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arXiv:2610.30581

Cluster property and entanglement in the superconducting state

Katsuhiko Higuchi, Masahiko Higuchi

cond-mat.supr-conquant-phSubmitted Oct 22, 2026• Announced Oct 23, 2026• 0 citations
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Abstract

We present the conditions for satisfying the cluster property in the superconducting state. Using these conditions, we show that the cluster property employed when deriving the order parameter of the superconducting state (OPSS) holds true, and that the OPSS is given by ${}_S\langle Ψ| \hatψ(x)\hatψ(x') | Ψ\rangle_S$, where $|Ψ\rangle_S$ is the superconducting state, and where $\hatψ(x)$ is the field operator of superconducting electrons. This provides a resolution to the long-standing concern raised by Leggett et al.\ regarding particle-number fluctuation in the superconducting state. In addition, by considering the entanglement between two particles composed of the maximum geminal of the BCS ground state [J.\ Phys. Commun. 5, 095003 (2021)], it is confirmed that the larger the deviation from the conditions for satisfying the cluster property becomes, the greater the magnitude of the entanglement becomes. This shows not only the effectiveness of the conditions for satisfying the cluster property but also the justification of the OPSS that has been commonly used.

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Journal of Physics Communications•Apr 1, 2024•0 citations
We present the conditions for satisfying the cluster property in the superconducting state. Using these conditions, we show that the cluster property employed when deriving the order parameter of the superconducting state (OPSS) holds true, and that the OPSS is given by S⟨Ψ∣ψ^(x)ψ^(x′)∣Ψ⟩S{}_S\langle Ψ| \hatψ(x)\hatψ(x') | Ψ\rangle_SS​⟨Ψ∣ψ^​(x)ψ^​(x′)∣Ψ⟩S​, where ∣Ψ⟩S|Ψ\rangle_S∣Ψ⟩S​ is the superconducting state, and where ψ^(x)\hatψ(x)ψ^​(x) is the field operator of superconducting electrons. This provides a resolution to the long-standing concern raised by Leggett et al.\ regarding particle-number fluctuation in the superconducting state. In addition, by considering the entanglement between two particles composed of the maximum geminal of the BCS ground state [J.\ Phys. Commun. 5, 095003 (2021)], it is confirmed that the larger the deviation from the conditions for satisfying the cluster property becomes, the greater the magnitude of the entanglement becomes. This shows not only the effectiveness of the conditions for satisfying the cluster property but also the justification of the OPSS that has been commonly used.
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 O(N).
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 O N .
A generic Hamiltonian, which incorporates the effect of the orbital contraction on the hopping amplitude between nearest sites, is studied both analytically at the weak coupling limit and numerically at the intermediate and strong coupling regimes for a finite atomic cluster. The effect of the orbital contraction due to hole localization at atomic sites is specified with two coupling parameters V and W (multiplicative and additive contraction terms). The singularity of the vertex part of the two-particle Green’s function determines the critical temperature Tc and the relaxation rate Γ(T) of the order parameter at temperature above Tc. Unlike the case in conventional BCS superconductors, Γ has a non-zero imaginary part which may influence the fluctuation conductivity of the superconductor above Tc. We compute the ground state energy as a function of the particle number and magnetic flux through the cluster, and show the existence of the parity gap Δ appearing at the range of system parameters consistent with the appearance of the Cooper instability. Numeric calculation of the Hubbard model (with U>0) at arbitrary occupation does not show any sign of superconductivity in a small cluster.
Spontaneous symmetry breaking is well known to be a macroscopic phenomenon in quantum physics in many body systems. The essential features of this phenomenon are the energy degeneracy and the existence of the local operator that can modify the degenerate states. Due to these properties the lowest energy state is not stable. In order to obtain the stable ground state we introduce the explicitly breaking interaction, by which it is difficult for the local operator to modify the ground state. In the superconducting qubit there exists the stable eigenstate with the definite charge number. We point out that in this system, the local operator can change this state to another state. As results the cluster property violates, which means that the events in the far distance are not irrelevant. In this paper we examine the cluster property in the superconducting qubit. Then we propose a method by cavity quantum electrodynamics with two cavities in order to observe the violation.
The entanglement between occupation numbers of different single particle basis states depends on coupling between different single particle basis states in the second-quantized Hamiltonian. Thus, in principle, interaction is not necessary for occupation-number entanglement to appear. However, in order to characterize quantum correlation caused by interaction , we use the eigenstates of the single-particle Hamiltonian as the single particle basis upon which the occupation-number entanglement is defined. Using this so-called proper single particle basis, if there is no interaction, the many-particle second-quantized Hamiltonian is diagonalized and thus cannot generate entanglement, while its eigenstates can always be chosen to be non-entangled. If there is interaction, entanglement in the proper single particle basis arises in energy eigenstates and can be dynamically generated. Using the proper single particle basis, we discuss occupation-number entanglement in important eigenstates, especially ground states, of systems of many identical particles, in exploring insights the notion of entanglement sheds on many-particle physics. The discussions on Fermi systems start with Fermi gas, the Hartree–Fock approximation and the electron–hole entanglement in excitations. In the ground state of a Fermi liquid, in terms of the Landau quasiparticles, entanglement becomes negligible. The entanglement in a quantum Hall state is quantified as − f ln f − (1 − f )ln(1 − f ), where f is the proper fractional part of the filling factor. For BCS superconductivity, the entanglement is a function of the relative momentum wavefunction of the Cooper pair g k , and is thus directly related to the superconducting energy gap, and vanishes if and only if superconductivity vanishes. For a spinless Bose system, entanglement does not appear in the Hartree–Gross–Pitaevskii approximation, but becomes important in the Bogoliubov theory, as a characterization of two-particle correlation caused by the weak interaction. In these examples, the interaction-induced entanglement as calculated is directly related to the macroscopic physical properties.
Abstract From the viewpoint of the Bose–Einstein condensation (BEC) of the fermion system, the maximum geminal of the second-order reduced density matrix of the superconducting state exactly corresponds to the Cooper pair. In this paper the entanglement entropy (EE) for the maximum geminal of the BCS ground state is evaluated. The EE behaves logarithmically with respect to the number of the maximum geminal. Furthermore, the disappearance point of superconductivity is defined on the basis of the fermion BEC. In the superconducting ground state, almost all electrons in the energy width of the gap parameter near the Fermi level are condensed as a maximum geminal. They suddenly change to normal electrons with a finite gap of the EE at the disappearance point like a first-order phase transition.