## Superconducting Phase in the BCS Model with Imaginary Magnetic Field. III. Non-Vanishing Free Dispersion Relations

J. Math. Sci. Univ. Tokyo
Vol. 28 (2021), No. 3, Page 399-556.

Kashima, Yohei
Superconducting Phase in the BCS Model with Imaginary Magnetic Field. III. Non-Vanishing Free Dispersion Relations
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Abstract:
We analyze a class of the BCS model, whose free dispersion relation is non-vanishing, under the influence of imaginary magnetic field at positive temperature. The magnitude of the negative coupling constant must be small but is allowed to be independent of the temperature and the imaginary magnetic field. The infinite-volume limit of the free energy density is characterized. A spontaneous symmetry breaking and an off-diagonal long range order are proved to occur only in high temperatures. This is because the gap equation in this model has a positive solution only if the temperature is higher than a critical value. The proof is based on a double-scale integration of the Grassmann integral formulation. In this scheme we integrate with the infrared covariance first and with the ultra-violet covariance afterwards, which is opposite to the previous schemes in [Kashima, Y., J. Math. Sci. Univ Tokyo $28$ (2021), 1-179], [Kashima, Y., J. Math. Sci. Univ. Tokyo $28$ (2021), 181-398] or [13], [14] in short. As the other focus, we study geometric properties of the phase boundaries, which are periodic copies of a closed curve in the two-dimensional space of the temperature and the real time variable. Here we adopt the real time variable in place of the temperature times the imaginary magnetic field by considering its relevance within contemporary physics of dynamical phase transition at positive temperature. As the main result, we show that for any choice of a non-vanishing free dispersion relation the representative curve of the phase boundaries has only one local minimum point, or in other words the phase boundaries do not oscillate with temperature, if and only if the minimum of the magnitude of the free dispersion relation over the maximum is larger than the critical value $\sqrt{17-12\sqrt{2}}$. Overall we use the same notational conventions as in [13], [14]. So this work is a continuation of these preceding papers.

Keywords: The BCS model, gap equation, phase boundaries, spontaneous symmetry breaking, off-diagonal long range order, Grassmann integral formulation.

Mathematics Subject Classification (2010): Primary 82D55; Secondary 81T28.
Mathematical Reviews Number: MR4313642