Abstract
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 Toverline{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 Toverline{T} -deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.
Highlights
Other solvable irrelevant deformations including the JTdeformation [15], deformation by higher spin irrelevant operators constructed from KdV currents [16, 17] and their various combinations [18, 19], which share these two features
We 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 T-deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore
Summary
We provide a short review of the solvable irrelevant deformations of QFT that are related to the discussion in this work; the reader is invited to consult the original papers for further details. For a given 2d QFT described by an action S0, consider two conserved currents Jμ(1) and Jμ(2) satisfying ∂μJμ(a) = 0. Using these currents we construct the following composite operator. The solvability of the quantum theory for the class of theory (2.5) is based on the factorization formula of the expectation value of the composite operator. The factorization formula leads to a flow equation for the deformed energy. Given some convenient expressions for the expectation values of the conserved currents, the above equation can be used to determine the deformed spectrum. Taking Oλ to be the T Toperator, the right hand side of (2.7) can be written in terms of the energy and momentum, so the flow equation reads. In the following we demonstrate that all these features can be generalised in a natural way to long range deformations of spin chains of bi-local type
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