Magnetism

Hamiltonian

\(H = -\mu\sum_{\mathbf{i}\sigma}c_{\mathbf{i}\sigma}^{\dagger}c_{\mathbf{i}\sigma} + t\sum_{\langle\mathbf{i}\mathbf{j}\rangle\sigma}c_{\mathbf{i}\sigma}^{\dagger}c_{\mathbf{j}\sigma} + J\sum_{\mathbf{i}\sigma}\left(\sigma_z\right)_{\sigma\sigma}c_{\mathbf{i}\sigma}^{\dagger}c_{\mathbf{i}\sigma}\)

Code

#include "TBTK/Property/DOS.h"
#include "TBTK/PropertyExtractor/Diagonalizer.h"
#include "TBTK/Range.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);

//Callback that allows for the Zeeman term (J) to be updated after the Model
//has been set up.
class JCallback : public HoppingAmplitude::AmplitudeCallback{
public:
    //Function that returns the HoppingAmplitude value for the given
    //Indices. The to- and from-Indices are identical in this example.
    complex<double> getHoppingAmplitude(
        const Index &to,
        const Index &from
    ) const{
        Subindex spin = from[2];
        return J*(1 - 2*spin);
    }

    //Set the value for J.
    void setJ(double J){
        this->J = J;
    }
private:
    double J;
};

int main(){
    //Initialize TBTK.
    Initialize();

    //Parameters.
    const unsigned int SIZE_X = 21;
    const unsigned int SIZE_Y = 21;
    const double t = -1;
    const double mu = 0;

    //Create a callback that returns the Zeeman term and that will be used
    //as input to the Model.
    JCallback jCallback;

    //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 spin = 0; spin < 2; spin++){
                model << HoppingAmplitude(
                    jCallback,
                    {x, y, spin},
                    {x, y, spin}
                );

                if(x+1 < SIZE_X){
                    model << HoppingAmplitude(
                        t,
                        {x+1, y, spin},
                        {x, y, spin}
                    ) + HC;
                }
                if(y+1 < SIZE_Y){
                    model << HoppingAmplitude(
                        t,
                        {x, y+1, spin},
                        {x, y, spin}
                    ) + HC;
                }
            }
        }
    }
    model.construct();
    model.setChemicalPotential(mu);

    //Number of iterations.
    const unsigned int NUM_ITERATIONS = 10;

    //Array to store the z-component of the magnetization in.
    Array<double> magnetizationZ({NUM_ITERATIONS});

    //Iterate over 100 values for J.
    Range j(0, 5, NUM_ITERATIONS);
    for(unsigned int n = 0; n < NUM_ITERATIONS; n++){
        //Update the callback with the current value of J.
        jCallback.setJ(j[n]);

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

        //Set up the PropertyExtractor.
        const double LOWER_BOUND = -8;
        const double UPPER_BOUND = 8;
        const unsigned int RESOLUTION = 1000;
        PropertyExtractor::Diagonalizer propertyExtractor(solver);
        propertyExtractor.setEnergyWindow(
            LOWER_BOUND,
            UPPER_BOUND,
            RESOLUTION
        );

        //Calculate the Magnetization.
        Property::Magnetization magnetization
            = propertyExtractor.calculateMagnetization(
                {{SIZE_X/2, SIZE_Y/2, IDX_SPIN}}
            );

        //Project the magnetization on the z-axis and store the value
        //in magnetizationZ.
        magnetizationZ[{n}] = Vector3d::dotProduct(
            magnetization({
                SIZE_X/2,
                SIZE_Y/2,
                IDX_SPIN
            }).getSpinVector(),
            {0, 0, 1}
        );

        //If J has its middle value, Calculate and plot the density of
        //states (DOS).
        if(n == NUM_ITERATIONS/2){
            //Calculate the DOS.
            Property::DOS dos = propertyExtractor.calculateDOS();

            //Calculate the DOS for spin up and down separately by
            //summing the LDOS over all lattice sites.
            Property::LDOS ldos = propertyExtractor.calculateLDOS(
                {{IDX_SUM_ALL, IDX_SUM_ALL, _a_}}
            );

            //Smooth the DOS.
            const double SMOOTHING_SIGMA = 0.2;
            const unsigned int SMOOTHING_WINDOW = 101;
            dos = Smooth::gaussian(
                dos,
                SMOOTHING_SIGMA,
                SMOOTHING_WINDOW
            );

            //Smooth the LDOS.
            ldos = Smooth::gaussian(
                ldos,
                SMOOTHING_SIGMA,
                SMOOTHING_WINDOW
            );

            //Plot the DOS.
            Plotter plotter;
            plotter.setTitle("DOS");
            plotter.setLabelX("Energy");
            plotter.setLabelY("DOS");
            plotter.plot(dos, {{"label", "Total"}});

            //Plot the LDOS.
            plotter.plot(
                {IDX_SUM_ALL, IDX_SUM_ALL, 0},
                ldos,
                {
                    {"color", "red"},
                    {"linestyle", "--"},
                    {"label", "Spin up"}
                }
            );
            plotter.plot(
                {IDX_SUM_ALL, IDX_SUM_ALL, 1},
                ldos,
                {
                    {"color", "blue"},
                    {"linestyle", "--"},
                    {"label", "Spin down"}
                }
            );

            //Save the plot.
            plotter.save("figures/DOS.png");
        }
    }

    //Plot the Magnetization along the z-axis.
    Plotter plotter;
    plotter.setTitle("Magnetization along the  z-axis");
    plotter.setLabelX("J");
    plotter.setLabelY("Magnetization");
    plotter.setAxes({
        {0, {0, 5}}
    });
    plotter.plot(magnetizationZ);
    plotter.save("figures/Magnetization.png");
}

Output