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In some cases, the Schrödinger equation can be solved analytically on a one-dimensional lattice of finite length [6] [7] using the theory of periodic differential equations. [8] The length of the lattice is assumed to be L = N a {\displaystyle L=Na} , where a {\displaystyle a} is the potential period and the number of periods N {\displaystyle ...
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.
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 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 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 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]
The Hubbard model is based on the tight-binding approximation from solid-state physics, which describes particles moving in a periodic potential, typically referred to as a lattice. For real materials, each lattice site might correspond with an ionic core, and the particles would be the valence electrons of these ions.
In the crystal, the potential has the periodicity, a, of the lattice while close to the surface it has to somehow attain the value of the vacuum level. The step potential (solid line) shown in Figure 1 is an oversimplification which is mostly convenient for simple model calculations. At a real surface the potential is influenced by image ...