Surface defects in 4d supersymmetric QFTs and integrable models

We study certain class of surface defects in 4d supersymmetric theories obtained as compactifications of 6d SCFTs.  In particular, we consider flows initiated by spacetime-dependent vacuum expectation values, which lead, in certain setups, to surface defects inserted into these theories. We compute a certain partition function of these theories, namely, the 4d supersymmetric index, with the insertion of a defect. The insertion of a defect is manifested in this computation as a difference operator acting on the index of the theory without a defect. The difference operators obtained in these computations can often be related to Hamiltonians of quantum integrable models. We derive difference operators for several theories and study their properties, which are implied from dualities. In some cases the operators derived are associated with known integrable models and in some cases they are novel. Moreover, based on few assumptions, we provide a recipe to compute eigenfunctions for the derived operators, and compute explicit eigenfunctions for several models, including Ruijssenaars-schneider and van Diejen models.