Temperature evolution of the quantum gap in CsNiCl₃

Neutron-scattering measurements on the one-dimensional gapped S = 1 antiferromagnet, CsNiCl3, have shown that excitation corresponding to the Haldane mass gap Delta at low temperatures persists as a resonant feature to high temperatures. We find that the strong upward renormalization of the gap excitation, by a factor of three between 5 and 70 K, is more than enough to overcome its decreasing lifetime. We find that the gap lifetime is substantially shorter than that predicted by the scaling theory of Damle and Sachdev based on a classical dispersion for the Haldane excitations, but when we include their expected relativistic dispersion, the theory, in its low-temperature range of validity, gives a good account of experiment. The upward gap renormalization agrees with the nonlinear sigma model at low temperatures and even up to T of order 2 Delta provided an upper momentum cutoff is included.