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 und Bosonen, 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.

Wechselwirkende Bosonen in Strukturen mit flachen Bändern zeigen eine Reihe von interessanten Phänomenen, u.a. die Bildung eines Wignerkristalls und Paarbildung.

## Ausgewählte Publikationen aus diesem Gebiet

#### Rebecca Pons, Andreas Mielke, Tobias Stauber: Flat-band ferromagnetism in twisted bilayer graphenePhys. Rev. B102, 235101 (2020) .

##### Abstract:
We discuss twisted bilayer graphene (TBG) based on a theorem of flat band ferromagnetism put forward by Mielke and Tasaki. According to this theorem, ferromagnetism occurs if the single particle density matrix of the flat band states is irreducible and we argue that this result can be applied to the quasi-flat bands of TBG that emerge around the charge-neutrality point for twist angles around the magic angle $\theta\sim1.05^\circ$. We show that the density matrix is irreducible in this case, thus predicting a ferromagnetic ground state for neutral TBG ($n=0$). We then show that the theorem can also be applied only to the flat conduction or valence bands, if the substrate induces a single-particle gap at charge neutrality. Also in this case, the corresponding density matrix turns out to be irreducible, leading to ferromagnetism at half filling ($n=\pm2$).

#### Andreas Mielke, Jacob Fronk: Localised pair formation in bosonic flat-band Hubbard modelspreprint, (2020) .

##### Abstract:
Using a generalised version of Gershgorin's circle theorem, rigorous boundaries on the energies of the lowest states of a broad class of line graphs above a critical filling are derived for hardcore bosonic systems. Also a lower boundary on the energy gap towards the next lowest states is established. Additionally, it is shown that the corresponding eigenstates are dominated by a subspace spanned by states containing a compactly localised pair and a lower boundary for the overlap is derived as well. Overall, this strongly suggests localised pair formationin the ground states of the broad class of line graphs and rigorously proves it for some of the graphs in it, including the inhomogeneous chequerboard chain as well as two novel examples of regular two dimensional graphs.

#### Andreas Mielke: Pair formation of hard core bosons in flat band systemsJ Stat Phys.171, 679-695 (2018) .

##### Abstract:
Hard core bosons in a large class of one or two dimensional flat band systems have an upper critical density, below which the ground states can be described completely. At the critical density, the ground states are Wigner crystals. If one adds a particle to the system at the critical density, the ground state and the low lying multi particle states of the system can be described as a Wigner crystal with an additional pair of particles. The energy band for the pair is separated from the rest of the multi-particle spectrum. The proofs use a Gerschgorin type of argument for block diagonally dominant matrices. In certain one-dimensional or tree-like structures one can show that the pair is localised, for example in the chequerboard chain. For this one-dimensional system with periodic boundary condition the energy band for the pair is flat, the pair is localised.

#### Moritz Drescher, Andreas Mielke: Hard-core bosons in flat band systems above the critical densityEur. Phys. J. B90, 217 (2017) .

##### Abstract:
We investigate the behaviour of hard-core bosons in one- and two-dimensional flat band systems, the chequerboard and the kagom\'e lattice and one-dimensional analogues thereof. The one dimensional systems have an exact local reflection symmetry which allows for exact results. We show that above the critical density an additional particle forms a pair with one of the other bosons and that the pair is localised. In the two-dimensional systems exact results are not available but variational results indicate a similar physical behaviour.

#### Petra Pudleiner, Andreas Mielke: Interacting bosons in two-dimensional flat band systemsEur. Phys. J. B88, 207 (2015) .

##### Abstract:
The Hubbard model of bosons on two dimensional lattices with a lowest flat band is discussed. In these systems there is a critical density, where the ground state is known exactly and can be represented as a charge density wave. Above this critical filling, depending on the lattice structure and the interaction strength, the additional particles are either delocalised and condensate in the ground state, or they form pairs. Pairs occur at strong interactions, e.g., on the chequerboard lattice. The general mechanism behind this phenomenon is discussed.

#### Andreas Mielke: Properties of Hubbard models with degenerate localised single particle eigenstatesEur. Phys. J. B85, (2012) .

##### Abstract:
We consider the repulsive Hubbard model on a class of lattices or graphs for which there is a large degeneracy of the single particle ground states and where the projector onto the space of single particle ground states is highly reducible. This means that one can find a basis in the space of the single particle ground states such that the support of each single particle ground state belongs to some small cluster and these clusters do not overlap. We show how such lattices can be constructed in arbitrary dimensions. We construct all multi-particle ground states of these models for electron numbers not larger than the number of localised single particle eigenstates. We derive some of the ground state properties, esp. the residual entropy, i.e. the finite entropy density at zero temperature.

#### Johannes Motruk, Andreas Mielke: Bose-Hubbard model on two-dimensional line graphsJ. Phys. A: Math. Gen45, 225206 (2012) .

##### Abstract:
We construct a basis for the many-particle ground states of the positive hopping Bose-Hubbard model on line graphs of finite 2-connected planar bipartite graphs at sufficiently low filling factors. The particles in these states are localized on non-intersecting vertex-disjoint cycles of the line graph which correspond to non-intersecting edge-disjoint cycles of the original graph. The construction works up to a critical filling factor at which the cycles are close-packed.

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

##### 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 statesJ. Phys. A, Math. Gen.32, 8411-8418 (1999) .

##### 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) .

##### 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 latticeJ. 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 modelJ. 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 graphsJ. 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 considerationsJ. 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: 11.4.2021. mielke@tphys.uni-heidelberg.de
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