How can quantum fluctuations lead to Spontaneous Symmetry Breaking?

Question

I'm studying Quantum field theory and I came across Spontaneous Symmetry breaking and the Weinberg-Coleman potential. My question is more conceptual. The way I understand it the Coleman-Weinberg potential gives the one-loop corrections, due to quantum fluctuations, in the classical action. This looks like this:

\begin{equation} \Gamma_{1-\text{loop}}[\phi]=S[\phi]+\frac{1}{2}\text{Trln}S^{(2)}[\phi] \end{equation}

These quantum fluctuations can induce a non-trivial minimum of the field $\phi$ which leads to Spontaneous Symmetry Breaking. This I cannot understand. I always compare Quantum field theory to statistical field theory and this is where my confusion comes from. In statistical field theory, starting from the SSB phase, thermal fluctuations can destroy the ordered phase and show that SSB doesn't occur. Or seen another way, starting from the disordered phase, thermal fluctuations lower the critical temperature to zero and there is again no SSB. Therefore the way I see it quantum fluctuations, which are the analogue to thermal fluctuations, should also destroy the ordered phase and not induce it. Am I understanding something wrong or is the analogy between QFT and SFT just not working in this situation?

Answer 1

The Coleman-Weinberg mechanism breaks a gauge symmetry, not a flavor one. Your intuition for flavor symmetries is perfectly right, but it is exactly the opposite for gauge symmetries. Fluctuations, whether thermal or quantum, tend to break gauge symmetries.

The short reason is that breaking a flavor symmetry gives you a massless particle (the NG boson) but breaking a gauge symmetry removes a massless particle, by the Higgs mechanism. So these two types of symmetry work in the opposite direction.

The longer answer is that gauge symmetries do not really exist, they are just a silly way we describe certain interactions. As such, it is not really meaningful to ask whether they are broken or restored.

Luckily for us, QED does have a flavor symmetry that can distinguish the Coulomb vs Higgs (a.k.a. conductor/superconductor) phases: this is the so-called magnetic one-form symmetry. In a rough sense, this flavor symmetry is broken if the gauge symmetry is unbroken, and unbroken if the gauge symmetry is broken. And now you can see how your intuition was actually correct from the beginning: quantum fluctuations, much like thermal ones, can restore symmetries. In this case, they restore the magnetic symmetry, which we (somewhat incorrectly) rephrase as the fact that they break the gauge symmetry.

Answer 2

I would like to point out that in statistical mechanics you can also have situations where quantum fluctuation induces order when thermal fluctuation does not. It is called order-by-disorder.

For example, consider the following Hamiltonian: \begin{equation} H=+\sum_{\langle i,j\rangle} Z_iZ_j ~-h\sum_{i}X_i, \end{equation} where the $Z$ and $X$ are Pauli operators of spin $s=1/2$, and the interactions $\langle i,j\rangle$ is over a triangular lattice. If you set $h=0$, the Hamiltonian is purely diagonal, and hence classical.

In that classical case, no long-range order is ever realized for any temperature including the zero-temperature limit $\beta\rightarrow\infty$. Intuitively, the system is "too frustrated" to order, and has exponential degeneracy even in the ground state.

Interestingly, it turns out that the moment you induce an infinitesimal amount of the transverse field $h=\epsilon>0$, the ground state becomes long-range ordered (it breaks the $D_3$ symmetry of the lattice) and is even robust against some thermal fluctuation (i.e. for $\beta>\beta_c(h)$). From the $h=0$ limit, introducing a finite $h$ is equivalent to introducing quantum fluctuations. Condensed matter physicists also had the idea that quantum fluctuation should be similar to classical fluctuation, so they were rather surprised by this phenomena that fluctuation can induce order, and dubbed it with a somewhat silly name order-by-disorder.

I remember that this can actually happen too with thermal fluctuation, since it will favor states with larger entropy, but can't come up with the right example just now. My main point is that it is actually not that uncommon where fluctuation actually induces order (SSB), especially with quantum fluctuations but also with classical. The above triangular AFM-TFIM is one of the simplest and nicest examples for that.