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# High-Temperature series for the $RP^{n-1}$ lattice spin model (generalized Maier-Saupe model of nematic liquid crystals) in two space dimensions and with general spin dimensionality n

Physical review. B, Condensed matter (Impact Factor: 3.66). 12/1992; 46(17):11141-11144. DOI: 10.1103/PhysRevB.46.11141

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**ABSTRACT:**We show how to compute the generating function of the self-avoiding polygons on a lattice by using the statistical mechanics Schwinger-Dyson equations for the correlation functions of the $N$-vector spin model on that lattice. -
##### Article: Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

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**ABSTRACT:**We compute the phase diagram in the N→∞ limit for lattice RPN−1, CPN−1 and QPN−1σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=∞ limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N→∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions. - [Show abstract] [Hide abstract]

**ABSTRACT:**We have extended through beta^{23} the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover the expansions for the nearest-neighbor correlation function, the susceptibility and the second correlation moment have been extended up to beta^{25}. Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyper-universal combinations of critical amplitudes such as the renormalized four-point coupling g_r or the quantity usually denoted by R^{+}_{xi}. We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent theta is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin. Comment: 35 pages, latex, 21 figures, to appear in Phys. Rev. B