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The diagonal Ising susceptibility. (English) Zbl 1118.82009
Summary: We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions $\chi ^{(1)}_{d}$ and $\chi ^{(2)}_{d}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution $\chi ^{(5)}_{d}$, but only modulo a given prime. We use these exact linear differential equations to show that not only the Russian-doll structure but also the direct sum structure on the linear differential operators for the n-particle contributions $\chi ^{(n)}_{d}$ are quite directly inherited from the direct sum structure on the form factors $f^{(n)}$. We show that the $n$th particle contributions $\chi ^{(n)}_{d}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh 2E_{v}/kT \sinh 2E_{h}/kT| = 1$ as $n \rightarrow \infty$.

82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
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