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Finite--Size Scaling Analysis of Generalized Mean--Field Theories

We investigate families of generalized mean--field theories that can be formulated using the Peierls--Bogoliubov inequality. For test--Hamiltonians describing mutually non--interacting subsystems of increasing size, the thermodynamics of these mean--field type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean--field critical points. Finite--size scaling analysis of this mean--field critical behaviour allows to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki, which is thus given a transparent interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls--Bogoliubov inequality. In the case of the 2--$d$ Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity of thermodynamic functions.