  In the presence of axial magnetic fields that can be realized in deliberately
buckled monolayer graphene, quasi-relativistic Dirac fermions may find
themselves in a variety of broken symmetry phases even for weak interactions.
Through a detailed Hartree self-consistent numerical calculation in finite
strained graphene with cylindrical and open boundaries we establish the
possibility of realizing a charge-density-wave order for the spinless fermions
in the presence of weak nearest-neighbor repulsion. Such an instability gives
rise to a staggered pattern of average fermionic density between bulk and
boundary of the system as well as among two sublattices of graphene, due to the
spatial separation of the zero energy states localized on opposite sublattices.
Although with fermions spin restored, an unconventional magnetic order driven
by the onsite repulsion, possibly leads to the dominant instability at the
Dirac point, the proposed charge-density-wave order can nevertheless be
realized at finite doping, which is always accompanied by a finite
ferromagnetic moment. Additionally, the charge-density-wave phase supports a
quantized charge or spin Hall conductivity when its formation away from the
Dirac point is further preceded by the appearance of topological anomalous or
spin Hall insulator respectively. The topological orders in strained graphene
can be supported by weak second neighbor repulsion, for example. Therefore,
depending on the relative strength of various short-range components of the
Coulomb interaction a number of broken symmetry phases can be realized within
the zero energy manifold in strained graphene.
