A method for computing low--temperature series for renormalized operators in the two--dimensional Ising model is proposed. Series for the renormalized magnetization and nearest--neighbor correlation function are given for the majority rule transformation on 2 \\Theta 2 blocks and random tie--breaker. These series are applied to the study at very low temperature of the first--order phase transition undergone by this model. We analyze how truncation in the renormalized Hamiltonian leads to spurious discontinuities of the Renormalization Group transformation. Keywords: Renormalization group, position--space renormalization--group transformations, Ising model, low--temperature expansions. E-mail: salas@mafalda.physics.nyu.edu 1 Introduction The behavior of the Renormalization Group (RG) in the vicinity of first order phase transitions has been a very controversial matter for the last 20 years. In 1975 Nienhuis and Nauenberg [1] proposed that the RG transformations behave near first--or...
We apply a recently proposed real-space renormalization-group method to the two-dimensional square Ising model in a transverse magnetic field at zero temperature. We do the calculation to first and second order in the intercell coupling; in both cases we find a nontrivial fixed point. We compare the obtained critical exponents with accepted values for the three-dimensional Ising model from high-temperature series expansions. In first order the agreement is poor, in second order we obtain good agreement with the expected values.