Vladimir's homepage
Some results of Vladimir
Korepin
Quantum field theory
Semiclassical quantization of sine Gordon: → φ_{tt } φ_{xx} +m^{2} 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
 Abovebarrier 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 antisoliton do not reflect.
 Quantum Theory of Solitons. by L.D. Faddeev and V.E. Korepin, Physics Reports vol 42 (1), pages 187, June 1978.
 Massive Thirring model is solved by Bethe ansatz :
Published in Theoretical and Mathematical Physics, vol 41, page 953967 in 1979.
 Massive Thirring model can be build with stationary pulses of light.
 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 YangBaxter 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 .
Integrable version of SineGordon was formlated on a lattice in clasical and quantum cases.
 Quantization of a nonAbelian Toda chain , published first in Zapiski Nauchnyh Seminarov LOMI in 1981.
 Solution of YangBaxter equation with nonAbelian spectral parameter [belongs to SU(2) group], see page 126 of the book
Quantum Inverse Scattering Method...
 Higher Conservation Laws for the Quantum NonLinear Schroedinger Equation , by B.Davis and V.E. Korepin,
Preprint CMAR3389 of the Center for Mathematical Analysis of Australian National University in Canberra, 1989
In LiebLiniger 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 onedimensional bosons and algebraic Bethe ansatz
A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6, page 283, 1982

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 spin½ 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
 SPECTRUM AND SCATTERING OF EXCITATIONS IN THE ONEDIMENSIONAL
ISOTROPIC HEISENBERG MODEL. by L. D. Faddeev and L. A. Takhtadzhyan. Translatetion from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo
Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 134178, 1981 published in JOURNAL OF MATHEMATICAL SCIENCES
Volume 24, Number 2 (1984), 241267
 in the paper Calculation of Norms of Bethe Wave Functions the author proved that
 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.
 Conformal dimensions in Bethe ansatz solvable models, Korepin , Izergin and Reshetikhin : Journal of Physics A: Math. Gen. vol 22 (year 1989), pages 26152620 .
 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, SpringerVerlag, 1985.
 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 sinhGordon → φ_{tt } φ_{xx} +m^{2} 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 LiebLiniger 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  x2 μ t  ln
 e^{μ 2  β } 1 
e^{μ 2  β } +1 
} .
This formula was confirmed and generalized by Subir Sachdev in this paper .
 Temperature Correlations of Quantum Spins
A.R.Its, A.G.Izergin, V.E.Korepin, N.A.Slavnov, Phys.Rev.Lett. 70 (1993) 17041708; Erratumibid. 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 DensityDensity Temperature Correlations of OneDimensional 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 53115316, 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 44434452, 2002
In XXX spin chain correlation functions are polynomial (with rational coefficients) of values of Riemann zeta function with odd arguments ζ(2n+1).
 FORM FACTORS IN THE FINITE VOLUME by V.E. Korepin and N.A. Slavnov.
International Journal of Modern Physics B, October 1999, Vol. 13, No. 24n25 : pp. 29332941
(doi: 10.1142/S0217979299002769)
 Critical exponents for the onedimensional Hubbard model by
H.Frahm and V. Korepin, Physical Review B
vol 42, number 16 page 10553 in 1990.

Correlation functions of the onedimensional 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 Fermisurface are evaluated.

THE ROLE OF QUASIONEDIMENSIONAL STRUCTURES IN HIGHTc 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 OneDimensional 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:hepth/9310158 .
 Norm of an eigenfunction is a determinant of linearized LiebWu equations.
F. Goehmann, V. E. Korepin , Phys.Lett. A263 (1999) 293298
 Formfactors in Hubbard , F. H.L. Essler, V. E. Korepin. Phys. Rev. B v 59, N 3, page 1734, 1999
 Thermodynamics and excitations of the onedimensional Hubbard model
by
T. Deguchi, K. Kusakabe, F. Goehmann, F. H. L. Essler, A. Klümper and V. E. Korepin: Physics Reports vol 331 (2000) 197281
 Correlations in 1D electron gas. F. Goehmann, V.E. Korepin , Phys.Lett. A260 (1999) 516521.

Universality of Entropy Scaling in 1D Gapless Models
V.E. Korepin, Physical Review Letters, vol 92, issue 9, electronic identifier 096402, 05 March 2004,
arXiv:condmat/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. 14, page 79, 2004 (submitted to arXiv.org on April 15 of 2003).
Isotropic XY model is considered: H=Σ_{n}(σ^{x}_{n}σ^{x}_{n+1} +
σ^{y}_{n}σ^{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 subleading term is calculated.
 Renyi entropy = → 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 29752990, 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 subexponentially as the eigenvalues diminishes. Quantum Information Processing: Vol 10, Issue 3 (2011), Page 325.
 Local Convertibility and the Quantum Simulation of Edge States in ManyBody Systems by Fabio Franchini, Jian Cui, Luigi Amico, Heng Fan, Mile Gu, Vladimir Korepin, Leong Chuan Kwek, and Vlatko Vedral
 Entanglement in a ValenceBondSolid 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 .
 ENTANGLEMENT IN VALENCEBONDSOLID STATES
LiebLiniger model of one dimensional anyons
 LargeDistance Asymptotic Behavior of the Correlation Functions of 1D Impenetrable
Anyons at Finite Temperatures
Ovidiu I. Patu, Vladimir E. Korepin, Dmitri V. Averin
 Nonconformal asymptotic behavior of the timedependent fieldfield 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

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 algebraic Bethe ansatz . It presents a method of calculations of correlation functions using Fredholm determinant representation, completely integrable differential equations and RiemannHilbert problem . Main example is LiebLiniger model also known as Tonks Girardeau gas or Bose gas with delta interaction, it is equivalent to quantum nonlinear Schroedinger equation : iψ_{t} =  ψ_{xx} + κ ψ^{2}ψ .

The OneDimensional Hubbard model by F.H.L. Essler, H.Frahm,
F. Goehmann, A. Kluemper and V.E. Korepin, Cambridge University Press , 2005
 Exactly Solvable Models of Strongly Correlated Electrons . Reprint volume, eds. F.H.L. Essler and V.E. Korepin, World Scientific, 1994
 Festschriff IJMPB proceedings also the Collection of Articles Written in Honor of the 60th Birthday of Vladimir Korepin
 SPECIAL ISSUE of IJMPB: Classical versus Quantum Correlations in Composite Systems
Former graduate students
Teaching