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.
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.
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 Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2O6. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8 particles, m1 to m6, and the two-particle (m1+m2) continuum in an applied transverse magnetic field of 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) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
Experimental Observation of High-Energy E8 Particles in the Ising Chain Ferromagnet CoNb2O6
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 Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb2O6. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six E8 particles, m1 to m6, and the two-particle (m1+m2) continuum in an applied transverse magnetic field of Bc1D=4.75 T, before the three-dimensional magnetic order is suppressed above Bc3D≈5.3 T. The observation of the higher-energy particles (m3 to m6) 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 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 in an applied transverse field. By comparing to an analytical calculation of the dynamical spin correlations, we identify E8 particles as well as their two-particle excitations.
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 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. 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 Lie algebra. Motivated by this experimental progress, we hereby revisit the spin excitations of CoNb2O6 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) Lie algebra instead of the E8 one. We further show that the 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.
Regarded as the closest realization of the quantum Ising model in a magnetic field, which features stable quasiparticles with an emergent E8 symmetry, CoNb2O6 keeps being a source of inspiration by manifesting intricacies beyond this model. A notable example is the recent finding of an unexpected band of localized gapless fermionic excitations around the field-induced quantum critical point (QCP), whose origin remains unclear. Despite differences in many aspects compared to CoNb2O6, BaCo2V2O8 in a magnetic field also exhibits the E8 physics, while allowing for better scrutiny of the critical excitations associated with its well-separated one- and three-dimensional QCPs. Here we study the low-lying magnetic excitations within the antiferromagnetic order of BaCo2V2O8 by performing ultralow-temperature heat transport measurements. The field-independent thermal conductivity at the lowest temperatures and the suppressed thermal conductivity with an increasing field at higher temperatures point to heat-carrying phonons scattered by magnetic excitations. For the magnetic excitations, importantly, we find no evidence for the emergence, around the one-dimensional QCP, of gapless fermionic excitations similar to those observed in CoNb2O6. Based on the contrasting cases of BaCo2V2O8 and CoNb2O6, we propose that the frustrated alignment of chains and the proximity of two QCPs in CoNb2O6 may put this system in a unique place to host the gapless fermionic excitations.
Quantum phase transitions occur at zero temperature upon variation of some nonthermal control parameters. The Ising chain in a transverse field is probably the most-studied model undergoing such a transition, from ferromagnetic to paramagnetic state1,2. This model can be exactly solved by using a Jordan-Wigner transformation, which transforms the spins into noninteracting spinless fermions1. At the quantum critical point, the magnetic excitations can carry arbitrarily low energy and dominate the low temperature properties. Here we report the unveiling of such quantum critical point in quasi-one-dimensional Ising ferromagnet CoNb2O6 by ultra-low-temperature thermal conductivity measurements. We find that in the paramagnetic state, phonons are scattered by the magnetic excitations above certain temperature Ts, which corresponds to a gap. As predicted by the theoretical model1, this gap linearly goes to zero with decreasing the transverse field, thus determining the quantum critical point of the Ising chain.