Convexity of the Phase Boundary in the BCS Model with Imaginary Magnetic Field
Vol. 31 (2024), No. 1, Page 1-75.
Kashima, Yohei
Convexity of the Phase Boundary in the BCS Model with Imaginary Magnetic Field
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Abstract:
We study geometric properties of the domain of the two parameters (inverse temperature, imaginary magnetic field) where the gap equation of the BCS model with imaginary magnetic field has a positive solution. If the interaction is weak and the free dispersion relation is non-vanishing, the domain is a disjoint union of periodic copies of one representative set in the plane of (inverse temperature, imaginary magnetic field). In this paper we provide a necessary and sufficient condition for the representative set to be convex as the main result. More precisely we prove the following. The representative set is convex for any weak coupling and non-vanishing free dispersion relation if and only if the minimum of the magnitude of the free dispersion relation over the maximum is larger than the critical value $\sqrt{9-4\sqrt{5}}$. In the context of dynamical quantum phase transition (DQPT) the imaginary magnetic field is considered as the real time variable. So this is an analysis of the phase boundary of a DQPT in the plane of (inverse temperature, real time). In particular convexity of the representative phase boundary is characterized by the critical constant $\sqrt{9-4\sqrt{5}}$. The gap equation rigorously derived in the preceding paper [Y. Kashima, J. Math. Sci. Univ. Tokyo $ {\bf 28} $ (2021), 399-556] is at the core of our analysis.
Keywords: The BCS model, dynamical quantum phase transition, phase boundary, convexity.
Mathematics Subject Classification (2020): Primary 82D26; Secondary 82C55.
Received: 2023-08-07
