3-point fermion-boson susceptibility in the Matsubara representation. More...

#include <ThreePointSusceptibility.hpp>

Inheritance diagram for Pomerol::ThreePointSusceptibility:

Collaboration diagram for Pomerol::ThreePointSusceptibility:

Public Member Functions
	ThreePointSusceptibility (Channel channel, StatesClassification const &S, Hamiltonian const &H, CreationOperator const &CX1, AnnihilationOperator const &C2, CreationOperator const &CX3, AnnihilationOperator const &C4, DensityMatrix const &DM)

void	prepare ()
	Select relevant parts of \(c^\dagger_1, c_2, c^\dagger_3, c_4\) and allocate resources for the parts. More...

std::vector< ComplexType >	compute (bool clear=false, FreqVec2 const &freqs={}, MPI_Comm const &comm=MPI_COMM_WORLD)

ParticleIndex	getIndex (std::size_t Position) const

ComplexType	operator() (ComplexType z1, ComplexType z2) const

ComplexType	operator() (long MatsubaraNumber1, long MatsubaraNumber2) const

bool	isVanishing () const
	Is this susceptibility identically zero? More...

Public Member Functions inherited from Pomerol::Thermal
	Thermal (RealType beta)

Public Member Functions inherited from Pomerol::ComputableObject
	ComputableObject ()=default

StatusEnum	getStatus () const
	Return the current computation status. More...

void	setStatus (StatusEnum Status_in)

Data Fields
RealType	PoleResolution = 1e-8
	Lehmann representation: Maximal distance between energy poles to be consider coinciding. More...

RealType	CoefficientTolerance = 1e-16
	Lehmann representation: Maximal magnitude of a term coefficient to be considered negligible. More...

Data Fields inherited from Pomerol::Thermal
RealType const	beta
	Inverse temperature \(\beta\). More...

ComplexType const	MatsubaraSpacing
	Spacing between (imaginary) Matsubara frequencies, \(i\pi/\beta\). More...

Protected Attributes
bool	Vanishing = true
	A flag that marks an identically vanishing susceptibility. More...

Protected Attributes inherited from Pomerol::ComputableObject
StatusEnum	Status = Constructed
	Current computation status. More...

Friends
class	ThreePointSusceptibilityContainer

Additional Inherited Members
Public Types inherited from Pomerol::ComputableObject
enum	StatusEnum { Constructed , Prepared , Computed }
	Computation status of the object. More...

Detailed Description

3-point fermion-boson susceptibility in the Matsubara representation.

The susceptibility can be defined in one of the following three channels.

Particle-particle channel:
\[\chi^{(3)}_{pp}(\omega_{n_1},\omega_{n_2}) = \int_0^\beta d\tau_1 d\tau_2 e^{-i\omega_{n_1}\tau_1} e^{-i\omega_{n_2}\tau_2} Tr[\mathcal{T}_\tau \hat\rho c^\dagger_1(\tau_1) c_2(0^+) c^\dagger_3(\tau_2) c_4(0)]\]

Particle-hole channel:
\[\chi^{(3)}_{ph}(\omega_{n_1},\omega_{n_2}) = \int_0^\beta d\tau_1 d\tau_2 e^{-i\omega_{n_1}\tau_1} e^{i\omega_{n_2}\tau_2} Tr[\mathcal{T}_\tau \hat\rho c^\dagger_1(\tau_1) c_2(\tau_2) c^\dagger_3(0^+) c_4(0)]\]

Crossed particle-hole channel:
\[\chi^{(3)}_{\bar{ph}}(\omega_{n_1},\omega_{n_2}) = \int_0^\beta d\tau_1 d\tau_2 e^{-i\omega_{n_1}\tau_1} e^{i\omega_{n_2}\tau_2} Tr[\mathcal{T}_\tau \hat\rho c^\dagger_1(\tau_1) c_2(0) c^\dagger_3(0^+) c_4(\tau_2)]\]

These susceptibilities can be interpreted as 3-point correlators of two fermionic operators \(\hat F_1(\omega_{n_1}), \hat F_2(\omega_{n_2})\) and one quadratic operator \(\hat B\).

PP channel: \(\hat F_1 = c^\dagger_1, \hat F_2 = c^\dagger_3, \hat B = \Delta_{24} = c_2 c_4 \);
PH channel: \(\hat F_1 = c^\dagger_1, \hat F_2 = c_2, \hat B = n_{34} = c^\dagger_3 c_4 \);
xPH channel: \(\hat F_1 = c^\dagger_1, \hat F_2 = c_4, \hat B = -n_{32} = -c^\dagger_3 c_2 \).

It is actually a container class for a collection of ThreePointSusceptibilityPart's (most of the real calculations take place in the parts).

Definition at line 67 of file ThreePointSusceptibility.hpp.

Constructor & Destructor Documentation

◆ ThreePointSusceptibility()

Pomerol::ThreePointSusceptibility::ThreePointSusceptibility	(	Channel	channel,
		StatesClassification const &	S,
		Hamiltonian const &	H,
		CreationOperator const &	CX1,
		AnnihilationOperator const &	C2,
		CreationOperator const &	CX3,
		AnnihilationOperator const &	C4,
		DensityMatrix const &	DM
	)

Constructor

Parameters

[in]	channel	Channel of the 3-point susceptibility.
[in]	S	Information about invariant subspaces of the Hamiltonian.
[in]	H	The Hamiltonian.
[in]	CX1	The creation operator \(c^\dagger_1\).
[in]	C2	The annihilation operator \(c_2\).
[in]	CX3	The creation operator \(c^\dagger_3\).
[in]	C4	The annihilation operator \(c_4\).
[in]	DM	Many-body density matrix \(\hat\rho\).

Member Function Documentation

◆ compute()

std::vector<ComplexType> Pomerol::ThreePointSusceptibility::compute	(	bool	clear = `false`,
		FreqVec2 const &	freqs = `{}`,
		MPI_Comm const &	comm = `MPI_COMM_WORLD`
	)

Compute the parts in parallel and fill the internal cache of precomputed values.

Parameters

[in]	clear	If true, computed ThreePointSusceptibilityPart's will be destroyed immediately after filling the precomputed value cache.
[in]	freqs	List of frequency duplets \((\omega_{n_1},\omega_{n_2})\).
[in]	comm	MPI communicator used to parallelize the computation.

Returns: A list of precomputed values.

Precondition: prepare() has been called.

◆ getIndex()

ParticleIndex Pomerol::ThreePointSusceptibility::getIndex ( std::size_t Position ) const

Returns the single particle index of one of the operators \(c^\dagger_1, c_2, c^\dagger_3, c_4\).

Parameters

[in] Position Position of the requested operator, 0–3.

◆ isVanishing()

bool Pomerol::ThreePointSusceptibility::isVanishing ( ) const

inline

Is this susceptibility identically zero?

Definition at line 150 of file ThreePointSusceptibility.hpp.

◆ operator()() [1/2]

ComplexType Pomerol::ThreePointSusceptibility::operator()	(	ComplexType	z1,
		ComplexType	z2
	)		const

inline

Return the value of the 3-point susceptibility calculated at a given complex frequency duplet. This method ignores the precomputed value cache.

Parameters

[in]	z1	First frequency \(z_1\).
[in]	z2	Second frequency \(z_2\).

Definition at line 162 of file ThreePointSusceptibility.hpp.

◆ operator()() [2/2]

ComplexType Pomerol::ThreePointSusceptibility::operator()	(	long	MatsubaraNumber1,
		long	MatsubaraNumber2
	)		const

inline

Return the value of the 3-point susceptibility calculated a given Matsubara frequency duplet.

Parameters

[in]	MatsubaraNumber1	Index of the first Matsubara frequency \(n_1\) ( \(\omega_{n_1}=\pi(2n_1+1)/\beta\)).
[in]	MatsubaraNumber2	Index of the second Matsubara frequency \(n_2\) ( \(\omega_{n_2}=\pi(2n_2+1)/\beta\)).

Definition at line 174 of file ThreePointSusceptibility.hpp.