Vertex Models and Spin Chains in Formulas and Pictures

Abstract

We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra ${\mathrm{U}}_q(\mathcal{L}({\mathfrak{sl}}_{l+1}))$ are given. The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method. Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems.

References in corpus (11)

• Baxter's Q-operators for supersymmetric spin chains
• Oscillator versus prefundamental representations II. Arbitrary higher ranks
• Oscillator versus prefundamental representations
• On the universal R-matrix for the Izergin-Korepin model
• Correlation functions of the integrable $SU(n)$ spin chain
• On the calculation of the correlation functions of the ${\mathfrak{sl}}_3$-model by means of the reduced qKZ equation
• Quantum groups, Verma modules and $q$-oscillators: General linear case
• Metaplectic Ice for Cartan Type C
• Solutions of the $U_q({\widehat{\mathfrak{sl}}}_N)$ reflection equations
• Symmetry classes of alternating sign matrices in the nineteen-vertex model
• Highest $\ell$-Weight Representations and Functional Relations

Cited by in corpus (5)

• A Q-operator for open spin chains I: Baxter's TQ relation
• Quantum groups and functional relations for arbitrary rank
• $\ell$-weights and factorization of transfer operators
• Khoroshkin-Tolstoy approach for quantum superalgebras
• On the properties of the density matrix of the ${\mathfrak{sl}}_{n+1}$-invariant model