Subexponentially Growing Hilbert Space and Nonconcentrating Distributions in a Constrained Spin Model
Motivated by recent experiments with two-component Bose–Einstein condensates, we study fully-connected spin models subject to an additional constraint. The constraint is responsible for the Hilbert space dimension to scale only linearly with the system size. We discuss the unconventional statistical physical and thermodynamic properties of such a system, in particular the absence of concentration of the underlying probability distributions. As a consequence, expectation values are less suitable to characterize such systems, and full distribution functions are required instead. Sharp signatures of phase transitions do not occur in such a setting, but transitions from singly peaked to doubly peaked distribution functions of an “order parameter” may be present.
KeywordsFully-connected spin model Canonical ensemble Concentration of measure
The authors acknowledge useful discussions with Markus Oberthaler and Hugo Touchette. J.R.W. acknowledges financial support through the internship programme of the National Institute for Theoretical Physics (South Africa). M.K. acknowledges financial support from the National Research Foundation of South Africa via the Competitive Programme for Rated Researchers.
- 4.Jensen, H.J., Pazuki, R.H., Pruessner, G., Tempesta, P.: Statistical mechanics of exploding phase spaces. arXiv:1609.02065