О векторах Бете gl(2|1)-инвариантных интегрируемых моделей тема диссертации и автореферата по ВАК РФ 01.01.03, кандидат наук Ляшик Андрей Николаевич

• Ляшик Андрей Николаевич
• кандидат наук
• 2020

Оглавление диссертации кандидат наук Ляшик Андрей Николаевич

Contents

Introduction

1 Quantum R-matrix structure

2 Spin chain as basic example

3 Algebraic Bethe ansatz for gl2

4 0[2|1-invariant Bethe vector

5 Scalar product of Bethe vectors

6 Norm of eigenvector

Appendix A. Multiple Actions of the Monodromy Matrix in gI(2|1)-Invariant Integrable Models

Appendix B. Scalar products of Bethe vectors in models with gl(2| 1) symmetry 1. Super-analog of Reshetikhin formula

Appendix C. Scalar products of Bethe vectors in models with gl(2| 1) symmetry 2. Determinant representation

Appendix D. Form factors of the monodromy matrix entries in g[(2|1)-invariant integrable models

Введение диссертации (часть автореферата) на тему «О векторах Бете gl(2|1)-инвариантных интегрируемых моделей»

Introduction

Quantum integrable models are a special class of physical models. These models describe non trivial systems of interacting particles and at the same time they can be studied accuracy using mathematical tools. They offer us a unique training ground for a deep study of non trivial physical phenomena explicitly.

A wide class of quantum integrable models is associated with higher rank algebras. Integrable models with symmetries of high rank appear in condensed matter physics, in particular in the 0Ím|n invariant XXX Heisenberg spin chain, in multi-component Bose/Fermi gas [25], and in the study of models of cold atoms (the Hubbard model [21], the t-J model [22, 23, 24]). Also spin chains of higher rank are interesting in the context of computing correlation functions in N =4 supersymmetric Yang-Mills theory [10, 11].

The role of the scalar product of Bethe vectors is extremely important in the study of correlation functions of local operators of the underlying quantum models [15, 17, 18]. One can reduce the problem of calculation of the form factors and the correlation functions of local operators to the calculation of the scalar products of the Bethe vectors [28, 29].

The study of integrable systems with high rank symmetry is still a challenging task. Until recently, such models have either not been studied at all, or have been studied under various simplifying hypotheses. The results presented in the thesis are the first in this direction.

My thesis presents the results of four articles in which I am one of the co-authors. The articles are devoted to the study of Bethe vectors and their scalar products in quantum integrable models with gi^-algebra symmetry. This research is the development of mathematical apparatus of the study of correlation functions of these systems. In fact, this thesis is completely devoted to the description of Bethe vectors and to study of their properties.

This section contains an overview of the results of the thesis, where we present generalization of Algebraic Bethe Anzats to the case gi^-invariant integrable models and scalar products of Bethe vectors in this case.

Заключение диссертации по теме «Математическая физика», Ляшик Андрей Николаевич

Conclusion

In this paper we obtained determinant representations for form factors of the monodromy matrix entries in integrable models described by fll(2| 1) and fll(112) superalgebras. The method is based on the determinant formula for a particular case of Bethe vectors scalar product [27]. This formula allows one to calculate form factors of the diagonal operators Tn. Further calculation of form factors of the off-diagonal operators Tik is based on the zero modes method [23].

The obtained results can be used for the calculation of form factors and correlation functions in the supersymmetric t-J model. For this model the solution of the quantum inverse scattering problem is known [11,28]. Therefore, form factors of local operators can be easily reduced to the ones considered in the present paper.

The calculation of form factors in models with gl(mln) symmetry remains to be done. Any results in this field would be desirable in view of their possible application to Hubbard model and supersymmetric gauge theories. It is clear that the zero modes method works in this case as well. Therefore, it would be enough to obtain a determinant formula for only one form factor. All other form factors would be achieved from the initial one as special limits of the Bethe parameters. However, the problem of calculating the initial form factor meets serious technical difficulties.

Acknowledgements

N.A.S. thanks LAPTH in Annecy-le-Vieux for the hospitality and stimulating scientific atmosphere, and CNRS for partial financial support. The work of A.L. has been funded by the Russian Academic Excellence Project 5-100 and by joint NASU-CNRS project F14-2016. The work of S.P. was supported in part by the RFBR grant 16-01-00562-a. N.A.S. was supported by the grants RFBR-15-31-20484-mol-a-ved and RFBR-14-01-00860-a.

Список литературы диссертационного исследования кандидат наук Ляшик Андрей Николаевич, 2020 год

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