Spin lattices

Framework

The theory of exactly integrable models is one of the main domain of the modern Theoretical and Mathematical Physics. 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 behavior of materials at nano-scale. Mathematically, the investigation of quantum spin chains gave rise to the quantum group theory.

Spin lattices are two-dimensional generalizations of spin chains. A spin lattice is a lattice of an arbitrary shape and size, such that each vertex of the lattice carries a local triplet of Pauli matrices (or some higher dimensional algebra of observables).

The spin chains and spin lattices may be considered as the discrete space approximations of quantum field theories. The terms "integrable spin chain" and "integrable spin lattice" mean that the spectra of involved quantum operators (higher Hamiltonians) are well defined when the spacing parameter tends to zero. In some sense, the integrability is the condition of the self-consistent definition of the path integration measure.

Example of integrable spin lattice was found in [1]. 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 [5]. Spectrum and structure of excitations are not known yet. This investigation is of great importance in the statistical physics and in the field theory.

The PhD projects include:

• Spectrum of excited states for the spin lattice. Intensive numerical tests at the first.
• Spin systems on the lattices with the higher symmetry (kagome, octagonal etc. lattices)
• Inhomogeneous spin lattices and their hypothetical phase properties.

Spin lattices correspond to the representation of the algebra of observables via the Pauli matrices. The other possible representation is the modular pair of Weyl algebras, result is some field theory. Its investigation is a separate stream of PhD projects.

Further reading:

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

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

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

[5]. S. Sergeev, Thermodynamic Limit for a Spin Lattice, J. Stat. Phys., 2006

Some history of multidimensional integrability may be found here