  We consider a system of interacting spinless fermions on a two-leg triangular
ladder with $\pi/2$ magnetic flux per triangular plaquette. Microscopically,
the system exhibits a U(1) symmetry corresponding to the conservation of total
fermionic charge, and a discrete $\mathbb{Z}_2$ symmetry -- a product of parity
transformation and chain permutation. Using bosonization, we show that, in the
low-energy limit, the system is described by the quantum double-frequency
sine-Gordon model. On the basis of this correspondence, a rich phase diagram of
the system is obtained. It includes trivial and topological band insulators for
weak interactions, separated by a Gaussian critical line, whereas at larger
interactions a strongly correlated phase with spontaneously broken
$\mathbb{Z}_2$ symmetry sets in, exhibiting a net charge imbalance and non-zero
total current. At the intersection of the three phases, the system features a
critical point with an emergent SU(2) symmetry. This non-Abelian symmetry,
absent in the microscopic description, is realized at low-energies as a
combined effect of the magnetic flux, frustration, and many-body correlations.
The criticality belongs to the SU(2)$_1$ Wess-Zumino-Novikov-Witten
universality class. The critical point bifurcates into two Ising critical lines
that separate the band insulators from the strong-coupling symmetry broken
phase. We establish an analytical connection between the low-energy description
of our model around the critical bifurcation point on one hand, and the
Ashkin-Teller model and a weakly dimerized XXZ spin-1/2 chain on the other. We
complement our field-theory understanding via tensor network simulations,
providing compelling quantitative evidences of all bosonization predictions.
Our findings are of interest to up-to-date cold atom experiments utilizing
Rydberg dressing, that have already demonstrated correlated ladder dynamics.

Strongly Correlated Electrons;Quantum Gases;Statistical Mechanics;Quantum Physics