%0 Book Section %1 statphys23_0139 %A Tasaki, H. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K ferromagnetism hubbard model statphys23 topic-1 %T Hubbard model and the origin of ferromagnetism %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=139 %X It has been believed since Heisenberg that ferromagnetism observed in nature is generated by quantum effects and Coulomb interaction between electrons. It is a challenging problem to confirm this scenario by proving that only quantum many-body effects (of fermions) and spin-independent Coulomb repulsion can generate ferromagnetism in a concrete setting. The Hubbard model is an idealized tight-binding model of itinerant electrons which is suitable for such studies. More precisely the Hubbard model consists of electrons which hop between lattice sites according to a prescribed hopping amplitude $t_x,y$. One gets extra energy $U>0$ when two electrons share a same site. If one denotes by $c_x,\sigma$ and $n_x,\sigma$ the standard annihilation and number operators, respectively, for an electron at site $x$ with spin $\sigma=\uparrow,\downarrow$, the Hamiltonian of the general Hubbard model is written as displaymath H=\sum_x,y_\sigma=\uparrow,\downarrowt_x,y c^\dagger_x,\sigma\,c_y,\sigma +U\sum_xn_x,\uparrow\,n_x,\downarrow. displaymath One first fixes $t_x,y$ (according to the lattice structure) and the electron number, and examine the properties of the model. Here we are mainly interested in the magnetism of the ground states. In the present talk we shall briefly review rigorous results about ferromagnetism in the Hubbard model (as summarized below), placing main emphasis on our own contributions. 1) Nagaoka-Thouless ferromagnetism:\/ In 1965, Nagaoka and independently Thouless proved that the Hubbard model on certain lattices exhibit ferromagnetism when there is exactly one hole, and the Coulomb $U$ is infinitely large. This result is interesting, but hardly extends to non-singular models. 2) flatband ferromagnetism:\/ In 1991, Mielke found a completely different class of rigorous examples in which the single-electron ground states are highly degenerate. These models exhibit ferromagnetism for any $U>0$. In 1992, Tasaki also found similar examples. 3) ferromagnetism in non-singular models:\/ In 1995, by extending his flatband models, Tasaki presented rigorous example of ferromagnetism in the Hubbard model which has finite density of states and finite (but sufficiently large) $U>0$. It was also proved that the model has a normal spin-wave excitation. This means that we have obtained a class of non-singular models of itinerant electrons (with only spin-independent interactions) whose low energy behaviors are rigorously proved to be that of a ``healthy'' insulating ferromagnet. 4) metallic ferromagnetism in three dimensions:\/ All the rigorous results discussed above correspond to insulating systems. Metallic ferromagnetism, in which same set of electrons contribute both the conduction and magnetism, is clearly more important and interesting. In 2006, Tanaka and Tasaki presented the first rigorous example of metallic ferromagnetism in the Hubbard model in any dimensions.

@incollection{statphys23_0139, abstract = {It has been believed since Heisenberg that ferromagnetism observed in nature is generated by quantum effects and Coulomb interaction between electrons. It is a challenging problem to confirm this scenario by {\em proving that only quantum many-body effects (of fermions) and spin-independent Coulomb repulsion can generate ferromagnetism} in a concrete setting. The Hubbard model is an idealized tight-binding model of itinerant electrons which is suitable for such studies. More precisely the Hubbard model consists of electrons which hop between lattice sites according to a prescribed hopping amplitude $t_{x,y}$. One gets extra energy $U>0$ when two electrons share a same site. If one denotes by $c_{x,\sigma}$ and $n_{x,\sigma}$ the standard annihilation and number operators, respectively, for an electron at site $x$ with spin $\sigma=\uparrow,\downarrow$, the Hamiltonian of the general Hubbard model is written as \begin{displaymath} H=\mathop{\sum_{x,y}}_{\sigma=\uparrow,\downarrow}t_{x,y} c^\dagger_{x,\sigma}\,c_{y,\sigma} +U\sum_xn_{x,\uparrow}\,n_{x,\downarrow}. \end{displaymath} One first fixes $t_{x,y}$ (according to the lattice structure) and the electron number, and examine the properties of the model. Here we are mainly interested in the magnetism of the ground states. In the present talk we shall briefly review rigorous results about ferromagnetism in the Hubbard model (as summarized below), placing main emphasis on our own contributions. {\em 1) Nagaoka-Thouless ferromagnetism:}\/ In 1965, Nagaoka and independently Thouless proved that the Hubbard model on certain lattices exhibit ferromagnetism when there is exactly one hole, and the Coulomb $U$ is infinitely large. This result is interesting, but hardly extends to non-singular models. {\em 2) flatband ferromagnetism:}\/ In 1991, Mielke found a completely different class of rigorous examples in which the single-electron ground states are highly degenerate. These models exhibit ferromagnetism for any $U>0$. In 1992, Tasaki also found similar examples. {\em 3) ferromagnetism in non-singular models:}\/ In 1995, by extending his flatband models, Tasaki presented rigorous example of ferromagnetism in the Hubbard model which has finite density of states and finite (but sufficiently large) $U>0$. It was also proved that the model has a normal spin-wave excitation. This means that {\em we have obtained a class of non-singular models of itinerant electrons (with only spin-independent interactions) whose low energy behaviors are rigorously proved to be that of a ``healthy'' insulating ferromagnet}. {\em 4) metallic ferromagnetism in three dimensions:}\/ All the rigorous results discussed above correspond to insulating systems. Metallic ferromagnetism, in which same set of electrons contribute both the conduction and magnetism, is clearly more important and interesting. In 2006, Tanaka and Tasaki presented the first rigorous example of metallic ferromagnetism in the Hubbard model in any dimensions.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Tasaki, H.}, biburl = {https://www.bibsonomy.org/bibtex/26fb6158a633e293a0699b7426d7bbad6/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {d36f6a7d3f7df399e092b476a915a354}, intrahash = {6fb6158a633e293a0699b7426d7bbad6}, keywords = {ferromagnetism hubbard model statphys23 topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:12.000+0200}, title = {Hubbard model and the origin of ferromagnetism}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=139}, year = 2007 }