N. M. Bogoliubov

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.

Correlation functions for a strongly correlated boson system

Abstract The correlation functions for a strongly correlated exactly solvable one-dimensional boson system on a finite chain as well as in the thermodynamic limit are calculated explicitly. This system, which we call the phase model, is the strong coupling limit of the integrable q -boson hopping model. The results are presented as determinants.

Critical exponents for integrable models

Abstract Phase transition in quantum systems in two space-time dimensions takes place at zero temperature. A general formula is obtained for the critical exponent describing the power decrease of zero-temperature correlation functions as long distances. This formula is valid for a large class of Bethe ansatz solvable models including the Heisenberg magnet and the one-dimensional Bose gas. The critical exponent is connected with the fractional charge; it is also expressed in terms of macroscopic characteristics of the models.

Boundary correlation functions of the six-vertex model

We consider the six-vertex model on an N × N square lattice with the domain wall boundary conditions. Boundary one-point correlation functions of the model are expressed as determinants of N × N matrices, generalizing the known result for the partition function. In the free fermion case explicit answers are obtained. The introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings.

Boxed plane partitions as an exactly solvable boson model

Plane partitions naturally appear in many problems of statistical physics and quantum field theory, for instance, in the theory of faceted crystals and of topological strings on Calabi–Yau threefolds. In this paper, a connection is made between the exactly solvable model with the boson dynamical variables and a problem of enumeration of boxed plane partitions–three-dimensional Young diagrams placed into a box of a finite size. The scalar product of the introduced model gives the MacMahon enumeration formula, while the correlation functions of the model may be considered as the generating functionals of the Young diagrams with the fixed heights of its certain columns. The evaluation of the scalar product and the correlation functions is based on the Yang–Baxter algebra. The analytical answers are obtained in terms of determinants and they can also be expressed through the Schur functions.

Exact solution of generalized Tavis-Cummings models in quantum optics

Quantum inverse methods are developed for the exact solution of models which describe N two-level atoms interacting with one mode of the quantized electromagnetic field containing an arbitrary number of excitations M. Either a Kerr-type nonlinearity or a Stark-shift term can be included in the model, and it is shown that these two cases can be mapped from one to the other. The method of solution provides a general framework within which many related problems can similarly be solved. Explicit formulae are given for the Rabi splitting of the models for some N and M, on- and off-resonance. It is also shown that the solution of the pure Tavis - Cummings model can be reduced to solving a homogeneous ordinary differential equation of second order. Generalization of the method to the case of several cavity modes is indicated.

The su(1,1) Tavis-Cummings model

A generic su(1,1) Tavis-Cummings model is solved both by the quantum inverse method and within a conventional quantum-mechanical approach. Examples of corresponding quantum dynamics including squeezing properties of the su(1,1) Perelomov coherent states for the multiatom case are given.

On the spectrum of the non-Hermitian phase-difference model

Abstract A modified version of the phase-difference model is introduced and diagonalized by means of the algebraic Bethe ansatz. The spectrum is determined for both small and large values of the coupling constant, and the low-lying excitations are shown to exhibit a conformal profile. Applications of the model to quantum optics and growth problems are briefly discussed.

THE ROLE OF QUASI-ONE-DIMENSIONAL STRUCTURES IN HIGH-Tc SUPERCONDUCTIVITY

The critical exponents describing the decrease of correlation functions on long distances for the one-dimensional Hubbard model is obtained. The behaviour of correlators shows that Cooper pairs of electrons are formed. The electron tunneling between the chains leads to the existence of the anomalous mean values and to the superconductive current. The anisotropy of the quasi-one-dimensional system leads to the rise of critical temperature Tc.

Correlators of the phase model

Abstract We introduce the phase model on a lattice and solve it using the algebraic Bethe ansatz. Time-dependent temperature correlation functions of phase operators and the “darkness formation probability” are calculated in the thermodynamical limit. These results can be used to construct integrable equations for the correlation functions and to calculate their asymptotics.