Vladimir Korepin

• Izergin-Korepin model
• Google scholar
• Springer
• List of early publications

Quantum Gravity and Field Theory

• 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. The diploma was cited in the Feynman lectures on gravitation . Here is English translation of the diploma.
• Reduction of оne-loop Feynman diagrams in massive scalar quantum field theory . 1979 in VESTNIK Leningradskogo Gosudarstvennogo Universiteta, 2, N16, page 17 in Russian language. Massive scalar quantum filed theory is considered. One loop Feynman diagrams evaluated. Theorem is proved that one loop diagram with many internal lines can be expressed in terms of one loop diagram with number of internal lines equal to the dimension of space-time [multiplied by tree diagrams], see English translation.
• Three instanton solution. Izvestiya Akademii Nauk. Ser. Mat. vol 48 , number 2, 1984; English translation is Math. USSR, Izvestiya, vol 24 no 2, 1985
• Entanglement entropy production in deep inelastic scattering. January first 2022 issue of Physical Review D (Vol. 105, No.1), DOI: 10.1103/PhysRevD.105.014002
• Yangian in QCD.

Quantization of sine-Gordon equation: φ_tt - φ_xx +m² sin(φ)= 0

• Scattering matrix in quantum Sine Gordon 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-barier reflections of solitons . JETP Letters , vol 23, page 201, 1976. The reflection coefficient is exponentially small in Plank constant
  Classically soliton and anti-soliton do not reflect.
• Quantum Theory of Solitons. 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.
• New Effects
• Sine-Gordon was experimentally implemented in an array of Josephson junctions in 1981. The field was developed, lectures were given in DPA of Stony Brook University. Now it is used for reversible computation .

Quantum inverse scattering method

• A Lattice Version of Nonlinear Schroedinger Equation : DOKLADY AKADEMII NAUK, 1981 , where quantum determinant was discovered, see also Quantum Groups and Quantun Determinants]. Antipode [ modification of the Cramer's rule ] was discovered in the same paper, see also Algebraic quantum groups.
  The latice version of nonlinear Schroedinger equation describes deep inelastic scattering .
• A Lattice Version of Quantum Field Theory Models in Two Dimensions : Nuclear Physics B 205 [FS5], 401, 1982 .
  Integrable version of Sine-Gordon was formlated on a lattice in clasical and quantum cases.
• Classical version of lattice nonlinear Schoedinger is gauge equivalent to Ablowitz-Ladik equation , also Hamiltonian structures are different.
• 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 end of the Section VI.4 of the book Quantum Inverse Scattering Method...
• Higher Conservation Laws for the Quantum Non-Linear Schroedinger Equation
  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 : Letter in Mathematical Physics vol 6, page 283, 1982
• ADIABATIC TRANSPORT PROPERTIES AND BERRY'S PHASE IN HEISBNBERG - ISING RING. : Int. J.our. of Mod. Phys. B . vol 5, no 3, (1991), 497.
• New Identity for the Scattering Matrx of Exactly Solvable Models. European Physics Journal B 5 page 555, 1998
• Fine Structure of the Bethe Ansatz for the spin-½ Heisenberg XXX Model . Journ. Phys. A25, 4115 (1992).
• Open Problems in Exactly Solvable Models Proceedings of SOLVAY workshop: Bethe ansatz: 75 years later, 2006
• SPECTRUM AND SCATTERING OF EXCITATIONS IN THE ONE-DIMENSIONALISOTROPIC HEISENBERG MODEL. Translation from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya MatematicheskogoInstituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 134-178, 1981 published in JOURNAL OF MATHEMATICAL SCIENCESVolume 24, Number 2 (1984), 241-267
• Chiral Potts J. Phys. A: Math. Theor. 49 (2016) 153/001
• R-matrix for Bullough-Dodd model. The differential equation was discovered in 1907 by M.G. Tzitzeica ; Sur une nouvelle classe de surfaces, Rendiconti del Circolo Matematico di Palermo 25 (1908) 180. The integrability was proved in 1979 by A.V. Zhiber and A.B. Shabat in Doklady Akademii Nauk SSSR vol 247, page 1103 and A.V. Mikhailov Pisma in Zh. Ex. Tr Fiz vol 30, page 443
• Coordinate Bethe Ansatz on a Quantum Computer , by Igor L. Aleiner
• Algebraic Bethe Ansatz on a Quantum Computer , by Alejandro Sopena, Max Hunter Gordon, Diego Garca-Martin, German Sierra, and Esperanza Lopez

Six vertex model and boundary conditions

• in the paper Calculation of Norms of Bethe Wave Functions the author proved Gaudin's formula:
  • 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 matrixes.
    Partition function of 6-vertex model with domain wall boundary conditions can be represented as determinant of a matrix of a size of the lattice.
    This formula generates determinant representation for scalar product. The Slavnov formula can be derived from Izergin-Korepin formula .
  • The free energy in thermodynamic limit for 6 vertex model with domain wall boundary conditions is different from the one for periodic b.c.
  • The bulk free energy in 6 vertex model with reflective boundary conditions is the same as with domain wall b.c.
  • Influence of boundary conditions on bulk properties of six-vertex model
  • The entropy of the six-vertex model with variety of different boundary conditions
  • Phase separation in the six-vertex model
  • The density profile of the six vertex model
  • Random Matrices and the Six-Vertex Model by Pavel Bleher and Karl Liechty
  • Proofs and Confirmations by M. Bressoud
  • SIX-VERTEX, LOOP AND TILING MODELS: INTEGRABILITY AND COMBINATORICS by Paul Zinn-Justin
• Conference on Six vertex
• Six vertex with reflecting boundary condition

• DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS,
  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 $$\displaystyle \mbox{Kernel}= \frac{\sum_ne_n(\lambda)E_n(\mu)}{\lambda-\mu}$$
  • 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.
• Conformal dimensions in Bethe ansatz solvable models: Journal of Physics A: Math. Gen. vol 22 (year 1989), pages 2615-2620 .
• QUANTUM INVERSE SCATTERING METHOD AND CORRELATION FUNCTIONS, Lecture Notes in Physics vol 242, page 220, Springer-Verlag, 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.
  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 Phys. Lett. A, 249 (1998) 117 .
• Space, time and temperature dependent correlation functions of Lieb-Liniger model were calculated in Nuclear Physics B in 1991 .
  When both space and time go to infinity then asymptotic of \[\langle\Psi(x,t)^\dagger\Psi(0,0)\rangle_T\rightarrow\exp\left\{(1/\pi)\int_{-\infty}^{\infty}{d\mu}|x-2\mu{t}|\ln\frac{|e^{\mu^2-\beta}-1|}{e^{\mu^2-\beta}+1}\right\}\]
• At large temperature correlation functions are Gaussian for Lieb-Liniger, see the last formula (A.4) of Nuclear PhysicsB257[FS14] (1985)766-778
• Temperature correlations of quantum spins can be described by Ablowitz-Ladik equation : Phys.Rev.Lett. 70 (1993) 1704-1708; Erratum-ibid. 70 (1993) 2357.
  
• Temperature Corrections to Conformal Field Theory : Europ. Phys. Journ. B 5, page 559, 1998
• Time Dependence of the Density-Density Temperature Correlations of One-Dimensional Bose Gas. Nuclear Physics B 340. page 759, 1990.
• Quantum Spin Chains and Riemann Zeta Function with Odd Arguments by 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 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).
  The efficient description is given by T. MIWA AND F. SMIRNOV
• FORM FACTORS IN THE FINITE VOLUME
  International Journal of Modern Physics B, October 1999, Vol. 13, No. 24n25 : pp. 2933-2941 (doi: 10.1142/S0217979299002769)
  The form factors in the finite volume can be represented in terms of determinants
• Auto-Correlation in Ising

Quantum entanglement

• Universality of Entropy Scaling in 1D Gap-less Models
  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 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: $$H\hspace{-1mm}=\hspace{-1mm}\sum_n({\sigma^x}_{n}{\sigma^x}_{n+1}\hspace{-1mm}+\hspace{-1mm}{\sigma^y}_{n}{\sigma^y}_{n+1}\hspace{-1mm}+\hspace{-1mm}h{\sigma^z}_{n})$$ here $\sigma$ 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 also behaves logarithmically with the size of the block $x\rightarrow\infty$, see formulae (3) and (5) $$\displaystyle \mbox{Renyi entropy}=\frac{\ln(\text{tr}\,\rho^\alpha)}{(1-\alpha)}\rightarrow\frac{(1+\alpha^{-1})}{6}\ln(x)$$ Here $\rho$ is density matrix of the block of spins and $\alpha$ is a positive real number.
• Entanglement in XY Spin Chain 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 Journal of Physics A: Math. Theor. 41 (2008) 025302
• Entanglement Spectrum for the XY Model in One Dimension . 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.
• Fredkin-gate Spin Chain , Rev. Math. Phys. 29(10) (2017) 1750031, 20 pp. and Abstract, DOI 10.1142/S0129055X17500313.
• Q-deformed Fredkin-gate Spin Chain, Deformed Fredkin Spin Chain with Extensive Entanglement : J. Stat. Mech. (2017) 063103.
• The Algebraic Bethe Ansatz and Tensor Networks Phys. Rev. B 86, 045125 (2012)
• Numerical Contraction of the Tensor Network generated by the Algebraic Bethe Ansatz Phys. Rev. B 86, 045125 (2012)
• Renyi entropy is calculated in highly entangled spin chains coming from information theory [Fredkin-gate and Motzkin chains]. It scales differently (with the size of the block) at different values of Renyi parameter $\alpha$ : Published in Strongly Coupled Field Theories for Condensed Matter and Quantum Information Theory Proceedings, International Institute of Physics, Natal, Brazil, 2–21 August 2015 Editors: Ferraz, A., Gupta, K.S., Semenoff, G.W., Sodano, P. (Eds.)
• Non-uniques of analytical continuation from positive integers in the complex plane.
• Riemann-Hilbert problem and entanglement in free fermionic models. Journal of Physics A: Mathematical and Theoretical, 2020, volume 53, number 34

Valence Bond Solid


• Entanglement in a Valence-Bond-Solid State , 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. Both von Neumann entropy and Renyi entropy of large block of spins is ln(4).
• Valence Bond Solid in Quasicrystals ALGEBRA and ANALYSIS , vol 1, issue 2, page 47, 1989 . A version of AKLT is constructed with unique VBS ground state on an arbitrary connected graph .
• ENTANGLEMENT IN VALENCE-BOND-SOLID STATES.
  The model has gap on an arbitrary connected graph.
• Entanglement in valence-bond-solid states on any symmetric graphs
• Block Spin Density Matrix of the Inhomogeneous AKLT Model
• Entanglement and Density Matrix of a Block of Spins in AKLT Model
• Entanglement spectra of q-deformed higher spin VBS states
• Entanglement spectra of the q-deformed Affleck-Kennedy-Lieb-Tasaki model and matrix product states
• Entanglement in SU(n) Valence-Bond-Solid state
• Entanglement of disjoint blocks in the one dimensional Spin 1 VBS , Journal of Phys. A: Math. Theor. 45 (2012) 125307
• Negativity for two blocks in the one dimensional Spin 1 AKLT model, Phys. Rev. A 84, 062307 (2011)
• Negativity in AKLT with higher spin

Lieb-Liniger model of one dimensional anyons

• Time and distance dependence of temperature correlations
• Large-Distance Asymptotic Behavior of the Correlation Functions of 1D Impenetrable Anyons at Finite Temperatures
  
• Non-conformal asymptotic behavior of the time-dependent field-field correlators of 1D anyons
  
• Quantum phase transition in a multicomponent anyonic Lieb-Liniger model
  
• Patu, Averin, Korepin

Fermi Hubbard model in one dimension

• Correlation functions of the one-dimensional Hubbard model in a magnetic field. 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
  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 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 .
  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 Physics Letters A vol 190 page 238, 1994, arXiv:hep-th/9310158 .
• Yangians by Denis Bernard
• Norm of an eigenfunction is a determinant of linearized Lieb-Wu equations. Phys.Lett. A263 (1999) 293-298
• Form-factors in Hubbard , Phys. Rev. B v 59, N 3, page 1734, 1999
• Thermodynamics and excitations of the one-dimensional Hubbard model : Physics Reports vol 331 (2000) 197-281
• Correlations in 1D electron gas. Phys.Lett. A260 (1999) 516-521.

Supersymmetric extension

• Spectrum of Low-Lying Excitations in a Supersymmetric Extended Hubbard Model
  An extension of the Hubbard model based on SU(2|2) symmetry was published in 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
• Higher conservation laws and algebraic Bethe Ansätze for the supersymmetric t-J model. . Phys. Rev. B 46, 9147–9162 (1992). See also on line version.

Books

• Quantum Inverse Scattering Method and Correlation Functions
• The One-Dimensional Hubbard model
• Exactly Solvable Models of Strongly Correlated Electrons
• 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

Publications

• Physics
• Spires

Conferences

• First quantum computing conference in the State of New York in 2003
• The program on quantum entanglement in 2016
• A workshop on Entanglement and Dynamical Systems in 2018
• my lecture at Banff International Research Station , Photo
• Lectures in the EIC sumer school in June of 2003
• The program on quantum entanglement in 2016
• A workshop on Entanglement and Dynamical Systems in 2018
• my lecture at Banff International Research Station , Photo

Former graduate students

• Samson Shatashvili
• Fabian Essler
• Vitaly Tarasov
• Nikita Slavnov
• Raul Santos
• Byung-Soo Choi
• Kun Zhang

Teaching

• Graduate Course in Quantum Information
• Statistical Mechanics I , PHY 540
• lecture in Simons Institute for Theory of Computing in Berkeley
• lecture in Dublin Institute for Advanced Studies

Synergistic activities

• International Journal of Modern Physics B, Editorial Board Member