Quantum geometry of three-dimensional lattices

Abstract

We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit.

Effective field theory and integrability in two-dimensional Mott transition

“…After reviewing some basic properties of integrability in statistical mechanics models, we discuss in the Section 3 the integrability of the specific two-dimensional fermion model considered in this paper. We show that in the strong coupling regime, the system defined by the ground and low-lying states of the model satisfies the Zamolodchikov tetrahedron equation, and is characterized by a novel family of solution to the tetrahedron equation recently found by Bazhanov et al [6,7]. We review these solutions for the sake of completeness and discuss the three-dimensional structure of an underlying quantum group algebraic structure.…”

“…Alternatively, one can say that due the projection-like property (3.43), the two-dimensional XXZ spin system may be decomposed in a consistent way into two one-dimensional chains, so that the entire system will have a quantum group symmetry (ðU q ð d slð2ÞÞÞ N ). The property that (2 + 1)-dimensional quantum systems in square lattice with periodic boundary conditions reduce to (1 + 1)-dimensional quantum chains, implying that the ððU q ð d slð2ÞÞÞ N Þ symmetry reduces to the U q ð d slð2ÞÞ quantum group symmetry, was first noted in [6,25].…”

“…It was shown in [6] that Eq. (3.31) defines a canonical transformation (automorphism) of the triple tensor product of the Poisson algebra.…”

“…Using a similar projection, Eq. (3.39) implies that there also exists another Yang-Baxter equation (for details see [6]):…”

“…Moreover, two remarkable properties of the quantum vertex model and the associated three-dimensional structure of quantum groups have been discussed in [6]: The quantum vertex model is stationary for L 12 ða; a y ; kÞ ¼ L 12 ða 0 ; a y 0 ; k 0 Þ, i.e., for the case when the patterns for two consecutive time slices are identical so that the third dimension has range n = 2. In this case, the operator L vb is: …”