Diploma
The diploma
was devoted to quantization of general relativity . It was written under advice of Ludvig Fadeev.
Cancellation of ultra-violet infinities in one loop quantum gravity on mass shell was proved simultaneously with G. t’Hooft and M. Veltman . The diploma was cited in the Feynman lectures on gravitation .
Here is English translation of the diploma.
Semi-classical quantization of sine Gordon: → φtt - φxx +m2 sin( φ )= 0.
- Scattering matrix in quantum Sine Gordon. by
I.Yu. Arefeva and V.E. Korepin. JETP Letters , vol 22, N10, page 312, 1974
- Above-barrier reflections of solitons . V.E. Korepin. JETP Letters , vol 23, page 201, 1976.
Classically soliton and anti-soliton do not reflect.
Reflection coefficient r is exponentially small in Plank constant : r ∼ exp(-1/h).
Corresponding complex trajectory and the reflection coefficient were found.
- Quantum Theory of Solitons. by L.D. Faddeev and V.E. Korepin, Physics Reports vol 42 (1), pages 1-87, June 1978.
- 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] : detq {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 TT(λ+i) σ y det-1q {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 .
An integrable version of lattice Sine Gordon is constructed in classical and quantum cases.
- 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.
Recently this became important for quantum optics, see formula (22) of the paper.
- 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 in momentum space was proved for models solvable by Bethe anzats .
- Massive Thirring model is solved by Bethe ansatz :
-
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.
- 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
J. Phys. A25, 4115 (1992).
Collapse of strings [bound states] is discussed in XXX spin chain.
- Open Problems in Exactly Solvable Models .V.Korepin and O. Patu.
Proceedings of SOLVAY workshop: Bethe Ansatz: 75 Years Later, 2006
- 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, V.E. Korepin , A.G.Izergin and N. Yu. Reshetikhin
Journal of Physics A: Math. Gen. vol 22 (year 1989), pages 2615-2620 .
Bethe Ansatz solvable models associated with higher Lie algebras are considered, including spin chains and non-linear Schrodinger equation [Bose gas with delta interaction with internal degrees of freedom. It also can be called a generalization of Lieb-Liniger model. ]
- 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, Springer-Verlag, 1985.
Average of operator eα Q(x) was introduced as generating function of
correlation functions of local densities
<ψ +(x)ψ (x) ψ+ (0) ψ (0) >. Here Q(x) is an operator of number of particles on space interval (0,x). The correlation functions of local densities were evaluated in Bose gas as a function of distance and temperature.
- 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. All quantum fields appearing in the expression for the Fredholm determinant belong to the same abelian subalgebra. 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 sinh-Gordon → φtt - φxx +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 F. Goehmann, A.R. Its, V.E. Korepin, Phys. Lett. A, 249 (1998) 117 . This approach was developed in the papers by V. V. Cheianov and M. B. Zvonarev
- Space, time and temperature dependent correlation functions of Lieb-Liniger 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 | x-2 μ t | ln
| | eμ 2 - β -1| |
| eμ 2 - β +1 |
}, see also page 469 of the textbook .
- 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 .
This work was extended by
Alexei Tsvelik and Subir Sachdev .
- 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 Density-Density Temperature Correlations of One-Dimensional 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 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 spin chain correlation functions are polynomial (with rational coefficients) of values of Riemann zeta function with odd arguments ζ(2n+1).
- Three instanton solution. by
S.L. Shatashvili and V.Korepin ; Izvestiya Akademii Nauk. Ser. Mat. vol 48 , nom 2, 1984; English translation is
Math. USSSR, Izvestiya, vol 24 no 2, 1985
ADHM construction is considered in the paper. The problem of parametrization is solved for three instantons.
- Critical exponents for the one-dimensional Hubbard model by
H.Frahm and V. Korepin, Physical Review B
vol 42, number 16 page 10553 in 1990. Conformal dimensions were expressed in terms of fractional charge. Below half filling the fractional charge is a matrix 2X2.
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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.
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THE ROLE OF QUASI-ONE-DIMENSIONAL STRUCTURES IN HIGH-Tc SUPERCONDUCTIVITY by N.M. Bogoliubov and V.E.Korepin
International Journal of Modern Physics B, vol 3, no 3, 427, 1989.
Critical exponents driving correlations functions are calculated for attractive Hubbard.
-
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
Scattering matrix solves of Yang-Baxter equation . It has SO(4) symmetry and Yangian symmetry. It demonstrates charge and spin separation
.
- Completeness of the SO(4) Extended Bethe Ansatz for the One-Dimensional 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
by D. B. Uglov and V.E. Korepin, Physics Letters A vol 190 page 238, 1994,
arXiv:hep-th/9310158 . The Yangian symmetry arises on infinite lattice.
- Norm of an eigenfunction is a determinant of linearized Lieb-Wu equations.
F. Goehmann, V. E. Korepin , Phys.Lett. A263 (1999) 293-298
- Form-factors in Hubbard , F. H.L. Essler, V. E. Korepin. Phys. Rev. B v 59, N 3, page 1734, 1999
- Thermodynamics and excitations of the one-dimensional Hubbard model
by T. Deguchi, K. Kusakabe, F. Goehmann, F. H. L. Essler, A. Klümper and V. E. Korepin Phys. Rep. 331 (2000) 197-281
- Correlations in 1D electron gas. F. Goehmann, V.E. Korepin , Phys.Lett. A260 (1999) 516-521. Correlation functions for dilute Fermi gas described explicitly.
-
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.
- 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 Lieb–Liniger Model
with delta interaction, spin chains and the 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).
- these ideas were developed by several authors: scopus and PRL 1,
PRL 2, PRL 3, PRL 4, PRL 5, PRL 6, PRL 7, PRE1, PRE2
-
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=Σn(σxnσxn+1 +
σynσyn+1+hσzn) 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 (x) , also behaves logarithmically with the size of the block x&rarr ∞ , see formulae (3) and (5).
Here &rho is density matrix of the block of spins and &alpha is a parameter.
- The results were generalized to several space intervals by
Jon Keating .
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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 )
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 .
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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
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.
- Entanglement Spectrum for the XY Model in One Dimension F. Franchini, A. R. Its, V. E. Korepin, L. A. Takhtajan . The theorem stating that the spectrum of reduced density matrix of large block of spins in the ground state of
XY model is exact geometric sequence is proved. The largest eigenvalue is explicitly evaluated. Degeneracy of individual eigenvalues increases sub-exponentially as the eigenvalue diminishes. Quantum Information Processing: Vol 10, Issue 3 (2011), Page 325.
as a resource for measurement based quantum computation
- Entanglement in a Valence-Bond-Solid 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 spin model is constructed with unique VBS ground state on a finite graph
- Entanglement in Valence-Bond-Solid States V. Korepin, Ying Xu.
The density matrix of a block of spins in VBS state has low rank , this is distinguishing property of AKLT even in two dimensions and on graphes.
- Entanglement in an SU(n) Valence-Bond-Solid State by Hosho Katsura, Takaaki Hirano, Vladimir E. Korepin .
- One-Dimensional Impenetrable Anyons in Thermal Equilibrium. IV. Large Time and Distance Asymptotic Behavior of the Correlation Functions
Ovidiu I. Patu, Vladimir E. Korepin, Dmitri V. Averin
- Large-Distance Asymptotic Behavior of the Correlation Functions of 1D Impenetrable
Anyons at Finite Temperatures
Ovidiu I. Patu, Vladimir E. Korepin, Dmitri V. Averin
- One-Dimensional Impenetrable Anyons in Thermal Equilibrium. II. Determinant
Representation for the Dynamic Correlation Functions
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 Impenetrable Anyons in Thermal Equilibrium. I. Anyonic Generalization of Lenard's Formula
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
-
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 ⌋ explains both coordinate and algebraic forms of 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 : iψt = −½ψxx + κ |ψ|2ψ .
At zero temperature correlation <ψ (x,t)+ ψ (0,0) > decays as a power θ of distance (and time), it is expressed in terms of dressed charge θ=2Z2(q).
Here Z(λ)-(1/2π)∫-qq K(λ, μ) Z(μ) dμ = 1 and q is value of spectral parameter on Fermi sphere, see Chapter XVIII, pages 508- 515.
For positive temperature
space, time dependent correlations <ψ (x,t)+ ψ (0,0) >T are calculated, see page 469.
- Exactly Solvable Models of Strongly Correlated Electrons . Reprint volume, eds. F.H.L. Essler and V.E. Korepin, World Scientific, 1994
ΞTeachingΞ
Former graduate students , under advice of Vladimir Korepin
◊ Conferences ♦
