| Summary: | © 2020 American Physical Society. In this paper, we propose strongly correlated gapless states (or critical states) of spin-1/2 electrons in 1+1 dimensions, such as the doped ferromagnetic and antiferromagnetic spin-1/2 Ising chains. We find that the metallic phases in the doped ferromagnetic and antiferromagnetic Ising chain are different strongly correlated gapless phases, despite the fact that the two phases have the same symmetry. The doped antiferromagnetic Ising chain has a finite energy gap for all charge-1 fermionic excitations even without pairing caused by the attractive interaction, resembling the pseudogap phase of underdoped high Tc superconductors. Applying a transverse field to the ferromagnetic and antiferromagnetic metallic phase can restore the Z2 symmetry, which gives rise to two distinct critical points despite the fact that the two transitions have exactly the same symmetry-breaking pattern. We also propose chiral metallic states. Some of these 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 noninteracting fermion systems with the same symmetry. Our nonperturbative results are obtained by noting that gapless quantum systems have emergent categorical symmetries (i.e., noninvertible gravitational anomalies), which are described by multicomponent 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.
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