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arXiv:2402.07171

Quantum geometric bound for saturated ferromagnetism

Junha Kang, Taekoo Oh, Junhyun Lee, Bohm‐Jung Yang

Published Feb 11, 2024• 0 citations
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Abstract

Despite its abundance in nature, predicting the occurrence of ferromagnetism in the ground state is possible only under very limited conditions such as in a flat band system with repulsive interaction or in a band with a single hole under infinitely large Coulomb repulsion, etc. Here, we propose a general condition to achieve saturated ferromagnetism based on the quantum geometry of electronic wave functions in itinerant electron systems. By analyzing the spin excitations of multi-band repulsive Hubbard models with an integer band filling, relevant to either ferromagnetic insulators or semimetals, we show that quantum geometry stabilizes the Goldstone mode in the strongly correlated limit. Our theory indicates the stability of ferromagnetism in a large class of insulators and semimetals other than the previously studied flat band systems and their variants. Moreover, we rigorously prove that saturated ferromagnetism is forbidden in any system with trivial quantum geometry, which includes every half-filled system. We believe that our findings reveal a profound connection between quantum geometry and ferromagnetism, which can be extended to various symmetry-broken ground states in itinerant electronic systems.

DOI
10.48550/arxiv.2402.07171

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Despite its abundance in nature, predicting the occurrence of ferromagnetism in the ground state is possible only under very limited conditions such as in a flat band system with repulsive interaction or in a band with a single hole under infinitely large Coulomb repulsion, etc. Here, we propose a general condition to achieve saturated ferromagnetism based on the quantum geometry of electronic wave functions in itinerant electron systems. By analyzing the spin excitations of multi-band repulsive Hubbard models with an integer band filling, relevant to either ferromagnetic insulators or semimetals, we show that quantum geometry stabilizes the Goldstone mode in the strongly correlated limit. Our theory indicates the stability of ferromagnetism in a large class of insulators and semimetals other than the previously studied flat band systems and their variants. Moreover, we rigorously prove that saturated ferromagnetism is forbidden in any system with trivial quantum geometry, which includes every half-filled system. We believe that our findings reveal a profound connection between quantum geometry and ferromagnetism, which can be extended to various symmetry-broken ground states in itinerant electronic 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 .
We construct a Hubbard model with a nearly flat band whose quantum geometry can be tuned independent of the energy dispersion and the Coulomb interaction. We show that, when the nearly flat band is half filled, the exact ground state of the model exhibits ferromagnetism and that this ferromagnetism is stabilized by the quantum metric through the spin stiffness. Furthermore, we demonstrate that tuning the quantum geometry alone drives a magnetic phase transition. Our nonperturbative results without resorting to mean-field approximations reveal the quantum-geometric origin of ferromagnetism and the underlying many-body physics in dispersive-band systems.
It is believed that strong ferromagnetic interactions in some solids are generated by subtle interplay between quantum many-body effects and spin-independent Coulomb interactions between electrons. It is a challenging problem to verify this scenario in the Hubbard model, which is an idealized model for strongly interacting electrons in a solid. Nagaoka's ferromagnetism is a well-known rigorous example of ferromagnetism in the Hubbard model. It deals with the limiting situation in which there is one fewer electron than in the half-filling and the on-site Coulomb interaction is infinitely large. There are relatively new rigorous examples of ferromagnetism in Hubbard models called flat-band ferromagnetism. Flat-band ferromagnetism takes place in carefully prepared models in which the lowest bands (in the single-electron spectra) are “flat.” Usually, these two approaches are regarded as two complimentary routes to ferromagnetism in the Hubbard model. In the present paper we describe Nagaoka's ferromagnetism and flat-band ferromagnetism in detail, giving all the necessary background as well as complete (but elementary) mathematical proofs. By studying an intermediate model called the long-range hopping model, we also demonstrate that there is indeed a deep relation between these two seemingly different approaches to ferromagnetism. We further discuss some attempts to go beyond these approaches. We briefly discuss recent rigorous example of ferromagnetism in the Hubbard model which has neither infinitely large parameters nor completely flat bands. We give preliminary discussion regarding possible experimental realizations of the (nearly-)flat-band ferromagnetism. Finally, we focus on some theoretical attempts to understand metallic ferromagnetism. We discuss three artificial one-dimensional models in which the existence of metallic ferromagnetism can be easily proved. We have tried to make the present paper as self-contained as possible, keeping in mind readers who are new to the field. Although the present paper is written as a review, it contains some material which appears for the first time.
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 study the Hubbard model on three coupled chains which has the saturated ferromagnetic ground states in a finite range of the electron filling factor . In the case that both the single-electron band gap between the highest and lower two bands and the on-site Coulomb interaction are infinitely large, we will rigorously show that the present model exhibits ferromagnetism for 0 < <(1/3), where 0 0 (1/6) depending on the band structure. We will also give numerical results, which indicate that, even in the case that the band gap and the Coulomb interaction are not so large, the present ferromagnetism will survive for suitable electron filling factors.