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arXiv:1602.00439 · Physical review. B./Physical review. B

Hubbard models with nearly flat bands: Ground-state ferromagnetism driven by kinetic energy

Patrick Müller, Johannes Richter, Oleg Derzhko

Published Apr 22, 2016• 12 citations

Abstract

We consider the standard repulsive Hubbard model with a flat lowest-energy band for two one-dimensional lattices (diamond chain and ladder) as well as for a two-dimensional lattice (bilayer) at half filling of the flat band. The considered models do not fall in the class of Mielke-Tasaki flat-band ferromagnets, since they do not obey the connectivity conditions. However, the ground-state ferromagnetism can emerge, if the flat band becomes dispersive. To study this kinetic-energy-driven ferromagnetism we use perturbation theory and exact diagonalization of finite lattices. We find as a typical scenario that small and moderate dispersion may lead to a ferromagnetic ground state for sufficiently large on-site Hubbard repulsion $U>{U}_{c}$, where ${U}_{c}$ increases monotonically with the acquired bandwidth. However, we also observe for some specific parameter cases, that (i) ferromagnetism appears at already very small ${U}_{c}$, (ii) ferromagnetism does not show up at all, (iii) the critical on-site repulsion ${U}_{c}$ is a nonmonotonic function of the bandwidth, or that (iv) a critical bandwidth is needed to open the window for ground-state ferromagnetism.

DOI: 10.1103/physrevb.93.144418

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We consider the standard repulsive Hubbard model with a flat lowest-energy band for two one-dimensional lattices (diamond chain and ladder) as well as for a two-dimensional lattice (bilayer) at half filling of the flat band. The considered models do not fall in the class of Mielke-Tasaki flat-band ferromagnets, since they do not obey the connectivity conditions. However, the ground-state ferromagnetism can emerge, if the flat band becomes dispersive. To study this kinetic-energy-driven ferromagnetism we use perturbation theory and exact diagonalization of finite lattices. We find as a typical scenario that small and moderate dispersion may lead to a ferromagnetic ground state for sufficiently large on-site Hubbard repulsion

U>{U}_{c}

, where

{U}_{c}

increases monotonically with the acquired bandwidth. However, we also observe for some specific parameter cases, that (i) ferromagnetism appears at already very small

{U}_{c}

, (ii) ferromagnetism does not show up at all, (iii) the critical on-site repulsion

{U}_{c}

is a nonmonotonic function of the bandwidth, or that (iv) a critical bandwidth is needed to open the window for ground-state ferromagnetism.

We investigate a mechanism to establish ground-state ferromagnetism in flat-band Hubbard systems based on a kind of order-from-disorder effect driven by dispersion. As a paradigm we consider a frustrated diamond chain, where for ideal diamond geometry the lowest one-electron band is flat, but the ground state remains paramagnetic for arbitrary on-site repulsion

U

. We focus on half-filling of the flat band. By using numerical and analytical arguments we present the ground-state phase diagram for a distorted diamond chain, i.e., the former flat band becomes dispersive. Driven by the interplay of dispersion and interaction the ground state is ferromagnetic if the interaction exceeds a critical value

{U}_{c}

.

The Hubbard model on the kagome lattice has highly degenerate ground states (the flat lowest band) in the corresponding single-electron problem and exhibits the so-called flat-band ferromagnetism in the many-electron ground states as was found by Mielke [J. Phys. A 24, L73 (1991)]]. Here we study the model obtained by adding extra hopping terms to the above model. The lowest single-electron band becomes dispersive, and there is no band gap between the lowest band and the other band. We prove that, at half filling of the lowest band, the ground states of this perturbed model remain saturated ferromagnetic if the lowest band is nearly flat.

The Hubbard model on line graphs has a flat band and highly degenerate ferromagnetic ground states. Here, we study the Hubbard model on a line graph of a planar bipartite graph by adding a special contribution to the kinetic energy which lifts the degeneracy of the lowest single particle state to a general Hamiltonian. We prove that, at half-filling of the lowest band and for sufficiently strong repulsion U , the ground states of this Hamiltonian remain saturated ferromagnetic for a class of line graphs of planar bipartite graphs.

We examined a one-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping with up to 32 sites by means of the density-matrix renormalization-group method. The model has two energy bands and includes the flat-band model studied by Tasaki as a special case. The model with finite band dispersion is found to show metallic (saturated) ferromagnetism for a sufficiently strong interaction. No partially polarized ground state is found. The interaction necessary to realize a ferromagnetic ground state becomes weak when the electron density is decreased from the quarter-filling.

Abstract We present rigorous results for the SU(n) Fermi–Hubbard models with finite-range hopping in d (

\ge 2

≥ 2 ) dimensions. The models are defined on a class of decorated lattices. We first study the models with flat bands at the bottom of the single-particle spectrum and prove that the ground states exhibit SU(n) ferromagnetism when the number of particles is equal to the number of unit cells. We then perturb the models by adding particular hopping terms and make the bottom bands dispersive. Under the same filling condition, it is proved that the ground states remain SU(n) ferromagnetic when the bottom bands are sufficiently flat and the Coulomb repulsion is sufficiently large.

We consider nearly-flat-band Hubbard models of a ferromagnet, that is the models that are weak perturbations of those flat-band Hubbard models whose ground state is ferromagnetic for any nonzero strength

U

of the Hubbard repulsion. In contrast to the flat-band case, in the nearly-flat-band case the ground state, being paramagnetic for

U

in a vicinity of zero, turns into a ferromagnetic one only if

U

exceeds some nonzero threshold value

U_{th}

. We address the question whether

U_{th}

of the considered models is in a physical range, therefore we attempt at obtaining possibly good estimates of the threshold value

U_{th}

. A rigorous method proposed by Tasaki is extended and the resulting estimates are compared with small-system, finite-size scaling results obtained for open- and periodic-boundary conditions. Contrary to suggestions in literature, we find the latter conditions particularly useful for our task.