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We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.[S0031-9007 (96)02173-4] PACS numbers: 05.50.+ q, 64.60.Cn, 75.10.Hk, 75.40.CxSince Onsager's famous solution of the square lattice Ising model in the absence of a magnetic field [1], the Ising model became a standard model for testing the scaling and universality hypotheses [2,3]. However, the Ising model in a magnetic field has not been solved exactly so far, although some exact results are known. Of particular interest is to determine the critical line in the ͑h, T ͒ plane, along which the free energy becomes singular. Yang and Lee [4] proved that for the Ising ferromagnet, the critical line is located at h 0. For the Ising antiferromagnet, the series expansion method was used to obtain related information [5]. Müller-Hartmann and Zittartz obtained the critical line by considering the interfacial tension [6,7]. Wu and coworkers [8,9] formulated the Ising model on the honeycomb lattice as an 8-vertex model and identified the critical line as a locus invariant under a generalized weak-graph transformation. In this paper we introduce a new approach by considering zeros of the partition function on an elementary cycle and using Griffiths' smoothness postulate [10]. A closed-form formula of the critical line of an Ising antiferromagnet is obtained, which is in good agreement with the numerical results [8,11,12].The partition function of the Ising model in the presence of a magnetic field is given bywhere S i 61 and K ij is the interaction strength. The sum over ͗ij͘ runs over the pairs of nearest neighbors on the lattices. Let us consider the ferromagnetic case, K ij . 0. The Ising partition function on an elementary cycle of the triangular, square, and honeycomb lattices (see Fig. 1) can be written, respectively, as z t 2͓e b͑K

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The results of renormalization-group (RG) analysis in reciprocal space are reported for solid-solid phase transitions. We list the Landau-Ginzburg-Wilson Hamiltonian densities for six-component order parameters. The resulting RG recursion relations are derived by means of symmetric products of basic invariants. A list of fixed points for the densities is given and the stable fixed points are noted. Only two of the effective Hamiltonian densities possess stable fixed points. %'e indicate those specific structural transitions which are allowed to be continuous as determined by RG methods. We give critical exponents for the stable fixed points.

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Applications of symmetry to the renormalization-group method are discussed. The six-dimensional representations of space groups and their associated Hamiltonians are investigated using the Toledano-Michel symmetry criterion for stability. It is found that only two potentials have stable fixed points. One of these arises from a newly identified space-group image.

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