# The “basic hamiltonian” of topological systems

I am currently studying topological insulators and repeatedly found the claim (e.g. here), that the "basic hamiltonian" of a topological system in $$d$$ spatial dimensions can be written using the elementary representation matrices of a Clifford algebra $$H(\vec{k}) = \vec{h}(\vec{k}) \cdot \vec{\gamma} = \sum_{i=0}^d h_i(\vec{k}) \gamma_i$$ where the $$\gamma_i$$ are generators of a Clifford algebra ($$\left\{ \gamma_i, \gamma_j \right\} = 2 \delta_{ij}$$).

For $$d=0$$ I understand that the $$\gamma_i$$ represent the presence / absence of time reversal symmetry $$T$$ and a particle number symmetry $$Q$$ and one sets $$\gamma_1 = T$$ and $$\gamma_2 = QT$$ and then wants to find another Clifford generator $$\gamma_0 = \tilde{H}$$, where $$\tilde{\cdot}$$ denotes spectral flattening. (See Kitaevs Paper on the details)

Now my question is: What happens for $$d > 0$$? Where do I get the additional $$d-1$$ generators from? And as a bonus question: What happens to $$\gamma_2 = QT$$ in the zero-dimensional case?

I guess that somehow the extra clifford generators come from the additional translational symmetries in $$d$$ space dimensions, but these are symmetries commuting with the hamiltonian instead of anti-commuting.

Edit: If I go to higher spatial dimensions, I get another symmetry that commutes with the hamiltonian, the translation $$P_i$$ by one unit cell in each spatial direction. If I set a $$\gamma_{i+1} = T P_i$$ I get another term that anti-commutes with the hamiltonian. But how do I also get it to anti-commute with $$T$$? Or does it already anti-commute with $$T$$?

Edit 2: Just figured out, how to extend a Clifford algebra, if I already have an even number of generatrs and a new operator that commutes with all of them and squares to 1.

Assume $$\{\gamma_i, \gamma_j\} = 2 \delta_{ij}$$ for $$i,j=1...2n$$. And we also have an operator $$g$$ s.t. $$[g, \gamma_i] = 0$$ for $$i=1...2n$$ and $$\gamma^2$$. Then we can set $$\gamma_{2n+1} = g \gamma_1 \cdots \gamma_{2n}$$ and easily verify that

$$\{\gamma_{2n+1}, \gamma_i\} = g \gamma_1 \cdots \gamma_{2n} \gamma_i + \gamma_i g \gamma_1 \cdots \gamma_{2n}\\ = g \gamma_1 \cdots \gamma_{2n} \gamma_i - g \gamma_1 \cdots \gamma_{2n} \gamma_i = 0$$

and $$\gamma_{2n+1}^2 = \pm 1$$ (might need to throw a factor of $$i$$ in to make this a +, but that doesn't change the general idea). But this only works if the Clifford algebra I start with has an even number of generators.