Matrix containing information about a spin. More...

#include <SpinMatrix.h>

Inheritance diagram for TBTK::SpinMatrix:

Public Member Functions
	SpinMatrix ()

	SpinMatrix (std::complex< double > value)

	~SpinMatrix ()

SpinMatrix &	operator= (std::complex< double > value)

SpinMatrix &	operator+= (const SpinMatrix &spinMatrix)

SpinMatrix &	operator-= (const SpinMatrix &spinMatrix)

double	getDensity () const

Vector3d	getSpinVector () const

std::string	toString () const

Public Member Functions inherited from TBTK::Matrix< std::complex< double >, 2, 2 >
	Matrix ()

	Matrix (const Matrix< std::complex< double >, ROWS, COLS > &matrix)

	Matrix (Matrix< std::complex< double >, ROWS, COLS > &&matrix)

	~Matrix ()

Matrix< std::complex< double >, ROWS, COLS > &	operator= (const Matrix< std::complex< double >, ROWS, COLS > &rhs)

Matrix< std::complex< double >, ROWS, COLS > &	operator= (Matrix< std::complex< double >, ROWS, COLS > &&rhs)

const std::complex< double > &	at (unsigned int row, unsigned int col) const

std::complex< double > &	at (unsigned int row, unsigned int col)

unsigned int	getNumRows () const

unsigned int	getNumCols () const

Friends
std::ostream &	operator<< (std::ostream &stream, const SpinMatrix &spinMatrix)

Detailed Description

Matrix containing information about a spin.

The SpinMatrix is a Matrix with DataType std::complex<double>. If the matrix has the form

\[ \frac{1}{2}\left[\begin{array}{cc} \rho + S_z & S_x + iS_y\\ S_x - iS_y & \rho - S_z \end{array}\right] \]

Then the SpinMatrix extends the Matrix with additional functions for extracting the density and spin-vector. Note that the SpinMatrix in principle can contain arbitrary complex entries in each position, but it is then undefined behavior to use any of the SpinMatrix functions. For such matrices, only those methods derived from Matrix<std::complex> should be used.

Note that the convention for the spin matrix is equal to \(\rho + S_x\sigma_x - S_y\sigma_y + S_z\sigma_z\) rather than \(\rho + S_x\sigma_x + S_y\sigma_y + S_z\sigma_z\). The convention follows from the SpinMatrix being used to store values of the form

\[ \left[\begin{array}{cc} \langle c_{\uparrow}^{\dagger}c_{\uparrow}\rangle & \langle c_{\uparrow}^{\dagger}c_{\downarrow}\rangle\\ \langle c_{\downarrow}^{\dagger}c_{\uparrow}\rangle & \langle c_{\downarrow}^{\dagger}c_{\downarrow}\rangle\\ \end{array}\right] \]

and these expectation values are related to \(\rho, S_x, S_y\), and \(S_z\) through

\begin{eqnarray*} \rho &=& \langle c_{\uparrow}^{\dagger}c_{\uparrow}\rangle + \langle c_{\downarrow}^{\dagger}c_{\downarrow}\rangle\\ S_x &=& \langle c_{\uparrow}^{\dagger}c_{\downarrow}\rangle + \langle c_{\downarrow}^{\dagger}c_{\uparrow}\rangle\\ S_y &=& \textrm{Im}\left( \langle c_{\uparrow}^{\dagger}c_{\downarrow}\rangle - \langle c_{\downarrow}^{\dagger}c_{\uparrow}\rangle\right)\\ S_z &=& \langle c_{\uparrow}^{\dagger}c_{\uparrow}\rangle - \langle c_{\downarrow}^{\dagger}c_{\downarrow}\rangle\\ \end{eqnarray*}

Example

#include "TBTK/SpinMatrix.h"
#include "TBTK/Streams.h"
#include "TBTK/TBTK.h"
 
#include <complex>
 
using namespace std;
using namespace TBTK;
 
int main(){
    Initialize();
 
    complex<double> i(0, 1);
    double rho = 1;
    double S_x = 0.1;
    double S_y = 0.2;
    double S_z = 0.3;
    SpinMatrix spinMatrix;
    spinMatrix.at(0, 0) = (rho + S_z)/2.;
    spinMatrix.at(1, 0) = (S_x - i*S_y)/2.;
    spinMatrix.at(0, 1) = (S_x + i*S_y)/2.;
    spinMatrix.at(1, 1) = (rho - S_z)/2.;
 
    Streams::out << spinMatrix << "\n";
    Streams::out << "Density: " << spinMatrix.getDensity() << "\n";
    Streams::out << "Spin vector: " << spinMatrix.getSpinVector() << "\n";
}

Output

SpinMatrix
    (0.65,0)    (0.05,0.1)
    (0.05,-0.1) (0.35,0)
Density: 1
Spin vector: (0.1, 0.2, 0.3)

Constructor & Destructor Documentation

◆ SpinMatrix() [1/2]

TBTK::SpinMatrix::SpinMatrix ( )

Constructor.

◆ SpinMatrix() [2/2]

TBTK::SpinMatrix::SpinMatrix ( std::complex< double > value )

Constructs a SpinMatrix with all elements set to the same value.

Parameters

value The value to assign each entry.

◆ ~SpinMatrix()

TBTK::SpinMatrix::~SpinMatrix ( )

Destructor.

Member Function Documentation

◆ getDensity()

double TBTK::SpinMatrix::getDensity ( ) const

Get desnity.

◆ getSpinVector()

Vector3d TBTK::SpinMatrix::getSpinVector ( ) const

Get spin direction.

◆ operator+=()

SpinMatrix& TBTK::SpinMatrix::operator+= ( const SpinMatrix & spinMatrix )

Addition equality operator.

◆ operator-=()

SpinMatrix& TBTK::SpinMatrix::operator-= ( const SpinMatrix & spinMatrix )

Addition equality operator.

◆ operator=()

SpinMatrix& TBTK::SpinMatrix::operator= ( std::complex< double > value )

Assignment operator for assigning a single value to every element of the matrix.

◆ toString()

std::string TBTK::SpinMatrix::toString ( ) const

Get string representation of the SpinMatrix.

Returns: A string representation of the SpinMatrix.

Friends And Related Function Documentation

◆ operator<<

std::ostream& operator<<	(	std::ostream &	stream,
		const SpinMatrix &	spinMatrix
	)

friend

Writes the SpinMatrix toString()-representation to a stream.