Can two electrons attract each other? Due to electrostatic repulsion the two electrons will repel each other as they both possess similar charges (lets leave gravitational attractive force out of the picture).   
My question is: can there be any attractive force(s) between the electrons? I know the reason that the particles stated above cannot share the same quantum state is due to the Pauli Exclusion Principle (PEP).
 A: you can check the discussion here. 
There is a certain case in which a phonon mediates attraction between two electrons. Indeed, acoustic phonons correspond to a slowly varying in-space
displacement of atoms which produces a charge. This charge, in turn, results in an electric potential for the electrons. This means that the electron distorts the crystal
lattice (builds up a positive charge around itself) which, at the end, attracts other electrons.
A: Two electrons when they move experience these forces
$$ F_{electrostatic repulsion } = \frac{ke^2}{r^2}$$
And,
$$ F_{magnetic attraction} = \frac{μ_0 . e^2 v^2}{4 \pi . r^2}$$
As you can see from the formulae for attraction there must be a velocity.
For the two forces to be the same the speed of the electrons must as fast as light, practically these two electrons will move in a double helix with increasing radius.
A: In certain scenarios there can be a magnetic attraction, but the electrostatic replusion will greatly overpower it.
A: A positive electric field exists at the center of a positively charged sphere. Electrons immersed within this field lose their mutual repulsion for each other and could touch each other due to gravitation.
