Quantized six-vertex model on a torus

TL;DR

The paper develops a three-dimensional integrable framework for the quantized six-vertex model on torus wiring diagrams, introducing ${\mathcal L}$ and ${\mathcal M}$, the tetrahedron equations, and two inversion relations to generate a two-parameter commuting family of layer transfer matrices $T_G(x,y)$. By distinguishing admissible diagrams and employing the $RLLL$ relation, it proves commuting transfer matrices for both ordinary square grids and broad classes of admissible graphs, including inhomogeneous generalizations and symmetry actions. It then connects the q-6v formalism to well-known integrable systems, yielding representations that realize free parafermions and the relativistic Toda chain, and shows a natural reformulation in terms of dimer models via Kasteleyn determinants. These results unify 3D integrable structures with 2D dimers and known spin chains, offering a versatile algebraic and combinatorial toolkit for analyzing integrable quantum lattice models on general graphs on the torus.

Abstract

We study the integrability of the quantized six-vertex model with four parameters on a torus. It is a three-dimensional integrable lattice model in which a layer transfer matrix, depending on two spectral parameters associated with the homology cycles of the torus, can be defined not only on the square lattice but also on more general graphs. For a class of graphs that we call admissible, we establish the commutativity of the layer transfer matrices by introducing four types of tetrahedron equations and two types of inversion relations. Expanding in the spectral parameters yields a family of commuting quantum Hamiltonians. The quantized six-vertex model can also be reformulated in terms of (quantized) dimer models, and encompasses known integrable systems as special cases, including the free parafermion model and the relativistic Toda chain.

Figures (14)

• Figure 1: The operator $\mathscr{L}(r,s,f,g;q)$.
• Figure 2: The operators $M^{ab}_{ij}$, where $p=-q^{-1}$.
• Figure 3: Trace construction (\ref{['tred']}). The diagram is a concatenation of the right one in (\ref{['fig:LM']}), where ${\mathcal{M}}^{ab}_{ij}$ is specialized to $M^{ab}_{ij}$. The green arrow is closed cyclically reflecting the trace.
• Figure 4: The square grid $G_{m,n}$ (left), and the corresponding vertex model (right). The red rectangle is a fundamental domain of a torus.
• Figure 5: Wiring diagrams on a torus. The left one is admissible, and the right one is not. The only difference is the orientation of the middle vertical wire.

Theorems & Definitions (23)

• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 2.7