  There have been lots of interest in two-dimensional (2D) fractional phases
with an emergent U (1) gauge field. However, many experimental realizations are
actually in three-dimensional (3D) systems with infinitely stacked 2D layers.
Then a natural question arises: starting from the decoupling limit with 2+1d U
(1) gauge field in each layer, how does the gauge field become 3+1d when
increasing inter-layer coupling? Here we propose a 2D to 3D transition through
condensing inter-layer exciton. The Goldstone mode of the condensation becomes
the missing az component in the 3D phase. As a simple example, we construct a
3D chiral spin liquid (CSL) from infinitely stacked 2D CSL. The 3D CSL has a
gapless photon mode with dispersion $\omega \sim q_z^2$ in the z-direction. The
same theory also applies to the fractional quantum Hall phase. At the 2D to the
3D transition point, there are gapless modes at each $q_z$ along a line $q =
(0,0,q_z)$ in momentum space, in contrast to a conventional critical point with
gapless mode only at one momentum. Meanwhile, the scaling dimension $\Delta
(q_z)$ has $q_z$ dependence, indicating a more non-trivial structure than a
simple decoupled fixed point. Our theory can also be generalized to a critical
point between a generic infinite component Chern-Simons-Maxwell theory (iCSM)
with both intra-layer and inter-layer Chern Simons term and a 3D gapless phase.
Certain iCSM theories have recently been shown to describe gapped non-foliated
fracton orders. Therefore we have a continuous transition between a gapped
fracton order and a 3D gapless phase.
