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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?

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  • $\begingroup$ Quick comment: but there is a definite difference between the thermal/statistical setting $s^{-\beta H}$ and the quantum setting $e^{i S}$ -- signs (or factors of $i$) on various calculations generically turn out different, especially for fermion loops! $\endgroup$
    – Siva
    Feb 6 at 7:49
  • $\begingroup$ It's true but after a wick rotation they look exactly the same! That's what also makes it (to me at least) so weird to me. $\endgroup$ Feb 6 at 10:55
  • $\begingroup$ Are you asserting that there is no SSB in statistical field theory? The picture of fluctuations breaking the long-range order is always true in one dimension and sometimes true in two dimensions from the Mermin-Wagner theorem, but there definitely exists SSB sometimes in statistical field theory, like for instance ferromagnets. $\endgroup$ Feb 6 at 14:36
  • $\begingroup$ No, I'm not. This is actually my point. In Mermin-Wagner fluctuations break long-range order therefore not allowing SSB. But in QFT fluctuations do the opposite and I have no intuition for this at all. $\endgroup$ Feb 6 at 20:36
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    $\begingroup$ Thermal fluctuations can also lead to order in classical statistical mechanics. A famous example is Rochelle salt, which exhibits an intermediate ferroelectric phase between two paraelectric phases. More examples of order from disorder have been discovered (here is another example nature.com/articles/s41567-022-01633-9 ). So in general, fluctuations do not always disorder systems. $\endgroup$ Feb 6 at 23:15

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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.

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  • $\begingroup$ (This is a really good question by the way, most QFT textbooks are very unclear about this) $\endgroup$ Feb 6 at 23:41
  • $\begingroup$ Is "flavor" symmetry the same as "global" symmetry? $\endgroup$ Feb 7 at 3:31
  • $\begingroup$ Could you perhaps elaborate a bit further on "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" ? Or provide some source I can read about this? Because I don't think I fully understand it. $\endgroup$ Feb 7 at 10:02
  • $\begingroup$ @flippiefanus Indeed $\endgroup$ Feb 8 at 21:08
  • $\begingroup$ @ΜπαμπηςΠοζουκιδης You may want to have a look at physics.stackexchange.com/q/257018/84967 and links therein. $\endgroup$ Feb 8 at 21:10
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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.

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Your confusion arises from a subtle but important difference in how fluctuations operate in quantum field theory (QFT) and statistical field theory (SFT), particularly in the context of spontaneous symmetry breaking (SSB).

In SFT, you're correct that thermal fluctuations can restore symmetry by destroying the ordered phase as the temperature increases. This is because thermal fluctuations tend to disorder the system, making it more energetically favorable for the system to be in a symmetric, disordered state at high temperatures. The critical temperature marks the transition between the ordered (SSB) phase and the disordered phase, with SSB occurring below this temperature.

In QFT, however, the situation with quantum fluctuations is quite different. Quantum fluctuations are inherent variations in the field's value, even at zero temperature. The Coleman-Weinberg potential accounts for these fluctuations by adding quantum corrections to the classical potential. Unlike thermal fluctuations, quantum fluctuations can induce a situation where a symmetric potential (with a single minimum at the origin, for example) is modified such that new minima appear away from the origin. This leads to SSB because the ground state of the system (the state of lowest energy) is no longer invariant under the symmetry of the original potential.

The key difference lies in the nature of the fluctuations:

  • Thermal fluctuations in SFT are driven by temperature and tend to disorder the system, potentially restoring symmetry in a previously broken symmetry phase as the temperature increases.
  • Quantum fluctuations in QFT are zero-point fluctuations present even at zero temperature. These can lead to the modification of the potential in such a way that it favors a new ground state (or states) that breaks the symmetry of the system, even if the classical potential did not allow for this.

The analogy between SFT and QFT does hold in the broad sense that fluctuations can profoundly affect the phase structure of a theory. However, the specific role and consequences of these fluctuations differ significantly between the two frameworks due to their different physical origins and how they influence the system's behavior. In QFT, rather than destroying order, quantum fluctuations can create conditions for new types of order (i.e., new ground states) to emerge, leading to SSB.

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