Finite q-differences and the Renormalization Group

Authors: Ayşe ERZAN

Abstract: We observe that the q-difference operator [1] and its inverse lend themselves naturally to describing systems with discrete dilatation symmetry, in particular, hierarchical lattices [2]. The homogeneity relation satisfied by the singular part of the free energy per bond can be written, D_\lambda f_{\rm sing}(u)=(\ell^d-1)f_{\rm sing}(u) /(\lambda -1) where u is the reduced inverse temperature, \lambda=\ell^y, y and d are the temperature critical index and effective dimensionality, respectively; \ell is the rescaling factor. Moreover, we show that the infinite sum which is the solution of this equation is simply given by f_{\rm sing}(u)= u^\psi \left[ {1\over 1-\lambda } \int_0^{\lambda u} {g_{\rm rem}(t) \over t^{1+\psi} } D_\lambda t +{1\over 1-\lambda^{-1}} \int_0^{\lambda^{-1} u} {g_{\rm rem}(t) \over t^{1+\psi} } D_{\lambda^{-1}} t\right] where g_{\rm rem} has the meaning in [3] and \psi=d/y. This provides a possibly new way to connect the q-differences to the generalized thermodynamics of Tsallis [4].

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