Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry

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Science  08 Jan 2010:
Vol. 327, Issue 5962, pp. 177-180
DOI: 10.1126/science.1180085


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.

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 (35) 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 E8 Lie group (2, 69). 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 low-energy description of superfluidity, superconductivity, and Bose-Einstein condensation (10, 11), to E8, the highest-order symmetry group discovered in mathematics (12), which has not yet been experimentally realized in physics.

The one-dimensional (1D) Ising chain in transverse field (10, 11, 13) is perhaps the most-studied theoretical paradigm for a quantum phase transition. It is described by the HamiltonianH=ΣiJSizSi+1zhSix(1)where a ferromagnetic exchange J > 0 between nearest-neighbor spin-½ magnetic moments Si arranged on a 1D chain competes with an applied external transverse magnetic field h. The Ising exchange J favors spontaneous magnetic order along the z axis (| or |), whereas the transverse field h forces the spins to point along the perpendicular +x direction (|). This competition leads to two distinct phases, magnetically ordered and quantum paramagnetic, separated by a continuous transition at the critical field hC = J/2 (Fig. 1A). Qualitatively, the magnetic field stimulates quantum tunneling processes between ↑ and ↓ spin states and these zero-point quantum fluctuations “melt” the magnetic order at hC (10).

Fig. 1

(A) Phase diagram of the Ising chain in transverse field (Eq. 1). Spin excitations are pairs of domain-wall quasiparticles (kinks) in the ordered phase below hC and spin-flip quasiparticles in the paramagnetic phase above hC. The dashed line shows the spin gap. (B) CoNb2O6 contains zigzag ferromagnetic Ising chains. (C) Intensity of the 3D magnetic Bragg peak as a function of applied field observed by neutron diffraction (27).

To explore the physics of Ising quantum criticality in real materials, several key ingredients are required: very good one-dimensionality of the magnetism to avoid mean-field effects of higher dimensions, a strong easy-axis (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 quasi-1D Ising ferromagnet CoNb2O6 (1416), where magnetic Co2+ ions are arranged into near-isolated zigzag chains along the c axis with strong easy-axis anisotropy due to crystal field effects from the distorted CoO6 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.

CoNb2O6 orders magnetically at low temperatures below TN1 = 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 long-range 3D magnetic order favored by the Ising exchange in a continuous phase transition at a critical field BC = 5.5 T.

Expected excitations for the model in Eq. 1 consist of (i) pairs of kinks, with the cartoon representation |z..., below BC, and (ii) spin-flip quasiparticles |x... above BC. The kinks interpolate between the two degenerate ground states with spontaneous magnetization along the +z or –z axis, respectively. Neutrons scatter by creating a pair of kinks (Fig. 2A). The results in Fig. 2, B and C, show that in the ordered phase below BC the spectrum is a bow tie–shaped continuum with strongly dispersive boundaries and large bandwidth at the zone center (L = 0), which we attribute to the expected two-kink states. This continuum increases in bandwidth and lowers its gap with increasing field, as the applied transverse field provides matrix elements for the kinks to hop, directly tuning their kinetic energy. Above BC a very different spectrum emerges (Fig. 2E), dominated by a single sharp mode. This is precisely the signature of a quantum paramagnetic phase. In this phase the spontaneous ferromagnetic correlations are absent, and there are no longer two equivalent ground states that could support kinks. Instead, excitations can be understood in terms of single spin reversals opposite to the applied field that cost Zeeman energy in increasing field. The fundamental change in the nature of quasiparticles observed here (compare Fig. 2, C and E) does not occur in higher-dimensional realizations of the quantum Ising model. The kinks are a crucial aspect of the physics in one dimension, and their spectrum of confinement bound states near the transition field will be directly related to the low-energy symmetry of the critical point.

Fig. 2

(A) Cartoon of a neutron spin-flip scattering that creates a pair of independently propagating kinks in a ferromagnetically ordered chain. (B to E) Spin excitations in CoNb2O6 near the critical field as a function of wave vector along the chain (in rlu units of 2π/c) and energy (18). In the ordered phase [(B) and (C)], excitations form a continuum due to scattering by pairs of kinks [as illustrated in (A)]; in the paramagnetic phase (E), a single dominant sharp mode occurs, due to scattering by a spin-flip quasiparticle. Near the critical field (D), the two types of spectra tend to merge into one another. Intensities in (E) are multiplied by Embedded Image to make them comparable to the other panels.

The very strong dimensionality effects in 3D systems stabilize sharp spin-flip quasiparticles in both the ordered and paramagnetic phases, as indeed observed experimentally in the 3D dipolar-coupled ferromagnet LiHoF4 (19, 20). In contrast, weak additional perturbations in the 1D Ising model, in particular a small longitudinal field hzΣiSiz, should lead to a rich structure of bound states (6, 7, 9). Such a longitudinal field, in fact, arises naturally in the case of a quasi-1D magnet: In the 3D magnetically ordered phase at low temperature, the weak couplings between the magnetic chains can be replaced in a first approximation by a local, effective longitudinal mean field (21), which scales with the magnitude of the ordered moment 〈Sz〉 [hz = ΣδJδSz〉 where the sum extends over all interchain bonds with exchange energy Jδ]. If the 1D Ising chain is precisely at its critical point (h = hC), then the bound states stabilized by the additional longitudinal field hz morph into the “quantum resonances” that are a characteristic fingerprint of the emergent symmetries near the quantum critical point. Nearly two decades ago, Zamolodchikov (2) proposed precisely eight “meson” bound states (the kinks playing the role of quarks), with energies in specific ratios given by a representation of the E8 exceptional Lie group (2). Before discussing the results near the QPT, we first develop a more sophisticated model of the magnetism in CoNb2O6 including confinement effects at zero field, where conventional perturbation theories are found to hold.

The zero-field 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 sub-leading 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 (69). 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) = λ|x|, where the “string tension” λ is proportional to the ordered moment magnitude 〈Sz〉 and the interchain coupling strength [λ = 2hzSz〉/c˜, where hz = ΣδJδSz〉 is the longitudinal mean field of the interchain couplings and c˜ = c/2 is the lattice spacing along the chain].

Fig. 3

Zero-field spin excitations in CoNb2O6. (A) In the 1D phase above TN1, a broad continuum occurs near the zone center (L = 0) due to scattering by pairs of unbound kinks. (B) The continuum splits into a Zeeman ladder of two-kink bound states deep in the ordered phase. (C and D) Model calculations (18) for hz = 0 and 0.02J to compare with data in (A) and (B), respectively. In (C) the thick dashed line is the kinetic two-kink bound state stable only outside the two-kink continuum (bounded by the dashed-dotted lines). Open symbols in (C) and (D) are peak positions from (A) and (B), respectively. (E) Energy scan at the zone center observing five sharp modes [red and blue circles are data from (B) and (A), respectively; solid line is a fit to Gaussians]. (F) Dynamical correlations Sxx(k,ω) (18) convolved with the instrumental resolution to compare with data in (B). (G) In the ordered phase, kink separation costs energy as it breaks interchain bonds J', leading to an effective linear “string tension” that confines kinks into bound states. (H) Observed and calculated bound-state energies.

The essential physics of confinement is apparent in the limit of small λ for two kinks near the band minimum, where the one-kink dispersion is quadratic: ε(k) = mο + ħ2k2/(2μ). In this case, the Schrödinger’s equation for the relative motion of two kinks in their center-of-mass frame is 2μd2ϕdx2+λ|x|ϕ=(m2m)ϕ(2) (69, 22), which has only bound-state solutions with energies (also called masses)mj=2m+zjλ2/3(2μ)1/3j=1,2,3,...(3)The bound states are predicted to occur above the threshold 2mο for creating two free kinks in a specific sequence given by the prefactors zn, the negative zeros of the Airy function Ai(–zn) = 0, zj = 2.33, 4.08, 5.52, 6.78, 7.94, etc. (18). The very nontrivial sequencing of the spacing between levels at the zone center agrees well with the measured energies of all five observed bound states (Fig. 3H), indicating that the weak confinement limit captures the essential physics.

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 short-range 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 short-range 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 BC in the paramagnetic phase as a result of the increase in Zeeman energy cost for spin-flip quasiparticles. In a quasi-1D system such as CoNb2O6 with finite interchain couplings, a complete gap softening is only expected (23) at the location of the 3D magnetic long-range 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.

Fig. 4

(A and B) Energy scans at the zone center at 4.5 and 5 T observing two peaks, m1 and m2, at low energies. (C) Softening of the two energy gaps near the critical field (above ~5 T the m2 peak could no longer be resolved). Points come from data as in Fig. 2, B to D; lines are guides to the eye. The incomplete gap softening is attributed to the interchain couplings as described in the text. (D) The ratio m2/m1 approaches the E8 golden ratio (dashed line) just below the critical field. (E) Expected line shape in the dominant dynamical correlations at the zone center Szz(k = 0,ω) for the case shown in (G) [vertical bars are quasiparticle weights (7) relative to m1]: two prominent modes followed by the 2m1 continuum (schematic dashed line), in strong resemblance to observed data in (A) and (B). (F) Gapless continuum of critical kinks (shaded area) predicted for the critical Ising chain. (G) E8 spectrum expected for finite hz. Lines indicate bound states; shaded area is the 2m1 continuum.

For the critical Ising chain, a gapless spectrum of critical kinks is predicted (Fig. 4F). Adding a finite longitudinal field hz generates a gap and stabilizes bound states (Fig. 4G). In the scaling limit sufficiently close to the quantum critical point (i.e., hz << J, h = hC), the spectrum is predicted to have eight particles with energies in specific ratios (given by a representation of the E8 Lie group) with the first mass at m1/J = C(hz/J)8/15, C ≈ 1.59 (2). The predicted spectrum for such an off-critical Ising chain to be observed by neutron scattering is illustrated in Fig. 4E for the dominant dynamical correlations Szz(k = 0,ω) for which quantitative calculations are available (7): Two prominent sharp peaks due to the first two particles m1 and m2 are expected at low energies below the onset of the continuum of two m1 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 m1 and m2 of the off-critical 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 m2/m1 = (1 + 5)/2 = 1.618 predicted for the E8 masses. We identify the field where the closest agreement with the E8 mass ratio is observed as the field BC1D where the 1D chains would have been critical in the absence of interchain couplings (25). Indeed, it is in this regime (21) that the special quantum critical symmetry theory would be expected to apply.

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

Materials and Methods


  • Present address: Helmholtz-Zentrum Berlin für Materialien und Energie, Lise Meitner Campus, Glienicker Str. 100, D-14109 Berlin, Germany.

References and Notes

  1. See supporting material on Science Online.
  2. The higher-energy particles m3 to m8 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 lower-boundary onset of the continuum scattering (see Fig. 4E).
  3. 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 E8 model would require extension of the theory to include how the mass ratio m2/m1 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 long-wavelength 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.
  4. The 3D magnetic ordering wave vector has a finite component in the interchain direction due to antiferromagnetic couplings between chains.
  5. 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 RII3-CT-2003-505925 (NMI3).
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