Communications in Mathematical Physics

, Volume 138, Issue 3, pp 487–505 | Cite as

Low temperature properties of the hierarchical classical vector model

  • Ricardo Schor
  • Michael O'Carroll


We obtain low temperature properties of the classical vector model in a hierarchical formulation in three or more dimensions. We consider the lattice model in a zero or non-zero magnetic field, where the single site spin variable ϕ∈Rv has a density proportional to\(e^{ - \lambda (\phi ^2 - 1)^2 } \) for large λ≦∞. Using renormalization group methods we obtain a convergent expansion for the free energy with zero magnetic field. For non-zero fields a shift formula is used to obtain the effective action generated by the renormalization group transformation (RGT). To obtain the pure state zero field free energy and spontaneous magnetization we take the thermodynamic limit together with the zero field limit at a specified rate. The spontaneous magnetization,m, is calculated, is non-zero and the pure state free energy coincides, as expected, with the zero field free energy. Also the sequence of zero field actions does not have a limit but we show that the sequence of actions generated from the original action shifted bym does; the limiting action corresponds to a non-canonical Gaussian fixed point of the RGT.


Pure State Spontaneous Magnetization Spin Variable Renormalization Group Method Zero Magnetic Field 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Ricardo Schor
    • 1
  • Michael O'Carroll
    • 1
  1. 1.Departamento de Física do ICExUniversidade Federal de Minas GeraisBelo HorizonteBrasil

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