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Physical review. B, Solid state

Exact thermodynamics of a generalized compressible Ising model in one dimension

Gregory S. Um, P. J. Gambardella, Max Dresden

Published Nov 1, 1976• 2 citations
Publisher

Abstract

We propose a generalized compressible Ising model and solve it exactly, using the renormalization-group method. We obtain exact expressions for free energy and spin-spin correlation functions for periodic and free boundary condition in a finite chain or in the thermodynamic limit. Through this method, we develop functional recursion relations. The corresponding fixed points are investigated with the correlation length. Our method is applicable to higher-order correlation functions.

DOI
10.1103/physrevb.14.4153

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Thermodynamic functions of an nnn-component magnetic system coupled to an isotropic elastic medium in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are calculated to leading order in $\ensuremath{\epsilon}$. Solutions to renormalization-group recursion relations correct to $O(\ensuremath{\epsilon})$ along with trajectory-integral methods are employed in these calculations. For n> 2n>~2n> 2 the finite compressibility modifies the critical-point behavior producing new corrections to scaling in the thermodynamic functions which are given. In the n=1n=1n=1 case renormalization-group predictions of a first-order transition are verified. Thermodynamic functions are given and analyzed in this case. In particular the transition temperature and discontinuity in order parameter are determined and the phase diagram is constructed.
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By explicit construction we show the existence of an exact renormalization group for triangular Ising models with only nearest-neighbor interactions. The recursion relations take the form of a set of three quasilinear first-order partial differential equations for the interactions. We determine a nontrivial fixed point and study the linearized flow around it. This yields the specific-heat exponent $\ensuremath{\alpha}=0$, in agreement with the Onsager and Houtappel solutions, and demonstrates universality. The free energy is expressed as the trajectory integral of an explicitly given function of the interactions.