Three-dimensional models

History

The 99.99% of a Zoo of integrable models of quantum and statistical mechanics in discrete space-time consist on so-called two dimensional models, for instance XXZ Heisenberg chain, nonlinear Schrodinger equation, eight vertex model, Chiral Potts Model and so on ...

The integrability is based on the Yang-Baxter (triangle) equation providing the commutativity of transfer-matrices. In the image below the triangle equation is the elementary event in the exchange of the blue and red transfer matrices. Transer matrices are polynomials of spectral parameter, their commutativity provides the existence of a big enough set of mutually commuting operators - this is the definition of Hamilltonian completely integrable system.

Contrary to this variety only two three dimensional integrable models of quantum and statistical mechanics are known. The term three dimensional means that they are either the models of statistical mechanics on three dimensional (e.g. cubic) lattice of they are quantum mechanical models (discrete quantum field theory) on two dimensional lattices. Models of classical dynamics can be viewed as classical limit of quantum systems.

During a long time the term three dimensional model implied an attempt to solve the Tetrahedron equation. Tetrahedron equation (TE) is the three dimensional generalization of the Yang-Baxter equation. In the same way as the Yang-Baxter (triangle) equation is the local integrability condition in two dimensions, the d+1 simplex equation and in particular 3-simplex (tetrahedron) equation is the local integrability condition for d+1 dimensional models of statistical mechanics or quantum field theory models [1].

In particular, tetrahedron equation can be considered as the local 3D structure of a single composed Yang-Baxter equation. In the image below the tetrahedron equation is the elementary event of the Yang-Baxter equation for red, blue and green effective two-dimensional vertices, the Yang-Baxter operators correspond to compactification of tetrahedral operators in "third'' hidden direction.

The first solution of the TE was invented by A. Zamolodchikov in 1980 [2]. He used the language of scattering of straight strings, where the TE is the factorization of the scattering of four strings. Later in 1983 R. Baxter had rewritten Zamolodchikov's S-matrix as Boltzmann weights of on elementary cube of the cubic lattice (so-called Interaction-Round-Cube or W-formulation) [3]. In 1992 V. Bazhanov and R. Baxter derived a generalization of W weights for an arbitrary number of spin states (originally Zamolodchikov's model had two spin states) [4]. Zamolodchikov-Bazhanov-Baxter model, being considered on a cubic lattice with n layers in the third direction, is equivalent to sl(n) generalized Chiral Potts model [5] and in particular to Chiral Potts model [6] for n=2.

Eventually in 1995 Sergeev, Mangazeev and Stroganov obtained the vertex formulation of Zamolodchikov-Bazhanov-Baxter model, its R-matrix was found in [7]. All S-matrix, W-weights and R-matrix are in some sense equivalent, they solve "S", "W" and "R" forms of TE, but to our opinion the fundamental object is R since it acts in a tensor cube of some vector space and therefore R has an algebraic origin.

Talking about known examples of R-matrices of TE, one should mention R-matrices of I. Korepanov [8], J. Hietarinta [9] and some earlier set of R-matrices found in [10]. All them appeared to be the particular limiting cases of Sergeev-Mangazeev-Stroganov R-matrix. A quite different series of R-matrices of constant TE (more precisely, solutions of decoupled 4-simplex equation) was obtained in an algebraic way in [11]. These R-matrices are the example of undue keenness of mathematical part of the problem: solution of TE exists but no integrable model is formulated.

The algebraic origin of Zamolodchikov-Bazhanov-Baxter R-matrix was found by Sergeev in 1998 [12]. Zamolodchikov-Bazhanov-Baxter model is related to some specific 3D zero-curvature representation. Zero curvature representation leads firstly to the quantum mechanical formulation of the model. Changing the representation of an algebra of observables, one can get a rich family of R-matrices and a rich family of integrable models. Secondly, the zero curvature representation leads to the integrability structure of three dimensional models. The last sentence may be clarified in the terms of two dimensional integrable models: of course, the Yang-Baxter equation provides the integrability, but solution of a model is the Bethe Ansatz - the integrability structure of two dimensional model. The integrability structure for the three dimensional model is only way co come finally to long time anticipated exact results for physical quantities [13].

Notwithstanding the variety of quantum integrable models corresponding to the zero curvature representation of [12], ranging from Zamolodchikov model to some strong-coupling field theory models corresponding to the infinite dimensional representation of the algebra of observables, we count all these models as the first known quantum integrable model since the zero curvature representation and the algebra of observables are the same.

The second quantum integrable in three dimensions was obtained quite recently [14]. It has been arisen from a different form of zero curvature representation and different algebra of observables. The new model now is in a tight investigation.

An essential amount of work is done since 1998 in the mathematics of multidimensional integrability, this subject however is not yet fully explored.

References.

[1] V. Bazhanov and Yu. Stroganov, Conditions of commutativity of transfer-matrices on a multidimensional lattice, Theor. Math. Phys. 52 (1982) 685-691

[2] A. B. Zamolodchikov, Tetrahedron equations and integrable systems in three dimensions, JETP 79 (1980) 641-664 (in russian);

Tetrahedron equations and the relativistic S matrix of straight strings in 2+1 dimensions'', Commun. Math. Phys. 79 (1981) 489-505

[3] R. J. Baxter, On Zamolodchikov's solution of the tetrahedron equation, Commun. Math. Phys. 88 (1983) 185-205

[4] V. V. Bazhanov and R. J. Baxter, New solvable lattice models in three dimensions, J. Stat. Phys. 69 (1992) 453-485

[5] V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev and Yu. G. Stroganov, Z_N^{\otimes(N-1)} - generalization of the chiral potts-model, Commun. Math. Phys. 138 (1991) 393-408

[6] R. J. Baxter, J. H. H. Perk, H. Au-Yang, New solutions of the star triangle relations for the Chiral Potts-model, Phys. Lett. A *128 (1988) 138-142

[7] S. M. Sergeev, V. V. Mangazeev and Yu. G. Stroganov, Vertex reformulation of the Bazhanov -- Baxter model, J. Stat. Phys. 82 (1996) 31-50

[8] I. G. Korepanov, Tetrahedral Zamolodchikov algebras corresponding to Baxter's L-operators, J. Stat. Phys. 71 (1993) 85-97

[9] J. Hietarinta, Labelling schemes for tetrahedron equations and dualities between them, J. Phys. A: Math. Gen. 27 (1994) 5727-5748

[10] V. V. Mangazeev, S. M. Sergeev, Yu. G. Stroganov, New solution of vertex type tetrahedron equations, Mod. Phys. Lett. A 10 (1995) 279-287

[11] R. M. Kashaev, S. M. Sergeev, On pentagon, ten-term, and tetrahedron relations, Commun. Math. Phys. 195 (1997) 211-216

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

Integrable three dimensional models in wholly discrete space-time, Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000) NATO Sci. Ser. II Math. Phys. Chem. 35 (2001) 293--304 Kluwer Acad. Publ;

Complex of three-dimensional solvable models, J. Phys. A: Math. Gen. 34 (2001) 10493--10503;

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

[13] S. Sergeev, Thermodynamic Limit for a Spin Lattice

[14] V. Bazhanov and S. Sergeev, Zamolodchikov tetrahedron equation and hidden structure of quantum groups