If ${L}$ is a line bundle over a complex manifold, what does the square root line bundle $L^{\frac{1}{2}}$ mean? After some google, I got to know that there are certain conditions for the existence of square root line bundle. In particular,I've following questions :

1. What is the square root of a line bundle, what are the conditions for its existence.

2. More importantly, how to think of the square root line bundle. Intuitively, It seems that a square root bundle $L^{\frac{1}{2}}$ is a line bundle s.t. the tensor bundle obtained by taking the tensor product of $L^{\frac{1}{2}}$ with itself gives the line bundle $L$.(correct if I am wrong).

I am particularly interested in square root of the Canonical Line bundle over a Riemann Sphere and the relation of square root bundle to Spinors in QFT. Please provide some references to look at.