Vladimir E. Korepin

Quantum inverse scattering method and correlation functions

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Calculation of norms of Bethe wave functions

A class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered. Eigenfunctions of the Hamiltonians are known for these models. Norms of these eigenfunctions in the finite box are calculated in the present paper. These models include in particular the quantum nonlinear Schrödinger equation and the HeisenbergXXZ model.

DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painleve transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrodinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.

Quantum theory of solitons

Abstract This paper describes the quantum theory of solitons — the localized solutions of the classical field equations. The scattering matrix for the processes with solitons is defined within the functional integral formalism. The Lorentz-invariant perturbation theory for solitons is consistently set up. The physical properties of solitons are calculated for two-dimensional scalar theories in the one-loop approximation.

The quantum inverse scattering method approach to correlation functions

The inverse scattering method approach is developed for calculation of correlation functions in completely integrable quantum models with theR-matrix of XXX-type. These models include the one-dimensional Bose-gas and the Heisenberg XXX-model. The algebraic questions of the problem are considered.

The Inverse Scattering Method Approach to the Quantum Shabat-Mikhailov Model

The Shabat-Mikhailov model is treated in the framework of the quantum inverse scattering method. The BaxtersR-matrix for the model is calculated.

Quantum spin chain, toeplitz determinants and the Fisher-Hartwig conjecture

We consider one-dimensional quantum spin chain, which is called XX model, XX0 model or isotropic XY model in a transverse magnetic field. We study the model on the infinite lattice at zero temperature. We are interested in the entropy of a subsystem [a block of L neighboring spins]. It describes entanglement of the block with the rest of the ground state. G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev showed that for large blocks the entropy scales logarithmically. We prove the logarithmic formula for the leading term and calculate the next term. We discovered that the dependence on the magnetic field interacting with spins is very simple: the magnetic field effectively reduce the size of the subsystem. We also calculate entropy of a subsystem of a small size. We also evaluated Renyi and Tsallis entropies of the subsystem. We represented the entropy in terms of a Toeplitz determinant and calculated the asymptotic analytically.We consider the one-dimensional quantum spin chain, which is called the XX model (XX0 model or isotropic XY model) in a transverse magnetic field. We are mainly interested in the entropy of a block of Lneighboring spins at zero temperature and of an infinite system. We represent the entropy in terms of a Toeplitz determinant and calculate the asymptotic analytically. We derive the first two terms of the asymptotic decomposition. Interestingly, these two terms of decomposition clearly show a length scale related to the field h.

Universality of Entropy Scaling in One Dimensional Gapless Models

We consider critical models in one dimension. We study the ground state in the thermodynamic limit (infinite lattice). We are interested in an entropy of a subsystem. We calculate the entropy of a part of the ground state from a space interval (0,x). At zero temperature it describes the entanglement of the part of the ground state from this interval with the rest of the ground state. We obtain an explicit formula for the entropy of the subsystem at any temperature. At zero temperature our formula reproduces a logarithmic formula, discovered by Vidal, Latorre, Rico, and Kitaev for spin chains. We prove our formula by means of conformal field theory and the second law of thermodynamics. Our formula is universal. We illustrate it for a Bose gas with a delta interaction and for the Hubbard model.

The One-Dimensional Hubbard Model: Index

The description of a solid at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism and the metalinsulator transition. Remarkably, the one-dimensional Hubbard model can be solved exactly using the Bethe ansatz method. The resulting solution has become a laboratory for theoretical studies of non-perturbative effects in strongly correlated electron systems. Many methods devised to analyse such effects have been applied to this model, both to provide complementary insight into what is known from the exact solution and as an ultimate test of their quality. This book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters develop a self-contained introduction to Bethe’s ansatz and its application to the one-dimensional Hubbard model, and will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more recent scientific results in the field of integrable models. The authors are distinguished researchers in the field of condensed matter physics and integrable systems, and have contributed significantly to the present understanding of the one-dimensional Hubbard model. Fabian Essler is a University Lecturer in Condensed Matter Theory at Oxford University. Holger Frahm is Professor of Theoretical Physics at the University of Hannover. Frank Göhmann is a Lecturer at Wuppertal University, Germany. Andreas Klümper is Professor of Theoretical Physics at Wuppertal University. Vladimir Korepin is Professor at the Yang Institute for Theoretical Physics, State University of New York at Stony Brook, and author of Quantum Inverse Scattering Method and Correlation Functions (Cambridge, 1993).

LATTICE VERSIONS OF QUANTUM FIELD THEORY MODELS IN TWO DIMENSIONS

Abstract The quantum inverse scattering method allows one to put quantum field theory models on a lattice in a way which preserves the dynamical structure. The trace identifies are discussed for these models.