  Recent years have seen tremendous progress in experimental techniques to
create uniaxial strain. Motivated by these advances we investigate the effect
of uniaxial strain on LaMnO$_3$ employing ab-initio dynamical mean-field
theory, and put it in contrast to biaxial strain that occurs in epitaxial
systems. Projecting on the low-energy subspace of Mn $3d$ states, and solving
multi-impurity problems, our approach emphasizes on local dynamic correlations
at Mn sites. At ambient pressures, LaMnO$_3$ crystallizes in an orthorhombic
unit cell, with in-plane lattice constants $a<b$, and shows an A-type
antiferromagnetic ground state. If we apply uniaxial compressive strain such
that the in-plane lattice becomes square with lattice constant $a$, we find a
ferromagnetic insulating state. This is in sharp contrast to DFT results using
various functionals like PBE, PBE+$U$, and hybrid functionals like HSE, which
all predict a half-metallic ferromagnetic behaviour. Interestingly, applying
uniaxial tensile strain, such that the in-plane lattice becomes square with the
longer lattice constant $b$, an antiferromagnetic insulating state is observed.
We trace back these results to the reduction in Jahn-Teller distortion in the
case of compressive strain, favoring a ferromagnetic state. This reduction is
absent in the tensile case, and the antiferromagnetic state therefore survives.
Our study shows that it is the flavour of the strain (compressive or tensile)
which is decisive for the magnitude of Jahn-Teller distortions and, hence, the
magnetic state.
