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American Journal of Physics

Mere coincidence?

Holger Friedrich

Published Sep 1, 1994• 0 citations
Publisher
DOI
10.1119/1.17457

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Scattering properties and time delays for general (nonsymmetric) potentials in terms of the respective S-matrices are discussed paradigmatically in one dimension and in comparison to symmetric potentials. Only for the latter the Wigner and Smith time delays coincide. Considering asymmetric potentials also reveals that only one version of S-matrices used in the literature (the one with reflection coefficients on the diagonal) generalizes to the asymmetric case. Finally, we give a criterion how to identify a potential with intrinsic symmetry that behaves like an asymmetric one if it is merely offset from the scattering center.
It is shown that for the spin 1/2 anisotropic Heisenberg model the result for the transition temperature is completely insensitive to the anisotropy within the Onsager reaction field theory, which yields a vanishing T_c in two dimensions, in total conflict with the expected finite T_c, increasing with anisotropy. This establishes that the breakdown of the Onsager reaction field theory in two dimensions is not limited to the Ising model, but actually extends over the whole range of anisotropy. Therefore, for the isotropic case, the results of vanishing T_c and exponential dependence of spin correlation length should be seen as mere coincidence.
We introduce a measure called reduced entropy of sublattice to quantify entanglement in spin, electron, and boson systems. By analyzing this quantity, we reveal an intriguing connection between quantum entanglement and quantum phase transitions in various strongly correlated systems: the local extremes of reduced entropy and its first derivative as functions of the coupling constant coincide, respectively, with the first and second order transition points. Exact numerical studies merely for small lattices reproduce several well-known results, demonstrating that our scenario is quite promising for exploring quantum phase transitions.