This small library helps with Coulomb interactions that don't follow a simple (as in computationally simple/fast) functional form, but are expensive to calculate. Currently, the only interaction profile that's implemented is the dual-gated Coulomb interaction [cf. Phys. Rev. B 86, 115447 (2012)], but with an $r=0$ regularization a la Ohno. Other profiles could be implemented trivially in the future.

The long-ranged part of the interaction is given as $V_\mathrm{long}(r)$:

where $\xi$ is the gate distance to both gates, s.t. the two gates have a distance $2\xi$ with the 2D sample in between. $K_0$ is a Bessel function of the second kind, and $V_0$ is an energy scale.

The short-ranged part $V_\mathrm{short}(r)$ must follow

where $U$ is the on-site interaction strength (Hubbard-$U$) and $a$ the "Ohno-parameter".

We make a smooth version of both profiles (i.e. a long- and short-ranged screened Coulomb interaction) by plugging the one into the other (one can check that the limits work out...). We arrive at

with the following constraints on $V_0$ and $a$:

The fine-structure constant $\alpha$ and the dielectric constant $\epsilon$ determine the interaction strength s.t. it follwos $\alpha/\epsilon r$ in intermediate distances. Note that $\epsilon$ should not carry any units, and $\alpha = 14.40\,\mathrm{eV}\text{Å}$ in SI units (inserted some $\hbar c$ to get those units).

see test/use.c and test/Makefile for how to use this library. It's fast as long as you have one handle (roughly as fast as evaluating exp(..)/sqrt(..)). Creation of the handle is what's slow.

If you don't like to use C++ and its library implementation of the Bessel function $K_0(x)$, you can use the DEFINE USE_BARE_C_BESSEL, i.e., put DEFINES += -DUSE_BARE_C_BESSEL in your Makefile.local/Makefile.inc.