Some Results of Vladimir Korepin

Formal profile from SUNY ; Google Scholar , papers refereeing to Korepin's results in Scopus .
Izergin-Korepin Model
H-index is 71 for Korepin + others; citation for 23826; cited publications 663

Diploma

The diploma was devoted to quantization of Einstein's theory of general relativity . The diploma was written under advice of Ludvig Fadeev. Cancellation of ultra-violet infinities in one loop quantum gravity on mass shell was proved simultaneously with G. t’Hooft and M. Veltman . The diploma was cited in the Feynman lectures on gravitation . Here is English translation of the diploma.

Semi-classical quantization of sine Gordon: → φ_tt- φ_xx +m² sin( φ )= 0.

Scattering matrix in quantum Sine Gordon. by I.Yu. Arefeva and V.E. Korepin. JETP Letters , v 22, N10, p 312, 1974
- Perturbative calculations show that there is no multi particle production on mass shell .
- Scattering matrix of fundamental bosons is calculated.
Above-barrier reflections of solitons . V.E. Korepin. JETP Letters , vol 23, page 201, 1976. Reflection coefficient r is exponentially small in Plank constant r ∼ exp(-1/h). Classically soliton and anti-soliton do not reflect:
Quantum Theory of Solitons. by L.D. Faddeev and V.E. Korepin, Physics Reports vol 42 (1), pages 1-87, June 1978.
- One loop calculation of mass of solitons and their bound states (exact masses at any coupling constant).
- Scattering matrix also was evaluated (in reflectionless cases scattering matrix was obtained exactly).

Quantum Inverse Scattering Method

A Lattice Version of Nonlinear Schroedinger Equation . by A.G.Izergin and V.E. Korepin, DOKLADY AKADEMII NAUK, 1981 .
Quantum Determinant is discovered [it is center of Yang-Baxter algebra] : det_q {T(λ)}= A(λ -i/2)D(λ +i/2)- B(λ -i/2)C(λ +i/2),
for R(λ -μ)= (λ -μ ) I- iΠ . Here A, B, C and D are entries of monodromy matrix T. Antipode was discovered in the same paper:
T^-1(λ) = σ ^y T^T(λ+i) σ ^y det^-1_q {T(λ +i/2)} . Quantum Determinant was generalized for monodromy matrices of higher dimensions by Evgeni Sklyanin .
A Lattice Version of Quantum Field Theory Models in Two Dimensions . A.G.Izergin and V.E. Korepin, Nuclear Physics B 205 [FS5], 401, 1982 .
Higher Conservation Laws for the Quantum Non-Linear Schroedinger Equation , by B.Davis and V.E. Korepin,
Preprint CMA-R33-89 of the Center for Mathematical Analysis of Australian National University in Canberra, 1989
In Lieb-Liniger model [also known as 1D Bose gas with delta interaction] higher conservation laws are constructed as explicit expression in terms of local fields. .
Pauli principle for one-dimensional bosons and algebraic Bethe ansatz A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6, page 283, 1982
Massive Thirring model is solved by Bethe ansatz :
- The fractional charge appears in the model during renormalization as a repulsion beyond cutoff.
- Massive Thirring model is equivalent to sine Gordon , for large coupling constant alternative quantization is possible.
ADIABATIC TRANSPORT PROPERTIES AND BERRY'S PHASE IN HEISBNBERG - ISING RING. Korepin and Wu, Int. J.our. of Mod. Phys. B . vol 5, no 3, (1991), 497.
New Identity for the Scattering Matrx of Exactly Solvable Models. V.Korepin and N.Slavnov. European Physics Journal B 5 page 555, 1998
Fine Structure of the Bethe Ansatz for the spin-½ Heisenberg XXX Model , F.H.L. Essler, V.E. Korepin, and K. Schoutens . Journ. Phys. A25, 4115 (1992).
Open Problems in Exactly Solvable Models .V.Korepin and O. Patu. Proceedings of SOLVAY workshop: Bethe Ansatz: 75 Years Later, 2006

Vertex models

in the paper Calculation of Norms of Bethe Wave Functions
the author proved that
- square of the norm of Bethe wave function is equal to the determinant of second derivatives of Yang's action.
- Domain wall boundary conditions for 6-Vertex model and recursion relations for the partition function were discovered .
  It is important for algebraic combinatorics, especially for enumeration of alternating sign matches.
The free energy in thermodynamic limit for 6 vertex model with domain wall boundary conditions is different from the one for periodic b.c. by P. Zinn-Justin and V. Korepin

19 vertex model was discovered in 1981 , now it is called Izergin-Korepin model.

Quasicrystals

Completely integrable models in quasicrystals
Critical properties of spin models in quasicrystals
Valence Bond Solid in Quasicrystals
Quasiperiodic tilings: generalized grid-projection method , V. E. Korepin, F. Gähler and J. Rhyner, Acta Crystallographica Section A (1988) A44, p 667-672

Correlation Functions

DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS, by A.R. Its , A.G. Izergin, V.E. Korepin and N.A. Slavnov ;
published in International Journal of Modern Physics vol B4, page 1003 in 1990.
- Completely integrable integral operators
  were discovered in this paper. It is Fredholm integral operators with a kernel=
  
  Σ_n e_n(λ)E_n(μ)
  
  λ -μ
- Emptiness formation probability
  was first introduced in the paper DIFFERENTIAL EQUATIONS ..., for spin chains it is a
  probability of formation of ferromagnetic string in anti-ferromagnetic ground state. At zero temperature it decays as a Gaussian
- Space and temperature dependent correlation function in Bose gas with delta interaction was calculated. The model also called Lieb–Liniger Model or Tonks–Girardeau gas.
Conformal dimensions in Bethe ansatz solvable models, Korepin , Izergin and Reshetikhin : Journal of Physics A: Math. Gen. vol 22 (year 1989), pages 2615-2620 .
QUANTUM INVERSE SCATTERING METHOD AND CORRELATION FUNCTIONS, N.M. Bolgoliubov, V.E. Korepin and A.G.Izergin.
Lecture Notes in Physics vol 242, page 220, Springer-Verlag, 1985. Average of operator e^{α Q(x)} was introduced as generating function of correlation functions of local densities
<ψ ⁺(x)ψ (x) ψ⁺ (0) ψ (0) >. Here Q(x) is an operator of number of particles on space interval (0,x).
Dual field formulation. Correlation function of Bethe ansatz solvable models can be represented as a Fredholm determinant away from free fermionic point, if one use quantum Bose fields in dual Fock space. Wick's theorem is used to remove scattering phase and reduce interacting fermions to free ones. This duality transform is similar to duality in string theory.
For sinh-Gordon → φ_tt- φ_xx +m² sinh( φ )= 0 closed expression for correlations was obtained in Journal of Physics A: Mathematical and General vol 31, page 9283, (1998).
Correlation functions in 1D electron gas satisfy simple differential equations F. Goehmann, A.R. Its, V.E. Korepin, Phys. Lett. A, 249 (1998) 117 .
Space, time and temperature dependent correlation functions of Lieb-Liniger model were calculated by Its, Izergin and Korepin in Nuclear Physics B in 1991 and Physica D .
When both space and time separation go to infinity then asymptotic of <ψ (x,t)⁺ ψ (0,0) >_T → exp { (1/π) ∫ d μ | x-2 μ t | ln

| e^{μ ² - β} -1|

e^{μ ² - β} +1

}, see also page 469 of the textbook .
Temperature Correlations of Quantum Spins A.R.Its, A.G.Izergin, V.E.Korepin, N.A.Slavnov, Phys.Rev.Lett. 70 (1993) 1704-1708; Erratum-ibid. 70 (1993) 2357.
Temperature Corrections to Conformal Field Theory . Fabian H. L. Essler, Vladimir E. Korepin, Franck T. Latremoliere. Europ. Phys. Journ. B 5, page 559, 1998
Time Dependence of the Density-Density Temperature Correlations of One-Dimensional Bose Gas. V.E. Korepin and N.A. Slavnov. Nuclear Physics B 340. page 759, 1990.
Quantum Spin Chains and Riemann Zeta Function with Odd Arguments
by H. E. Boos and V. E. Korepin. Journal of Phys. A Math. and General, vol 34, pages 5311-5316, 2001
The discovery of factorization of multiple integrals, representing correlation functions into a sum of products of single integrals .
Quantum Correlations and Number Theory
by H. E. Boos, V. E. Korepin, Y. Nishiyama, M. Shiroishi. Journal of Physics A Math. and General, vol 35, pages 4443-4452, 2002
In XXX spin chain correlation functions are polynomial (with rational coefficients) of values of Riemann zeta function with odd arguments ζ(2n+1).

Instantons

Three instanton solution. by S.L. Shatashvili and V.Korepin ; Izvestiya Akademii Nauk. Ser. Mat. vol 48 , nom 2, 1984; English translation is Math. USSSR, Izvestiya, vol 24 no 2, 1985

Hubbard Model

Critical exponents for the one-dimensional Hubbard model by H.Frahm and V. Korepin, Physical Review B vol 42, number 16 page 10553 in 1990.
Correlation functions of the one-dimensional Hubbard model in a magnetic field H.Frahm and V. Korepin, Physical Review B vol 43, number 7 page 5653 in 1991.
Critical exponents describing distribution of electrons close to Fermi-surface are evaluated.
THE ROLE OF QUASI-ONE-DIMENSIONAL STRUCTURES IN HIGH-Tc SUPERCONDUCTIVITY by N.M. Bogoliubov and V.E.Korepin
International Journal of Modern Physics B, vol 3, no 3, 427, 1989. Critical exponents are calculated for attractive Hubbard.
Scattering Matrix of 1D Hubbard model by Essler and Korepin. PRL vol 72, n. 6, p. 908, 1994. It describes scattering of spinons and holons , which form the basis of all exited stated at half filled band. The scattering matrix has the same symmetry as the Hamiltonian: the Yangian.
Completeness of the SO(4) Extended Bethe Ansatz for the One-Dimensional Hubbard Model . Fabian H.L. Essler, Vladimir E. Korepin
and Kareljan Schoutens, Phys. Rev. Lett. vol 67, number 27, page 3848, 1991 and Nucl .Phys. B. vol 384, no 3, page 431, 1992
Yangian Symmetry of 1D Hubbard model D. B. Uglov and V.E. Korepin, Physics Letters A vol 190 page 238, 1994, arXiv:hep-th/9310158 .
Norm of an eigenfunction is a determinant of linearized Lieb-Wu equations. F. Goehmann, V. E. Korepin , Phys.Lett. A263 (1999) 293-298
Form-factors in Hubbard , F. H.L. Essler, V. E. Korepin. Phys. Rev. B v 59, N 3, page 1734, 1999
Thermodynamics and excitations of the one-dimensional Hubbard model by
T. Deguchi, K. Kusakabe, F. Goehmann, F. H. L. Essler, A. Klümper and V. E. Korepin Phys. Rep. 331 (2000) 197-281
Correlations in 1D electron gas. F. Goehmann, V.E. Korepin , Phys.Lett. A260 (1999) 516-521.

Supersymmetric extension

Spectrum of Low-Lying Excitations in a Supersymmetric Extended Hubbard Model
F. H.L. Essler, V. E. Korepin . Int. Journ. Modern Phys. B vol 8, N 23, p 3243, 1994
Eta-pairing as a mechanism of superconductivity in models of strongly correlated electrons
J. de Boer, V. Korepin and A. Schadschneider. An extended Hubbard model is constracted, for which η pairs are in the ground state .

Entanglement

Universality of Entropy Scaling in 1D Gap-less Models
V.E. Korepin, Physical Review Letters, vol 92, issue 9, electronic identifier 096402, 05 March 2004, arXiv:cond-mat/0311056
Critical models are considered in one dimension with central charge c and Fermi velocity v.

At zero temperature logarithmic scaling of the entropy is derived form the second law of thermodynamics.
The entropy of a subsystem is calculated for Lieb–Liniger Model with delta interaction, spin chains and fermi Hubbard model.
Entropy of electrons on a space interval calculated for positive temperature Τ : S(x)= (c/3) ln{ (v/π T) sin [π Tx/v] }

, see formula (14).

Quantum Spin Chain, Toeplitz Determinants and Fisher -Hartwig Formula .
B.-Q.Jin, V.E.Korepin, Journal of Statistical Physics , vol 116, Nos. 1-4, page 79, 2004 (submitted to arXiv.org on April 15 of 2003).
Isotropic XY model is considered: H=Σ_n(σ^x_nσ^x_n+1 + σ^y_nσ^y_n+1+hσ^z_n) here σ are Pauli matrices and h is magnetic field.
- Logarithmic formula for the leading term on entanglement entropy is proven and sub-leading term is calculated.
- Renyi entropy =
  
  ln (tr ρ^α)
  
  (1-α )
  
  →
  
  (1+α^-1)
  
  6
  
  ln (x) , also behaves logarithmically with the size of the block x→ ∞ , see formulae (3) and (5).
  Here ρ is density matrix of the block of spins and α is a parameter.
Entanglement in XY Spin Chain A. R. Its, B.-Q. Jin, V. E. Korepin, Journal Phys. A: Math. Gen. vol 38, pages 2975-2990, 2005 (submitted to arXiv.org on September 3 of 2004 ).
Renyi Entropy of the XY Spin Chain by F. Franchini , A. R. Its, V. E. Korepin, Journal of Physics A: Math. Theor. 41 (2008) 025302
Entanglement Spectrum for the XY Model in One Dimension F. Franchini, A. R. Its, V. E. Korepin, L. A. Takhtajan . The spectrum of reduced density matrix of large block of spins in the ground state of XY model is geometric sequence. The largest eigenvalue is explicitly evaluated. Degeneracy of eigenvalues increases sub-exponentially as the eigenvalues diminishes. Quantum Information Processing: Vol 10, Issue 3 (2011), Page 325.

AKLT and VBS

Entanglement in a Valence-Bond-Solid State H. Fan, V. Korepin, V. Roychowdhury, Physical Review Letters, vol 93, issue 22, 227203, 2004
The reduced density matrix of a continuous block of spins [of arbitrary length] has rank four. In the limit of large block the density matrix is proportional to a projector to degenerated ground state of a 'block' Hamiltonian [a part of original Hamiltonian describing interaction of spins inside of the block]. Both von Neumann entropy and Renyi entropy of large block of spins is ln(4).

Valence Bond Solid in Quasicrystals by A. Kirillov and V. Korepin, ALGEBRA and ANALYSIS , vol 1, issue 2, page 47, 1989
A version of AKLT is constructed with unique VBS ground state on a finite graph .

Quantum Search Algorithms

Simple Algorithm for Partial Quantum Search by L. Grover and V. Korepin
Optimization of Partial Search

Lieb-Liniger Model of One Dimensional Anyons

Large-Distance Asymptotic Behavior of the Correlation Functions of 1D Impenetrable Anyons at Finite Temperatures
Ovidiu I. Patu, Vladimir E. Korepin, Dmitri V. Averin
Non-conformal asymptotic behavior of the time-dependent field-field correlators of 1D anyons
Ovidiu I. Patu, Vladimir E. Korepin, Dmitri V. Averin
One dimensional abelian anyons was realized experimentally in quantum optics in Centre for Nanoscience and Quantum Information in University of Bristol , see J. C. F. Matthews, A. Politi, A. Stefanov, J. L. O'Brien , “Manipulation of multiphoton entanglement in waveguide quantum circuits”, Nature Photonics , 3, 346 - 350 (2009) and Matthews, J. C. F., Poulios, K., Meinecke, J. D. A., Politi A., Peruzzo, A., Ismail, N., Worhoff, K., Thompson, M. G. & O'Brien, J. L. Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermion.

Books

The One-Dimensional Hubbard model by F.H.L. Essler, H.Frahm, F. Goehmann, A. Kluemper and V.E. Korepin, Cambridge University Press , 2005 The text book describes theory of fermi Hubbard model .
Quantum Inverse Scattering Method and Correlation Functions by V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Cambridge University Press, 1993.
The text book explains Bethe ansatz . It presents a method of calculations of correlation functions using Fredholm determinant representation, completely integrable differential equations and Riemann-Hilbert problem . Main example is Lieb-Liniger model also known as Tonks–Girardeau gas or Bose gas with delta interaction, it is equivalent to quantum nonlinear Schroedinger equation : iψ_t = −½ψ_xx + κ |ψ|²ψ .
Exactly Solvable Models of Strongly Correlated Electrons . Reprint volume, eds. F.H.L. Essler and V.E. Korepin, World Scientific, 1994