Scaling with a Parameter in Spin Systems near the Critical Point. I*)

The Hamiltonian of the form.!] { =.!] { 0 +)...!] { 1 is discussed, where).. is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter).. is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(s,)..) =S2-aF()..js<P) for the singular part of the free energy near the critical point Tc (S = (T- Tc) I Tc), the expression (or explicit value) of the critical exponent ¢ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation ¢=r is found for change of dimensionality, where r is the critical exponent of the suscepti-bility in the unperturbed Hamiltonian.!]{0 • An extension to dynamical critical phenomena is also discussed briefly, particularly in connection with the critical slowing down. A possibility is suggested to derive this generalized scaling with the parameter).. by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter)..