SCALING RELATIONS - OLD AND NEW

Ralph Kenna

AMRC, Coventry University

By the early 1960's theoretical and experimental advances in Statistical Mechanics had established the existence of a range of universality classes for systems with second-order phase transitions. These universality classes are characterized by critical exponents which are different to the mean-field (classical) ones. The next crucial theoretical achievement was the discovery (advanced especially by Essam and Fisher) of four (now famous) scaling relations between the six critical exponents describing second-order criticality. These scaling relations are of fundamental importance and now form a cornerstone of Statistical Mechanics and other areas. In the first part of these lectures, the scaling relations will be introduced and a history of their discovery will be given. Separately, in the 1950's Lee and Yang introduced a theoretical way to conceptualize phase transitions. This involves allowing the parameters controlling the system (the temperature or magnetic field) to become complex. In the 1960's and 1970's, Abe and Suzuki used the fact that the even (temperature) and odd (magnetic) scaling fields can be linked by Lee-Yang zeros to rederive the scaling relations. In the second part of these lectures, the theory of complex Lee-Yang zeros will be discussed and Abe's and Suzuki's achievements will be highlighted. Very recently, new theories concerning scaling relations have been established. These theories were established through the medium of Lee-Yang analyses. These theories will be outlined and a famous long-standing puzzle of Statistical Mechanics will be resolved.
Historical Articles:
[1] J.W. Essam, M.E. Fisher. Pade approximant studies of the lattice gas and Ising ferromagnet below the critical point. J. Chem. Phys. 38 (1963) 802.
[2] C.N. Yang, T.D. Lee. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87 (1952) 404; Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. ibid. 410.
[3] R. Abe. Note on the critical behavior of Ising ferromagnets. Prog. Theor. Phys. 38 (1967) 72.
[4] M. Suzuki. A theory on the critical behaviour of ferromagnets. Prog. Theor. Phys. 38 (1967) 289; A theory of the second order phase transitions in spin systems. II: Complex magnetic field. ibid. 1225.
Reviews:
[5] M.E. Fisher. Renormalization group theory: its basis and formulation in statistical physics. Rev. Mod. Phys. 70 (1998) 653.
[6] F. Ravndal. Scaling and Renormalization Groups. Nordisk Inst. for Teoretisk Atomfysik, 1976.
Recent Developments:
[7] W. Janke, D.A. Johnston, R. Kenna. Properties of higher-order phase transitions. Nucl. Phys. B, 736 (2006) 319.
[8] R. Kenna, D.A. Johnston, W. Janke. Scaling Relations for Logarithmic Corrections. Phys. Rev. Lett. 96 (2006) 115701.
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