Gapless foliated-exotic duality

A bstract In this work, we construct a new foliated quantum field theory equivalent to the exotic ϕ -theory — a fractonic gapless scalar field theory described by tensor gauge fields and exhibiting U(1) × 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 ϕ -theories. Furthermore, we also investigate the foliated-exotic duality in the $$ \hat{\phi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> -theory, which is dual to the ϕ -theory, and construct the foliated $$ \hat{\phi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> -theory. These are the first examples of the foliated-exotic duality in gapless theories.