Finite temperature crossovers of the quantum Ising chain

The Ising chain in a transverse field is one of the simplest models which exhibits a quantum critical point. The finite temperature crossovers in the vicinity of the critical point can be described rather precisely, both for static (paper 1) and dynamic (paper 3) properties, as outlined in the figure below.

Finite temperature crossovers of the Ising chain in a transverse field. The coupling g is a measure of the transverse field strength. For large g, all spins point along the direction of the transverse field (+x) in the ground state; the excited states consist of "particles" of flipped (-x) spins moving in this background. For small g, the spins are ferromagnetically aligned by their exchange interactions, and the "particles" are now domain walls separating regions of oppositely aligned spins. In the quantum critical region, neither description is valid, and non-trivial scaling arguments must be used.


  1. Universal, finite temperature, crossover functions of the quantum transition in the Ising chain in a transverse field, S. Sachdev, Nuclear Physics B 464, 576 (1996); cond-mat/9509147.
  2. Finite temperature correlations in the one-dimensional quantum Ising model, A. Leclair, F. Lesage, S. Sachdev, and H. Saleur, Nuclear Physics B 482, 579 (1996); cond-mat/9606104.
  3. Low temperature relaxational dynamics of the Ising chain in a transverse field, S. Sachdev and A.P. Young, Physical Review Letters 78, 2220 (1997); cond-mat/9609185.
  4. Finite temperature correlations of the Ising chain in transverse field, S. Sachdev, Symposium on Exactly Soluble Models in Statistical Mechanics: Historical Perspectives and Current Status, C. King and F.Y. Wu eds., International Journal of Modern Physics B 11, 57 (1997).
  5. 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/9705206.
  6. Finite temperature dynamics near quantum phase transitions, S. Sachdev, keynote talk at the 11th International Conference on Recent Progress in Many-Body Theories, UMIST, Manchester UK, 9-13 July, 2001, edited by Raymond F. Bishop, Tobias Brandes, Klaus A. Gernoth, Niels R. Walet and Yang Xian, World Scientific, Singapore (2002); cond-mat/0110161.
  7. Quantum Fluctuations of the Transverse Field Ising Chain, A. W. Kinross, M. Fu, T. J. Munsie, H. A. Dabkowska, G. M. Luke, S. Sachdev, and T. Imai, Physical Review X 4, 031008 (2014); arXiv:1401.6917.