Superconducting vortex (Caroli-de Gennes-Matricon)

Hamiltonian

\(H = -\mu\sum_{\mathbf{i}}\left(c_{\mathbf{i}\uparrow}^{\dagger}c_{\mathbf{i}\uparrow} - c_{\mathbf{i}\downarrow}c_{\mathbf{i}\downarrow}^{\dagger}\right) + t\sum_{\langle\mathbf{i}\mathbf{j}\rangle}\left(c_{\mathbf{i}\uparrow}^{\dagger}c_{\mathbf{j}\uparrow} - c_{\mathbf{i}\downarrow}c_{\mathbf{j}\downarrow}^{\dagger}\right) + \sum_{\mathbf{i}}\left(\Delta(\mathbf{i}) c_{\mathbf{i}\uparrow}^{\dagger}c_{\mathbf{i}\downarrow}^{\dagger} + \Delta^{*}(\mathbf{i})c_{\mathbf{i}\downarrow}c_{\mathbf{i}\uparrow}\right)\)

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 = 31;
    const unsigned int SIZE_Y = 31;
    const double t = -1;
    const double mu = -2;
    const double Delta = 0.5;

    //Set up the Model.
    Model model;
    for(unsigned int x = 0; x < SIZE_X; x++){
        for(unsigned int y = 0; y < SIZE_Y; y++){
            for(unsigned int ph = 0; ph < 2; ph++){
                model << HoppingAmplitude(
                    -mu*(1. - 2*ph),
                    {x, y, ph},
                    {x, y, ph}
                );

                if(x+1 < SIZE_X){
                    model << HoppingAmplitude(
                        t*(1. - 2*ph),
                        {x+1, y, ph},
                        {x, y, ph}
                    ) + HC;
                }
                if(y+1 < SIZE_Y){
                    model << HoppingAmplitude(
                        t*(1. - 2*ph),
                        {x, y+1, ph},
                        {x, y, ph}
                    ) + HC;
                }
            }

            double X = x - SIZE_X/2.;
            double Y = y - SIZE_Y/2.;
            double R = sqrt(X*X + Y*Y);
            model << HoppingAmplitude(
                Delta*exp(i*atan2(Y, X))*tanh(R),
                {x, y, 1},
                {x, y, 0}
            ) + HC;
        }
    }
    model.construct();

    //Set up the Solver.
    Solver::Diagonalizer solver;
    solver.setModel(model);
    solver.run();

    //Set up the PropertyExtractor.
    const double LOWER_BOUND = -1.5;
    const double UPPER_BOUND = 1.5;
    const unsigned int RESOLUTION = 1000;
    PropertyExtractor::Diagonalizer propertyExtractor(solver);
    propertyExtractor.setEnergyWindow(
        LOWER_BOUND,
        UPPER_BOUND,
        RESOLUTION
    );

    //Calculate the density of states (DOS).
    Property::LDOS ldos = propertyExtractor.calculateLDOS({
        {_a_, SIZE_Y/2, IDX_SUM_ALL}
    });

    //Smooth the LDOS.
    const double SMOOTHING_SIGMA = 0.01;
    const unsigned int SMOOTHING_WINDOW = 201;
    ldos = Smooth::gaussian(ldos, SMOOTHING_SIGMA, SMOOTHING_WINDOW);

    //Plot the DOS.
    Plotter plotter;
    plotter.setNumContours(100);
    plotter.plot(
        {_a_, SIZE_Y/2, IDX_SUM_ALL},
        ldos
    );
    plotter.save("figures/LDOS.png");
}

Output