Two-dimension to three-dimension transition of chiral spin liquid and fractional quantum Hall phases

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.