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Register a new userThe solution to the BCS gap equation for superconductivity and its temperature dependence
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Authors
Shuji Watanabe
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From the viewpoint of operator theory, we deal with the temperature dependence of the solution to the BCS gap equation for superconductivity. When the potential is a positive constant, the BCS gap equation reduces to the simple gap equation. We first show that there is a unique nonnegative solution to the simple gap equation, that it is continuous and strictly decreasing, and that it is of class $C^2$ with respect to the temperature. We next deal with the case where the potential is not a constant but a function. When the potential is not a constant, we give another proof of the existence and uniqueness of the solution to the BCS gap equation, and show how the solution varies with the temperature. We finally show that the solution to the BCS gap equation is indeed continuous with respect to both the temperature and the energy under a certain condition when the potential is not a constant.
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One of long-standing problems in mathematical studies of superconductivity is to show that the solution to the BCS gap equation is continuous in the temperature. In this paper we address this problem. We regard the BCS gap equation as a nonlinear integral equation on a Banach space consisting of continuous functions of both $T$ and $x$. Here, $T (geq 0)$ stands for the temperature and $x$ the kinetic energy of an electron minus the chemical potential. We show that the unique solution to the BCS gap equation obtained in our recent paper is continuous with respect to both $T$ and $x$ when $T$ is small enough. The proof is carried out based on the Banach fixed-point theorem.
In the preceding work cite{watanabe3}, it is shown that the solution to the BCS gap equation for superconductivity is continuous with respect to both the temperature and the energy under the restriction that the temperature is very small. Without this restriction, we show in this paper that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotonically decreasing with respect to the temperature.
We show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation for superconductivity. Here the temperature belongs to the closed interval $[0,, tau]$ with $tau>0$ nearly equal to half of the transition temperature. We show that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotone decreasing with respect to the temperature. Moreover, we show that the solution is partially differentiable with respect to the temperature twice and the second-order partial derivative is continuous with respect to both the temperature and the energy, or that the solution is approximated by such a smooth function.
For the BCS equation with local two-body interaction $lambda V(x)$, we give a rigorous analysis of the asymptotic behavior of the critical temperature as $lambda to 0$. We derive necessary and sufficient conditions on $V(x)$ for the existence of a non-trivial solution for all values of $lambda>0$.
We present a state-of-the-art x-ray diffraction study of the charge density wave order in 1T-TaS2 as a function of temperature and pressure. Our results prove that the charge density wave, which we characterize in terms of wave vector, amplitude and the coherence length, indeed exists in the superconducting region of the phase diagram. The data further imply that the ordered charge density wave structure as a whole becomes superconducting at low temperatures, i. e, superconductivity and charge density wave coexist on a macroscopic scale in real space. This result is fundamentally different from a previously proposed separation of superconducting and insulating regions in real space and, instead, provides evidence that the superconducting and the charge density wave gap exist in separate regions of reciprocal space.
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