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arXiv:2502.15463 · Journal of High Energy Physics

Mesons in a quantum Ising ladder

Yunjing Gao, Yunfeng Jiang, Jianda Wu

Published Jul 4, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep07(2025)072

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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^2Isingh2​ 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.
Weakly coupled Ising chains provide a condensed-matter realization of confinement. In these systems, kinks and antikinks bind into mesons due to an attractive interaction potential that increases linearly with the distance between the particles. While single mesons have been directly observed in experiments, the role of the multiparticle continuum and bound states of mesons in the excitation spectrum is far less clear. Using time-dependent density-matrix renormalization group methods, we study the dynamical structure factors of one- and two-spin operators in a transverse-field two-leg Ising ladder in the ferromagnetic phase. The propagation of time-dependent correlations and the two-spin excitation spectrum reveal the existence of interchain bound states, which are absent in the one-spin dynamical structure factor. We also identify two-meson bound states that appear at higher energies, above the thresholds of several two-meson continua.
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.
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 Heisenberg–Ising spin ladder is one of the few short-range models showing confinement of elementary excitations without the need of an external field, neither transverse nor longitudinal. This feature makes the model suitable for an experimental realization with ultracold atoms. In this paper, we combine analytic and numerical techniques to precisely characterize its spectrum in the regime of Hamiltonian parameters showing confinement. We find two kinds of particles, which we dub intrachain and interchain mesons, that correspond to bound states of kinks within the same chain or between different ones, respectively. The ultimate physical reasons leading to the existence of two families of mesons is a residual double degeneracy of the ground state: the two types of mesons interpolate either between the same vacuum (intrachain) or between the two different ones (interchain). While the intrachain mesons can also be qualitatively assessed through an effective mean field description and were previously known, the interchain ones are new and they represent general features of spin ladders with confinement.
Emergent Isingh2{\mathrm{Ising}}_{h}^{2}Isingh2​ integrability is anticipated in a quantum Ising ladder composed of two weakly coupled critical transverse field Ising chains. The system is remarkable for including eight types of massive relativistic particles, whose scattering matrix and mass spectrum are characterized by the D8(1){\mathcal{D}}_{8}^{(1)}D8(1)​ Lie algebra. In this article, by computing the spin dynamical structure factors using analytical form-factor approach, we clearly identify dispersive single-particle excitations of (anti) soliton and breathers as well as their multiparticle continua in the spectra, which is further confirmed by numerical simulations. We show that the selection rule inherent in the parity and topological charge of the theory causes a significant result: charge-parity-odd particles, termed dark particles, cannot be directly excited from the ground state through any local or quasilocal operations. This, in turn, suggests the long lifetime of the lightest dark particle.
Abstract The Heisenberg–Ising spin ladder is one of the few short-range models showing confinement of elementary excitations without the need of an external field, neither transverse nor longitudinal. This feature makes the model suitable for an experimental realization with ultracold atoms. In this paper, we combine analytic and numerical techniques to precisely characterize its spectrum in the regime of Hamiltonian parameters showing confinement. We find two kinds of particles, which we dub intrachain and interchain mesons, that correspond to bound states of kinks within the same chain or between different ones, respectively. The ultimate physical reasons leading to the existence of two families of mesons is a residual double degeneracy of the ground state: the two types of mesons interpolate either between the same vacuum (intrachain) or between the two different ones (interchain). While the intrachain mesons can also be qualitatively assessed through an effective mean field description and were previously known, the interchain ones are new and they represent general features of spin ladders with confinement.