# Why doesn't topological phase transition break any symmetry? Hidden symmetry?

This question may be superficial. However why all people saying this without a proof? Just like the "hidden variables" assumption in quantum mechanics, can one disproof that there is no hidden symmetry unknown that is actually breaking when have the so-called topological phase transition.

• Not an answer, but just a clarification - we do not "assume" hidden variables do not exist. They have been proven repeatedly to NOT exist, unless the theory is nonlocal (Bell's theorem and experimental results). This is a strong constraint, in many people's opinion. – levitopher Oct 18 '14 at 1:38
• @levitopher Thank you for the clarification, I know that agonist explanation of quantum mechanics had proven wrong by experiment. The reason I mention this is to analogy this with my thought. I want to know whether no symmetry breaking is really true when we have topological phase transition(like QHE), or this statement is made just because people haven't found some hidden symmetries. – an offer can't refuse Oct 18 '14 at 3:27

To grasp the relevant physics at a sloppy level, perhaps you simply need a few examples. You know a concept is commonly constructed by the manner you refer to it together with other concepts. Symmetry breaking usually results in ground state degeneracy and long range order. Order parameter field aids you in identifying degenerate sectors with the symmetries broken by the order. And such order is commonly reflected by correlation function, e.g., $C(\vec{r}_i,\vec{r}_j)=\langle \vec{S}(\vec{r}_i)\cdot\vec{S}(\vec{r}_j)\rangle$. So you should calculate it out.

• Berezinsky-Kosterlitz-Thouless Transition
Consider the quantum XY model $$\mathcal{H}=-\frac{1}{2C}\sum_i{\frac{\partial^2}{{\partial\theta_i}^2}}-J\sum_{\langle ij\rangle}{\cos{(\theta_i-\theta_j)}}$$ defined on a $d$-dimensional lattice and the temperature is not too high. You can expand around $\langle\theta\rangle$ and proceed to partition function via path integral method.
Among other things, you get $\langle \cos{\theta}\rangle=\langle \sin{\theta}\rangle=0$ when $d$ is lower than certain critical dimensions $d_c$ ($d\le2,T>0$ and $d\le1,T=0$). Therefore, order parameter is 0 and this is called Mermin-Wagner theorem. More importantly, correlation function $C(\vec{r}_i-\vec{r}_j)$ shows power-law decay at $d_c$ (correlation length $\xi=\infty$) while the high-temperature limit of this model only gives you exponential decay $$C(\vec{r}_i-\vec{r}_j)\sim\exp{\left[-\ln\left(\frac{2}{\beta J}\right)\left|\vec{r}_i-\vec{r}_j\right|\right]}.$$ Thus, obviously some phase transition takes place from high $T$ to low $T$ for $d=2$ scenario. However, you've got no order parameter in hand.
Moreover, the classical version of this model can be mapped onto a dual model composed of spin-wave DOFs and a 2D Coulomb gas of vortex DOF (topological defects). The BKT transition is mapped onto a metal-insulator transition.
The DOF in the model is so simplistic. It is undoubtedly a topological phase transition without any symmetry beraking.

• $\mathbb{Z}_2$ topological fliud
This may be a more "topological" example. Ising ($\mathbb{Z}_2$) gauge theory on square lattice is defined by $$\mathcal{H}=-g\sum_{\vec{x},j}{\sigma_j^x(\vec{x})-\frac{1}{g}\sum_\vec{x}\sigma_1^z(\vec{x})\sigma_2^z(\vec{x}+e_1)\sigma_1^z(\vec{x}+e_2)\sigma_2^z(\vec{x})},$$ in which $\sigma_j^{x/z}(\vec{x})$ is Pauli matrix defined on the link $(\vec{x},\vec{x}+\hat{e}_j)$. Let's consider a deconfining phase ($g<g_c$) on a torus geometry. The ground state has $4$-fold degeneracy. More generally, it has $4^q$-fold degeneracy for a closed surface with $q$ handles (genus). Just feel the topology :)
In stark contrast to ordinary symmetry breaking aforementioned, where degenerate sectors have no idea about topology, here in the transition from confined phase ($g>g_c$) to deconfined phase, nothing is associated with spontaneous breaking of any symmetry. That is to say, the label you stick to ground state degenerate sectors is completely changed, from broken symmetry to topological index. No local parameter is able to distinguish the degeneracy, except the magnetic holonomy 't Hooft operators defined on possible non-contractible loops.
In a more picturesque description, the deconfined phase contains "electric loops" proliferated to the extent that they wind around two non-contractible (large and nonlocal) loops of a torus, while ground state in the confined phase is unique and dominated by typically short "electric loops". This no longer sounds exotic once you recall this is a gauge theory, which normally can borrow some jargons from $U(1)$-gauge electromagnetism, e.g., "electric" and "magnetic".

As for the faint philosophical facet, I'd rather say you might redefine symmetry by incorporating novel things. The point is what physics you want to extract in the context. In the above contexts, I think there's no ambiguity. "Hidden variable" has been falsified by experimental tests of various Bell inequalities. It's good to continue arguing over loopholes therein or whatever as some serious researcher do. Note that it'd better be on some new stage of understanding. See this paper by Prof. Leggett for instance.