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# Welcome to the documentation for TBTK!

TBTK is an open-source C++ framework for modeling and solving problems formulated using the language of second quantization. It can be used to set up general models with little effort and provides a variety of native solution methods.

To get started, see the installation instructions, the manual, and the tutorials. Also, see the blog posts and other resources collected on second-tech.com.

Download TBTK from GitHub. See the installation instructions to make sure you checkout the right version before installation.

## Core strengths

• The speed of a low-level language with the syntax of a high-level language.
• Results in readable code that puts emphasis on the physics.
• Allows for a wide variety of models and solution methods to be combined in different ways.
• Focus on your own task, while still benefiting from the work of others.
• Gives method developers complete freedom to optimize their solvers without having to worry about model-specific details.
• A versioning system that ensures that results are reproducible forever.

# Example

## Problem formulation

Consider a two-dimensional substrate of size 51x51 described by the Hamiltonian

$$H_{S} = U_S\sum_{\mathbf{i}\sigma}c_{\mathbf{i}\sigma}^{\dagger}c_{\mathbf{i}\sigma} - t\sum_{\langle\mathbf{i}\mathbf{j}\rangle\sigma}c_{\mathbf{i}\sigma}^{\dagger}c_{\mathbf{j}\sigma}$$.

Here $$\mathbf{i}$$ is a two-dimensional index, $$\sigma$$ is a spin index, and $$\langle\mathbf{i}\mathbf{j}\rangle$$ denotes summation over nearest-neighbors. Further, consider a magnetic impurity on top of the substrate described by the Hamiltonian

$$H_{Imp} = (U_{Imp} - J)d_{\uparrow}^{\dagger}d_{\uparrow} + (U_{Imp} + J)d_{\downarrow}^{\dagger}d_{\downarrow}.$$

Finally, the impurity connects to the site (25, 25) in the substrate through the term

$$H_{Int} = \delta\sum_{\sigma}c_{(25,25)\sigma}^{\dagger}d_{\sigma} + H.c.$$

The total Hamiltonian is

$$H = H_{S} + H_{Imp} + H_{Int}$$.

Question: What is the spin-polarized LDOS in the substrate as a function of $$U_S, U_{Imp}, t, J$$, and $$\delta$$?

## Numerical solution

//Parameters.
const int SIZE_X = 51;
const int SIZE_Y = 51;
double U_S = 1;
double U_Imp = 1;
double t = 1;
double J = 1;
double delta = 1;
//Create model.
Model model;
//Setup substrate.
for(int x = 0; x < SIZE_X; x++){
for(int y = 0; y < SIZE_Y; y++){
for(int spin = 0; spin < 2; spin++){
model << HoppingAmplitude(
U_S,
{0, x, y, spin},
{0, x, y, spin}
);
if(x+1 < SIZE_X){
model << HoppingAmplitude(
-t,
{0, x+1, y, spin},
{0, x, y, spin}
) + HC;
}
if(y+1 < SIZE_Y){
model << HoppingAmplitude(
-t,
{0, x, y+1, spin},
{0, x, y, spin}
) + HC;
}
}
}
}
for(int spin = 0; spin < 2; spin++){
//Setup impurity.
model << HoppingAmplitude( U_Imp, {1, spin}, {1, spin});
model << HoppingAmplitude(-J*(1-2*spin), {1, spin}, {1, spin});
//Add coupling between the substrate and impurity.
model << HoppingAmplitude(
delta,
{0, SIZE_X/2, SIZE_Y/2, spin},
{1, spin}
) + HC;
}
//Construct model.
model.construct();
//Setup Solver
Solver::Diagonalizer solver;
solver.setModel(model);
solver.run();
//Extract the spin-polarized LDOS in the substrate.
PropertyExtractor::Diagonalizer propertyExtractor(solver);
propertyExtractor.setEnergyWindow(-1, 1, 1000);
Property::SpinPolarizedLDOS spinPolarizedLDOS
= propertyExtractor.calculateSpinPolarizedLDOS(
{{0, _a_, _a_, IDX_SPIN}}
);