What is the basic distinction between a Chern Insulator and a Topological Insulator? Right now I know that a Chern Insulator has "topologically non-trivial band structure" and that a Topological Insulator has "symmetry protected surface states".
[Physics] Chern insulator vs topological insulator
topological-insulatorstopological-ordertopological-phase
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The 'topological' in topological order means 'robust against ANY local perturbations'.
According to such a definition, topological insulator is not 'topological' since its properties are not robust against ANY local perturbations, such as the perturbation that break the U(1) and time reversal symmetry. So a more proper name for topological insulator is 'U(1) and time-reversal symmetry protected insulator', which is one example of SPT order.
Some example of topologically ordered states (in the sense of 'robust against ANY local perturbations'):
1) $\nu=\frac{1}{3}$ FQH state
2) $Z_2$ spin liquid state
3) $\nu=1$ IQH state
4) $E_8$ bosonic QH state
The example 3) and 4) have no non-trivial topological quasi-particles (ie no non-trivial statistics, no non-trivial topological degeneracy), but have gapless edge state that is 'robust against ANY local perturbations'.
-- Edit -- (I lifted some discussions below to here):
There are two kinds of topology in math. The "topology" in "topological order" is directly related to the first kind of topology in mathematics, as in algebraic topology, homology, cohomology, tensor category. The "topology" in "topological order" is different from the "topology" in "topological insulator". The "topology" in "topological insulator" is related to the second kind of topology in mathematics, as in mapping class, homotopy, K-theory, etc. The first kind of topology is algebraic, while the second kind of topology is related to the continuous manifold of finite dimensions. We may also say that the first kind of topology is "quantum", while the second kind of topology is "classical".
The correct way to describe any gapped phases (such as topological orders and topological insulators) is to use the first kind of topology -- "quantum" topology, because the gapped phases are usually interacting. The second kind of topology -- the "classical" topology -- can be used to describe the one-body physics (include free fermion systems). The "classical" topology cannot be used to describe interacting many-body systems, which need "quantum topology".
One needs to go beyond "filling energy level" picture to understand topological order (the first kind of topology). Our education in traditional condensed matter physics (or traditional many-body physics) is almost all about "filling energy levels" (such as Landau Fermi liquid theory, band theory, etc), which is a trap that limit our imagination. The second kind of topology (the "topology" in "topological insulator") can be understood within the framework of "filling energy level" picture.
To answer the question What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this? I like to say that topological order is algebraic, not geometric. So the topological invariants of topological order are very different from Chern numbers. The robust ground state degenercy and the robust non-Abelian geometric phases of the degenerate ground states are the topological invariants of topological order (which are the analogues of the Chern number).
Let me first answer your question "is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?"
(1) Topological superconductors, by definition, are free fermion states that have time-reversal symmetry but no U(1) symmetry (just like topological insulator always have time-reversal and U(1) symmetries by definition). Topological superconductor are not p+ip superconductors in 2+1D. But it can be p-wave superconductors in 1+1D.
(2) 1+1D topological superconductor is a SET state with a Majorana-zero-mode at the chain end. But time reversal symmetry is not important. Even if we break the time reversal symmetry, the Majorana-zero-mode still appear at chain end. In higher dimensions, topological superconductors have no topological order. So they cannot be SET states.
(3) In higher dimensions, topological superconductors are SPT states.
The terminology is very confusing in literature:
(1) Topological insulator has trivial topological order, while topological superconductors have topological order in 1+1D and no topological order in higher dimensions.
(2) 3+1D s-wave superconductors (or text-book s-wave superconductors which do not have dynamical U(1) gauge field) have no topological order, while 3+1D real-life s-wave superconductors with dynamical U(1) gauge field have a Z2 topological order. So 3+1D real-life topological superconductors (with dynamical U(1) gauge field and time reversal symmetry) are SET states.
(3) p+ip BCS superconductor in 2+1D (without dynamical U(1) gauge field) has a non-trivial topological order (ie LRE) as defined by local unitary (LU) transformations. Even nu=1 IQH state has a non-trivial topological order (LRE) as defined by LU transformations. Majorana chain is also LRE (ie topologically ordered). Kitaev does not use LU transformation to define LRE, which leads to different definition of LRE.
Best Answer
I don't think the provided comment gives the right answer. Topological insulators is the bigger group and Chern insulator are a subgroup of that. This means that every Chern insulator is a topological insulator, but not every topological insulator is a Chern insulator. Can maybe someone confirm that this is indeed true?
In general a topological insulator is a material that has gapped bulk, but conducting edge states that are protected by some symmetry. The surface Hamiltonian is gapless and cannot be gapped by perturbations that do not break the symmetry that protects the edge states, people say that the edge states are topologically protected.
A Chern insulator is 2-dimensional insulator with broken time-reversal symmetry. (If you have for example a 2-dimensional insulator with time-reversal symmetry it can exhibit a Quantum Spin Hall phase). The topological invariant of such a system is called the Chern number and this gives the number of edge states. So, when you have a non-trivial Chern insulator this means that it has edge states. The edge states of a Chern insulator are chiral meaning that in one channel the electrons only go one way and in the other channel the electrons go the other way. This may remind you of the Integer Quantum Hall Effect, which also has chiral edge states. You can see a Chern insulator as a 2D lattice version of the IQHE. (It is also called the Quantum Anomalous Hall Effect). You can go from the trivial phase to the topological phase by changing parameters in the lattice model such as the on-site or hopping energy.
The first Chern insulator was the Haldane model for graphene, where time-reversal symmetry is broken by introducing complex second nearest neighbour hopping but inversion symmetry still survives. This gave the chiral edge states characteristic of the now called Chern insulators.