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

概要

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The Hamiltonian of the form h=h_0+λH_1 is discussed, where λ is a parameter to change symmetry, dimensionality, or potential range. Scaling with the parameter λ is studied for thermodynamic quantities such as the free energy. By assuming the scaled form F(ε,λ)=ε^<2-α>F(λ/ε^φ) for the singular part of the free energy near the critical point T_c(ε=(T-T_c)/T_c), the expression (or explicit value) of the critical exponent φ is obtained in each case of change of symmetry, dimensionality and potential range. In particular, the universal relation φ=γ is found for change of dimensionality, where δ is the critical exponent of the susceptibility in the unperturbed Hamiltonian H_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 λ by applying the usual scaling law to each term of the perturbational expansion with respect to the parameter λ.
理論物理学刊行会の論文
1971-10-25