40 research outputs found
Rigorous results concerning the Holstein--Hubbard model
The Holstein model has been widely accepted as a model comprising electrons
interacting with phonons; analysis of this model's ground states was
accomplished two decades ago. However, the results were obtained without
completely taking repulsive Coulomb interactions into account. Recent progress
has made it possible to treat such interactions rigorously; in this paper, we
study the Holstein--Hubbard model with repulsive Coulomb interactions. The
ground state properties of the model are investigated; in particular, the
ground state of the Hamiltonian is proven to be unique for an even number of
electrons on a bipartite connected lattice. In addition, we provide a rigorous
upper bound on charge susceptibility.Comment: 42 pages, Title changed, Major improvements, to appear, Annales Henri
Poincar
Quantum Griffiths inequalities
We present a general framework of Griffiths inequalities for quantum systems.
Our approach is based on operator inequalities associated with self-dual cones
and provides a consistent viewpoint of the Griffiths inequality. As examples,
we discuss the quantum Ising model, quantum rotor model, Bose--Hubbard model,
Hubbard model, and spin-1/2 quantum Heisenberg model. We present a
model-independent structure that governs the correlation inequalities.Comment: Revised and improved version, to appear in J. Stat. Phy
Note on the Retarded van der Waals Potential within the Dipole Approximation
We examine the dipole approximated Pauli-Fierz Hamiltonians of the
nonrelativistic QED. We assume that the Coulomb potential of the nuclei
together with the Coulomb interaction between the electrons can be approximated
by harmonic potentials. By an exact diagonalization method, we prove that the
binding energy of the two hydrogen atoms behaves as , provided that the
distance between atoms is sufficiently large. We employ the Feynman's
representation of the quantized radiation fields which enables us to
diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture
by Casimir and Polder
Spectral Analysis of the Semi-relativistic Pauli-Fierz Hamiltonian
We consider a charged particle, spin 1/2, with relativistic kinetic energy
and minimally coupled to the quantized Maxwell field. Since the total momentum
is conserved, the Hamiltonian admits a fiber decomposition as , P\in
\BbbR^3. We study the spectrum of .
In particular we prove that, for non-zero photon mass, the ground state is
exactly two-fold degenerate and separated by a gap, uniformly in , from the
rest of the spectrum.Comment: 32 page
Stability of good quantum numbers in ground states
Let be a self-adjoint operator, bounded from below and let be a
bounded self-adjoint operator with purely discrete spectrum. Suppose that (i)
is a simple eigenvalue, and (ii) strongly
commutes with . Let be the eigenvector associated with . By
the assumptions (i) and (ii), is an eigenvector of :
. In the context of quantum mechanics, is called
a good quantum number. In this note, we examine the stability of under
perturbations of from a viewpoint of the order theory.Comment: Typos corrected, 13 page