Talk:Periodic table of topological invariants

Article needs to be rewritten in its entirety

I was able to learn so little about what the subject of this article even means, that I am not sure whether or not this article is a hoax!

I have the following suggestions, just in case the article is not a hoax and someone who is knowedgeable on the subject would like to improve it:

1. Make sure that each statement that is purely a true, proven, published mathematical statement be stated clearly as a mathematical statement. Mathematical statements should use as much universal mathematical notation and terminology as possible. This enables the statements to be understood by as wide an audience as possible.

2. Likewise, it should be made clear to the reader that the purely physics statements (or statements that use math that was already mentioned ... are essentially statements about physics.

3. Then, there will probably be a third category of statement that arises: a statement that is essentially about both mathematics and physics — a statement that connects the two disciplines to each other. This is perhaps the most interesting kind of statement. So this type of statement, too, should be made clear to the reader when it is made.

4. Besides keeping those categories clear: Make sure that if you use a word for the first time, then explain it to the reader. This is most important in the introduction. The current introduction is a train wreck.

This is how the article begins:

The periodic table of topological invariants is an application of topology to physics. It indicates the group of topological invariant for topological insulators and superconductors in each dimension and in each discrete symmetry class.[1]

Discrete symmetry classes

There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian ${\hat {H}}=\sum _{i,j}H_{ij}c_{i}^{\dagger }c_{j}$, (where $c_{i}$, and $c_{i}^{\dagger }$, are the annihilation and creation operators of mode $i$, in some arbitrary spatial basis) : time reversal symmetry, particle hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry.

Chiral symmetry is a unitary operator $S$, that acts on $c_{i}$, as a unitary rotation ($Sc_{i}S^{-1}=(U_{S})_{ij}c_{j}$,) and satisfies $S^{2}=1$,. A Hamiltonian $H$ possesses chiral symmetry when $S{\hat {H}}S^{-1}=-{\hat {H}}$, for some choice of $S$ (on the level of first-quantised Hamiltonians, this means $U_{S}$ and $H$ are anticommuting matrices).

Time reversal is an antiunitary operator $T$, that acts on $\alpha c_{i}$, (where $\alpha$, is an arbitrary complex coefficient, and $^{*}$, denotes complex conjugation) as $T\alpha c_{i}T^{-1}=\alpha ^{*}{(U_{T})}_{ij}c_{j}$,. It can be written as $T=U_{T}{\mathcal {K}}$ where ${\mathcal {K}}$ is the complex conjugation operator and $U_{T}$ is a unitary matrix. Either $T^{2}=1$ or $T^{2}=-1$. A Hamiltonian with time reversal symmetry satisfies $T{\hat {H}}T^{-1}={\hat {H}}$, or on the level of first-quantised matrices, $U_{T}H^{*}U_{T}^{-1}=H$, for some choice of $U_{T}$.

Charge conjugation $C$ is also an antiunitary operator which acts on $\alpha c_{i}$ as $C\alpha c_{i}C^{-1}=\alpha ^{*}(U_{C}^{\dagger })_{ji}c_{j}$, and can be written as $C=U_{C}{\mathcal {K}}$ where $U_{C}$ is unitary. Again either $C^{2}=1$ or $C^{2}=-1$ depending on what $U_{C}$ is. A Hamiltonian with particle hole symmetry satisfies $C{\hat {H}}C^{-1}=-{\hat {H}}$, or on the level of first-quantised Hamiltonian matrices, $U_{C}H^{*}U_{C}^{-1}=-H$, for some choice of $U_{C}$.

Despite a strong math background, I cannot understand one bit of this. (For example, what is c_i?) Nothing is defined, nothing is explained.

No article should be so badly written. It really needs to be rewritten in its entirety. (Unless it's a hoax.)2600:1700:E1C0:F340:2C49:6FF1:574B:2144 (talk) 21:56, 4 November 2018 (UTC)

Error in second-quantised form of discrete symmetry classes?

I believe that charge conjugation and chiral symmetries map between creation and annihilation operators. Otherwise there is no difference between the two anti-unitary classes. Not confident enough to make these changes myself, the reference I'm looking at, also contains inconsistencies. Link below:

Ludwig, Andreas W W (2016-12-01). "Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond". Physica Scripta. T168: 014001. doi:10.1088/0031-8949/2015/T168/014001. Retrieved 2023-11-28. 2001:1BA8:401:7E:CD5E:CD38:DD29:AFEB (talk) 10:42, 6 December 2023 (UTC)