[Paper Review] Some rigorous results on the Holstein-Hubbard model

Tadahiro Miyao|arXiv (Cornell University)|Feb 21, 2014

Quantum and electron transport phenomena4 citations

TL;DR

This paper rigorously analyzes the Holstein-Hubbard model by incorporating repulsive Coulomb interactions, a previously neglected aspect in earlier Holstein model studies. It proves the uniqueness of the ground state for an even number of electrons on a bipartite connected lattice and establishes an infrared bound on the two-point function, addressing significant technical challenges introduced by electron-electron repulsion.

ABSTRACT

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.

Motivation & Objective

• To extend the Holstein model by rigorously incorporating repulsive Coulomb interactions, which were previously neglected in ground state analyses.
• To investigate the ground state properties of the Holstein-Hubbard model under these more realistic electron-electron interaction conditions.
• To overcome technical challenges arising from the inclusion of repulsive Coulomb interactions in the many-body Hamiltonian framework.
• To establish rigorous mathematical results on ground state uniqueness and correlation functions in the model.

Proposed method

• The study employs rigorous many-body quantum field theory techniques to analyze the Hamiltonian of the Holstein-Hubbard model.
• It uses the bipartite lattice structure to exploit symmetry properties that aid in proving ground state uniqueness.
• A variational approach combined with spectral theory is applied to derive bounds on the two-point correlation function.
• The analysis includes a detailed treatment of the electron-phonon coupling and the on-site Coulomb repulsion within a second-quantized formalism.
• Infrared bounds are derived using functional integral methods and estimates on the Green's function behavior at low energies.
• The proof of ground state uniqueness relies on the absence of degeneracy in the Hamiltonian spectrum for even electron numbers on bipartite lattices.

Experimental results

Research questions

• RQ1 Does the inclusion of repulsive Coulomb interactions lead to a unique ground state in the Holstein-Hubbard model on a bipartite lattice?
• RQ2 How do repulsive Coulomb interactions affect the two-point correlation function in the model?
• RQ3 What mathematical techniques are required to rigorously treat electron-phonon coupling and electron-electron repulsion simultaneously?
• RQ4 Can an infrared bound be established for the two-point function in the presence of both electron-phonon and Coulomb interactions?
• RQ5 What structural properties of the lattice and electron number are essential for proving ground state uniqueness?

Key findings

• The ground state of the Holstein-Hubbard Hamiltonian is proven to be unique when the number of electrons is even and the lattice is bipartite and connected.
• An infrared bound is rigorously established for the two-point correlation function, indicating controlled long-wavelength behavior.
• The inclusion of repulsive Coulomb interactions introduces significant technical challenges that require advanced functional analytic techniques.
• The proof of ground state uniqueness relies on the interplay between electron-phonon coupling and the lattice's bipartite symmetry.
• The results demonstrate that electron-electron repulsion does not destroy the uniqueness of the ground state under the specified conditions.
• The two-point function's infrared bound implies stability of long-range correlations under the model's dynamics.