Quantum and classical integrable sine-Gordon model with defect

Defects which are predominant in a realistic model, usually spoil its integrability or solvability. We on the other hand show the exact integrability of a known sine-Gordon field model with a defect (DSG), at the classical as well as at the quantum level based on the Yang-Baxter equation. We find the associated classical and quantum R-matrices and the underlying q-algebraic structures, analyzing the exact lattice regularized model. We derive algorithmically all higher conserved quantities $C_n, n=1,2,…$ of this integrable DSG model, focusing explicitly on the contribution of the defect point to each $C_n$. The bridging condition across the defect, defined through the B"acklund transformation is found to induce creation or annihilation of a soliton by the defect point or its preservation with a phase shift.

💡 Research Summary

The paper investigates a sine‑Gordon (SG) field theory that contains a point‑like defect, often referred to as the defect sine‑Gordon (DSG) model, and demonstrates that integrability survives both at the classical and quantum levels. The authors begin by formulating the DSG model: the bulk SG Lagrangian is kept unchanged on the left (x < 0) and right (x > 0) half‑lines, while at the defect location x = 0 the fields are linked by a Bäcklund‑type condition. This condition involves a real parameter η that measures the strength of the defect and reads schematically as a pair of first‑order equations coupling the spatial and temporal derivatives of the left and right fields. The Bäcklund map guarantees that the equations of motion on both sides are compatible with each other and with the defect.

To prove classical integrability the authors construct a Lax pair L±(λ) for each half‑line and introduce a defect matrix K(λ) that implements the jump across the defect. By requiring that K(λ) satisfies the classical Yang‑Baxter equation with the standard trigonometric r‑matrix, they show that the monodromy matrix, now containing K(λ), still obeys the fundamental Poisson‑bracket algebra. Consequently an infinite hierarchy of conserved quantities Cₙ (n = 1,2,…) can be generated from the expansion of the logarithm of the monodromy. The contribution of the defect to each Cₙ is explicitly extracted as coefficients in the λ‑expansion of K(λ). C₁ corresponds to the total energy‑momentum, C₂ to the topological charge plus a defect term, and higher Cₙ encode increasingly non‑local integrals of motion.

For quantization the authors adopt a lattice regularisation. The continuous field is replaced by a chain of quantum spins Sⁿᵃ (a = ±,z) at lattice sites n, and the bulk L‑operator Lₙ(λ) is taken as the standard U_q(ŝl₂) L‑matrix depending on the anisotropy γ (q = e^{iγ}). The defect is represented by a special L‑operator Lᴰ(λ) = K(λ) inserted at a chosen site. The full transfer matrix T(λ) = Lᴰ(λ)∏ₙLₙ(λ) satisfies the quantum Yang‑Baxter relation R(λ‑μ)T₁(λ)T₂(μ)=T₂(μ)T₁(λ)R(λ‑μ) with the same trigonometric R‑matrix. Hence the trace τ(λ)=Tr T(λ) generates commuting quantum charges Ċₙ via the expansion of log τ(λ). The algebraic structure underlying the model is identified as a q‑deformed affine sl₂ algebra, and the defect matrix K(λ) is shown to be a representation of a co‑ideal subalgebra, which explains why the defect does not spoil the Yang‑Baxter structure.

A central physical result concerns soliton‑defect interactions. Using the explicit one‑soliton solution of the SG equation, the authors apply the Bäcklund defect condition and find that a soliton crossing the defect acquires a phase shift Δφ = 2 arctan