Ruprecht-Karls-Universität Heidelberg
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Andreas Mielke
Institut für Theoretische Physik
Ruprecht-Karls Universität
Philosophenweg 19
D-69120 Heidelberg
Germany
Tel.: ++49 6221 549431 (Sekretariat)
Fax: ++49 6221 549331
e-mail: mielke@tphys.uni-heidelberg.de

Korrelierte Fermionen, Hubbardmodell

Ferromagnetismus ist eine der am längsten untersuchten Eigenschaften des Hubbardmodell. Man kann nur für sehr wenige Fälle exakte Resultate herleiten. Eine große Klasse von Modellen, die Ferromagnetismus zeigen, sind Modelle mit mehreren Bändern, von denen eines dispersionslos ist. Diesen sogenannten flat band Ferromagnetismus haben wir seit 1991 ausführlich untersucht. In den letzten Jahren konnten diese Resultate auf Modelle mit einem teilweise flachen Band verallgemeinert werden. Dies eröffnet einen möglichen Weg zu metallischem Ferromagnetismus im Hubbardmodell.

Ausgewählte Publikationen aus diesem Gebiet

Andreas Mielke: Ferromagnetism in single band Hubbard models with a partially flat band
Phys. Rev. Lett.82, 4312-4315 (1999) .

Archiv: cond-mat/9904258
Abstract:
A Hubbard model with a single, partially flat band has ferromagnetic ground states. It is shown that local stability of ferromagnetism implies its global stability in such a model: The model has only ferromagnetic ground states if there are no single spin-flip ground states. Since a single-band Hubbard model away from half filling describes a metal, this result may open a route to metallic ferromagnetism in single band Hubbard models.

Andreas Mielke: Stability of ferromagnetism in Hubbard models with degenerate single-particle ground states
J. Phys. A, Math. Gen.32, 8411-8418 (1999) .

Archiv: cond-mat/9910385
Abstract:
A Hubbard model with a \( N_{d} \)-fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less or equal to \( N_{d} \). It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: The model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to \( N_{d} \), it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result.

Andreas Mielke: Ferromagnetism in the Hubbard model and Hund's rule
Phys. Lett. A174, 443-448 (1993) .

Abstract:
We investigate the Hubbard model with a $N_{\rm d}$-fold degenerate single particle ground state. If the number of electrons satisfies $N_{\rm e}<N_{\rm d}$, the model has ferromagnetic multiparticle ground states. We give a necessary and sufficient condition for the ground state to be unique $N_{\rm e}=N_{\rm d}$. It is ferromagnetic with spin $S=\frac12N_{\rm e}$. As a corollary, we obtain Hund's rule for the general Hubbard model with degenerate single particle eigenstates on translationally invariant lattices in the special case, where each of the degenerate single particle states if filled with one electron.

Andreas Mielke, Hal Tasaki: Ferromagnetism in the Hubbard model - Examples from Models with Degenerate Single-Electron Ground States
Commun. Math. Phys.158, 341-371 (1993) .

Archiv: cond-mat/9305026
Abstract:
Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has $N_{\rm d}$-fold degenerate ground states. The degeneracy $N_{\rm d}$ is proportional to the total number of sites $\abs{\Lambda}$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $\rho_0=N_{\rm d}/(2\abs{\Lambda})$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.

Andreas Mielke: Exact ground states for the Hubbard model on the kagomé lattice
J. Phys. A: Math. Gen.25, 4335-4345 (1992) .

Abstract:
The author gives a complete and rigorous description of the ground states of the Hubbard model on the Kagome lattice for electron densities n>or=5/3 and U>0. If 11/6>n>or=5/3 the system shows a ferromagnetic behaviour at zero temperature. If n is above 11/6 the system is paramagnetic. The proof of these results uses some graph-theoretic methods. The results are applicable to all line graphs of planar lattices, of which the Kagome lattice is an example.

Andreas Mielke: Exact results for the $U=\infty$ Hubbard model
J. Phys. A: Math. Gen.25, 6507-6515 (1992) .

Abstract:
The author investigates the U= infinity Hubbard model on a large class of lattices which are line graphs. The most interesting lattices in this class are line graphs of regular bipartite lattices with Ns sites and coordination number k>or=4. The ground state energy and some ground states are given. If the number of electrons N satisfies Ns>or=N>or=2Ns/k-2, the ground state energy is -4 mod t mod (Ns-N). The ground states have no magnetic ordering, they are projections of the ground states at U=0 onto the subspace of states without doubly occupied sites.

Andreas Mielke: Ferromagnetic ground states for the Hubbard model on line graphs
J. Phys. A: Math. Gen.24, L73-L77 (1991) .

Abstract:
The author discusses some of the properties of the Hubbard model on a line graph with n vertices. It is shown that the model has ferromagnetic ground states if the interaction is repulsive (U)0) and if the number of electrons N satisfies 2n>or=N>or=M. M is a natural number that depends on the line graph. For example, the Kagome lattice is a line graph, it has M=5n/3-1.

Andreas Mielke: Ferromagnetism in the Hubbard model on line graphs and further considerations
J. Phys. A: Math. Gen.24, 3311-3321 (1991) .

Abstract:
Let L(G) be the line graph of a graph G=(V,E). The Hubbard model on L(G) has ferromagnetic ground states with a saturated spin if the interaction is repulsive (U>0) and if the number of electrons N satisfies N>or=M. M= mod E mod + mod V mod -1 if G is bipartite and M= mod E mod + mod V mod otherwise. The author shows that the ferromagnetic ground state is unique if N=M. Further he gives a sufficient condition for the existence of other ground states if N>M. The results are valid also for a multi-band Hubbard model on a bipartite graph. In the case of a periodic lattice, the results are related to the existence of a flat energy band.

Letzte Änderung: 26.4.2012. mielke@tphys.uni-heidelberg.de