What is the difference between topological order and Landau's order

Question

I have thought about topological order for a long time, but I am still confused by it.

Roughly speaking in my understanding, the topological state is the eigenstate of some special symmetry such time reversal symmetry and space inversion symmetry, and distinguished from each other by different eigenvalues. Some people say this state has topological order and is protected by the symmetry.

I want to know what occurs during topological transition.

Is it a phase transition? Is there any universal class?

I think it is better to understand what topological order is first. So, I compare the topological order with Landau's order.

The Landau's order jumps from zero to finite value when breaking symmetry in a system. It is a well-known conclusion. I can imagine what occurs during phase transition. Some part of the system breaks its symmetry at first and has its local Landau's order. However, at the same time, the other part of the system has not broken the symmetry yet and its local order is zero. Hence, on average, the whole system has its Landau's order parameter by summing all the local orders in the system. This is why the Landau theory is some kind of mean field theory in my opinion.

However, when I want to use the analogy to try to understand the topological order, I am in trouble. The topological properties of a system are global, not local. Hence, I cannot imagine what happens when the system undergoes a topological transition. It looks the topological transition suddenly appears and the system changes its eigenvalue at that moment. This process makes me very confused...

I want to know, what is exactly the topological order of a system?

Does it jump from zero to a finite value or other similar cases when topology changes?

How to determine its value or the level of topological transition?

Comments

Hi @qfzklm, welcome to Physicsoverflow! I edited your question to improve the wordings a bit. You can see the edits by clicking on the time(displayed as blue) in "edited (time) ago by Jia Yiyang" at the bottom of your question. You can also edit it for yourself by clicking "edit" button at the bottom.

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As another tip, you can use the search box on the upper right corner of the webpage to see if there have been similar questions or answers. For example, after a brief search for "topological order", this post seems to be quite related to yours.  

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To understand the difference between topological order and Landau's order, one may start with the wiki article on topological order.

Answer 1

To understand topological order, it is helpful to forget about Landau's description of phase transitions in terms of local order parameters and broken symmetries — it does not apply here.

Note that in topological phases of matter, you have to distinguish between several concepts:

1. Topological Order

2. Symmetry-protected Topological Phase (STP)

They are defined as follows:

1. A Hamiltonian/system is said to have topological order it it has some feature, for instance the number of degenerate ground states or the entanglement of the ground state, that stays invariant under local perturbations of the Hamiltonian. For example, adding an impurity at a single site cannot change the dimension of the lowest energy eigenspace.

2. STP is a relative notion. Two gapped Hamiltonians $H(0)$ and $H(1)$ are in the same STP if they can be connected by a continuous path $t \mapsto H(t)$ such that each intermediate Hamiltonian respects the symmetries and is still gapped. Unlike for topoloical order, this path is not required to add only local perturbations to the Hamiltonian.

This means that any continuous path between Hamiltonians from different phases will, at some point, have one Hamiltonian with a zero-energy eigenstate. If you consider two systems spatially next to each other, then the interface between can be interpreted as a continuous path between the two, and you obtain that the interface must host gapless modes.

Note that this classification is not unique and very much depends on the symmetries you consider! Two Hamiltonians can be in the same phase with respect to one collection of symmetries, and be in different phases with respect to another. For instance, the quantum spin Hall effect (QSHE) is in the trivial class if you consider only the $U(1)$ gauge symmetry, but it is in a non-trivial class if you consider the symmetry group $U(1)\times \mathbb{Z}_2$ corresponding to charge conservation and time-reversal symmetry. In this sense, being in an STP is not a property of the system per se, but a property of the system in relation to other systems.