Higher Order Phase Transitions in the BCS Model with Imaginary Magnetic Field
Vol. 30 (2023), No. 2, Page 125–203.
Kashima, Yohei
Higher Order Phase Transitions in the BCS Model with Imaginary Magnetic Field
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Abstract:
In the BCS model with imaginary magnetic field at positive temperature we provide necessary and sufficient conditions for existence of a higher order phase transition driven by temperature. We define the order of the phase transition by regularity of the extended free energy density with temperature. More precisely we prove the following. There exist a non-vanishing free dispersion relation and a weak coupling constant such that a temperature-driven phase transition of order $n\in 4\mathbb{N}+2$ $(=\{6,10,14,\cdots\})$ occurs if and only if the minimum of the magnitude of the free dispersion relation over the maximum is less than or equal to the critical value $\sqrt{17-12\sqrt{2}}$. These statements are also proved to be equivalent to that there exist a non-vanishing free dispersion relation and a weak coupling constant such that the phase boundary varying with the inverse temperature has a stationary point of inflection. Moreover, it follows that for any non-vanishing free dispersion relation and weak coupling constant the temperature-driven phase transition is of 2nd order if and only if the minimum of the magnitude of the free dispersion relation over the maximum is larger than $\sqrt{17-12\sqrt{2}}$. We apply some key lemmas established in Section 2 of [Y. Kashima, J. Math. Sci. Univ. Tokyo $\bf 28$ (2021), 399--556]. So this work is a continuation of the section of the preceding paper.
Keywords: The BCS model, non-Hermitian Hamiltonian, higher order phase transition, stationary point of inflection.
Received: 2022-03-07
