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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 computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices.The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: lattice problems are an example of NP-hard problems which have been shown to be average-case hard, providing a test case for the security of cryptographic ...
Gold deposited on a stepped Si(553) surface has shown evidence of two simultaneous Peierls transitions. The lattice period is distorted by factors of 2 and 3, and energy gaps open for nearly 1/2-filled and 1/3–1/4 filled bands. The distortions have been studied and imaged using LEED and STM, while the energy bands were studied with ARP. [9]
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]
Transfer-matrix methods have been critical for many exact solutions of problems in statistical mechanics, including the Zimm–Bragg and Lifson–Roig models of the helix-coil transition, transfer matrix models for protein-DNA binding, as well as the famous exact solution of the two-dimensional Ising model by Lars Onsager.
The Hamiltonian of the one-dimensional Ising model on a lattice of L sites with free boundary conditions is = =, …, +, where J and h can be any number, since in this simplified case J is a constant representing the interaction strength between the nearest neighbors and h is the constant external magnetic field applied to lattice sites.
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]
In a one-dimensional lattice the number of reciprocal lattice vectors that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands E n ( k ) {\displaystyle E_{n}(\mathbf {k} )} increases more rapidly ...