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Exceptional E$_8$ symmetry in spin dynamics of quasi-one-dimensional antiferromagnet BaCo$_2$V$_2$O$_8$

Haiyuan Zou, Yuliang Cui, X. Wang, Zhao Zhang, Jianwei Yang, Guangyong Xu, Akira Okutani, Masayuki Hagiwara, Masaaki Matsuda, Gang Wang

Published May 27, 2020• 1 citation
OpenAlex

Abstract

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 E$_8$ Lie algebra. Here we show an unambiguous experimental realization of the massive E$_8$ phase in the quasi-1D antiferromagnetic material BaCo$_2$V$_2$O$_8$, 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 E$_8$ 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.

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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.
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.
Combining inelastic neutron scattering and numerical simulations, we study the quasi-one-dimensional Ising anisotropic quantum antiferromagnet BaCo_{2}V_{2}O_{8} in a longitudinal magnetic field. This material shows a quantum phase transition from a Néel ordered phase at zero field to a longitudinal incommensurate spin density wave at a critical magnetic field of 3.8 T. Concomitantly, the excitation gap almost closes and a fundamental reconfiguration of the spin dynamics occurs. These experimental results are well described by the universal Tomonaga-Luttinger liquid theory developed for interacting spinless fermions in one dimension. We especially observe the rise of mainly longitudinal excitations, a hallmark of the unconventional low-field regime in Ising-like quantum antiferromagnetic chains.
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.
Spin dynamics of the E8E_8E8​ particles
In this article, we report on inelastic neutron scattering measurements on a quasi-1D antiferromagnet BaCo2_22​V2_22​O8_88​ under a transverse magnetic field applied along the (0,1,0) direction. Combining results of inelastic neutron scattering experiments, analytical analysis, and numerical simulations, we precisely studied the E8E_8E8​ excitations appearing in the whole Brillouin zone at Bc1D≈4.7B_c^{1D}\approx 4.7Bc1D​≈4.7 T. The energy scan at Q=(0,0,2)Q=(0,0,2)Q=(0,0,2) reveals a match between the data and the theoretical prediction of energies of multiple E8E_8E8​ excitations. Furthermore, dispersions of the lightest three E8E_8E8​ particles have been clearly observed, confirming the existence of the E8E_8E8​ particles in BaCo2_22​V2_22​O8_88​. Our results lay down a concrete ground to systematically study the physics of the exotic E8E_8E8​ particles.
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.
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.