Coordinate space wave function from the Algebraic Bethe Ansatz for the inhomogeneous six-vertex model

We derive the coordinate space wave function for the inhomogeneous six-vertex model from the Algebraic Bethe Ansatz. The result is in agreement with the result first obtained long time ago by Yang and Gaudin in the context of the problem of one-dimensional fermions with delta- function interaction.

💡 Research Summary

The paper presents a rigorous derivation of the coordinate‑space wave function for the inhomogeneous six‑vertex (XXZ) model using the Algebraic Bethe Ansatz (ABA). The authors begin by recalling the standard R‑matrix of the six‑vertex model and introduce site‑dependent inhomogeneity parameters ξ₁,…,ξ_N, which modify each L‑operator as L_i(u)=R_{0i}(u−ξ_i). The monodromy matrix T(u)=L_N(u)…L_1(u) thus carries the full set of inhomogeneities, and its entries A(u), B(u), C(u), D(u) obey the usual RTT relations.

In the ABA framework a reference (pseudo‑vacuum) state |0⟩ with all spins down is chosen. The creation operator B(u) generates spin‑up excitations when acting on |0⟩. An M‑particle Bethe state is written as |{v}⟩=B(v₁)…B(v_M)|0⟩, and the Bethe equations ensuring that this state is an eigenstate of the transfer matrix read
∏{i=1}^N b(v_j−ξ_i)/c(v_j−ξ_i)=∏{k≠j}^M b(v_j−v_k)/c(v_j−v_k) ,
where b and c are the usual six‑vertex weight functions.

To obtain the coordinate representation ψ(x₁,…,x_M) the authors expand each B‑operator in terms of the local spin‑raising operators σ_i⁺. The expansion reads
B(v)=∑_{i=1}^N