Bibliography

It has been shown that there is a phase transition for sufficiently large $\lambda$ , see J. Glimm, A. Jaffe and T. Spencer, ``Phase Transitions for $\varphi_{2}^{4}$ Quantum Fields'', Commun. Math. Phys. 45, (1975), 203-216. This, plus the fact that the exact solution for critical point of the two-dimensional Ising model is known, guarantees that there is a critical curve starting at some sufficiently large value of $\lambda$ . There are also expansions that are both convergent and analytical near the Gaussian point, see J. Glimm, A. Jaffe and T. Spencer, ``A Convergent Expansion about Mean Field Theory I. The Expansion'', Ann. Physics, 101, (1976), 610-630, and J. Glimm, A. Jaffe and T. Spencer, ``A Convergent Expansion about Mean Field Theory II. Convergence of the Expansion'', Ann. Physics, 101, (1976), 631-669. This indicates that there is no critical behavior of the model in a neighborhood of the Gaussian point, and that therefore the critical curve does not connect with that point.

29

For the standard treatment in statistical mechanics of the one-site and several-site approximations for ferromagnetic systems, see: J. S. Smart, ``Effective Field Theories of Magnetism'', Saunders, (1966), and references therein. For a discussion of the method in the context of quantum field theory, see: E. Brézin, J. C. Le Guillou and J. Zinn-Justin, ``Field Theoretical Approach to Critical Phenomena'', in ``Phase Transitions and Critical Phenomena'', volume 6, C. Domb and M. S. Green, editors, Academic Press, (1976).

30

The basic idea here is that the partition function of finite systems is an analytic function and that, therefore, the systems cannot display the behaviors of non-differentiability and even of discontinuity that are characteristic of phase transitions, see: C. N. Yang and T. D. Lee, ``Statistical Theory of Equations of State and Phase Transitions I. Theory of Condensation'', Phys. Rev. 87, (1952), 404-409; T. D. Lee and C. N. Yang, ``Statistical Theory of Equations of State and Phase Transitions II. Lattice Gas and Ising Model'', Phys. Rev. 87, (1952), 410-419.

31

In one dimension, see: D. Rouelle, ``Statistical Mechanics of a One-Dimensional Lattice Gas'', Commun. Math. Phys. 9, (1968), 267-278; F. Dyson, ``Non-Existence of Spontaneous Magnetization in a One-Dimensional Ising Ferromagnet'', Commun. Math. Phys. 12, (1969), 212-215. In two dimensions, see: D. Mermin and H. Wagner, ``Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models'', Phys. Rev. Lett. 17, (1966), 1133-1136; R. L. Dobrushin and S. B. Shlosman, ``Absence of Breakdown of Continuous Symmetry in Two-Dimensional Models of Statistical Physics'', Commun. Math. Phys. 42, (1975), 31-40.

32

Ibid.[2], chapter 2, section 2.

33

Ibid.[2], chapter 2, section 8.

34

Ibid.[2], chapter 2, section 11.

35

Ibid.[2], chapter 6, section 2.

36

The number quoted was obtained from M. N. Barber, R. B. Pearson, D. Toussaint and J. L. Richardson, Phys. Rev. B32 (1985), 1720; it can also be found in N. Ito and M. Suzuki, J. of the Phys. Soc. of Japan 60 (1991), 1978-1987.

37

The number quoted is $\ln\!\sqrt{1+\sqrt{2}}$ and was obtained from T. L. Hill, ``Statistical Mechanics'', McGraw-Hill, (1956, reprinted by Dover, 1987), page 329.

38

``Numerical Solution for the Mean-Field Critical Curves of the $SO(\mathfrak{N})$ Polynomial $\lambda \varphi ^{4}$ Models'', http://latt.if.usp.br/scientific-pages/nsmfccsonpm/

39