Cosmic string in real space

Question

I am considering $U(1)$ global cosmic string.

In the case of wine-bottle potential with real field ($\phi$), it is known that there exists a static solution that corresponds to cosmic string (or vortices). If you calculate the stress-energy tensor and look at the 00 component ($T^0_0$), the energy is highly concentrated at $\phi=0$. So, I would think that the core of the string will have $\phi\approx 0$.

Now, in real space (In lattice simulations) once you connect the points with $\phi \approx 0$, it forms a loop or an infinite line. I don't understand why this is the case.

I would appreciate any intuitive explanation or a reference.

Answer 1

If I'm not misunderstanding, the region where $\phi\approx 0$ resembles a cylinder or a string of finite width. Inside this volume, connecting the lattice points where $\phi\approx 0$ might reasonably form a loop or an infinite line.