Abstract
The aim of the current paper is to illustrate, in a simple example, our recent, very general, rigorous results [1, 2] on the dynamical properties of fermions and quantum-spin systems with long-range, or mean-field, interactions, in infinite volume. We consider here the strong-coupling BCS-Hubbard model studied in [3, 4], because this example is very pedagogical and, at the same time, physically relevant for it highlights the impact of the (screened) Coulomb repulsion on (\(s\)-wave) superconductivity.
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Notes
The positivity of \({{\gamma }_{{k,q}}}\) imposes constraints on the choice of the function \(f\).
In the weak \(^{ * }\) topology.
In the weak \(^{ * }\) topology.
More precisely, it converges to the barycenter of a Choquet measure.
This implies that any solution \(\left| d \right|\) to the variational problem (8) must have the same absolute value.
For instance, \(\left( {1,0,0,0} \right)\) is the vacuum; \(\left( {0,1,0,0} \right)\) and \(\left( {0,0,1,0} \right)\) correspond to one fermion with spin \( \uparrow \) and \( \downarrow \), respectively; \(\left( {0,0,0,1} \right)\) refers to two fermions with opposite spins.
The product state \({{\rho }^{{(L)}}}\) is (well-) defined by \({{\rho }^{{(L)}}}({{\alpha }_{{{{x}_{1}}}}}({{A}_{1}}) \cdots {{\alpha }_{{{{x}_{n}}}}}({{A}_{n}}))\) = \(\rho ({{A}_{1}}) \cdots \rho ({{A}_{n}})\) for all \({{A}_{1}}, \ldots ,{{A}_{n}} \in B\left( {{{F}_{{\{ 0\} }}}} \right)\) and all \({{x}_{1}}, \ldots ,{{x}_{n}} \in {{\Lambda }_{L}}\) such that \({{x}_{i}} \ne {{x}_{j}}\) for \(i \ne j\), where \({{\alpha }_{{{{x}_{j}}}}}({{A}_{j}}) \in B\left( {{{F}_{{\{ {{x}_{j}}\} }}}} \right)\) is the \({{x}_{j}}\)-translated copy of \({{A}_{j}}\) for all \(j \in \{ 1, \ldots ,n\} \).
Even states are the physically relevant ones. Even means that the expectation value of any odd monomials in \({{\left\{ {a_{{0,s}}^{ * },{{a}_{{0,s}}}} \right\}}_{{s \in \{ \uparrow , \downarrow \} }}}\) with respect to the on-site state \(\rho \) is zero.
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Funding
This work is supported by CNPq (308337/2017-4), FAPESP (2017/22340-9), as well as by the Basque Government through the grant IT641-13 and the BERC 2018-2021 program, and by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718, MTM2017-82160-C2-2-P.
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Bru, JB., de Siqueira Pedra, W. Macroscopic Dynamics of the Strong-Coupling BCS-Hubbard Model. Phys. Part. Nuclei 51, 802–806 (2020). https://doi.org/10.1134/S106377962004019X
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DOI: https://doi.org/10.1134/S106377962004019X