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arXiv:1008.3534 · Physical Review B

Bound states and E8 symmetry effects in perturbed quantum Ising chains

Jonas A. Kjäll, Frank Pollmann, Joel E. Moore

Published Jan 13, 2011• 60 citations
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Abstract

In a recent experiment on ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$, R. Coldea et al. [Science 327, 177 (2010)] found experimental evidence of the exceptional Lie algebra ${E}_{8}$. 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 ${\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}$ 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 ${E}_{8}$ 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.

DOI
10.1103/physrevb.83.020407

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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.
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the E8{E}_{8}E8​ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8{E}_{8}E8​ particles, m1{\mathrm{m}}_{1}m1​ to m6{\mathrm{m}}_{6}m6​, and the two-particle (m1+m2)({\mathrm{m}}_{1}+{\mathrm{m}}_{2})(m1​+m2​) continuum in an applied transverse magnetic field of Bc1D=4.75T{B}_{c}^{\text{1D}}=4.75 \mathrm{T}Bc1D​=4.75T, before the three-dimensional magnetic order is suppressed above ${B}_{c}^{\text{3D}}\ensuremath{\approx}5.3 \mathrm{T}$. The observation of the higher-energy particles (m3tom6)({\mathrm{m}}_{3} \mathrm{to} {\mathrm{m}}_{6})(m3​tom6​) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
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.
Experimental Observation of High-Energy E8E_8E8​ Particles in the Ising Chain Ferromagnet CoNb2_22​O6_66​
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the E8E_8E8​ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2_22​O6_66​. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8E_8E8​ particles, m1\mathbf{m}_1m1​ to m6\mathbf{m}_6m6​, and the two-particle (m1+m2\mathbf{m}_1+\mathbf{m}_2m1​+m2​) continuum in an applied transverse magnetic field of Bc1D=4.75B_c^{1D}=4.75Bc1D​=4.75 T, before the three-dimensional magnetic order is suppressed above Bc3D≈5.3B_c^{3D}\approx 5.3Bc3D​≈5.3 T. The observation of the higher-energy particles (m3\mathbf{m}_3m3​ to m6\mathbf{m}_6m6​) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
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.
Near the transverse-field-induced quantum critical point of the Ising chain, an exotic dynamic spectrum consisting of exactly eight particles was predicted, which is uniquely described by an emergent quantum integrable field theory with the symmetry of the E8{E}_{8}E8​ Lie algebra, but rarely explored experimentally. Here we use high-resolution terahertz spectroscopy to resolve quantum spin dynamics of the quasi-one-dimensional Ising antiferromagnet BaCo2V2O8\mathrm{BaC}{\mathrm{o}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}BaCo2​V2​O8​ in an applied transverse field. By comparing to an analytical calculation of the dynamical spin correlations, we identify E8{E}_{8}E8​ particles as well as their two-particle excitations.