Dynamics of one-dimensional Heisenberg antiferromagnets

Numerous finite temperature properties of quantum spin chains were computed, and many of these have since been quantitatively tested in a number of experiments.The dynamics of half-integer spin chains can be probed by NMR experiments, and predictions for NMR relaxation rates were made in paper 2; these have been tested in experiments of Takigawa et al (Physical Review Letters 76, 4612 (1996)).

Integer spin chains and even-leg spin ladders, which have an energy gap, were considered in papers 4 and 5. It was argued that their low temperature dynamics permits a quasi-classical description in a "beads on a string" model sketched below.

Quasiclassical model for the real time dynamics of gapped spin chains and ladders at low temperatures. The horizontal axis represents spatial position and the vertical axis is time. The lines are the trajectories of thermally excited spin 1 "excitons", and the three colors represent the Sz quantum number. Notice that there is perfect reflection of the Sz quantum number at each collision.

This model allowed computation of the low temperature values of the spin-1 exciton lifetime and the spin diffusivity (these results are believed to be exact in the limit of vanishing temperature), which led to predictions for NMR and neutron scattering experiments. The results imply that the ratio of the  "dynamical energy gap" (measured by the nuclear relaxation rate) to the "static energy gap" (measured by the magnetic suscepibility) is exactly 3/2 at low temperatures: this is consistent with tends noted by Itoh and Yasuoka (Journal of the Physical Society of Japan 66, 334 (1997)) across a number of observations on gapped spin chains as shown below.

Experimental values of "dynamical" and "static" energy gaps of a number of spin gap compounds. The blue triangles are one-dimensional systems, and cluster around the theoretically predicted slope of 1.5. The red diamonds are isolated spin dimer compounds and, as expected, these do not show the one-dimensional anomaly..

The predictions for the field and temperature dependence of the NMR relaxation rate explain well the observations of Takigawa et al (Physical Review Letters 76, 2173 (1996)).

The lifetime of the spin 1 excitons can be measured in neutron scattering experiments. Such experiments have been carried out by M. Kenzelmann et al, Physical Review B 63, 134417 (2001) and Physical Review B 66, 174412 (2001), and the results are in good quantitative agreement with our predictions, which had no free parameters. There is also good agreement with the neutron scattering measurements of Xu, Broholm et al (Science 317, 1049 (2007)).

Paper 5 also predicted the existence of certain two magnon bound states in the two-leg ladder, and these appear to have been observed by Windt et al. (Physical Review Letters 87, 127002 (2001)) and Grueninger et al., cond-mat/0109524.


  1. Finite temperature properties of quantum antiferromagnets in a uniform magnetic field in one and two dimensions, S. Sachdev, T. Senthil, and R. Shankar, Physical Review B 50, 258 (1994); cond-mat/9401040.
  2. NMR relaxation in half-integer spin chains, S. Sachdev, Physical Review B 50, 13006 (1994); cond-mat/9411079.
  3. Multicritical crossovers near the dilute Bose gas quantum critical point, K. Damle and S. Sachdev, Physical Review Letters 76, 4412 (1996); cond-mat/9602073.
  4. Low temperature spin diffusion in the one-dimensional quantum O(3) non-linear sigma model, S. Sachdev and K. Damle, Physical Review Letters 78, 943 (1997); cond-mat/9610115.
  5. Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets ar nonzero temperatures, K. Damle and S. Sachdev, Physical Review B 57, 8307 (1998); cond-mat/9711014.
  6. Intermediate temperature dynamics of one-dimensional Heisenberg antiferromagnets, C. Buragohain and S. Sachdev, Physical Review B 59, 9285 (1999); cond-mat/9811083.
  7. Dynamics and transport near quantum-critical points, S. Sachdev, Dynamical properties of unconventional magnetic systems, A. Skjeltorp and D. Sherrington eds., NATO ASI Series E: Applied Sciences, vol 349, Kluwer Academic, Dordrecht (1997); cond-mat/9705266.
  8. Comment on "Spin Transport of the quantum one-dimensional non-linear sigma model", S. Sachdev and K. Damle, Journal of the Physical Society of Japan 69, 2712 (2000); cond-mat/9906054.
  9. Universal relaxational dynamics of gapped one dimensional models in the quantum sine-Gordon universality class, K. Damle and S. Sachdev, Phys. Rev. Lett. 95, 187201 (2005); cond-mat/0507380.