2D square lattice

Hamiltonian

\(H = -\mu\sum_{\mathbf{i}}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}} + t\sum_{\langle \mathbf{i}\mathbf{j}\rangle}c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}\)

Code

#include "TBTK/Property/DOS.h"
#include "TBTK/PropertyExtractor/Diagonalizer.h"
#include "TBTK/Smooth.h"
#include "TBTK/Solver/Diagonalizer.h"
#include "TBTK/TBTK.h"
#include "TBTK/Visualization/MatPlotLib/Plotter.h"
 
#include <complex>
 
using namespace std;
using namespace TBTK;
using namespace Visualization::MatPlotLib;
 
complex<double> i(0, 1);
 
int main(){
    //Initialize TBTK.
    Initialize();
 
    //Parameters.
    const unsigned int SIZE_X = 30;
    const unsigned int SIZE_Y = 30;
    const double t = -1;
    const double mu = 0;
 
    //Set up the Model.
    Model model;
    for(unsigned int x = 0; x < SIZE_X; x++){
        for(unsigned int y = 0; y < SIZE_Y; y++){
            if(x+1 < SIZE_X){
                model << HoppingAmplitude(
                    t,
                    {x+1, y},
                    {x, y}
                ) + HC;
            }
            if(y+1 < SIZE_Y){
                model << HoppingAmplitude(
                    t,
                    {x, y+1},
                    {x, y}
                ) + HC;
            }
        }
    }
    model.construct();
    model.setChemicalPotential(mu);
 
    //Set up the Solver.
    Solver::Diagonalizer solver;
    solver.setModel(model);
    solver.run();
 
    //Set up the PropertyExtractor.
    const double LOWER_BOUND = -5;
    const double UPPER_BOUND = 5;
    const unsigned int RESOLUTION = 1000;
    PropertyExtractor::Diagonalizer propertyExtractor(solver);
    propertyExtractor.setEnergyWindow(
        LOWER_BOUND,
        UPPER_BOUND,
        RESOLUTION
    );
 
    //Calculate the density of states (DOS).
    Property::DOS dos = propertyExtractor.calculateDOS();
 
    //Smooth the DOS.
    const double SMOOTHING_SIGMA = 0.1;
    const unsigned int SMOOTHING_WINDOW = 101;
    dos = Smooth::gaussian(dos, SMOOTHING_SIGMA, SMOOTHING_WINDOW);
 
    //Plot the DOS.
    Plotter plotter;
    plotter.plot(dos);
    plotter.save("figures/DOS.png");
 
    //Calculate the wave functions.
    Property::WaveFunctions waveFunctions
        = propertyExtractor.calculateWaveFunctions(
            {{_a_, _a_}},
            {_a_}
        );
 
    //Plot the wave function for state 37.
    plotter.clear();
    plotter.plot({_a_, _a_}, 37, waveFunctions);
    plotter.save("figures/WaveFunction.png");
}

Output