# Where is the connection between $U(1)$ gauge field and $\mathbb{Z}_2$ gauge theory?

I am a graduate student in condensed matter physics and today I was reading the Wikipedia article Topological Order. There is the part:

Note that superconductivity can be described by the Ginzburgâ€“Landau theory with dynamical U(1) EM gauge field, which is a Z2 gauge theory, that is, an effective theory of Z2 topological order.

Where is the connection between the U(1) gauge field and the Z2 gauge Theory? Or in other words is the Z2 topological oder a consequence of the U(1) gauge field?

This answer requires understanding of various things, so I am just gonna drop googleable names of theories and concepts to keep the text concise and readable.

TL;DR
It's all about classifying the superconducting phase transition. Real-life superconductivity does not break any symmetries, hence the symmetry classification (Ginzburg-Landau formalism) is not applicable.
Topological phase transitions do not need symmetry breaking. Real-life superconductors are hence labelled by their (non-trivial) topological order.

Classifying phase transitions

We can all agree that there is a definite difference between a normal metallic phase ($$T \gg T_{\mathrm{c}}$$) and a superconducting phase ($$T \leq T_{\mathrm{c}}$$). Namely, the lack of electrical resistance and hence unhindered flow of charge carriers.

A very powerful way of classifying phase transition is looking that which (global) symmetry is broken at the transition. Global symmetries are real symmetries, and breaking them has physical effects, namely the emergence of Goldstone modes, which are massless and gapless.
Typical examples are the breaking of the rotational invariance of a paramagnet to a ferromagnet, at the Curie temperature. For $$T, the magnetisation is non-zero and (spontaneously, in the absence of an external magnetic field) chooses a direction. The symmetry of the system goes from $$SO(3)$$ to $$SO(2)$$ with two Goldstone bosons generated, called spin-waves or magnons.
Or a liquid ($$SO(3)$$) becoming a solid (no continuous symmetry), generating three phonons.

This is nicely quantified by the Ginzburg-Landay formalism, where the potential energy usually looks like: $$V \propto a\phi^2 + b\phi^4,$$ with $$a= a_0(T-T_{\mathrm{c}})$$ such that for $$T the potential doesnâ€™t have a minimum at $$\phi=0$$ anymore but rather a ring of minima at $$\phi = -a_0/b$$. Because the state chooses a particular state out of this degenerate ring, the symmetry is broken:

Symmetry breaking is only a way of classifying phase transitions, in particular is only applicable to second-order (or "continuous") phase transitions.

Other transitions do not break any symmetries. But the phases are "definitely different" anyway. So how do you quantify this order? Sometimes you can classify it as a topological order, that is you can find a topological invariant which is different in either phase.

Supefluidity
("fake" superconductivity, but sometimes referred to as "textbook", since it's the easiest to show)

Consider an interacting field theory with a global $$U(1)$$ symmetry. According to Noether's theorem, this is associated with the conservation of particle number.

Breaking this symmetry leads to one Goldstone boson, with a linear (low-energy) dispersion $$E \propto |\mathbf{p}|$$. This results in the presence of a critical velocity $$v_p$$ below which the fluid experiences no viscosity.

This is a superfluid. Viscous-free flow. Somehow reminiscent of superconducting flow eh?

Real-life superconductivity

Superfluid is a scalar complex field theory (hence the $$U(1)$$ symmetry) with a global (real) symmetry. In the non-interacting limit, it's just Bose-Einstein condensation.

Real-life superconductors are not just "Bose-Einstein condensates" of Cooper pairs, as sometimes found in the literature.
By real-life I still mean the most basic form of superconductivity, that is $$s$$-wave and within the BCS theory.

There is still a $$U(1)$$ symmetry involved. But it's not a real one. It is a local $$U(1)$$ symmetry, which is nothing else than a redundancy. Hence it's also called a gauge symmetry, and it cannot be broken, and it cannot lead to Goldstone bosons. So we can't classify superconductivity as a symmetry-breaking transition!

Some people call the $$U(1)$$ for superfluidity "static", and the one here "dynamical". Local/dynamical gauge symmetries are coupled to gauge fields $$\mathbf{A}$$ such as the electro-magnetic field. So a local $$U(1)$$ scalar complex field theory describes a charged (bosonic, spin-$$0$$) system.

This is the "dynamical U(1) EM gauge field" mentioned in your quote.

There is still an order parameter for the transition. This is not the number of condensed particles (like in a superfluid or a BEC), but it's the superconducting bandgap $$\Delta$$.

However, as mentioned, local symmetries cannot be broken. Hence there is no symmetry-breaking that one can use to classify the transition.

How to classify the two phases then?

It turns out you can calculate a topological invariant in the superconducting phase, and it's the Pfaffian of the Hamiltonian. It is just a sign and can therefore only take two values, $$\pm 1$$, which lead to the $$\mathbb{Z}_2$$ theory that you talk about. This invariant is only defined in the superconducting phase, when the superconducting gap $$\Delta \neq 0$$. For the metallic phase, the Hamiltonian has the usual particle number conservation: if $$\Delta=0$$ the numbers of filled electron and hole states is conserved, and so the invariant once again becomes just the number of filled states.

The $$\mathbb{Z}_2$$ order stems from the particle-hole symmetry of the superconducting Hamiltonian, and is discussed more here .