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.
The Berezinskii–Kosterlitz–Thouless (BKT) transition is a phase transition of the two-dimensional (2-D) XY model in statistical physics. It is a transition from bound vortex-antivortex pairs at low temperatures to unpaired vortices and anti-vortices at some critical temperature. The transition is named for condensed matter physicists Vadim Berezinskii, John M. Kosterlitz and David J. Thouless. BKT transitions can be found in several 2-D systems in condensed matter physics that are approximated by the XY model, including Josephson junction arrays and thin disordered superconducting granular films. More recently, the term has been applied by the 2-D superconductor insulator transition community to the pinning of Cooper pairs in the insulating regime, due to similarities with the original vortex BKT transition.
Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on:
In quantum field theory and statistical mechanics, the Hohenberg–Mermin–Wagner theorem or Mermin–Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost, and since they increase the entropy, they are favored.
Barry Malcolm McCoy is an American physicist, known for his contributions to classical statistical mechanics, integrable models and conformal field theories.
Tai Tsun Wu is a Chinese-born American physicist and writer well known for his contributions to high-energy nuclear physics and statistical mechanics.
Jürg Martin Fröhlich is a Swiss mathematician and theoretical physicist. He is best known for introducing rigorous techniques for the analysis of statistical mechanics models, in particular continuous symmetry breaking, and for pioneering the study of topological phases of matter using low-energy effective field theories.
In statistical mechanics, the Griffiths inequality, sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality, named after Robert B. Griffiths, is a correlation inequality for ferromagnetic spin systems. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.
The Kibble–Zurek mechanism (KZM) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate. It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation through cosmological phase transitions in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.
Quantum simulators permit the study of a quantum system in a programmable fashion. In this instance, simulators are special purpose devices designed to provide insight about specific physics problems. Quantum simulators may be contrasted with generally programmable "digital" quantum computers, which would be capable of solving a wider class of quantum problems.
Antti Kupiainen is a Finnish mathematical physicist.
Jorge V. José is a Mexican/American physicist born in Mexico City. Currently the James H. Rudy Distinguished Professor of Physics at Indiana University. He has made seminal contributions to research in a variety of disciplines, including condensed matter physics, nonlinear dynamics, quantum chaos, biological physics, computational neuroscience and lately precision psychiatry. His pioneering work on the two-dimensional x-y model has been exceedingly influential in many areas of physics and has garnered many citations. He edited the book on the “40 Years of Berezinskii-Kosterlitz-Thouless Theory”, on two-dimensional topological phase transitions in 2013. Three years later KT were awarded the 2016 Nobel Physics Prize.
David Chandos Brydges is a mathematical physicist.
Tetsuji Miwa is a Japanese mathematician, specializing in mathematical physics.
Gian Michele Graf is a Swiss mathematical physicist.
The KTHNY-theory describes the melting of crystals in two dimensions (2D). The name is derived from the initials of the surnames of John Michael Kosterlitz, David J. Thouless, Bertrand Halperin, David R. Nelson, and A. Peter Young, who developed the theory in the 1970s. It is, beside the Ising model in 2D and the XY model in 2D, one of the few theories, which can be solved analytically and which predicts a phase transition at a temperature .
The quantum clock model is a quantum lattice model. It is a generalisation of the transverse-field Ising model. It is defined on a lattice with states on each site. The Hamiltonian of this model is
Giovanni Felder is a Swiss mathematical physicist and mathematician, working at ETH Zurich. He specializes in algebraic and geometric properties of integrable models of statistical mechanics and quantum field theory.
Krzysztof Gawędzki was a Polish mathematical physicist, a graduate of the University of Warsaw and professor at the École normale supérieure de Lyon. He was primarily known for his research on quantum field theory and statistical physics. In 2022, he shared the Dannie Heineman Prize for Mathematical Physics with Antti Kupiainen.
