Martinez et al. (2015) gives an example of a neutron star with mass as low as $1.17M_\odot$ (solar masses).
There is also the recent discovery of this candidate neutron star which is apparently only $0.77M_\odot$.
Yet the Classical Chandrasekhar limit for a white dwarf is $1.44M_\odot$, while the "true Chandrasekhar limit" "true Chandrasekhar limit" is around 1.37 to 1.38 What is the reason for this discrepancy?
Well because they have different formation pathways. White dwarfs don't (often) become neutron stars, so there is no obvious reason why neutron stars should exceed the Chandrasekhar mass for carbon/oxygen white dwarfs.
However, there is reasonable cause to suppose that neutron stars should be heavier than the Chandrasekhar mass for an "iron white dwarf". That is because the main pathway to create a neutron star is a core collapse supernova. This is triggered by an instability (caused by photodisintegration or electron capture) that occur as the electron-degenerate iron core accretes mass and approaches its Chandrasekhar limit.
The Chandrasekhar mass is composition dependent. The "classic" limit derived by Chandrasekhar is given by $$M_c = 1.44\left(\frac{\mu_e}{2}\right)^{-5/3}\ ,$$ where $\mu_e$ is the number of mass units per electron. For carbon and oxygen, $\mu_e= 2$, for iron $\mu_e = 56/26$ and $M_c = 1.27 M_\odot$.
In practice, the real $M_c$ is lower than the classic limit because of (i) General Relativity, which leads to instability at a finite density and lower mass and (ii) electron capture which occurs once the electron Fermi energy reaches a threshold energy of about 13 MeV for carbon, but only 3.7 MeV for iron. This means the real Chandrasekhar mass for carbon/oxygen white dwarfs is about $1.38 M_\odot$ but for iron only around $1.11 M_\odot$ (Rotondo et al. 2011).
Thus, roughly speaking, we expect neutron stars to be above the Chandrasekhar mass for an "iron white dwarf" and usually a bit bigger because the core is hot and because some mass can be added after the core collapse.
There is no route to produce a subsolar mass neutron star via core collapse. The possible $0.7 M_\odot$ neutron star you refer to must have a bizarre story if true.
