Solves a Model using the Chebyshev method. More...

#include <ChebyshevExpander.h>

Inheritance diagram for TBTK::Solver::ChebyshevExpander:

Public Types
enum	Type { Advanced, Retarded, Principal, NonPrincipal }

Public Member Functions
	ChebyshevExpander ()

virtual	~ChebyshevExpander ()

void	setScaleFactor (double scaleFactor)

double	getScaleFactor ()

void	setNumCoefficients (int numCoefficients)

int	getNumCoefficients () const

void	setBroadening (double broadening)

double	getBroadening () const

void	setEnergyResolution (int energyResolution)

int	getEnergyResolution () const

void	setLowerBound (double lowerBound)

double	getLowerBound () const

void	setUpperBound (double upperBound)

double	getUpperBound () const

void	setCalculateCoefficientsOnGPU (bool calculateCoefficientsOnGPU)

bool	getCalculateCoefficientsOnGPU () const

void	setGenerateGreensFunctionsOnGPU (bool generateGreensFunctionsOnGPU)

bool	getGenerateGreensFunctionsOnGPU () const

void	setUseLookupTable (bool useLookupTable)

bool	getUseLookupTable () const

std::vector< std::vector< std::complex< double > > >	calculateCoefficients (std::vector< Index > &to, Index from)

std::vector< std::complex< double > >	calculateCoefficients (Index to, Index from)

std::vector< std::complex< double > >	generateGreensFunction (const std::vector< std::complex< double >> &coefficients, Type type=Type::Retarded)

Public Member Functions inherited from TBTK::Solver::Solver
	Solver ()

virtual	~Solver ()

virtual void	setModel (Model &model)

Model &	getModel ()

const Model &	getModel () const

Public Member Functions inherited from TBTK::Communicator
	Communicator (bool verbose)

void	setVerbose (bool verbose)

bool	getVerbose () const

Additional Inherited Members
Static Public Member Functions inherited from TBTK::Communicator
static void	setGlobalVerbose (bool globalVerbose)

static bool	getGlobalVerbose ()

Detailed Description

Solves a Model using the Chebyshev method.

The ChebyshevExpander can be used to calculate Green's function for a given Model using an expansion of the form

\( G_{\mathbf{i}\mathbf{j}}(E) = \frac{1}{\sqrt{s^2 - E^2}} \sum_{m=0}^{\infty} \frac{b_{\mathbf{i}\mathbf{j}}^{(m)}}{1 + \delta_{0m}} F(m\textrm{acos}(E/s)), \)

where \(F(x)\) is one of the functions \(\cos(x)\), \(\sin(x)\), \(e^{ix}\), and \(e^{-ix}\). The implementation is based on PhysRevLett.105.167006. The ChebyshevExpander can be run on CPU, GPU, or a mixture of both. The calculation of Chebyshev coefficients scales as \(O(n)\) with each of the following: dimension of the Hilbert space and number of Chebyshev coefficients. The generation of Green's functions scales as \(O(n)\) with the following: Number of coefficients, energy resolution, and the number of Green's functions.

Use the PropertyExtractor::ChebyshevExpander to calculate Properties.

Example

#include "TBTK/Model.h"
#include "TBTK/PropertyExtractor/ChebyshevExpander.h"
#include "TBTK/Solver/ChebyshevExpander.h"
#include "TBTK/Streams.h"
#include "TBTK/TBTK.h"
 
using namespace TBTK;
 
int main(){
    Initialize();
 
    Model model;
    for(int x = 0; x < 10; x++){
        for(int y = 0; y < 10; y++){
            model << HoppingAmplitude(1, {x+1, y}, {x, y}) + HC;
            model << HoppingAmplitude(1, {x, y+1}, {x, y}) + HC;
        }
    }
    model.construct();
 
    Solver::ChebyshevExpander solver;
    solver.setModel(model);
    solver.setScaleFactor(10);
    solver.setCalculateCoefficientsOnGPU(false);
    solver.setGenerateGreensFunctionsOnGPU(false);
    solver.setUseLookupTable(true);
    solver.setNumCoefficients(10);
 
    const double LOWER_BOUND = -1;
    const double UPPER_BOUND = 1;
    const int RESOLUTION = 10;
    PropertyExtractor::ChebyshevExpander propertyExtractor(solver);
    propertyExtractor.setEnergyWindow(
        LOWER_BOUND,
        UPPER_BOUND,
        RESOLUTION
    );
    Property::GreensFunction greensFunction
        = propertyExtractor.calculateGreensFunction({
            {{5, 5}, {5, 5}},
            {{2, 4}, {2, 4}}
        });
    for(unsigned int n = 0; n < greensFunction.getResolution(); n++)
        Streams::out << greensFunction({{5, 5}, {5, 5}}, n) << "\t";
    Streams::out << "\n";
    for(unsigned int n = 0; n < greensFunction.getResolution(); n++)
        Streams::out << greensFunction({{2, 4}, {2, 4}}, n) << "\t";
    Streams::out << "\n";
}

Output

(-0.0885539,-0.358951) (-0.0715352,-0.364819) (-0.0540579,-0.369434) (-0.0362332,-0.372757) (-0.0181752,-0.374762) (5.57054e-17,-0.375432) (0.0181752,-0.374762) (0.0362332,-0.372757) (0.0540579,-0.369434) (0.0715352,-0.364819)

(-0.0885604,-0.358958) (-0.0715405,-0.364827) (-0.054062,-0.369442) (-0.036236,-0.372766) (-0.0181766,-0.374771) (5.57097e-17,-0.375442) (0.0181766,-0.374771) (0.036236,-0.372766) (0.054062,-0.369442) (0.0715405,-0.364827)

Member Enumeration Documentation

◆ Type

enum TBTK::Solver::ChebyshevExpander::Type

strong

Enum class describing the type of Green's function to calculate.

Constructor & Destructor Documentation

◆ ChebyshevExpander()

TBTK::Solver::ChebyshevExpander::ChebyshevExpander ( )

Constructs a Solver::ChebyshevExpander.

◆ ~ChebyshevExpander()

virtual TBTK::Solver::ChebyshevExpander::~ChebyshevExpander ( )

virtual

Destructor.

Member Function Documentation

◆ calculateCoefficients() [1/2]

std::vector< std::complex< double > > TBTK::Solver::ChebyshevExpander::calculateCoefficients	(	Index	to,
		Index	from
	)

inline

Calculates the Chebyshev coefficients for \( G_{ij}(E)\), where \(i = \textrm{to}\) and \(j = \textrm{from}\).

Parameters

to	'To'-index, or \(i\).
from	'From'-index, or \(j\).
coefficients	Pointer to array able to hold numCoefficients coefficients.
numCoefficients	Number of coefficients to calculate.
broadening	Broadening to use in convolusion of coefficients to remedy Gibb's osciallations.

◆ calculateCoefficients() [2/2]

std::vector<std::vector<std::complex<double> > > TBTK::Solver::ChebyshevExpander::calculateCoefficients	(	std::vector< Index > &	to,
		Index	from
	)

Calculates the Chebyshev coefficients for \( G_{ij}(E)\), where \(i = \textrm{to}\) is a set of indices and \(j = \textrm{from}\).

Parameters

to	vector of 'to'-indeces, or \(i\)'s.
from	'From'-index, or \(j\).
coefficients	Pointer to array able to hold numCoefficients \(\times\)toIndeices.size() coefficients.
numCoefficients	Number of coefficients to calculate for each to-index.
broadening	Broadening to use in convolusion of coefficients to remedy Gibb's osciallations.

◆ generateGreensFunction()

std::vector< std::complex< double > > TBTK::Solver::ChebyshevExpander::generateGreensFunction	(	const std::vector< std::complex< double >> &	coefficients,
		Type	type = `Type::Retarded`
	)

inline

Genererate Green's function. Uses lookup table generated by ChebyshevExpander::generateLookupTable. Runs on CPU.

Parameters

greensFunction	Pointer to array able to hold Green's function. Has to be able to hold energyResolution elements.
coefficients	Chebyshev coefficients calculated by ChebyshevExpander::calculateCoefficients.

numCoefficients and energyResolution are here the values specified in the call to ChebyshevExpander::generateLookupTable

◆ getBroadening()

double TBTK::Solver::ChebyshevExpander::getBroadening ( ) const

inline

Get the broadening to use in convolusion of coefficients to remedy Gibb's osciallations.

Returns: The broadening parameter used.

◆ getCalculateCoefficientsOnGPU()

bool TBTK::Solver::ChebyshevExpander::getCalculateCoefficientsOnGPU ( ) const

inline

Get whether Chebyshev coefficients are set to be calculated on GPU.

Returns: True if GPU is used.

◆ getEnergyResolution()

int TBTK::Solver::ChebyshevExpander::getEnergyResolution ( ) const

inline

Get the number of energy point to use when generating Green's functions.

Returns: The number of energy points to use.

◆ getGenerateGreensFunctionsOnGPU()

bool TBTK::Solver::ChebyshevExpander::getGenerateGreensFunctionsOnGPU ( ) const

inline

Get whether Green's functions are set to be generated on GPU.

Returns: True if GPU is used.

◆ getLowerBound()

double TBTK::Solver::ChebyshevExpander::getLowerBound ( ) const

inline

Get the lower bound to use for the energy when generating the Green's function.

Returns: The lower bound for the energy.

◆ getNumCoefficients()

int TBTK::Solver::ChebyshevExpander::getNumCoefficients ( ) const

inline

Get the number of Chebyshev coefficients to use during calculations.

Returns: The number of Chebyshev coefficients.

◆ getScaleFactor()

double TBTK::Solver::ChebyshevExpander::getScaleFactor ( )

inline

Get scale factor.

Returns: The scale factor that is used to rescale the Hamiltonian.

◆ getUpperBound()

double TBTK::Solver::ChebyshevExpander::getUpperBound ( ) const

inline

Get the upper bound to use for the energy when generating the Green's function.

Returns: The upper bound for the energy.

◆ getUseLookupTable()

bool TBTK::Solver::ChebyshevExpander::getUseLookupTable ( ) const

inline

Set whether a lookup table should be used to when generating Green's functions.

Returns: True if a lookup table is used.

◆ setBroadening()

void TBTK::Solver::ChebyshevExpander::setBroadening ( double broadening )

inline

Set the broadening to use in convolusion of coefficients to remedy Gibb's osciallations.

Parameters

broadening The broadening parameter to use.

◆ setCalculateCoefficientsOnGPU()

void TBTK::Solver::ChebyshevExpander::setCalculateCoefficientsOnGPU ( bool calculateCoefficientsOnGPU )

inline

Set whether Chebyshev coefficients should be calculated on GPU. The default value is false.

Parameters

calculateCoefficientsOnGPU True to use GPU, false to use CPU.

◆ setEnergyResolution()

void TBTK::Solver::ChebyshevExpander::setEnergyResolution ( int energyResolution )

inline

Set the number of energy point to use when generating Green's functions. The default value is 1000.

Parameters

energyResolution The number of energy points to use.

◆ setGenerateGreensFunctionsOnGPU()

void TBTK::Solver::ChebyshevExpander::setGenerateGreensFunctionsOnGPU ( bool generateGreensFunctionsOnGPU )

inline

Set whether Green's functions should be generated on GPU. The default value is false.

Parameters

generateGreensFunctionsOnGPU True to use GPU, false to use CPU.

◆ setLowerBound()

void TBTK::Solver::ChebyshevExpander::setLowerBound ( double lowerBound )

inline

Set the lower bound to use for the energy when generating the Green's function. The default value is -1.

Parameters

lowerBound The lower bound for the energy.

◆ setNumCoefficients()

void TBTK::Solver::ChebyshevExpander::setNumCoefficients ( int numCoefficients )

inline

Set the number of Chebyshev coefficients to use during calculations. The default value is 1000.

Parameters

numCoefficients The number of Chebyshev coefficients.

◆ setScaleFactor()

void TBTK::Solver::ChebyshevExpander::setScaleFactor ( double scaleFactor )

inline

Overrides Solver::setModel(). Sets the scale factor that rescales the Hamiltonian to ensure that the energy spectrum of the Hamiltonian is bounded on the interval (-1, 1).

Parameters

scaleFactor The scale factor.

◆ setUpperBound()

void TBTK::Solver::ChebyshevExpander::setUpperBound ( double upperBound )

inline

Set the upper bound to use for the energy when generating the Green's function. The default value is 1.

Parameters

upperBound The upper bound for the energy.

◆ setUseLookupTable()

void TBTK::Solver::ChebyshevExpander::setUseLookupTable ( bool useLookupTable )

inline

Set whether a lookup table should be used to when generating Green's functions.

Parameters

useLookupTable True to use a lookup table.

The documentation for this class was generated from the following file:

/TBTK/Lib/include/TBTK/Solver/ChebyshevExpander.h

Public Types

Public Member Functions

Additional Inherited Members

Detailed Description

Example

Output

Member Enumeration Documentation

◆ Type

Constructor & Destructor Documentation

◆ ChebyshevExpander()

◆ ~ChebyshevExpander()

Member Function Documentation

◆ calculateCoefficients() [1/2]

◆ calculateCoefficients() [2/2]

◆ generateGreensFunction()

◆ getBroadening()

◆ getCalculateCoefficientsOnGPU()

◆ getEnergyResolution()

◆ getGenerateGreensFunctionsOnGPU()

◆ getLowerBound()

◆ getNumCoefficients()

◆ getScaleFactor()

◆ getUpperBound()

◆ getUseLookupTable()

◆ setBroadening()

◆ setCalculateCoefficientsOnGPU()

◆ setEnergyResolution()

◆ setGenerateGreensFunctionsOnGPU()

◆ setLowerBound()

◆ setNumCoefficients()

◆ setScaleFactor()

◆ setUpperBound()

◆ setUseLookupTable()