Some Results and Publication of Vladimir Korepin

• Diploma of V.Korepin of 1974 [in Russian] and its English translation ↵
  Cancellation of ultra-violet infinities in one loop gravity on mass shell ∞ Cancellation of ultra-violet infinities in one loop gravity on mass shell
  
• Valence Bond Solid in Quasicrystals
  A. Kirillov and V. Korepin, ALGEBRA and ANALYSIS, vol 1, issue 2, page 47, 1989
  A version of AKLT spin model is constructed with unique VBS ground state on a finite graph
• A Lattice Version of Nonlinear Schroedinger Equation
  A.G.Izergin and V.E. Korepin, DOKLADY AKADEMII NAUK, 1981 .
  Quantum Determinant is discovered [it is central for quantum groups] as well as anti-pod.
• 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 .
  Lattice Sine-Gordon is constructed [with the same R matrix as continuous one] in classical and quantum cases.
• Calculation of Norms of Bethe Wave Functions
  V.E. Korepin, Communications in Mathematical Physics, vol 86 , page 191, 1982.
  Domain Wall Boundary Conditions for 6-Vertex model are discovered [see Appendix D, page 413 ] &
  First proof of determinant formula for norm of Bethe wave function.
• Several publications are devoted to Six vertex model with domain wall boundary conditions
  
• Izergin-Korepin model was discovered in their paper published in Communications in Mathematical Physics vol 79, page 303 in 1981
  Since then several papers were writen about Izergin-Korepin model
  
• Quantum Theory of Solitons ©
  L.D. Faddeev and V.E. Korepin, Physics Reports vol 42 (1), pages 1-87, June 1978.
  Semi-classical calculation and one loop corrections to mass of solitons [including bound states] and Scattering matrix.
  Main example is Sine-Gordon.
• Completely integrable integral operators were discovered in the paper
  DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS, ∂
  by A.R. Its, A.G. Izergin, V.E. Korepin and N.A. Slavnov;
  International Journal of Modern Physics vol B4, page 1003 in 1990
• Temperature Correlations of Quantum Spins Τ
  A.R.Its, A.G.Izergin, V.E.Korepin, N.A.Slavnov, Phys.Rev.Lett. 70 (1993) 1704-1708; Erratum-ibid. 70 (1993) 2357.
  Space, time and temperature dependent correlation function is calculated in isotropic version of XY spin chain.
  The correlation function decays exponentially with time and space separation. The rate of exponential decay is evaluated explicitly .
• ADIABATIC TRANSPORT PROPERTIES AND BERRY'S PHASE IN HEISBNBERG-ISING RING
  V.E. Korepin and A.C.T.Wu, International Journal of Modern Physics B . vol 5, no 3, (1991), 497.
  Change of boundary conditions in spin chain generates adiabatic process. We follow low energy levels and calculate the Berry's phase .
• Correlation Functions in 1D Hubbard model
  H.Frahm and V. Korepin in Physical Review B vol 42, number 16 page 10553.
  Correlation Functions are described by means of conformal field theory . The correlation functions demonstrate charge and spin separation.
  Central charge of each Virasoro algebra is equal to 1.
• 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.
• Scattering Matrix of 1D Hubbard model >
  F.H.L. Essler and V.E. Korepin in Phys. Rev. Letters vol 72 number 6 page 908, 1994
  At half filled band and zero magnetic field the model has only four different excitations: holon, anti-holon and a spinon.
  Scattering matrix is 16X16 solution of Yang-Baxter equation, it is explicitly evaluated.
  It has Yangian symmetry & demonstrate charge and spin separation.
• 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
  Non-abelian symmetry arises on infinite lattice: it is infinite dimensional quantum group.
  Explicit expression for Yangian generators commuting with the Hamiltonian is presented.
• Spectrum of Low-Lying Excitations in a Supersymmetric Extended Hubbard Model
  F. H.L. Essler, V. E. Korepin.
  A model of strongly correlated electrons is solved by algebraic Bethe Ansatz .
  It is derived from SU(2|2) super-symmetric solution of Yang-Baxter equation
• Eta-pairing as a mechanism of superconductivity in models of strongly correlated electrons
  J. de Boer, V. Korepin, A. Schadschneider
  An extended Hubbard model is constracted, for which η pairs are in the ground state .
• 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. Entanglement entropy is evaluated. At zero temperature logarithmic scaling of the entropy is derived form the second law of thermodynamics. The entropy of a subsystem is calculated for Bose gas with delta interaction, spin chains and the Hubbard model. Entropy of electrons on a space interval also calculated for positive temperature Τ
• 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
  Isotropic XY model in a transverse magnetic field is considered. Entanglement entropy is calculated. Logarithmic formula for the leading term is proven and sub-leading term is calculated. Renyi entropy also evaluated !
• 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 also arXiv:quant-ph/0409027 , 2004
  This is the first analytical calculation of limiting entanglement entropy in XY spin chain. The entropy of block of spins in the ground state of the XY model is expressed in terms of elliptic functions. It has essential singularity at multi-critical point.
• Renyi Entropy of the XY Spin Chain
  F. Franchini, A. R. Its, V. E. Korepin, Journal of Physics A: Math. Theor. 41 (2008) 025302
  Limiting entropy [large block] is represented in terms of Klein's elliptic lambda - function. The Renyi entropy is essentially an automorphic function of the parameter α. Renyi entropy is equivalent to zeta function of reduced density matrix of the block, also to replica trick.
• Fisher-Hartwig Formula and Entanglement entropy
  Toeplitz matrices can be used for calculation of entanglement entropy in isotropic XY model. Block Toeplitz matrices are useful for evaluation of the entropy in unisotropic XY model. The spectrum of the density matrix can be extracted from Renyi entropy. The spectrum is exact geometric sequence, but different eigenvalues are degenerated differently. Degeneracy increases sub-exponentially as eigenvalue diminishes.
  
• Entanglement in a Valence-Bond-Solid State
  H. Fan, V. Korepin, V. Roychowdhury, Physical Review Letters, vol 93, issue 22, 227203, 2004
  Entanglement in AKLT quantum spin chain, consisting of spin 1 is studied. The authors proved that 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 on degenerated ground state of a 'block' Hamiltonian [a part of original Hamiltonian describing interaction of spins inside of the block]—
• Entanglement of Valence-Bond-Solid on an Arbitrary Graph ∴
  Y. Xu, V. E Korepin, Journal of Physics A: Math. Theor. 41 505302 (2008)
  AKLT spin model is constructed on arbitrary connected graph ∴ The ground state has unique VBS state. The graph is cut into two disconnected parts: block and environment. We consider reduced density matrix of the block and prove that many of its eigenvalues vanish. We describe eigenvectors of the density matrix corresponding to non-zero eigenvalues. They form a degenerate ground state of a 'block' Hamiltonian: a part of original AKLT Hamiltonian describing interaction of spins inside of the block.
• Entanglement in an SU(n) Valence-Bond-Solid State >
  H. Katsura, T. Hirano, V. E. Korepin, Journal of Physics A: Math & Theor. 41, 135304 (2008)
  Exact expression for the reduced density matrix of a block of continuous spins is obtained as well as von Neumann and Renyi entropies.
  Renyi entropy coincides with von Neumann entropy and equal to 2ln(n)
• The One-Dimensional Hubbard model ©
  F.H.L. Essler, H.Frahm, F. Goehmann, A. Kluemper and V.E. Korepin, Cambridge University Press, 2005
  ⌊ The text book ⌋ describes complete theory { at the date } of the Hubbard model in one dimension.
• Quantum Inverse Scattering Method and Correlation Functions ©
  V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Cambridge University Press, 1993.
  ⌊ The text book ⌋ presents a method of calculations of correlation functions of Bethe Ansatz solvable models using Fredholm determinant representation, completely integrable differential equations and Riemann-Hilbert problem. Main example is Bose gas with delta interaction δ [quantum nonlinear Schroedinger equation].
• Recent Papers ↵ by Vladimir Korepin
• Early Papers ← mainly in Russian

Graduate Course on Statistical Mechanics ♠