Quantum Integrable model in 2+1 dimensional space-time

Description of the Projects

A new class of integrable models of quantum mechanics in 2+1 dimensional discrete space-time, discovered recently by S. Sergeev and V. Bazhanov, describes an ensemble of interacting q-oscillators situated in vertices of a two dimensional lattice of arbitrary shape and size. The structure of interaction and the algebra of observables differ essentially from the previously known spin-lattice systems (see the projects Spin Lattice ).

The model can be re-formulated in more traditional terms of quantum mechanics in 1+1 dimensional space-time with a high rank symmetry group. Recall, the method of completely integrable quantum mechanics in 1+1 dimensional space-time is called the Quantum Inverse Scattering Methods, the symmetry groups are called the quantum groups.

From this point of view, the 3D model produces in a very evident and simple way to obtain the whole representation theory of quantum groups and quantum inverse scattering method for U_q(sl(n)) (the deformation parameter q of oscillators becomes the deformation parameter q of the quantum group) -- the representations, L-operators, R-matrices -- all them have the amazingly simple forms in 3D framework.

It is my pleasure to propose several Ph.D. projects in this field.

• A systematic investigation of the representation theory in the framework of its 3D unification. In particular, it is interesting to obtain a modification of 3D scheme leading to the other quantum algebras. Possible way to solve this problem is the so-called tetrahedral reflection equation. The form of equation is known, no one solution of it is still obtained.
• The search of higher level 3D models. The idea of the project is based on an observation that several structures, derived in the framework of existing 3D method, are related to the representation theory of U_q(gl(n|1)). Hypothetically, there must exist a k-level generalization of the method, leading to U_q(gl(n|k)).
• An elliptic deformation of existing 3D scheme. Representation theory of quantum group U_q(sl(n)) as well as its R-matrices have an elliptical deformation to Belavin's R-matrix and Sklyanin's algebra. Their 3D unification would have a great importance e.g. for the string theory.
• Spectral equations for evolution operators.
• Two-dimensional lattice Bose gas: intensive numerical tests.
• Quantum curve and nested Bethe Ansatz equations for kagome lattice.
• q-oscillator model at root of unity
• q-oscillator model in the strong coupling regime
• ...

In the beginning, a student must have the comprehensive knowledge of analysis, complex analysis, linear algebra and quantum mechanic -- all these subjects are the "arithmetic', the basic skills. The ability to manipulate the elementary functions (e.g. the hypergeometric function; various popular functions with a series or an integral definition; orthogonal polynomials; Bessel, Weierstrass, Jacobi etc. functions) is also the basic skill. The knowledge of analytical mechanics, field theory, statistical mechanics, quantum statistical physics and group theory is expected as well, but we can overtake it during the project's duration, together with the most modern subjects: conformal field theory, quantum group theory, quantum inverse scattering method, theory of classical integrable systems etc.

Further reading:

[1] V. Bazhanov and S. Sergeev, Zamolodchikov's tetrahedron equation ...

[2] S. Sergeev, Ansatz of Hans Bethe for a two-dimensional lattice Bose gas

Some history of multidimensional integrability may be found here