The Holstein model has been widely accepted as a model of electrons interacting with the phonons. Analysis of its ground states was accomplished decades ago. However obtained results did not completely take account of the repulsive Coulomb interactions. Recent progress has made it possible to treat such interactions rigorously. In this paper, we study the Holstein-Hubbard mdoel with the repulsive Coulomb interactions. Ground state properties of the model are investigated. Especially, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on bipartite connected lattice. In addition, an infrared bound on the two-point function is given. The effects of the repulsive Coulomb interaction induce several technical difficulties.
The Holstein model has been widely accepted as a model of electrons interacting with the phonons. Analysis of its ground states was accomplished decades ago. However obtained results did not completely take account of the repulsive Coulomb interactions. Recent progress has made it possible to treat such interactions rigorously. In this paper, we study the Holstein-Hubbard mdoel with the repulsive Coulomb interactions. Ground state properties of the model are investigated. Especially, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on bipartite connected lattice. In addition, an infrared bound on the two-point function is given. The effects of the repulsive Coulomb interaction induce several technical difficulties.
Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. However, when a direct Coulomb interaction between two electrons is also present, this interaction is only partially accounted for. In this work we use a well-defined Hamiltonian, the Hubbard-Holstein model, to examine this competition more rigorously, using exact diagonalization on a two-site system. We find that the direct electron-electron repulsion between two electrons has a significantly more harmful effect on pairing than suggested through the standard treatment of this interaction.
Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. However, when a direct Coulomb interaction between two electrons is also present, this interaction is only partially accounted for. In this work we use a well-defined Hamiltonian, the Hubbard-Holstein model, to examine this competition more rigorously, using exact diagonalization on a two-site system. We find that the direct electron-electron repulsion between two electrons has a significantly more harmful effect on pairing than suggested through the standard treatment of this interaction.
Abstract Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. However, when a direct Coulomb interaction between two electrons is also present, this interaction is only partially accounted for. In this work we use a well-defined Hamiltonian, the Hubbard-Holstein model, to examine this competition more rigorously, using exact diagonalization on a two-site system. We find that the direct electron-electron repulsion between two electrons has a significantly more harmful effect on pairing than suggested through the standard treatment of this interaction.
Abstract Eliashberg theory provides a theoretical framework for understanding the phenomenon of superconductivity when pairing between two electrons is mediated by phonons, and retardation effects are fully accounted for. However, when a direct Coulomb interaction between two electrons is also present, this interaction is only partially accounted for. In this work we use a well-defined Hamiltonian, the Hubbard-Holstein model, to examine this competition more rigorously, using exact diagonalization on a two-site system. We find that the direct electron-electron repulsion between two electrons has a significantly more harmful effect on pairing than suggested through the standard treatment of this interaction.
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.