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Experimental Observation of High-Energy $E_8$ Particles in the Ising Chain Ferromagnet CoNb$_2$O$_6$

Kirill Amelin, Johannes Engelmayer, J. Viirok, U. Nagel, T. Rõõm, T. Lorenz, Zhe Wang

Published Jun 23, 2020• 0 citations
OpenAlex

Abstract

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 $E_8$ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb$_2$O$_6$. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six $E_8$ particles, $\mathbf{m}_1$ to $\mathbf{m}_6$, and the two-particle ($\mathbf{m}_1+\mathbf{m}_2$) continuum in an applied transverse magnetic field of $B_c^{1D}=4.75$ T, before the three-dimensional magnetic order is suppressed above $B_c^{3D}\approx 5.3$ T. The observation of the higher-energy particles ($\mathbf{m}_3$ to $\mathbf{m}_6$) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.

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Journal of Physics A Mathematical and Theoretical•Nov 29, 2022•16 citations
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.
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.
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.
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.
Kink bound states in the one-dimensional ferromagnetic Ising chain compound CoNb2O6 have been studied using high-resolution time-domain terahertz spectroscopy in zero applied magnetic field. When magnetic order develops at low temperature, nine bound states of kinks become visible. Their energies can be modeled exceedingly well by the Airy function solutions to a 1D Schrödinger equation with a linear confining potential. This sequence of bound states terminates at a threshold energy near 2 times the energy of the lowest bound state. Above this energy scale we observe a broad feature consistent with the onset of the two particle continuum. At energies just below this threshold we observe a prominent excitation that we interpret as a novel bound state of bound states--two pairs of kinks on neighboring chains.
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.
Abstract We report on high-resolution terahertz spectroscopic studies of quantum spin dynamics in the quasi-one-dimensional Ising-like ferromagnet CoNb2O6 and antiferromagnet BaCo2V2O8 as a function of an applied transverse magnetic field. In the ordered phases stabilized by inter-chain couplings, we reveal characteristics for confined spinon excitations, E 8 dynamical spectrum, and field-induced quantum phase transitions. The connections between these characteristic dynamical features are found in the field-dependent evolution of the excitation spectra.