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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 ...
However, to date, no three-dimensional (3D) problem has had a solution that is both complete and exact. [4] Over the last ten years, Aranovich and Donohue have developed lattice density functional theory (LDFT) based on a generalization of the Ono-Kondo equations to three-dimensions, and used the theory to model a variety of physical phenomena.
This form is universal (at least for non-nested systems), with the momentum and scattering functions being model-dependent. The Yang–Baxter equation guarantees consistency of the construction. The Pauli exclusion principle is valid for models solvable by the Bethe ansatz, even for models of interacting bosons. The ground state is a Fermi sphere.
The main problem of quantum many-body physics is the fact that the Hilbert space grows exponentially with size. In other words if one considers a lattice, with some Hilbert space of dimension on each site of the lattice, then the total Hilbert space would have dimension , where is the number of sites on the 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.
This models a one-dimensional lattice of fixed particles with spin 1/2. A simple version (the antiferromagnetic XXX model) was solved, that is, the spectrum of the Hamiltonian of the Heisenberg model was determined, by Hans Bethe using the Bethe ansatz . [ 2 ]
If the lattice is even, the form has level 1, and if the lattice is odd the form has Γ 0 (4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣ n /24⎦ + 1.