J.-B. Bru, W. de Siqueira Pedra, and A.-S. D mel
A microscopic two-band model for the electron-hole asymmetry in high-$T_c$ superconductors and reentering behavior
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ABSTRACT. To our knowledge there is no rigorously analyzed microscopic model explaining the electron--hole asymmetry of the
critical temperature seen in high--$T_{c}$ cuprate superconductors -- at least no model not breaking artificially this symmetry.
We present here a microscopic two--band model based on the structure of energetic levels of holes in
$\mathrm{CuO}_{2}$ conducting layers of cuprates. In particular,
our Hamiltonian does not contain \emph{ad hoc} terms implying -- explicitly -- different masses for electrons and holes.
We prove that two energe\-tically near--lying interacting bands can explain
the electron--hole asymmetry.
Indeed, we rigorously
analyze the phase diagram of the model and show that
the critical temperatures
for fermion densities below half--filling can manifest a very different behavior as compared to the case of densities above half--filling. This fact results from the inter--band interaction
and intra--band Coulomb repulsion in interplay with thermal fluctuations between two energetic levels. So, if the energy difference between
bands is too big (as compared to the energy scale defined by the critical temperatures of
superconductivity) then the asymmetry disappears.
Moreover, the critical temperature turns out to be a non--monotonic function
of the fermion density and the phase diagram of our model shows
``superconducting domes'' as in high--$T_{c}$ cuprate
superconductors. This explains why
the maximal critical temperature is attained at donor densities away from the
maximal one. Outside the superconducting phase and for fermion densities
near half--filling the thermodynamics
governed by our Hamiltonian corresponds, as in real high--$T_c$ materials,
to a Mott--insulating phase.
The nature of the inter--band interaction can be
electrostatic (screened Coulomb interaction), magnetic
(for instance some Heisenberg--type one--site spin--spin interaction),
or a mixture of both.
If the inter--band interaction is predominately
magnetic then -- additionally to the electron--hole asymmetry -- we observe a reentering behavior meaning that the superconducting phase can only occur in a finite interval
of temperatures. This phenomenon is rather rare, but has also been observed in the so--called magnetic superconductors.
The mathematical results here
are direct consequences of [J.-B. Bru, W. de Siqueira Pedra, Rev.
Math. Phys. 22, 233--303 (2010)] which is reviewed in the introduction.