Functional relations for the six vertex model with domain wall boundary conditions

In this work we demonstrate that the Yang-Baxter algebra can also be employed in order to derive a functional relation for the partition function of the six vertex model with domain wall boundary conditions. The homogeneous limit is studied for small lattices and the properties determining the partition function are also discussed.

💡 Research Summary

In this paper the authors present a novel derivation of a functional relation for the partition function of the six‑vertex model with domain‑wall boundary conditions (DWBC) by exploiting the Yang‑Baxter algebra, i.e. the Quantum Inverse Scattering Method (QISM). Traditionally the partition function Z_N for an N×N lattice with DWBC has been expressed through the Izergin‑Korepin determinant, a compact but rather involved formula that requires careful handling of normalization factors and limits. The present work shows that the same object can be obtained directly from the algebraic relations among the monodromy‑matrix entries, without invoking determinant identities.

The construction starts from the standard R‑matrix of the six‑vertex model, R(λ−μ)=(\begin{pmatrix}a(λ−μ)&0&0&0\0&b(λ−μ)&c&0\0&c&b(λ−μ)&0\0&0&0&a(λ−μ)\end{pmatrix}), and the associated L‑operators L_j(λ)=R_{0j}(λ−θ_j) where θ_j are inhomogeneity parameters. The monodromy matrix T(λ)=L_N(λ)…L_1(λ) is written in the usual 2×2 block form T(λ)=(\begin{pmatrix}A(λ)&B(λ)\ C(λ)&D(λ)\end{pmatrix}). For DWBC the reference state |0⟩ (all spins up) and its dual ⟨0| (all spins down) are eigenvectors of A and D with eigenvalue 1, while B and C act as creation and annihilation operators respectively. The partition function is then defined as the vacuum expectation value Z_N(λ_1,…,λ_N; μ_1,…,μ_N)=⟨0| C(μ_1)…C(μ_N) B(λ_1)…B(λ_N) |0⟩.

The core of the derivation is the RTT relation R(λ−μ) (T(λ)⊗T(μ)) = (T(μ)⊗T(λ)) R(λ−μ), which yields the fundamental commutation rule between B and C: B(λ) C(μ) = C(μ) B(λ) + g(λ,μ)