Original proofs of Mermin-Wagner-Hohenberg theorem
The Mermin-Wagner-Hohenberg theorem (Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interaction in one and two dimensions. In this talk, I will give the original proofs of Mermin-Wagner-Hohenberg theorem based on the Bogoliubov inequality. First, I will show that the Bogoliubov inequality may be used to rule out the existence of long-range order in Bose and Fermi systems for one and two dimensions at finite temperature. The results depend on the f-sum rule which restricts the spectrum and interaction of Bose and Fermi systems. Then, I will show that at any nonzero temperature, a one or two dimensional isotropic spin-S Heisenberg model with finite-range exchange interaction can be neither ferromagnetic or antiferromagnetic by the Bogoliubov inequality.
 P.C. Hohenberg (1967), Phys. Rev., 158(2): 383
 N. D. Mermin and H. Wagner (1966), Phys. Rev. Lett., 17(22): 1133-1136
 S. Coleman(1973), Commun. math. Phys. 31(4):259-264