Vladimir Korepin
Encyclopedia
Vladimir Korepin is a Russian
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
-American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
physicist and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He is a professor at the C. N. Yang Institute of Theoretical Physics of the State University of New York
State University of New York
The State University of New York, abbreviated SUNY , is a system of public institutions of higher education in New York, United States. It is the largest comprehensive system of universities, colleges, and community colleges in the United States, with a total enrollment of 465,000 students, plus...
at Stony Brook. Korepin has notable results in several areas and has published about 187 papers in theoretical and mathematical physics.
Background
Korepin completed his undergraduate study at Saint Petersburg State UniversitySaint Petersburg State University
Saint Petersburg State University is a Russian federal state-owned higher education institution based in Saint Petersburg and one of the oldest and largest universities in Russia....
, graduating with a diploma in theoretical physics in 1974. In that same year he was employed by the Mathematical Institute of Academy of Sciences of Russia
St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences
The St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences for "Петербургское отделение Математического института", Petersburg Department of the Mathematical Institute) is a mathematical research institute in St. Petersburg, part of the Russian Academy of...
. He complted his graduate and postdoctoral studies in this Institute and worked there until 1989. Korepin obtained his PhD degree in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
in 1977 from the Mathematical Institute of the Academy of Sciences of Russia
Steklov Institute of Mathematics
Steklov Institute of Mathematics or Steklov Mathematical Institute is a research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. It was established April 24, 1934 by the decision of the General Assembly of the Academy of Sciences of the USSR in...
in Moscow. His scientific adviser was Ludwig Faddeev. In 1985, he got a degree of doctor of sciences in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
from the Council of Ministers of the Russian Federation.
Contributions to physics
One of Korepin's results in quantum gravityQuantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
(cancellation of ultra-violet infinities
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
in one loop gravity on mass shell
On shell and off shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell....
) was cited in the Feynman lectures on gravitation by Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
, edited by B. Hatfield, R. B. Morinigo and W. G. Wagner.
His papers on 1D Hubbard model
Hubbard model
The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...
(the central model of strongly correlated electrons
Strongly correlated material
Strongly correlated materials are a wide class of electronic materials that show unusual electronic and magnetic properties, such as metal-insulator transitions or half-metallicity...
) has citations in condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
. He has written a textbook with F. H. L. Essler, H. Frahm, F. Goehmann and A. Kluemper on the exact solution
Bethe ansatz
In physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
of the Hubbard model
Hubbard model
The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...
in one dimension.
Korepin has contributed results in 1D Bose gas
Bose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
with a delta potential interactions. He has written a book [with Bogoliubov and Izergin] for Cambridge University Press
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
in 1993 about the Bose gas
Bose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
es with delta interactions. The book can also be used as a textbook on quantum inverse scattering method
Quantum inverse scattering method
Quantum inverse scattering method relates two different approaches:1) Inverse scattering transform is a method of solving classical integrable differential equations of evolutionary type.Important concept is Lax representation....
and the algebraic Bethe ansatz
Bethe ansatz
In physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
.
Korepin presented a solution of the massive Thirring model
Thirring model
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in two dimension.-Definition:The Thirring model is given by the Lagrangian density...
in one space and one time dimension using Bethe ansatz
Bethe ansatz
In physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
; in particular, the exact calculation of the mass spectrum
Mass spectrum
A mass spectrum is an intensity vs. m/z plot representing a chemical analysis. Hence, the mass spectrum of a sample is a pattern representing the distribution of ions by mass in a sample. It is a histogram usually acquired using an instrument called a mass spectrometer...
and the scattering matrix. The paper with this solution was first published in Russian in 1979, and was translated into English in Theoretical and Mathematical Physics
Theoretical and Mathematical Physics
Theoretical and Mathematical Physics is a Russian scientific journal. It was founded in 1969 by Nikolai Bogolubov. Currently handled by the Russian Academy of Sciences, it appears in 12 issues per year...
.
He has also made a semiclassical calculation and one loop corrections
One-loop Feynman diagram
In physics, a one-loop Feynman diagram is a connected Feynman diagram with only one cycle . Such a diagram can be obtained from a connected tree diagram by taking two external lines of the same type and joining them together into an edge....
to the mass of soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
s (including bound states) and the scattering matrix (in reflectionless cases scattering matrices were obtained exactly). The main example used in his published work is the sine-Gordon model.
He discovered, in collaboration with Anatoly Izergin, the 19-vertex model (sometimes called the Izergin-Korepin model).
In 1993 Korepin together with A.R.Its, A.G.Izergin and
N.A.Slavnov calculated space, time and temperature dependent
correlation functions
Correlation function (statistical mechanics)
In statistical mechanics, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function, and describes how microscopic variables at different positions are correlated....
in isotropic version of the XY spin chain using the Heisenberg model
Heisenberg model (quantum)
The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
. The correlation function decays exponentially with time and space separation. The rate of exponential decay is evaluated explicitly.
Contributions to mathematics
In 1982, Korepin introduced domain wall boundary conditions for the six vertex model, published in Communications in Mathematical PhysicsCommunications in Mathematical Physics
Communications in Mathematical Physics is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in Analysis related to condensed matter physics, statistical mechanics and quantum field theory, and in...
. The result plays a role in diverse fields of mathematics such as algebraic combinatorics
Algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....
, Alternating sign matrix
Alternating sign matrix
In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices arise naturally when using Dodgson condensation to compute a determinant...
es, domino tiling
Domino tiling
A domino tiling of a region in the Euclidean plane is a tessellation of the region by dominos, shapes formed by the union of two unit squares meeting edge-to-edge...
, Young diagrams and plane partitions. In the same paper the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
formula was proved for the square of the norm of the Bethe ansatz wave function. It can be represented as a determinant of linearized system of Bethe equations. It can also be represented as a matrix determinant of second derivatives of the Yang action.
The so called "Quantum Determinant" was discovered in 1981 by A.G. Izergin and V.E. Korepin. It is the center of the Yang–Baxter algebra.
The study of differential equations for quantum correlation functions led to the discovery of a special class of Fredholm integral operators
Fredholm integral equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm.-Equation of the first kind :...
. Now they are referred to as completely integrable integral operators. They have multiple applications not only to quantum exactly solvable models
Integrable system
In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems.In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical...
, but also to random matrices
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
and algebraic combinatorics
Algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....
.
Contributions to quantum information
Vladimir Korepin has produced important results in the evaluation of the entanglement entropy of different dynamical models, such as interacting spins, Bose gasBose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
es, and the Hubbard model
Hubbard model
The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...
. He considered models with a unique ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
s, so that the entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
of the whole ground state is zero. The ground state is partitioned
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
into two spatially separated parts: the block and the environment. He calculated the entropy of the block as a function of its size and other physical parameters. In a series of articles, Korepin was the first to compute the analytic formula for the entanglement entropy
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
of the XX (isotropic) and XY Heisenberg model
Heisenberg model (quantum)
The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...
s. He used Toeplitz Determinants
Toeplitz matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...
and Fisher
Michael Fisher
Michael Ellis Fisher is an English physicist, as well as chemist and mathematician, known for his many seminal contributions...
-Hartwig Formula for the calculation. In the Valence-Bond-Solid states (which is the ground state of the Affleck-Kennedy-Lieb
Elliott H. Lieb
Elliott H. Lieb is an eminent American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis....
-Tasaki model of interacting spins), Korepin evaluated the entanglement entropy and studied the reduced density matrix. He also worked on quantum search algorithms
Grover's algorithm
Grover's algorithm is a quantum algorithm for searching an unsorted database with N entries in O time and using O storage space . It was invented by Lov Grover in 1996....
with Lov Grover
Lov Grover
Lov Kumar Grover is an Indian-American computer scientist. He is the originator of the Grover database search algorithm used in quantum computing. He obtained his undergraduate degree at Indian Institute of Technology, Delhi...
. Many of his publications on entanglement and quantum algorithms can be found on arxiv.org.
In May 2003, Korepin helped organize a conference on quantum
Quantum computer
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from traditional computers based on transistors...
and reversible computations in Stony Brook. Another conference is scheduled for November 15-18th 2010, entitled the "Simons Conference on New Trends in Quantum Computation".
Honours
In 1996 Korepin was elected fellow of the American Physical Society. He is also a fellow of the International Association of Mathematical Physics and the Institute of Physics. He is an editor of Reviews in Mathematical PhysicsReviews in Mathematical Physics
- External links :*...
, the International Journal of Modern Physics
International Journal of Modern Physics
The International Journal of Modern Physics is a series of 5 journals published by World Scientific.-B:The International Journal of Modern Physics B was founded in 1987...
and Theoretical and Mathematical Physics
Theoretical and Mathematical Physics
Theoretical and Mathematical Physics is a Russian scientific journal. It was founded in 1969 by Nikolai Bogolubov. Currently handled by the Russian Academy of Sciences, it appears in 12 issues per year...
. His 60's birthday was celebrated by Institute of Advanced Studies in Singapore in 2011.