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Description
This work is devoted to the study of the scaling function $\Phi(\eta)$, associated with the free energy of the Ising model in the region $T \rightarrow T_{c}$, $H \rightarrow 0$, using numerical data obtained by the truncated free fermion space approach method (TFFSA method). Using this method, an estimate of some parameters of this function is provided, including the position of the Yang-Lee singularity and the leading amplitudes of the associated singular expansion. The analyticity of the scaling function and the proof of the “extended analyticity” hypothesis were tested.
The numerical TFFSA method is implemented using the Python programming language. This is the technical part of this research work.
Also, the work provides a full justification for why the proof of this analyticity is an important issue. A brief rationale is presented below.
As is known, in the classical theory of phase transitions, which does not take into account fluctuation effects, the Langer feature of low-temperature free energy at $H = 0$ is absent. Instead, the analytic continuation of the classical free energy at low T shows a singularity with a branching at some finite negative value of $H$, $H = -H_{SP}$, known as the "spinodal point". This feature arises because at sufficiently large $H$ $(|H| > H_{SP})$ the metastable state becomes unstable with respect to classical decay. When fluctuations are taken into account, the free energy becomes singular (even weakly singular) at $H = 0$, since the metastable phase is always prone to decay through nucleation. But what happens to the spinodal feature? This is a question that still remains open, and the authors of this research work, upon completion, hope to answer it.
