  Recently, it has been shown that the ground state of quantum chromodynamics
(QCD) in sufficiently strong magnetic fields and at moderate baryon number
chemical potential carries a crystalline condensate of neutral pions: the
chiral soliton lattice (CSL). While the result was obtained in a
model-independent manner using effective field theory techniques, its
realization from first principles using lattice Monte Carlo simulation is
hampered by the infamous sign problem. Here we show that CSL, or a similar
inhomogeneous phase, also appears in the phase diagram of a class of
vector-like gauge theories that do not suffer from the sign problem even in the
presence of a baryon chemical potential and external magnetic field. We also
show that the onset of nonuniform order manifests itself already in the
adjacent homogeneous Bose-Einstein-condensation phase through a characteristic
roton-like minimum in the dispersion relation of the lowest-lying quasiparticle
mode. Last but not least, our work gives a class of explicit counterexamples to
the long-standing conjecture that positivity of the determinant of the Dirac
operator (that is, absence of the sign problem) in a vector-like gauge theory
precludes spontaneous breaking of translational invariance, and thus implies
the absence of inhomogeneous phases in the phase diagram of the theory.
