RealTimeTransport::AndersonDot Class Reference

#include <RealTimeTransport/Models/AndersonDot.h>

This class implements the Anderson quantum dot model (single level with spin). The Hamiltonian of the quantum dot is given by

\[ H = \epsilon (n_\uparrow + n_\downarrow) + \frac{B}{2} (n_\uparrow - n_\downarrow) + U n_\uparrow n_\downarrow. \]

It is coupled to the reservoirs via the tunneling Hamiltonian

\[ H_T = \sum_{r\sigma} \int d\omega \sqrt{\frac{\Gamma_{r\sigma}}{2\pi}} (d^\dagger_\sigma a_{r\sigma}(\omega) + a^\dagger_{r\sigma}(\omega) d_\sigma). \]

All operators are represented in the basis \( \ket{0}, \ket{\uparrow}=d^\dagger_\uparrow \ket{0}, \ket{\downarrow}=d^\dagger_\downarrow \ket{0}, \ket{\uparrow\downarrow} = d^\dagger_\uparrow d^\dagger_\downarrow \ket{0} \).

Public Member Functions
	AndersonDot () noexcept=default
	Default constructor.
	AndersonDot (AndersonDot &&other) noexcept=default
	Default move constructor.
	AndersonDot (const AndersonDot &other)=default
	Default copy constructor.
AndersonDot &	operator= (AndersonDot &&other) noexcept=default
	Default move assignment operator.
AndersonDot &	operator= (const AndersonDot &other)=default
	Default copy assignment operator.
	AndersonDot (SciCore::Real epsilon, SciCore::Real B, SciCore::Real U, const SciCore::RealVector &T, const SciCore::RealVector &mu, const SciCore::RealVector &Gamma)
	Constructs a new Anderson dot object.
	AndersonDot (SciCore::Real epsilon, SciCore::Real B, SciCore::Real U, const SciCore::RealVector &T, const SciCore::RealVector &mu, const SciCore::RealVector &GammaUp, const SciCore::RealVector &GammaDown)
	Constructs a new Anderson dot object.
SciCore::Real	epsilon () const noexcept
	Returns the energy of the level.
SciCore::Real	B () const noexcept
	Returns the magnetic field.
SciCore::Real	U () const noexcept
	Returns the Coulomb interaction.
const SciCore::RealVector &	GammaUp () const noexcept
	Returns the couplings \(\Gamma_{r\uparrow}\).
const SciCore::RealVector &	GammaDown () const noexcept
	Returns the couplings \(\Gamma_{r\downarrow}\).
int	dimHilbertSpace () const noexcept override
	The Hilbert space dimension is 4.
int	numStates () const noexcept override
	The single particle states are \(\uparrow\) and \(\downarrow\), thus returns 2.
int	numChannels () const noexcept override
	Each reservoir is spinful, thus returns 2.
int	numReservoirs () const override
	Returns the number of reservoirs the system is connected to.
OperatorType	H () const override
	Returns the Hamiltonian.
OperatorType	d (int l) const override
	Returns the annihilation operator of the single particle state indexed by \( l=\uparrow\equiv 0 \), \( \downarrow\equiv 1 \).
SciCore::Complex	coupling (int r, int nu, int l) const override
	Returns the coupling coefficient in the tunneling Hamiltonian, given by \(\sqrt{\lambda}_{r \nu} t_{r \nu l}\).
const SciCore::RealVector &	temperatures () const noexcept override
	Returns the temperatures of the reservoirs the system is connected to.
const SciCore::RealVector &	chemicalPotentials () const noexcept override
	Returns the chemical potentials of the reservoirs the system is connected to.
SupervectorType	vectorize (const OperatorType &op) const override
	Vectorizes an operator into its supervector form.
OperatorType	operatorize (const SupervectorType &supervector) const override
	Transforms a supervector back into operator form.
const std::vector< int > &	blockDimensions () const noexcept override
	The memory kernel and propagator are block diagonal. This function returns a reference to a vector containing the dimensions of these blocks.
std::unique_ptr< Model >	copy () const override
	Returns a deep copy of the model.
bool	isEqual (const Model &other) const override
	Returns true if other is the same as *this, otherwise false.
Public Member Functions inherited from RealTimeTransport::Model
bool	operator== (const Model &other) const
	Equality comparison operator.
bool	operator!= (const Model &other) const
	Inequality comparison operator.
virtual OperatorType	P () const
	Returns the parity operator.

Additional Inherited Members
Public Types inherited from RealTimeTransport::Model
using	OperatorType = SciCore::Matrix
	Type of ordinary operators.
using	SupervectorType = SciCore::Vector
	Type of vectorized operators.
using	SuperRowVectorType = SciCore::RowVector
	Type of adjoint of vectorized operators.
using	SuperoperatorType = SciCore::Matrix
	Type of superoperators.

Constructor & Destructor Documentation

AndersonDot() [1/2]

RealTimeTransport::AndersonDot::AndersonDot	(	SciCore::Real	epsilon,
		SciCore::Real	B,
		SciCore::Real	U,
		const SciCore::RealVector &	T,
		const SciCore::RealVector &	mu,
		const SciCore::RealVector &	Gamma )

Parameters

epsilon	Level energy
B	Magnetic field
U	Coulomb interaction
T	Temperatures of the reservoirs
mu	Chemical potentials of the reservoirs
Gamma	Spin-independent dot-reservoir couplings \(\Gamma_r=\Gamma_{r\uparrow}=\Gamma_{r\downarrow}\)

AndersonDot() [2/2]

RealTimeTransport::AndersonDot::AndersonDot	(	SciCore::Real	epsilon,
		SciCore::Real	B,
		SciCore::Real	U,
		const SciCore::RealVector &	T,
		const SciCore::RealVector &	mu,
		const SciCore::RealVector &	GammaUp,
		const SciCore::RealVector &	GammaDown )

Parameters

epsilon	Level energy
B	Magnetic field
U	Coulomb interaction
T	Temperatures
mu	Chemical potentials
GammaUp	Dot-reservoir coupling \(\Gamma_{r\uparrow}\)
GammaDown	Dot-reservoir coupling \(\Gamma_{r\downarrow}\)