet BaCo_{2}V_{2}O_{8} under transverse field along the [010] direction. The scaling behavior of the spin-lattice relaxation rate above the Néel temperatures unveils a 1D quantum critical point (QCP) at H_{c}^{1D}≈4.7 T, which is masked by the 3D magnetic order. With the aid of accurate analytical analysis and numerical calculations, we show that the zone center INS spectrum at H_{c}^{1D} is precisely described by the pattern of the 1D quantum Ising model in a magnetic field, a class of universality described in terms of the exceptional E_{8} Lie algebra. These excitations are nondiffusive over a certain field range when the system is away from the 1D QCP. Our results provide an unambiguous experimental realization of the massive E_{8} phase in the compound, and open a new experimental route for exploring the dynamics of quantum integrable systems as well as physics beyond integrability.