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In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice.The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice.
In physics, the Bethe ansatz is an ansatz for finding the exact wavefunctions of certain quantum many-body models, most commonly for one-dimensional lattice models. It was first used by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic isotropic (XXX) Heisenberg model. [1]
The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz. [1] In the algebraic formulation, these are related to particular quantum affine algebras and elliptic quantum groups in the XXZ and XYZ cases respectively. [2] Other approaches do so without Bethe ansatz. [3]
The Toda lattice, introduced by Morikazu Toda , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian
In condensed matter physics, the Su–Schrieffer–Heeger (SSH) model or SSH chain is a one-dimensional lattice model that presents topological features. [1] It was devised by Wu-Pei Su, John Robert Schrieffer, and Alan J. Heeger in 1979, to describe the increase of electrical conductivity of polyacetylene polymer chain when doped, based on the existence of solitonic defects.
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics , where the atoms of a crystal automatically form a lattice.
In 1925, Ising [2] gave an exact solution to the one-dimensional (1D) lattice problem. In 1944 Onsager [3] was able to get an exact solution to a two-dimensional (2D) lattice problem at the critical density. However, to date, no three-dimensional (3D) problem has had a solution that is both complete and exact. [4]
A simplified illustration of the spin chain model. The spin of the ith site can interact with the spins from the i - 1 and i + 1 sites. A spin chain is a type of model in statistical physics. Spin chains were originally formulated to model magnetic systems, which typically consist of particles with magnetic spin located at fixed sites on a lattice.