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
  
• 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 center of Yang-Baxter algebra] . See formulae (11).
• 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 .
  
• Pauli principle for one-dimensional bosons and algebraic Bethe ansatz
  A. Izergin and V. Korepin, Letter in Mathematical Physics vol 6, page 283, 1982
  Pauli principle proved for models solvable by Bethe Bethe anzats .
• Massive Thirring model is solved by Bethe ansatz : ultraviolet renormalization done in the frame of Bethe ansatz,
  spectrum of masses and Zamilodchikov scattering matrix derived dynamically
  
• in the paper Calculation of Norms of Bethe Wave Functions
  Domain Wall Boundary Conditions for 6-Vertex model were discovered and recursion relations for the partition function derived &
  First proof of determinant formula for norm of Bethe wave function was published.
• Several publications are devoted to Six vertex model with domain wall boundary conditions
  
• The bulk free energy for 6 vertex model with domain wall boundary conditions is different from the one with periodic boundary conditions
  
• 19 vertex model (Izergin-Korepin model) was discovered in 1981
  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 (in reflectionless cases scattering matrix were obtained exactly). 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
  Fredholm integral operators of a special form are discovered in the paper Emptiness formation probability was first introduced in this paper
  (for spin chains it is a probability of formation of ferromagnetic string in anti-ferromagnetic ground state).
• 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.
  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 factorize into charge and spin parts.
  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 SO(4) symmetry and Yangian symmetry (it demonstrates 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 .
• 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 Heisenberg spin chain [infinite chain, zero temperature, no magnetic field] correlation functions are polynomial (with rational coefficients) of values of Riemann zeta function with odd arguments.
• 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.
• The Fisher-Hartwig Formula and Generalized Entropies in XY Spin Chain
  by A. R. Its and V. E. Korepin
  Spectrum of the density matrix of a large block of spins in the ground state of XY model is calculated, see formula (117) as well as degeneracy of individual eigenvalues, see (126) .
• 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]
• Open Problems in Exactly Solvable Models
  Following open problems related to Bethe anzats are formulated: 1) norms in Hubbard; 2) spin correlations and number theory ; 3) time and temperature dependent correlations in XXZ ; 4) entanglement entropy in XXZ .
  
• Publications in Communications in Mathematical Physics
• 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
• Papers related to condensed matter physics
• Papers related to high energy physics
• Early Papers ← mainly in Russian

Teaching

• Graduate Course in Quantum Information ♦
• Graduate Course on Bethe Ansatz solvable models ♣
• Graduate Course on Advanced Solid State PHY 556 ♠