Quantum Electrodynamics in $2+1$ Dimensions, Confinement, and the Stability of $U(1)$ Spin Liquids

Compact quantum electrodynamics in $2+1$ dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when $N>N_c=36/{\pi }^3\approx 1.161$, where $N$ is the number of fermion replica. For $N<N_c$, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as $N\to N_c^{-}$. Our results imply the stability of a spin liquid at the physical value $N=2$ for Mott insulators.

Figure 1

Schematic flow diagram for the case $N=1$.