  We study quantum phases and phase transitions in a one-dimensional
interacting fermion system with a Lieb-Schultz-Mattis (LSM) type anomaly.
Specifically, the inversion symmetry enforces any symmetry-preserving gapped
ground state of the system to be a Kitaev chain, following a
Lieb-Schultz-Mattis type theorem that we prove. Alternatively, via the
Jordan-Wigner transformation, this system describes a spin system whose gapped
ground states must break either the inversion or the Ising symmetry associated
with fermion parity. We obtain a phase diagram using analytical methods and
variational matrix product state simulations, and study the critical behaviors
of the quantum phase transitions therein using entanglement entropy, energy
variance and finite size scaling of order parameters. In particular, we observe
continuous phase transitions between different ordered phases that are beyond
the Ginzburg-Landau-Wilson paradigm, in analogy to the deconfined quantum
critical points in two spatial dimensions. We show this type of 1D deconfined
quantum critical point is described by the Tomonaga-Luttinger liquid theory,
and extract the Luttinger parameter and critical exponents. We also identify a
gapless phase between two ordered phases, which cannot be described by a U(1)
Luttinger liquid.
