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Mattis and E. H. Lieb, Rev. Mod. , 36 (1964) 856. 2 R. Peierls, Proc. Cambridge Phil. , 32 (1936) 477. , to an external field of order 1/ N ). At zero temperature, the only contribution to the partition function comes from a configuration in which all spins point up. At higher temperatures, some bubbles of down spins will appear. A bubble of linear dimension L has a frontier with about L d−1 spins on its edge, in a lattice of dimension d. The interaction energy between two up spins being the same as for two down spins, the energy ‘cost’ of making such a bubble is of order 2J L d−1 .
68) Therefore, like in the spin models, we have to find the eigenvalues of the transfer matrix T . 69) Derivation of the GKS inequalities 49 in which p is the usual momentum operator, up to a multiplicative constant d/2 −iπd/4 m e which has to be included in a definition of the measure for 2π the integral over the intermediate qn . 71) p2 with the Hamiltonian H = 2m + V (q). The Schrödinger equation, which gives the eigenvalues of the Hamiltonian, is thus a means to diagonalize the transfer matrix for this path integral.
Therefore, bubbles are favoured and the ordered zero-temperature state is destroyed for any positive temperature. In dimensions greater than one, the energy cost wins compared to the entropy and the system will remain ordered up to some finite temperature. I shall now give a real proof of the existence of a spontaneous magnetization in the two-dimensional case below some finite temperature (following Ginibre’s presentation of Peierls’ proof). Again, we consider configurations in which the spins are frozen and ↑ on the boundary.