Quantum decoherence is the loss of quantum coherence. Quantum decoherence has been studied to understand how quantum systems convert to systems which can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have confirmed some of the key issues. Quantum computing relies on quantum coherence and is the primary practical applications of the concept.
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. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory to prevent divergences or to perform numerical computations. An example of a continuum theory that is widely studied by lattice models is the QCD lattice model, a discretization of quantum chromodynamics. However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics of polymers.
The Ising model, named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states. The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.
The classical XY model is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's n-vector model for n = 2.
In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles. While sometimes called an exchange force, or, in the case of fermions, Pauli repulsion, its consequences cannot always be predicted based on classical ideas of force. Both bosons and fermions can experience the exchange interaction.
The Bose–Hubbard model gives a description of the physics of interacting spinless bosons on a lattice. It is closely related to the Hubbard model that originated in solid-state physics as an approximate description of superconducting systems and the motion of electrons between the atoms of a crystalline solid. The model was introduced by Gersch and Knollman in 1963 in the context of granular superconductors. The model rose to prominence in the 1980s after it was found to capture the essence of the superfluid-insulator transition in a way that was much more mathematically tractable than fermionic metal-insulator models.
In solid-state physics, the t-J model is a model first derived by Józef Spałek to explain antiferromagnetic properties of Mott insulators, taking into account experimental results about the strength of electron-electron repulsion in this materials.
In non-equilibrium physics, the Keldysh formalism or Keldysh–Schwinger formalism is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields. Historically, it was foreshadowed by the work of Julian Schwinger and proposed almost simultaneously by Leonid Keldysh and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel.
The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos and quantum chaos. It describes a free rotating stick in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the Hamiltonian
Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in density functional theory and usual band structure calculations, breaks down. Dynamical mean-field theory, a non-perturbative treatment of local interactions between electrons, bridges the gap between the nearly free electron gas limit and the atomic limit of condensed-matter physics.
In condensed matter physics, an AKLT model, also known as an Affleck-Kennedy-Lieb-Tasaki model is an extension of the one-dimensional quantum Heisenberg spin model. The proposal and exact solution of this model by Ian Affleck, Elliott H. Lieb, Tom Kennedy and Hal Tasaki provided crucial insight into the physics of the spin-1 Heisenberg chain. It has also served as a useful example for such concepts as valence bond solid order, symmetry-protected topological order and matrix product state wavefunctions.
In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, , on the spatial slice, and its conjugate momentum variable related to the extrinsic curvature, ,. These are the metric canonical coordinates.
The Lieb–Robinson bound is a theoretical upper limit on the speed at which information can propagate in non-relativistic quantum systems. It demonstrates that information cannot travel instantaneously in quantum theory, even when the relativity limits of the speed of light are ignored. The existence of such a finite speed was discovered mathematically by Elliott H. Lieb and Derek W. Robinson in 1972. It turns the locality properties of physical systems into the existence of, and upper bound for this speed. The bound is now known as the Lieb–Robinson bound and the speed is known as the Lieb–Robinson velocity. This velocity is always finite but not universal, depending on the details of the system under consideration. For finite-range, e.g. nearest-neighbor, interactions, this velocity is a constant independent of the distance travelled. In long-range interacting systems, this velocity remains finite, but it can increase with the distance travelled.
In computational solid state physics, Continuous-time quantum Monte Carlo (CT-QMC) is a family of stochastic algorithms for solving the Anderson impurity model at finite temperature. These methods first expand the full partition function as a series of Feynman diagrams, employ Wick's theorem to group diagrams into determinants, and finally use Markov chain Monte Carlo to stochastically sum up the resulting series.
In statistical mechanics, Lee–Yang theory, sometimes also known as Yang–Lee theory, is a scientific theory which seeks to describe phase transitions in large physical systems in the thermodynamic limit based on the properties of small, finite-size systems. The theory revolves around the complex zeros of partition functions of finite-size systems and how these may reveal the existence of phase transitions in the thermodynamic limit.
Hamiltonian truncation is a numerical method used to study quantum field theories (QFTs) in spacetime dimensions. Hamiltonian truncation is an adaptation of the Rayleigh–Ritz method from quantum mechanics. It is closely related to the exact diagonalization method used to treat spin systems in condensed matter physics. The method is typically used to study QFTs on spacetimes of the form , specifically to compute the spectrum of the Hamiltonian along . A key feature of Hamiltonian truncation is that an explicit ultraviolet cutoff is introduced, akin to the lattice spacing a in lattice Monte Carlo methods. Since Hamiltonian truncation is a nonperturbative method, it can be used to study strong-coupling phenomena like spontaneous symmetry breaking.
Phase space crystal is the state of a physical system that displays discrete symmetry in phase space instead of real space. For a single-particle system, the phase space crystal state refers to the eigenstate of the Hamiltonian for a closed quantum system or the eigenoperator of the Liouvillian for an open quantum system. For a many-body system, phase space crystal is the solid-like crystalline state in phase space. The general framework of phase space crystals is to extend the study of solid state physics and condensed matter physics into phase space of dynamical systems. While real space has Euclidean geometry, phase space is embedded with classical symplectic geometry or quantum noncommutative geometry.
The Aubry–André model is a toy model of a one-dimensional crystal with periodically varying onsite energies. The model is employed to study both quasicrystals and the Anderson localization metal-insulator transition in disordered systems. It was first developed by Serge Aubry and Gilles André in 1980.
The quantum boomerang effect is a quantum mechanical phenomenon whereby wavepackets launched through disordered media return, on average, to their starting points, as a consequence of Anderson localization and the inherent symmetries of the system. At early times, the initial parity asymmetry of the nonzero momentum leads to asymmetric behavior: nonzero displacement of the wavepackets from their origin. At long times, inherent time-reversal symmetry and the confining effects of Anderson localization lead to correspondingly symmetric behavior: both zero final velocity and zero final displacement.
Quantum computational chemistry is an emerging field that exploits quantum computing to simulate chemical systems. Despite quantum mechanics' foundational role in understanding chemical behaviors, traditional computational approaches face significant challenges, largely due to the complexity and computational intensity of quantum mechanical equations. This complexity arises from the exponential growth of a quantum system's wave function with each added particle, making exact simulations on classical computers inefficient.
