In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying force laws as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle.
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.
In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change i.e. a minimum in electrical resistivity with temperature. The cause of the effect was first explained by Jun Kondo, who applied third-order perturbation theory to the problem to account for scattering of s-orbital conduction electrons off d-orbital electrons localized at impurities. Kondo's calculation predicted that the scattering rate and the resulting part of the resistivity should increase logarithmically as the temperature approaches 0 K. Experiments in the 1960s by Myriam Sarachik at Bell Laboratories provided the first data that confirmed the Kondo effect. Extended to a lattice of magnetic impurities, the Kondo effect likely explains the formation of heavy fermions and Kondo insulators in intermetallic compounds, especially those involving rare earth elements such as cerium, praseodymium, and ytterbium, and actinide elements such as uranium. The Kondo effect has also been observed in quantum dot systems.
In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. One GL-type superconductor is the famous YBCO, and generally all cuprates.
In the physical theory of spin glass magnetization, the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction models the coupling of nuclear magnetic moments or localized inner d- or f-shell electron spins through conduction electrons. It is named after Malvin Ruderman, Charles Kittel, Tadao Kasuya, and Kei Yosida, the physicists who first proposed and developed the model.
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard.
In quantum physics, the spin–orbit interaction is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is at the origin of magnetocrystalline anisotropy and the spin Hall effect.
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.
Spin-polarized scanning tunneling microscopy (SP-STM) is a type of scanning tunneling microscope (STM) that can provide detailed information of magnetic phenomena on the single-atom scale additional to the atomic topography gained with STM. SP-STM opened a novel approach to static and dynamic magnetic processes as precise investigations of domain walls in ferromagnetic and antiferromagnetic systems, as well as thermal and current-induced switching of nanomagnetic particles.
In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927. In its linearized form it is known as Lévy-Leblond equation.
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.
The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals. It is often applied to the description of Kondo effect-type problems, such as heavy fermion systems and Kondo insulators. In its simplest form, the model contains a term describing the kinetic energy of the conduction electrons, a two-level term with an on-site Coulomb repulsion that models the impurity energy levels, and a hybridization term that couples conduction and impurity orbitals. For a single impurity, the Hamiltonian takes the form
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as quantum triviality. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions, but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics, the question of triviality in Higgs models is of great importance.
The numerical renormalization group (NRG) is a technique devised by Kenneth Wilson to solve certain many-body problems where quantum impurity physics plays a key role.
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.
The Rashba effect, also called Bychkov–Rashba effect, is a momentum-dependent splitting of spin bands in bulk crystals and low-dimensional condensed matter systems similar to the splitting of particles and anti-particles in the Dirac Hamiltonian. The splitting is a combined effect of spin–orbit interaction and asymmetry of the crystal potential, in particular in the direction perpendicular to the two-dimensional plane. This effect is named in honour of Emmanuel Rashba, who discovered it with Valentin I. Sheka in 1959 for three-dimensional systems and afterward with Yurii A. Bychkov in 1984 for two-dimensional systems.
In quantum mechanics, the Schrieffer–Wolff transformation is a unitary transformation used to determine an effective Hamiltonian by decoupling weakly interacting subspaces. Using a pertubative approach, the transformation can be constructed such that the interaction between the two subspaces vanishes up to the desired order in the perturbation. The transformation also perturbatively diagonalizes the system Hamiltonian to first order in the interaction. In this, the Schrieffer–Wolff transformation is an operator version of second-order perturbation theory. The Schrieffer–Wolff transformation is often used to project out the high energy excitations of a given quantum many-body Hamiltonian in order to obtain an effective low energy model. The Schrieffer–Wolff transformation thus provides a controlled perturbative way to study the strong coupling regime of quantum-many body Hamiltonians.
Electric dipole spin resonance (EDSR) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.
Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.
In particle physics, the Cornell potential is an effective method to account for the confinement of quarks in quantum chromodynamics (QCD). It was developed by Estia J. Eichten, Kurt Gottfried, Toichiro Kinoshita, John Kogut, Kenneth Lane and Tung-Mow Yan at Cornell University in the 1970s to explain the masses of quarkonium states and account for the relation between the mass and angular momentum of the hadron. The potential has the form:
