Gapless Foliated-Exotic Duality

In this work, we construct a new foliated quantum field theory equivalent to the exotic $\phi$-theory -- a fractonic gapless scalar field theory described by tensor gauge fields and exhibiting $U(1) \times U(1)$ subsystem global symmetry. This subsystem symmetry has an 't Hooft anomaly, which is captured by a subsystem symmetry-protected topological (SSPT) phase in one dimension higher via the anomaly inflow mechanism. By analyzing both the anomaly inflow structure and the foliated-exotic duality in the SSPT phases, we establish the foliated-exotic duality in the $\phi$-theories. Furthermore, we also investigate the foliated-exotic duality in the $\hat\phi$-theory, which is dual to the $\phi$-theory, and construct the foliated $\hat\phi$-theory. These are the first examples of the foliated-exotic duality in gapless theories.