We model conducting pentagon chains with a multi-orbital Hubbard model and prove that well below half filling, exact ferromagnetic ground states appear. The rigorous method we use is based on the transformation of original hamiltonian into positive semidefinite form. This technique is independent of the spatial dimension and does not require integrability of the model. The obtained ferromagnetism is connected to dispersionless bands but in a much broader sense than flat-band ferromagnetism requires, where on every site a Hubbard term is present. In our case, only a small percentage of, even randomly distributed, sites are only interacting.
Nagaoka ferromagnetism (NF) is a long-predicted example of itinerant ferromagnetism (IF) in the Hubbard model that has been studied theoretically for many years. The condition for NF, an infinite on-site Coulomb repulsion and a single hole in a half-filled band, does not arise naturally in materials. NF was only realized recently for the first time in experiments on a
2×2 array of gated quantum dots. Dopant arrays and gated quantum dots in Si allow for engineering controllable systems with complex geometries. This makes quantum dot arrays good candidates to study NF in different geometries through analog quantum simulation. Here we present theoretical simulations done for
3×3 arrays and larger
N×N arrays, and predict the emergence of different forms of ferromagnetism in different geometries. We find NF in perfect
3×3 arrays, and in
N×N arrays for one hole doping of a half-filled band. The ratio of the hopping
t to Hubbard on-site repulsion
U that defines the onset of NF scales as
1/N4 as
N increases, approaching the bulk limit of infinite
U for large
N. Additional simulations are done for geometries made by removing sites from
N×N arrays. Different forms of ferromagnetism are found for different geometries. Loops show ferromagnetism, but only for three electrons. For loops, the critical
t/U for the onset of ferromagnetism scales as
N as the loop length increases. We show that the different dependences on size for loops and
N×N arrays can be understood by scaling arguments that highlight the different energy contributions to each form of ferromagnetism. Our results show how analog quantum simulation with small arrays can elucidate the role of effects including wavefunction connectivity; system geometry, size and symmetry; bulk and edge sites; and kinetic energy in determining quantum magnetism of small systems.
We investigate the magnetization of the ground state of the one-dimensional flat-band Hubbard model as a function of the electron filling, using exact diagonalization techniques. Our calculation shows that the systems with odd n holes added to half-filling bands for a sufficiently large U (the on-site Coulomb repulsion energy) have a saturated ferromagnetic ground state, which is nondegenerate except for the trivial spin degeneracy. Dependences of the critical value of U on the size and on λ (a parameter of the system) are also investigated. For all values of λ, it is found that the ground state is a saturated ferromagnetic state if U is sufficiently large. Moreover, we find that the critical U for the saturated ferromagnetic ground state is almost independent on the system size, at least, for the system near quarter-filling. These facts indicate that even if the system size go to infinity, the ferromagnetic ground state can be exist for a finite U .
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
Nd-fold degenerate ground states. The degeneracy
Nd is proportional to the total number of sites $\absΛ$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $ρ_0=N_{\rm d}/(2\absΛ)$, 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.
We present a Hubbard model with a special hopping matrix, which has highly degenerate single-electron energy eigenstates but has different properties than the flat-band Hubbard model. Our model has a paramagnetic ground state when the electron filling factor is less than or equal to quarter-filling. In the case of quarter-filling with one extra electron, the present model exhibits saturated ferromagnetism in the ground states for a sufficiently large value of on-site Coulomb interaction but does not for a small one.
Nagaoka showed that ferromagnetism can arise from a basic Hubbard model of certain lattices under special conditions. Motivated by recent experimental advances in simulating such systems using small plaquettes of a few quantum dots, this work investigates what geometrical arrangement of these dots and what number of electrons are necessary to lead to nontrivial ferromagnetic ground states. This includes many situations not covered by Nagaoka's theorem, which is only valid under specific constraints.