Excitation spectrum for a spin lattice

Description of the Project

The theory of exactly integrable models is one of the main domain of the modern Theoretical and Mathenatical Phyics. The significant part of it consists of the various models of spin chains. Spin chains have great importance for the Physics since they describe precisely the behavour of materials at nano-scale. Mathematically, the investigation of quantum spin chains gave rise to the quantum group theory.

The spin lattice is a dimensional generalization of a spin chain. It is a lattice of an arbitrary shape and size, such that each vertex of the lattice carries a local triplet of Pauli matrices (they form the algebra of observables) and some extra numerical parameters. It is known how [1] to produce the set of operators with the following features:

• Each operator is a polynomials on the algebra of observables

• All operators commute

• Operators are Hermitian (i.e. they are Hamiltonians)

• Set of operators is complete

A system with these features is by definition the quantum integrable model. Some steps toward the complete theory of spin lattices were done in [2,3]. The eigenvalues of the Hamiltonians may be found from the known functional equation [1,4] (analogue of the Bethe ansatz equations for a spin chain), which allows one to find the density of Hamiltonians for the ground state in the thermodynamic limit. But neither the structure of ground eigenstate nor a spectrum and structure of excitations are not known yet. This investigation is of great importance in the statistical physics and in the field theory.

Further reading:

[1]. S. M. Sergeev, "Auxiliary transfer matrices for three-dimensional integrable models", Theoretical and Mathematical Physics 124 (2000) 391--409

[2]. S. M. Sergeev, "Quantum 2+1 evolution model", J. Phys. A: Math. Gen. 32 (1999) 5693--5714

[3]. S. M. Sergeev, Quantum integrable models in discrete 2+1 dimensional space-time: auxiliary linear problem on a lattice, zero curvature representation, isospectral deformation of the Zamolodchikov-Bazhanov-Baxter model , Particles and Nuclei 35 (2004) 1051-1111 (Chapters 1,2,3).

[4]. S. M. Sergeev, "Evidence for a phase transition in three dimensional lattice models", Theoretical and Mathematical Physics 138 (2004) 310-321