# Basics of topological order and its relation to entanglement

What is a topological order that drives a topological phase transition?

How is it different from say magnetic ordering or the superfluid ordering?

What is its relation with entanglement?

Please consider me a novice in this subject and explain me the very basics of it. Since I wish to start reading some serious stuff related to this suggest some beginner level references too.

The origin of people's interests on the quantum phases, is from the fact that, in certain systems, even though two ground states share the same symmetry, they still belong to different phases (cannot be continuously tuned to each other through a smooth deformation of Hamiltonian).

Firstly, as you mentioned, conventional phases, like superfluid phases and magnetic orders, can be distinguished by their symmetries. And the Landau's theory of phase transition can therefore be successfully applied -- the symmetry would be related to the order parameter.

To connect to the modern language of phases and transitions, we now rephrase the above conventional problems in the following way: we start from a gapped Hamiltonian, which means the ground state is gapped, then turn on some symmetric perturbations -- you don't break the symmetry by hand otherwise is trivial. If at certain point, the perturbation term makes the spectrum gapless, then this defines a phase transition point; keep increasing the amplitude of this term, the spectrum of the whole system would become gapped again, at the same time the symmetry of the system is also different, which means it enters another phases. Therefore we can define the phases using the system's spectrum: every time the gap closes, we could identify a transition point. This is also consistent with the conventional definition of phases, where one can calculate the correlation function and find the divergence at a phase transition.

Now, you are careful enough, you would have noticed that when I said: "the spectrum of the whole system would become gapped again, at the same time the symmetry of the system is also different", I didn't really rigorously relate these two. And in fact, this is not guaranteed: one can actually cross a gapless transition point without changing the symmetry of the system. And this is exactly one way to define the quantum phases and quantum phase transitions. Well-known examples includes the fractional quantum hall systems (FQH), and symmetry protected topological phases (SPT).

However, people would not be satisfied if the two phases cannot be distinguished by an explicit "order parameter". So, what would be the closest analogy to the context we used in the Landau's conventional theory? That's why people propose the idea of quantum order. And topological order (TO) is one type of quantum orders. An important thing to clarify is that: quantum phases can be distinguished by quantum order, but not all phases have non-trivial topological order: the above definition of topological phases concerning spectrum gap could also provide a classification for SPT phases, which have no topological orders at all. Classic examples are Chern insulators, where one can identify (improperly) an "order parameter": the Chern number of the system; however, this is only a quantum order, and different classes with different Chern numbers all have zero topological order, according to the modern definition. The most classic example of TO would be the $$Z_2$$ toric code, where the topological order is defined as the ground state degeneracy.

The question then comes into one's mind: what exactly gives the topological order, or, what is "topological" exactly in quantum phases? It turns out that in the systems with nonzero topological order, the low energy physics (degree of freedom) could be captured by topological quantum field theories (TQFT) -- that's where the "topology" lies on.

So, besides the ground state degeneracy, is there other ways to define (quantify) the topological order? The answer turns out to be related to quantum entanglement. At 2004, two important papers on "Topological entanglement entropy" (TEE) proposed the TEE as another way to quantify TO. Now how to connect the two perspectives: TQFT and entanglement? One possible way is: the entanglement is related to the true degree of freedom of the system, and the most classic paper is on the Minimum Entangled State (MES), where the authors related eigenstates of quasi-particle operators to the states with the minimum (in the sense that topological entangled entropy is negative) entanglement entropy.

It then becomes more comprehensive to use the entanglement concept after people apply some quantum information approaches to tackle the quantum phase problems, and the important difference compared with previous studies on topological phases is that people start using pure wave-function language to formulate all questions, which then makes entanglement very necessary. One of the most useful concepts is "quantum circuit", where a state (wave-function) evolves into another through certain unitary (also local in a lot of cases) operator layer. To cross a quantum phase transition, the depth of the evolution layers required explodes; and at the same time, by looking at quantum circuits, one can ask the question about the quantum entanglement: long or short range entangled, and so on.

• "The origin of people's interests on the topological phases, is from the fact that, in certain systems, even though two states share the same symmetry, they still belong to different phases." This statement appears to be not quite true because liquids and gases also share the same symmetry. Are they different topological phases? – SRS Mar 27 at 5:23
• @SRS You are right, I should be more careful: Landau's phase transition theory with a continuous order parameter is mostly designed for second order transition. Back to the liquid/gas example you mentioned. Firstly, this is a 1st-order transition, and there is indeed no symmetry breaking. Secondly, quantum phase transitions study systems in the ground state of a model; and, if you like, you can map the liquid/gas to a Ising model, and then you notice the Hamiltonian explicitly break the Ising symmetry (an external field) -- there is no symmetry at the beginning, and no need to consider that. – Kite.Y Mar 27 at 12:40
• @SRS Now I modified the answer a little bit and explicitly focus on the quantum phase transition. (Note: quantum phase transition just means zero-temperature (ground state) transitions, and does not imply a quantum order, e.g. the examples in your original question.) – Kite.Y Mar 27 at 12:49