Let us now look at all the possible symmetry classes in dimensions from \(0\) to \(3\), and see what kind of topological insulators are possible.

There are quite a few, here is the full list:

\[ \require{color} {} \newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{c|cccc} \hline \textrm{class} & d=0 & 1 & 2 & 3\T\\ \hline \textrm{A} & \mathbb{Z} & & \mathbb{Z} & \T\\ \textrm{AI} & \mathbb{Z} & & & \T\\ \textrm{AII} & 2\mathbb{Z} & & \mathbb{Z}_2 & \mathbb{Z}_2\T\\ \textrm{AIII} & & \mathbb{Z} & & \mathbb{Z}\T\\ \textrm{BDI} & \mathbb{Z}_2 & \mathbb{Z} & & \T\\ \textrm{C} & & & 2\mathbb{Z} & \T\\ \textrm{CI} & & & & 2\mathbb{Z}\T\\ \textrm{CII} & & 2\mathbb{Z} & & \mathbb{Z}_2\T\\ \textrm{D} & \mathbb{Z}_2 & \mathbb{Z}_2 & \mathbb{Z} & \T\\ \textrm{DIII} & & \mathbb{Z}_2 & \mathbb{Z}_2 & \mathbb{Z}\\ \hline \end{array} \]

This table has a lot of logic in it, but to you it most likely looks no better than this:

\[ \require{color} \definecolor{70482580}{rgb}{0.39,0.67,0.94} \definecolor{49770034}{rgb}{0.85,0.31,0.52} \definecolor{62598958}{rgb}{0.44,0.23,0.53} \definecolor{33537890}{rgb}{0.91,0.46,0.43} \definecolor{65621068}{rgb}{0.94,0.78,0.72} \definecolor{12177369}{rgb}{0.8,0.09,0.52} \definecolor{36361699}{rgb}{0.87,0.83,0.83} \definecolor{46816101}{rgb}{0.27,0.06,0.67} \definecolor{15206575}{rgb}{0.59,0.67,0.41} \definecolor{66457491}{rgb}{0.2,0.29,0.14} \definecolor{20900298}{rgb}{0.78,0.41,0.03} \definecolor{87300451}{rgb}{0.62,0.66,0.3} \definecolor{39061320}{rgb}{0.45,0.22,0.07} \definecolor{55504490}{rgb}{0.47,0.1,0.9} \definecolor{66792844}{rgb}{0.12,0.52,0.08} \definecolor{12830100}{rgb}{0.92,0.91,0.3} \definecolor{80294043}{rgb}{0.58,0.57,0.61} \definecolor{61044772}{rgb}{0.96,0.26,0.23} \definecolor{17642411}{rgb}{0.53,0.95,0.49} \definecolor{87708611}{rgb}{0.54,0.77,0.05} \definecolor{38052502}{rgb}{0.14,0.79,0.03} \definecolor{50209852}{rgb}{0.88,0.54,0.45} \definecolor{57663530}{rgb}{0.89,0.38,0.54} \definecolor{85994069}{rgb}{0.65,0.36,0.57} \definecolor{19161031}{rgb}{0.64,0.13,0.69} \definecolor{21635154}{rgb}{0.65,0.35,0.76} \definecolor{65731139}{rgb}{0.36,0.75,0.88} \definecolor{24916795}{rgb}{0.01,0.5,0.07} \definecolor{73933025}{rgb}{0.79,0.06,0.36} \definecolor{66644735}{rgb}{0.94,0.38,0.76} \definecolor{29509734}{rgb}{0.77,0.3,0.77} \definecolor{89214602}{rgb}{0.15,0.58,0.01} \definecolor{13510191}{rgb}{0.71,0.47,0.76} \definecolor{59405972}{rgb}{0.47,0.27,0.83} \definecolor{50015480}{rgb}{0.55,0.07,0.47} \definecolor{52425900}{rgb}{0.74,0.19,0.46} \definecolor{68793716}{rgb}{0.23,0.51,0.21} \definecolor{73569480}{rgb}{0.05,0.52,0.17} \definecolor{43297786}{rgb}{0.4,0.11,0.51} \definecolor{14154401}{rgb}{0.1,0.29,0.23} \definecolor{46476335}{rgb}{0.97,0.28,0.23} \definecolor{11383989}{rgb}{0.09,0.57,0.42} \definecolor{36782930}{rgb}{0.37,0.81,0.29} \definecolor{64260075}{rgb}{0.72,0.61,0.43} \definecolor{78065885}{rgb}{0.75,0.43,0.43} \definecolor{28768905}{rgb}{0.36,0.15,0.94} \definecolor{30323684}{rgb}{0.92,0.78,0.62} \definecolor{42927158}{rgb}{0.04,0.65,0.13} \definecolor{77314792}{rgb}{0.29,0.36,0.27} \definecolor{50962049}{rgb}{0.07,0.15,0.16} \definecolor{47087454}{rgb}{0.94,0.37,0.05} \definecolor{26539388}{rgb}{0.74,0.3,0.2} \definecolor{11376156}{rgb}{0.98,0.89,0.76} \definecolor{75840297}{rgb}{0.65,0.04,0.81} \definecolor{14701575}{rgb}{0.76,0.46,0.53} \newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{c|cccc} \hline \color{70482580}{\textrm{class}} & \color{49770034}{d=0} & \color{62598958}{1} & \color{33537890}{2} & \color{65621068}{3}\T\\ \hline \color{12177369}{\textrm{AI}} & \color{36361699}{\mathbb{Z}} & \color{46816101}{} & \color{15206575}{} & \color{66457491}{}\T\\ \color{20900298}{\textrm{CI}} & \color{87300451}{} & \color{39061320}{} & \color{55504490}{} & \color{66792844}{2\mathbb{Z}}\T\\ \color{12830100}{\textrm{AII}} & \color{80294043}{2\mathbb{Z}} & \color{61044772}{} & \color{17642411}{\mathbb{Z}_2} & \color{87708611}{\mathbb{Z}_2}\T\\ \color{38052502}{\textrm{D}} & \color{50209852}{\mathbb{Z}_2} & \color{57663530}{\mathbb{Z}_2} & \color{85994069}{\mathbb{Z}} & \color{19161031}{}\T\\ \color{21635154}{\textrm{A}} & \color{65731139}{\mathbb{Z}} & \color{24916795}{} & \color{73933025}{\mathbb{Z}} & \color{66644735}{}\T\\ \color{29509734}{\textrm{BDI}} & \color{89214602}{\mathbb{Z}_2} & \color{13510191}{\mathbb{Z}} & \color{59405972}{} & \color{50015480}{}\T\\ \color{52425900}{\textrm{AIII}} & \color{68793716}{} & \color{73569480}{\mathbb{Z}} & \color{43297786}{} & \color{14154401}{\mathbb{Z}}\T\\ \color{46476335}{\textrm{CII}} & \color{11383989}{} & \color{36782930}{2\mathbb{Z}} & \color{64260075}{} & \color{78065885}{\mathbb{Z}_2}\T\\ \color{28768905}{\textrm{C}} & \color{30323684}{} & \color{42927158}{} & \color{77314792}{2\mathbb{Z}} & \color{50962049}{}\T\\ \color{47087454}{\textrm{DIII}} & \color{26539388}{} & \color{11376156}{\mathbb{Z}_2} & \color{75840297}{\mathbb{Z}_2} & \color{14701575}{\mathbb{Z}}\\ \hline \end{array} \]

But don’t worry, we are going to learn exactly what these tables mean.

First of all, let’s review the meaning of the entries in the table. Each entry gives the topological classification of a system with a given combination of symmetries and dimensionality. In other words, it gives us the possible values that the topological invariant \(Q\) of such a system can take.

An empty entry means that the system does not have a topological phase. In other words, all gapped Hamiltonians with dimension and symmetries corresponding to an empty entry can be deformed into each other, without ever closing the bulk gap and without breaking any existing symmetry.

A \(\mathbb{Z}\) entry tells us that the topological invariant is an integer number, \(Q=0, \pm 1, \pm 2, \dots\) An example of such a system is the quantum Hall effect, for which the topological invariant is the Chern number. The value of \(Q\) specifies the number of chiral edge states and their chirality, which is opposite for opposite signs of \(Q\).

A \(2\mathbb{Z}\) entry is much like a \(\mathbb{Z}\) entry, except that the invariant may only take even numbers, \(Q=0,\pm2, \pm4, \dots\), because of some doubling of the degrees of freedom. An example is a quantum dot with spinful time-reversal symmetry, for which the topological invariant is the number of filled energy levels. It may only be an even number because of Kramers degeneracy.

A \(\mathbb{Z}_2\) entry means that there are only two distinct topological phases, with \(Q=\pm 1\) (or \(Q=0, 1\), depending on convention). An example we know is the Majorana chain, with the Pfaffian topological invariant, which distinguishes between the two phases with or without unpaired Majorana modes and the ends. Another example we know are the time-reversal invariant topological insulators in two and three dimensions.

Now that we have attached a meaning to each entry in the table, let’s try to understand the table as a whole. The first thing to do is to understand why it has ten and only ten rows.