KuboFormulas - crowlogic/arb4j GitHub Wiki

The Kubo formulas are a fundamental part of the theoretical framework of linear response theory. They enable us to compute transport coefficients such as electrical conductivity (σ), thermal conductivity (κ), and what we formerly referred to as 'susceptibility', now renamed as 'propensity' (χ). The Kubo formulas serve as a bridge, linking these macroscopic transport coefficients with microscopic quantum mechanical quantities.

Here is a properly formatted form of the Kubo formula:

$$\sigma(\omega) = \frac{1}{\hbar}\int_0^{\infty} \langle [j(t), j(0)] \rangle e^{i\omega t} dt$$

In this equation, $\sigma(\omega)$ is the frequency-dependent conductivity, $\omega$ is the frequency of the applied field, $j(t)$ is the current density operator in the Heisenberg picture, and $\langle[j(t), j(0)]\rangle$ denotes the quantum mechanical expectation value of the commutator of $j(t)$ with itself at different times.

For calculating specific transport coefficients like electrical conductivity (σ), the current operator j would be substituted with the appropriate quantum mechanical operator for the system under examination.

Bear in mind that the Kubo formulas are derived on the basis of linear response theory, which posits that the system's response is proportional to the intensity of the adjustment. The KMS (Kubo-Martin-Schwinger) theorem, which states that for a system in thermal equilibrium, the correlation functions satisfy the KMS condition, plays a crucial role in deriving the Kubo formulas.

These formulas offer us an invaluable tool in understanding a range of phenomena in condensed matter physics and beyond.