  In this paper, we propose some new strongly correlated gapless states (or
critical states) of spin-1/2 electrons in 1+1-dimensions, such as doped
anti-ferromagnetic spin-1/2 Ising chain. We find doped anti-ferromagnetic Ising
chain to be a different metallic phase from the doped ferromagnetic Ising
chain, despite the two have identical symmetry. The doped anti-ferromagnetic
Ising chain has a finite energy gap for all charge-1 fermionic excitations even
without pairing caused by attractive interactions, resembling the pseudo-gap
phase of underdoped high Tc superconductors. Applying a transverse field to the
ferromagnetic and anti-ferromagnetic metallic phases can restore the $Z_2$
symmetry, which gives rise to two distinct critical points despite that the two
transitions have exactly the same symmetry breaking pattern. We also propose
new chiral metallic states. All those new gapless states are strongly
correlated in the sense that they do not belong to the usual Tomonaga-Luttinger
phase of fermions, i.e., they cannot be smoothly deformed into the
non-interacting fermion systems of the same symmetry. Our non-perturbative
results are obtained by noticing that gapless quantum systems have emergent
categorical symmetries, i.e., non-invertible gravitational anomalies), which
are described by multi-component partition functions that are modular
covariant. This allows us to calculate the scaling dimensions and quantum
numbers of all the low energy operators for those strongly correlated gapless
states. This demonstrates an application of emergent categorical symmetries in
determining low energy properties of strongly correlated gapless states, which
are hard to obtain otherwise.
