# Charles Kane, Quantized Nonlinear Conductance in Ballistic Metals 

:::{.remark}
Applications of Chern-Simons to condensed matter physics.
Intro: relation to quantized Hall effect, and a related quantized response in 1D: the Landauer formula.
How generalize: the Euler characteristic of the Fermi sea.
:::

:::{.remark}
Integer quantized Hall effect: can confine electrons to a 2D plane.
Standard experiment: measure conductance as a function of the magnetic field.
Remarkably, nearly a sum of step functions!
Measures fundamental constant $h/e^2$ extremely accurately, NIST declared it an *exact* number which in turn defines the Ohm and kilogram.
Low energy produces a Chern-Simons TFT.
:::

:::{.remark}
Topological band theory: a mean field theory that reduces QM particles to understanding single particles.
Band structure: collection of energy eigenvalues/eigenvectors parametrized by momenta ($S^1$ or more generally a torus due to periodicity).
If occupied/empty states are separated by a gap, the $N$ occupied bands forum a $\U_n$ bundle over $T^d$ a torus.
Problem: classify vector bundles over $T^d$.
In dimension 2, the first Chern class $c_1$ yields a number: $n = {1\over 2\pi }\int_{T^2} \Tr(F), F = da + A \wedge A$ for $A = i\inner{u_i}{du_j}$ the Berry connection.
:::

:::{.remark}
In metals, electron states are occupied for $E<E_F$ a constant, so define Fermi sea as the region in momentum space bounded by the associated Fermi surface.
Why topology shows up: integrate the connection associated to the above Chern class to get something gauge invariant.
This is something that can be measured in a lab!
Chern numbers measured how twisted the vector bundle is.
:::

:::{.remark}
Seemingly tough problem (to me): put a sphere inside of a regular polyhedron.
Puncture "symmetrically" corresponding to opposite faces, add a tube passing through the face, and then glue the resulting faces together.
What surface do you get?
Shows that Fermi surfaces can have nontrivial genus.
:::

:::{.remark}
For 1D: see Landauer conductance.
With assumptions (no backscattering of electrons, no reflectance), can get exact model 
\[
I = {1\over 2\pi } \int \mathrm{d}k \, ev_k = {e\over h}\int \mathrm{d}E = \qty{e^2\over h}V
.\]
What is the homeomorphism type of the 1D Fermi sea?
Use quantized Landauer conductance experiments to probe $\size \pi_0$.
Idea: $\chi_F$ carries more information than $\chi_{\bd F}$ for $F$ the Fermi sea and $\bd F$ its boundary surface.
For $F$, there is a natural Morse functional given by measuring electron energy.
Passing through critical points is like a phase transition?
:::

:::{.remark}
How to explain the quantization in the Landauer formula: consider dimension $D=1$.
Apply a voltage impulse to move electrons:

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/999_Chern/figures/2021-11-17_09-42.xoj -->

![](figures/2021-11-17_11-42-54.png)

See chiral anomaly.
Use Morse theory to compute $\chi$.

:::

:::{.remark}
See conformal field theories, e.g. $1+1$ boson for Fermi liquids?
Where else might $\chi_F$ show up?
Generalizing chiral anomalies?
A universal way to characterize quantum entanglement?
Generalizations to higher dimensions or non-Fermi liquids?
:::