Surface defect in N=4 SYM and integrability

In the N=4 super Yang-Mills theory, it is well-known that the one-loop anomalous dimension operator for the single trace operators is equivalent to an integrable spin chain. Recent works have extended the application of integrability to scenarios involving a BPS boundary or defects such as 't Hooft line. One can describe the correlators of the single trace operators as an overlap between the Bethe state and the corresponding defect state. This overlap can be exactly calculated if the defect state is a so-called integrable state. We show that the state corresponding to the Gukov-Witten surface defect is integrable. We also calculate the tree-level one-point function of the single trace operators and set up the perturbation calculation in this defect background for one-loop corrections.