An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism
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An Extension of the Cell-Construction Method for the Flat-Band Ferromagnetism Akinori Tanaka1 Received: 26 April 2020 / Accepted: 5 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present an extension of the cell-construction method for the flat-band ferromagnetism. In a rather general setting, we construct Hubbard models with highly degenerate single-electron ground states and obtain a formal representation of these single-electron ground states. By our version of the cell-construction method, various types of flat-band Hubbard models, including the one on line graphs, can be designed and shown to have the unique ferromagnetic ground states when the electron number is equal to the degeneracy of the single-electron ground states. Keywords Hubbard model · Ferromagnetism · Flat band · Cell construction
1 Introduction Rigorous results on quantum many-body systems, even if they are obtained with some special conditions, provide us with an understanding of mechanisms for phenomena arising from the interplay between the quantum mechanical motion of particles and the interactions among them. One of the examples is flat-band ferromagnetism found in a class of Hubbard models with highly degenerate single-electron ground states [1,2]. Flat-band ferromagnetism, which was first discovered by Mielke [3–5] and Tasaki [6], explains clearly how the spinindependent Coulomb repulsion combined with the Pauli exclusion principle for electrons generates ferromagnetism. Furthermore the mechanism turns out to work in more general settings. In fact, it is shown that flat-band ferromagnetism is stable against perturbations which change the flat band into a dispersive band [7–13]. Examples of Hubbard models which exhibit metallic ferromagnetism are also derived by taking into account the mechanism of flat-band ferromagnetism [14,15]. The idea has been further applied to more fascinating problems of topological systems [16,17].
Communicated by Hal Tasaki.
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Akinori Tanaka [email protected] Department of General Education, National Institute of Technology, Ariake College, Omuta, Fukuoka 836-8585, Japan
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A. Tanaka
The class of flat-band ferromagnetism studied by Mielke was found in the flat-band Hubbard models defined on line graphs. The result of Mielke was obtained by using some ideas from graph theory and is summarized in the theorem in which the structure of graphs is related to the occurrence of ferromagnetism in the models [4,5]. On the other hand, the class of Tasaki’s flat-band models was constructed by using the cell-construction method [6,18]. Lattices of Tasaki’s flat-band models are constructed by assembling cells each of which has one internal site and several external sites. Some external sites from different cells are identified and are regarded as a single site to form the whole lattice. It is noted that every internal site belongs to exactly one cell in Tasaki’s cell-construction. Due to this property one can obtain explicit expressions for the single-electron gro
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