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Abstract
We present the explicit forms of the maximum eigenvalue and the corresponding eigenfunction for the second-order reduced density matrix (RDM2) of the BCS superconducting state (SS). Using these quantities, we deal with two topics in the present paper. As the first topic, it is shown that the cluster decomposition principle holds in the BCS-SS. This proof gives a theoretical foundation that the abnormal density can be chosen as the order parameter of the SS. As the second topic, it is shown that such an eigenfunction is spin singlet and spatially extends isotopically, and further that the mean distance of two electrons which consists of the above eigenfunction is in a good agreement with Pippard’s coherence length. This means that maximum geminal of the RDM2 of the BCS-SS can be regarded as the Cooper pair itself which are condensed to the same energy level in a number of
Highlights
In a lot of first-principles calculations of the superconductivity, the abnormal density (r ) (r ) is generally regarded as an order parameter of the superconducting state (SS), where (r ) is the field operator of electrons and denotes a SS [1, 2, 3, 4]
Cluster decomposition principle in the BCS superconducting state First, we shall prove that the cluster decomposition principle holds in the BCS-SS
Discussions and concluding remarks In this paper we deal with a long-standing suspicion as to whether the abnormal density is appropriate as the order parameter of SS or not
Summary
In a lot of first-principles calculations of the superconductivity, the abnormal density (r ) (r ) is generally regarded as an order parameter of the superconducting state (SS), where (r ) is the field operator of electrons and denotes a SS [1, 2, 3, 4]. If the cluster decomposition principle holds in a state , the expectation value of the RDM2 is expressed as the product of expectation values of abnormal densities. Since the magnitude of the eigenvalue for the RDM2 means the occupation number of two-particle states [37, 38], and since the magnitude of the eigenvalue for equation (13) is O(N) (see Appendix B), vk 2 uk 2 corresponds to the maximum eigenvalue k of the RDM2 of the BCS-SS.
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