$Tbar T$-deformation and long range spin chains

We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the $T\bar T$-deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of $\mathcal{N} = 4$ super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the $T\bar T$-deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.

💡 Research Summary

The paper establishes a deep and precise correspondence between two seemingly unrelated families of integrable deformations: the $T\bar T$ deformation of two‑dimensional quantum field theories (QFTs) and a class of long‑range integrable deformations of quantum spin chains originally introduced in the context of planar $\mathcal N=4$ super‑Yang‑Mills theory. The authors first review the salient features of solvable irrelevant deformations in QFT, emphasizing that the $T\bar T$ operator is built from an antisymmetric bilinear of two conserved currents, $O=\epsilon^{\mu\nu}J^{(1)}\mu J^{(2)}\nu$. Zamolodchikov’s factorisation theorem shows that the expectation value of $O$ factorises into a product of current expectation values, which leads to a flow equation for the spectrum (the inviscid Burgers equation) and to a simple CDD‑type phase factor multiplying the factorised S‑matrix.

Turning to lattice models, the authors consider integrable spin chains with local Hamiltonians $H=\sum_x h(x)$, such as the SU(N) Heisenberg XXX chain and its generalisations. These models possess infinitely many conserved charges $Q_\alpha=\sum_x q_\alpha(x)$ and associated lattice currents $J_\alpha(x)$ defined by discrete continuity equations. By introducing a boost operator $B