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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".

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  • $\begingroup$ Chern insulators must break time-reversal symmetry and topological insulators require time-reversal symmetry for protection. Both of them have nontrivial band structures and edge/surface states, the difference being that TI is nontrivial only with time-reversal symmetry (and the edge states are protected by the symmetry). $\endgroup$ – Meng Cheng Dec 30 '15 at 20:33
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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.

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    $\begingroup$ When you say "every Chern insulator is a topological insulator", you need to define what is "topological insulator". According to a standard definition, "topological insulator" is an fermionic insulator with time reversal symmetry and U(1) symmetry. According to this definition, a Chern insulator is NOT a topological insulator. $\endgroup$ – Xiao-Gang Wen Mar 13 '19 at 0:21
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2+1d Chern insulator (CI):

1) belongs to the classes of systems realizing Integer Quantum Hall states on the lattice without external magnetic field. It belongs to the long-ranged entangled Topological Order by the definition of X.G.Wen of MIT, but it is still a part of theory of Invertible Topological Quantum Field Theory (of Dan Freed, see References and papers by him) with its partition function $|Z|=1$ on a closed manifold, and can be gapped by coupling to its time-reversal partner (with opposite sign of the Chern number). It is however short-ranged entangled by the definition of A Kitaev of Caltech.

2) The Chern insulator on the lattice without external magnetic field realizes so-called the Anomalous Quantum Hall Effect.

3) And CI is characterized by the Chern number $C_1$ under the $\mathbb{Z}$ class of topological invariances: $$ C_1=\frac{1}{2\pi}\int_{\mathbf{k} \in \text{BZ}} d^2\mathbf{k}\; \epsilon^{\mu \nu } \partial_{k_\mu} \langle \psi(\mathbf{k}) | -i \partial_{k_\nu} | \psi(\mathbf{k}) \rangle \in \mathbb{Z}. $$

2+1d and 3+1d Topological insulator (TI):

1) belongs to the classes of systems realizing Symmetric-protected topological states, needed to be protected by time reversal and U(1) charge symmetry. The TI has NO bulk intrinsic topological orders.

2) The 2+1d and 3+1d free (quadratic Hamiltonian) TI both are characterized by the $\mathbb{Z}_2$ invariance instead of $\mathbb{Z}$ invariance. See the paper of Fu and Kane. The can also be characterized by the $\Theta=\pi$ for the probed bulk U(1) field action (see Qi, Hughes, Zhang, Phys Rev B 2008.) $$S_{3+1d\;bulk}= \frac{\Theta}{8 \pi} \int F \wedge F$$ term with some proper normalization. Namely the 3+1d $\mathbb{Z}_2$ for free TI is $$ \Theta=0, \pi \in \mathbb{Z}_2, $$ which respect the time reversal symmetry of TI.

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A topological insulator has a non-trivial (non-zero) topological invariant.

Chern number is one such topological invariant. If the Chern number is non-zero, then the system is a Chern insulator. Hence, a Chern insulator is a subgroup of topological insulators.

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you cannot discuss classification of topology without symmetry.

Any gapped system with nontrival edge states can be called as topology insulator. Haldane model defined in honeycomb lattice is an example of chern insulator which has non-zero chern-number and doesn't need any symmetry to protect it from adiabatic transform (without gap close) to a trival insulator.

Z2 topology insulator is a subset of chern insulator but its chern-number is zero. However, it still has nontrival edge mode which need to be described by a new topology index called Z2 index. The reason why it belong to the subset of chern insulator is that it need time reversal symmetry to protect it from adiabatic connected to an phase which has different Z2 number.

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