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arXiv:2402.11229 · Physical review. B./Physical review. B

Spin dynamics and dark particle in a weak-coupled quantum Ising ladder with D 8 ( 1 ) spectrum

Yunjing Gao, Xiao Wang, Ning Xi, Yunfeng Jiang, Rong Yu, Jianda Wu

Published May 27, 2025• 4 citations
arXivHTMLPDF

Abstract

Emergent ${\mathrm{Ising}}_{h}^{2}$ 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 ${\mathcal{D}}_{8}^{(1)}$ 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.

DOI
10.1103/physrevb.111.l201117

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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.
The Isingh2{\mathrm{Ising}}_{h}^{2}Isingh2​ integrable field theory emerges when two quantum critical Ising chains are weakly coupled. This theory possesses eight types of relativistic particles, among which the lightest one (B1{B}_{1}B1​) has been predicted to be a dark particle, which cannot be excited from the ground state through (quasi)local operations. The stability on the one hand highlights its potential for applications, and on the other hand makes it challenging to be observed. Here, we point out that the mass of the B1{B}_{1}B1​ dark particle mB1{m}_{{B}_{1}}mB1​​ appears as a thermally activated gap extracted from local spin dynamical structure factors at low frequency ($\ensuremath{\omega}\ensuremath{\ll}{m}_{{B}_{1}}$) and low temperatures ($T\ensuremath{\ll}{m}_{{B}_{1}}$). We then further propose that this gapped behavior can be directly detected via the NMR relaxation rate measurement in a proper experimental setup. Our results provide a practical criterion for verifying the existence of dark particles.
When the quantum critical transverse-field Ising chain is perturbed by a longitudinal field, a quantum integrable model emerges in the scaling limit with massive excitations described by the exceptional E8{E}_{8}E8​ Lie algebra. Using the corresponding analytical form factors of the quantum E8{E}_{8}E8​ integrable model, we systematically study the spin dynamic structure factor of the perturbed quantum critical Ising chain, where particle channels with total energy up to 5m15{m}_{1}5m1​ (m1{m}_{1}m1​ being the mass of the lightest E8{E}_{8}E8​ particle) are exhausted. In addition to the significant single-particle contributions to the dynamic spectrum, each two-particle channel with different masses is found to exhibit an edge singularity at the threshold of the total mass and decays with an inverse square root of energy, which is attributed to the singularity of the two-particle density of states at the threshold. The singularity is absent for particles with equal masses due to a cancellation mechanism involving the structure of the form factors. As a consequence, the dynamic structure factor displays a cascade of bumping peaks in the continuum region with clear singular features which can serve as a solid guidance for the material realization of the quantum E8{E}_{8}E8​ model.
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 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.
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.
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.