Superconductivity in the Fibonacci Chain
arxiv(2024)
Abstract
Superconductivity was recently reported in several quasicrystalline systems.
These are materials which are structurally ordered, but since they are not
translationally invariant, the usual BCS theory does not apply. At the present
time, the underlying mechanism and the properties of the superconducting phase
are insufficiently understood. To gain a better understanding of quasiperiodic
superconductors, we consider the attractive Hubbard model on the Fibonacci
chain, and examine its low-temperature superconducting phase in detail using
the Bogoliubov-de Gennes mean-field approach. We obtain superconducting
solutions as a function of the parameters controlling the physical properties
of the system: the strength of the Hubbard attraction U, the chemical
potential μ, and the strength of the modulation of the Fibonacci
Hamiltonian, w. We find that there is a bulk transition at a critical
temperature that obeys a power law in U, where all sites become
superconducting at once. All sites with the Fibonacci chain have the same
superconducting gap width. The local superconducting order parameter is
self-similar both in real and perpendicular space. The interplay between the
Hubbard attraction and the intrinsic gaps of the Fibonacci chain results in a
complex zero-temperature μ-U phase diagram with insulating domes
surrounded by superconducting regions. Finally, we show that tuning w from
weak to strong quasicrystalline modulation gives rise to qualitatively
different thermodynamic behaviors as could be observed by measuring the
specific heat.
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