there is something called the Altland Zirnbauer classification of topological insulators which is related to both the random matrices and Bott periodicity

https://golem.ph.utexas.edu/category/2014/07/the_tenfold_way.html

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures (Altland + Zirnbauer)

• Periodic table for topological insulators and superconductors (Kitaev)

Attempt at Discussion

As typical in QFT there is a difficult Hamiltonian (this is Hartree-Fock approximation of some electric field):

$$ \hat{H} = \int d^d x \; \Bigg( \sum_{\sigma, \tau \in \uparrow ,\downarrow} \psi^\dagger_\sigma h_{\sigma \tau}\psi_\tau + \Delta \psi^\dagger_\uparrow \psi^\dagger_\downarrow + \Delta^\ast \psi_\uparrow \psi_\downarrow \Bigg)$$

for reasons one might challenge this can be written in terms of creation an anihilation operators:

$$ \hat{H} = \sum_{\alpha \beta} \big( h_{\alpha\beta} c^\dagger_\alpha c_\beta + \frac{1}{2} \Delta_{\alpha\beta} c_\alpha^\dagger c_\beta^\dagger + \frac{1}{2} \Delta_{\alpha\beta}^\ast c_\alpha c_\beta \big) $$

The indices instead of ranging from 1 thru N, both indices $\alpha, \beta$ range over the sites of a lattice. This Hamiltonian is tell us what happens at the edges.

In layman's terms: it's too complicated! So we will approximate with random matrix ensemble. (to be continued...)

there is something called the Altland Zirnbauer classification of topological insulators which is related to both the random matrices and Bott periodicity

https://golem.ph.utexas.edu/category/2014/07/the_tenfold_way.html

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures (Altland + Zirnbauer)

• Periodic table for topological insulators and superconductors (Kitaev)

Attempt at Discussion

As typical in QFT there is a difficult Hamiltonian (this is Hartree-Fock approximation of some electric field):

$$ \hat{H} = \int d^d x \; \Bigg( \sum_{\sigma, \tau \in \uparrow ,\downarrow} \psi^\dagger_\sigma h_{\sigma \tau}\psi_\tau + \Delta \psi^\dagger_\uparrow \psi^\dagger_\downarrow + \Delta^\ast \psi_\uparrow \psi_\downarrow \Bigg)$$

for reasons one might challenge this can be written in terms of creation an anihilation operators:

$$ \hat{H} = \sum_{\alpha \beta} \big( h_{\alpha\beta} c^\dagger_\alpha c_\beta + \frac{1}{2} \Delta_{\alpha\beta} c_\alpha^\dagger c_\beta^\dagger + \frac{1}{2} \Delta_{\alpha\beta}^\ast c_\alpha c_\beta \big) $$

The indices instead of ranging from 1 thru N, both indices $\alpha, \beta$ range over the sites of a lattice. This Hamiltonian is tell us what happens at the edges.

In layman's terms: it's too complicated! So we will approximate with random matrix ensemble.

$$ \hat{H} = \frac{1}{2} ( \begin{array}{cc} \mathbf{c}^\dagger & \mathbf{c} \end{array} ) \left( \begin{array}{cc} h & \Delta \\ -\Delta^\ast & h \end{array} \right) \left( \begin{array}{c} \mathbf{c}^\dagger \\ \mathbf{c} \end{array} \right) \equiv \frac{1}{2} ( \begin{array}{cc} \mathbf{c}^\dagger & \mathbf{c} \end{array} ) \; \mathcal{H} \;\left( \begin{array}{c} \mathbf{c}^\dagger \\ \mathbf{c} \end{array} \right) $$

This is how Altland+Zirnbauer define the "Bogliubov-de Gennes Hamiltonian" $\mathcal{H}$ which acts on the space of spins at all points $\mathbf{C}^{2N} \times \mathbb{C}^2$.

Then we can examine each symmetry class (A, AIII, AI, D, C, CI, etc) and check for various types of time-reversal or spin-rotation or charge-conjugation. And he will obtain various generalized GUE/GOE/GSE etc.

(to be continued...)

K-theory

Kitaev seems to feel the Altland-Zirnbauer classification was incomplete in dimensions > 3. He is also the first to mention Bott periodicity.

Here K-theory is read literally as "linear algebra over matrices with elements that are functions of the base space". Classical random matrix theory is a 0-dimensional case where the base space is $\{ pt \} $. In higher dimensions these become related to the Clifford algebras.

(More to say as I learn.)

there is something called the Altland Zirnbauer classification of topological insulators which is related to both the random matrices and Bott periodicity

https://golem.ph.utexas.edu/category/2014/07/the_tenfold_way.html

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures (Altland + Zirnbauer)

• Periodic table for topological insulators and superconductors (Kitaev)

there is something called the Altland Zirnbauer classification of topological insulators which is related to both the random matrices and Bott periodicity

https://golem.ph.utexas.edu/category/2014/07/the_tenfold_way.html

• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures (Altland + Zirnbauer)

• Periodic table for topological insulators and superconductors (Kitaev)

Attempt at Discussion

As typical in QFT there is a difficult Hamiltonian (this is Hartree-Fock approximation of some electric field):

$$ \hat{H} = \int d^d x \; \Bigg( \sum_{\sigma, \tau \in \uparrow ,\downarrow} \psi^\dagger_\sigma h_{\sigma \tau}\psi_\tau + \Delta \psi^\dagger_\uparrow \psi^\dagger_\downarrow + \Delta^\ast \psi_\uparrow \psi_\downarrow \Bigg)$$

for reasons one might challenge this can be written in terms of creation an anihilation operators:

$$ \hat{H} = \sum_{\alpha \beta} \big( h_{\alpha\beta} c^\dagger_\alpha c_\beta + \frac{1}{2} \Delta_{\alpha\beta} c_\alpha^\dagger c_\beta^\dagger + \frac{1}{2} \Delta_{\alpha\beta}^\ast c_\alpha c_\beta \big) $$

The indices instead of ranging from 1 thru N, both indices $\alpha, \beta$ range over the sites of a lattice. This Hamiltonian is tell us what happens at the edges.

In layman's terms: it's too complicated! So we will approximate with random matrix ensemble. (to be continued...)