Finite-Temperature Spin Dynamics in a Perturbed Quantum Critical Ising Chain with an $E_8$ Symmetry

A spectrum exhibiting $E_8$ symmetry is expected to arise when a small longitudinal field is introduced in the transverse-field Ising chain at its quantum critical point. Evidence for this spectrum has recently come from neutron scattering measurements in cobalt niobate, a quasi-one-dimensional Ising ferromagnet. Unlike its zero-temperature counterpart, the finite-temperature dynamics of the model has not yet been determined. We study the dynamical spin structure factor of the model at low frequencies and nonzero temperatures, using the form factor method. Its frequency dependence is singular, but differs from the diffusion form. The temperature dependence of the nuclear magnetic resonance (NMR) relaxation rate has an activated form, whose prefactor we also determine. We propose NMR experiments as a means to further test the applicability of the $E_8$ description for ${\mathrm{CoNb}}_2{\mathrm{O}}_6$.

Figure 1

The NMR relaxation rate as a function of temperature. The frequency is chosen to be $\omega /{\mathrm{\Delta }}_a=0.001$. The temperature dependence is well described by ${\mathrm{\Delta }}_{a}S(T)=631e^{-{\mathrm{\Delta }}_a/T}$. The inset picture shows that channels other than $a-a$ give negligible contributions.

Figure 2

The local dynamical structure factor as a function of frequency at a fixed temperature $T/{\mathrm{\Delta }}_a=0.05$. The $\omega$ dependence is well described by $10^7{\mathrm{\Delta }}_{a}S(\omega )=-5.28+2.48\ln ({\mathrm{\Delta }}_a/\omega )$.