Abstract
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 E_{8} 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–onedimensional Ising ferromagnet CoNb_{2}O_{6} (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spinflips 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 E_{8} spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.
Symmetry is present in many physical systems and helps uncover some of their fundamental properties. Continuous symmetries lead to conservation laws; for example, the invariance of physical laws under spatial rotation ensures the conservation of angular momentum. More exotic continuous symmetries have been predicted to emerge in the proximity of certain quantum phase transitions (QPTs) (1, 2). Recent experiments on quantum magnets (3–5) suggest that quantum critical resonances may expose the underlying symmetries most clearly. Remarkably, the simplest of systems, the Ising chain, promises a very complex symmetry, described mathematically by the E_{8} Lie group (2, 6–9). Lie groups describe continuous symmetries and are important in many areas of physics. They range in complexity from the U(1) group, which appears in the lowenergy description of superfluidity, superconductivity, and BoseEinstein condensation (10, 11), to E_{8}, the highestorder symmetry group discovered in mathematics (12), which has not yet been experimentally realized in physics.
The onedimensional (1D) Ising chain in transverse field (10, 11, 13) is perhaps the moststudied theoretical paradigm for a quantum phase transition. It is described by the Hamiltonian
To explore the physics of Ising quantum criticality in real materials, several key ingredients are required: very good onedimensionality of the magnetism to avoid meanfield effects of higher dimensions, a strong easyaxis (Ising) character, and a sufficiently low exchange energy J of a few meV that can be matched by experimentally attainable magnetic fields (10 T ~ 1 meV) to access the quantum critical point. An excellent model system to test this physics is the insulating quasi1D Ising ferromagnet CoNb_{2}O_{6} (14–16), where magnetic Co^{2+} ions are arranged into nearisolated zigzag chains along the c axis with strong easyaxis anisotropy due to crystal field effects from the distorted CoO_{6} local environment (Fig. 1B). Large single crystals can be grown (17), which is an essential precondition for measurement of the crucial spin dynamics with neutron scattering.
CoNb_{2}O_{6} orders magnetically at low temperatures below T_{N1} = 2.95 K, stabilized by weak interchain couplings. The chains order ferromagnetically along their length with magnetic moments pointing along the local Ising direction, contained in the crystal (ac) plane (18). To tune to the critical point, we apply an external magnetic field along the b axis, transverse to the local Ising axis. Figure 1C shows that the external field suppresses the longrange 3D magnetic order favored by the Ising exchange in a continuous phase transition at a critical field B_{C} = 5.5 T.
Expected excitations for the model in Eq. 1 consist of (i) pairs of kinks, with the cartoon representation
The very strong dimensionality effects in 3D systems stabilize sharp spinflip quasiparticles in both the ordered and paramagnetic phases, as indeed observed experimentally in the 3D dipolarcoupled ferromagnet LiHoF_{4} (19, 20). In contrast, weak additional perturbations in the 1D Ising model, in particular a small longitudinal field
The zerofield data in Fig. 3A reveal a gapped continuum scattering at the ferromagnetic zone center (L = 0) due to kink pairs, which are allowed to propagate even in the absence of an external field. This is caused by subleading terms in the spin Hamiltonian. Upon cooling to the lowest temperature of 40 mK, deep in the magnetically ordered phase, the continuum splits into a sequence of sharp modes (Fig. 3B). At least five modes can be clearly observed (Fig. 3E), and they exist over a wide range of wave vectors and have a quadratic dispersion (open symbols in Fig. 3D). These data demonstrate the physics of kink confinement under a linear attractive interaction (6–9). In the ordered phase, kink propagation upsets the bonds with the neighboring chains (Fig. 3G) and therefore requires an energy cost V(x) that grows linearly with the kink separation x, V(x) = λx, where the “string tension” λ is proportional to the ordered moment magnitude 〈S^{z}〉 and the interchain coupling strength [λ = 2h_{z}〈S^{z}〉/
The essential physics of confinement is apparent in the limit of small λ for two kinks near the band minimum, where the onekink dispersion is quadratic: ε(k) = m_{ο} + ħ^{2}k^{2}/(2μ). In this case, the Schrödinger’s equation for the relative motion of two kinks in their centerofmass frame is
A full modeling of the data throughout the Brillouin zone can be obtained (18) by considering an extension of Eq. 2 to finite wave vectors and adding a shortrange interaction between kinks, responsible for stabilizing the observed bound state near the zone boundary L = –1. Interestingly, this is a kinetic bound state; that is, it is stabilized by virtue of the extra kinetic energy gained by two kinks if they hop together as a result of their shortrange interaction, as opposed to the Zeeman ladder of confinement bound states (near L = 0), stabilized by the potential energy V(x). The good agreement with the dispersion relations of all the bound states observed (Fig. 3D), as well as the overall intensity distribution (compare Fig. 3, B and F), shows that an effective model of kinks with a confinement interaction can quantitatively describe the complete spin dynamics.
Having established the behavior at zero field, we now consider the influence of the QPT at high field. Figure 4C shows that the excitation gap decreases upon approaching the critical field (as quantum tunneling lowers the energy of the kink quasiparticles), then increases again above B_{C} in the paramagnetic phase as a result of the increase in Zeeman energy cost for spinflip quasiparticles. In a quasi1D system such as CoNb_{2}O_{6} with finite interchain couplings, a complete gap softening is only expected (23) at the location of the 3D magnetic longrange order Bragg peaks, which occur at a finite interchain wave vector q_{┴} that minimizes the Fourier transform of the antiferromagnetic interchain couplings; the measurements shown in Fig. 4C were in a scattering plane where no magnetic Bragg peaks occur, so an incomplete gap softening would be expected here, as indeed was observed.
For the critical Ising chain, a gapless spectrum of critical kinks is predicted (Fig. 4F). Adding a finite longitudinal field h_{z} generates a gap and stabilizes bound states (Fig. 4G). In the scaling limit sufficiently close to the quantum critical point (i.e., h_{z} << J, h = h_{C}), the spectrum is predicted to have eight particles with energies in specific ratios (given by a representation of the E_{8} Lie group) with the first mass at m_{1}/J = C(h_{z}/J)^{8/15}, C ≈ 1.59 (2). The predicted spectrum for such an offcritical Ising chain to be observed by neutron scattering is illustrated in Fig. 4E for the dominant dynamical correlations S^{zz}(k = 0,ω) for which quantitative calculations are available (7): Two prominent sharp peaks due to the first two particles m_{1} and m_{2} are expected at low energies below the onset of the continuum of two m_{1} particles (24).
The neutron data taken just below the critical field (Fig. 4, A and B) are indeed consistent with this highly nontrivial prediction of two prominent peaks at low energies, which we identify with the first two particles m_{1} and m_{2} of the offcritical Ising model. Figure 4D shows how the ratio of the energies of those peaks varies with increasing field and approaches closely (near 5 T just below the 3D critical field of 5.5 T) the golden ratio m_{2}/m_{1} = (1 +
Our results show that the exploration of continuous quantum phase transitions can open up avenues to experimentally realize otherwise inaccessible (1, 26) correlated quantum states of matter with complex symmetries and dynamics.
Supporting Online Material
www.sciencemag.org/cgi/content/full/327/5962/177/DC1
Materials and Methods
References

↵† Present address: HelmholtzZentrum Berlin für Materialien und Energie, Lise Meitner Campus, Glienicker Str. 100, D14109 Berlin, Germany.
References and Notes
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 ↵ See supporting material on Science Online.
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 ↵ The higherenergy particles m_{3} to m_{8} are expected to produce much smaller features in the total scattering line shape, as they carry a much reduced weight and are overlapping or are very close to the lowerboundary onset of the continuum scattering (see Fig. 4E).
 ↵ The small offset between the estimated 1D and 3D critical fields is attributed to the interchain couplings, which strengthen the magnetic order. We note that a more precise quantitative comparison with the E_{8} model would require extension of the theory to include how the mass ratio m_{2}/m_{1} depends on the interchain wave vector q_{┴}, as the data in Fig. 4D were collected slightly away from the 3D Bragg peak positions; the already good agreement with the longwavelength prediction expected to be valid near the 3D Bragg wave vector may suggest that the mass ratio dispersion is probably a small effect at the measured wave vectors.
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 ↵ The 3D magnetic ordering wave vector has a finite component in the interchain direction due to antiferromagnetic couplings between chains.
 We thank G. Mussardo, S. T. Carr, A. M. Tsvelik, M. Greiter, and in particular F. H. L. Essler and L. Balents for very useful discussions. Work at Oxford, Bristol, and ISIS was supported by the Engineering and Physical Sciences Research Council (UK) and at HZB by the European Commission under the 6th Framework Programme through the Key Action: Strengthening the European Research Area, Research Infrastructures, contract RII3CT2003505925 (NMI3).