Some Results of Vladimir Korepin

• Izergin-Korepin Model
• Six vertex model with domain wall boundary conditions was first introduced by Korepin in 1982.
• Google Scholar
• SPRINGER
• List of early publications
• Math-Net.ru in russian
• ResearchGate

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).
• Massive Thirring model is solved by Bethe ansatz : Published in Theoretical and Mathematical Physics, vol 41, page 953-967 in 1979.
• 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.
• Massive Thirring model can be build with stationary pulses of light.

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 .
  Integrable version of Sine-Gordon was formlated on a lattice in clasical and quantum cases.
• Quantization of a non-Abelian Toda chain , published first in Zapiski Nauchnyh Seminarov LOMI in 1981.
• Solution of Yang-Baxter equation with non-Abelian spectral parameter [belongs to SU(2) group], see page 126 of the book Quantum Inverse Scattering Method...
• 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
• 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
• SPECTRUM AND SCATTERING OF EXCITATIONS IN THE ONE-DIMENSIONAL 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. 134-178, 1981 published in JOURNAL OF MATHEMATICAL SCIENCES Volume 24, Number 2 (1984), 241-267

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.

• 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 ρα)

    (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.

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

• 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 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 . The djvu file of the book can be converted into pdf using instructions from website .
• The One-Dimensional 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