Experimental observation of quantum many-body excitations of $E_8$ symmetry in the Ising chain ferromagnet ${\mathrm{CoNb}}_2{\mathrm{O}}_6$

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

Figure 1

Illustration of phase diagram for a quasi-one-dimensional (quasi-1D) ferromagnetic Ising-chain system in an applied transverse field. For a 1D ferromagnet, a long-range order is formed only at zero temperature, whereas a 3D order can be stabilized at a finite temperature $T_C$ in the presence of interchain couplings. The 1D and 3D long-range orders can be suppressed by an applied transverse field at $B_c^{\text{1D}}$ and $B_c^{\text{3D}}$, respectively. When $B_c^{\text{1D}}<B_c^{\text{3D}}$ the $E_8$ dynamic spectrum could be realized around $B_c^{\text{1D}}$ as illustrated by the dashed area. Inset shows the zigzag spin chain constituted by edge-shared ${\mathrm{CoO}}_6$ octahedra in ${\mathrm{CoNb}}_2{\mathrm{O}}_6$.

Figure 2

Absorption spectra measured in zero field at 0.25 K (below $T_c$) with unpolarized terahertz radiation, and at 4 K (above $T_c$) for the terahertz polarizations ($e^{\omega }\parallel c$, $h^{\omega }\parallel a$) and ($e^{\omega }\parallel a$, $h^{\omega }\parallel c$). Inset shows that the eigenenergies of the modes $M_1$ to $M_7$ observed at 0.25 K follow a linear dependence on ${\zeta }_i$ which are the negative zeros of the Airy function $\text{Ai}(-{\zeta }_i)=0$. The linear dependence is expected for confined-spinon excitations. The different nomenclature for these zero-field excitations is used in order to discriminate them from the $E_8$ particles close to the 1D critical field $B_c^{\text{1D}}=4.75$ T (see below).

Figure 3

(a) Absorption spectra of ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ measured at 0.25 K in various applied transverse magnetic fields, $B\parallel b$. $\mathrm{\Delta }\alpha$ and $\omega$ denote absorption coefficient and wave number, respectively, in the unit of ${\mathrm{cm}}^{-1}$. The arrows indicate the modes ${\mathrm{m}}_1$, ${\mathrm{m}}_2,...,{\mathrm{m}}_6$ at 4.75 T and their field-dependent evolution. The asterisk ($*$) marks the onset of the (${\mathrm{m}}_1+{\mathrm{m}}_2$) continuum. At 4.75 T the circles ($\circ$) mark the satellite peaks. The spectra in higher fields are shifted upward by a constant for clarity. (b) Absorption spectra measured at 5 T for 0.25 and 4 K below and above $T_C$, respectively. For 0.25 K the arrows indicate those modes marked by arrows in (a). The black arrow at the 4 K spectrum marks a broader band observed due to the zone-folding effect. The down-pointing triangle ($▿$) indicates a mode which similar to $m_1$ is present both above and below $T_C$. The 0.25 K spectrum is shifted upward for clarity.

Figure 4

Field dependence of the energies of the observed modes ${\mathrm{m}}_1,...,{\mathrm{m}}_6$ and of the maxima for (${\mathrm{m}}_1+{\mathrm{m}}_1$) and (${\mathrm{m}}_1+{\mathrm{m}}_2$), normalized to the ${\mathrm{m}}_1$ energy in each field. All the ratios exhibit a monotonic increase with increasing field. The predicted ratios (dashed lines; see Refs. [9, 10]) for the $E_8$ particles are simultaneously reached at $B_c^{\text{1D}}=4.75$ T, evidencing the observation of the high-energy $E_8$ particles in ${\mathrm{CoNb}}_2{\mathrm{O}}_6$.