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Journal of Physics Condensed Matter

Theory of ferrimagnetism in the Hubbard model on bipartite lattices with spectral symmetry

Xue Yang, Jing He, Xing‐Hai Zhang, Su-Peng Kou

Published Aug 9, 2015• 0 citations
Publisher

Abstract

The Hubbard model is one of the most important models in condensed matter physics. In this paper, we developed a theory of ferrimagnetism in the Hubbard model on bipartite lattices with spectral symmetry. By taking three models as examples, we studied the ferrimagnetic orders that emerge from three typical fermionic systems--metal, semi-metal and (Chern) insulator. In particular, we found that there may exist various ferrimagnetic orders and explored the universal features.

DOI
10.1088/0953-8984/27/35/356001

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The Hubbard model is one of the most important models in condensed matter physics. In this paper, we developed a theory of ferrimagnetism in the Hubbard model on bipartite lattices with spectral symmetry. By taking three models as examples, we studied the ferrimagnetic orders that emerge from three typical fermionic systems--metal, semi-metal and (Chern) insulator. In particular, we found that there may exist various ferrimagnetic orders and explored the universal features.
In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When NA{\mathit{N}}_{\mathit{A}}NA​>NB{\mathit{N}}_{\mathit{B}}NB​(NA{\mathit{N}}_{\mathit{A}}NA​ and NB{\mathit{N}}_{\mathit{B}}NB​ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.
The Hubbard model is a `highly oversimplified model' for electrons in a solid which interact with each other through extremely short-ranged repulsive (Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two parts: which describes quantum mechanical hopping of electrons, and which describes non-linear repulsive interaction. Either or alone is easy to analyse, and does not favour any specific order. But their sum is believed to exhibit various non-trivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting `universality classes' of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results relating to the Hubbard model which shed light on the `physics' of this fascinating model. We mainly concentrate on the magnetic properties of the model in its ground states. We discuss the Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on the decay of correlations at finite temperatures in two dimensions, the Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one dimension, Lieb's important theorem for the half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to non-experts by giving basic definitions and describing elementary materials in detail.
The Hubbard model is a "highly oversimplified model" for electrons in a solid which interact with each other through extremely short ranged repulsive (Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two pieces; H_hop which describes quantum mechanical hopping of electrons, and Hint which describes nonlinear repulsive interaction. Either H_hop or H_int alone is easy to analyze, and does not favor any specific order. But their sum H=H_hop+H_int is believed to exhibit various nontrivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting "universality classes" of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results on the Hubbard model which shed light on "physics" of this fascinating model. We mainly concentrate on magnetic properties of the model at its ground states. We discuss Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on decay of correlations at finite temperatures in two-dimensions, Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one-dimension, Lieb's important theorem for half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to nonexperts by describing basic definitions and elementary materials in detail.
We investigate the Hubbard Hamiltonian on ladders where the number of sites per rung alternates between two and three. These geometries are bipartite with nonequal or equal number of sites on the two sublattices. Thus they share a key feature of the Hubbard model in a class of lattices which Lieb has shown analytically to exhibit long-range ferrimagnetic order while being amenable to powerful numeric approaches developed for quasi-one-dimensional geometries. The density matrix renormalization group (DMRG) method is used to obtain the groundstate properties, e.g., excitation gaps, charge and spin densities as well as their correlation functions at half filling. We show the existence of long-range ferrimagnetic order in the one-dimensional ladder geometries. Our work provides detailed quantitative results which complement the general theorem of Lieb for generalized bipartite lattices. It also addresses the issue of how the alternation between quasi-long-range order and spin liquid behavior for uniform ladders with odd and even numbers of legs might be affected by a regular alternation pattern.
We study the metal-insulator transition and magnetic ordering in the Hubbard model using the particle-hole mapping. The analysis simplifies near the ferromagnetic limit. We find that the two-dimensional Hubbard model has a charge excitation gap at half filling for any finite U in this region on both the bipartite square lattice and the nonbipartite triangular lattice. In some cases, the system goes through a first-order phase transition to become a paramagnetic metal as Sz{\mathit{S}}_{\mathit{z}}Sz​ is lowered. We also discuss the extension to the doped case. We find that in the large U limit, a single doped hole has a bandwidth of order of J rather than t at Sz{\mathit{S}}_{\mathit{z}}Sz​=0.