938 lines
33 KiB
Fortran
938 lines
33 KiB
Fortran
!
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!=======================================================================
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!
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MODULE LANDAU_PARAM
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!
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! This module provides the standard dimensionless Landau parameters
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!
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USE ACCURACY_REAL
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!
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CONTAINS
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!
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!
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!=======================================================================
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!
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SUBROUTINE LANDAU_PARAMETERS_3D(X,XC,U0,W,D,RS,LANDAU, &
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F0S,F0A,F1S,F1A,F2S,F2A)
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!
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! This subroutine computes the standard dimensionless Landau parameters
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! as a function of the Hubbard-like bare coupling constant, or the
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! hard-sphere radius, in the 3D case
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!
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * XC : dimensionless cut-off --> XC = q_c / (2 * k_F)
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! * U0 / A : bare interaction constant / hard sphere radius (in SI)
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! * W : half bandwidth for bare particle (ref. 7)
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! * D : filling (dopant concentration) in ref. 7
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * LANDAU : model chosen for the calculation of the parameters
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! LANDAU = 'ANBR' Anderson-Brinkman model
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! LANDAU = 'CHEN' Chen's approach
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! LANDAU = 'GUTZ' Gutzwiller model
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! LANDAU = 'GVYO' Giuliani-Vignale parametrization of
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! LANDAU = 'IWPI' Iwamoto-Pines model (hard-sphere)
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! Yasuhara-Ousaka approach
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! LANDAU = 'RASC' Rayleigh-Schrödinger expansion
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! LANDAU = 'SBOH' slave-boson one-band Hubbard model
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!
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!
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! Output parameters:
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!
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! * F0S,F0A,F1S,F1A,F2S,F2A
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Jun 2020
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!
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IMPLICIT NONE
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!
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CHARACTER (LEN = 4) :: LANDAU
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!
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REAL (WP) :: X,XC,U0,A,W,D,RS
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REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
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!
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IF(LANDAU == 'ANBR') THEN !
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!
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CALL ANBR_LP_3D(U0,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'CHEN') THEN !
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!
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CALL CHEN_LP_3D(U0,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'GUTZ') THEN !
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!
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CALL GUTZ_LP_3D(U0,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'GVYO') THEN !
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!
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CALL GVYO_LP_3D(RS,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'IWPI') THEN !
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!
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A=U0 !
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CALL IPWI_LP_3D(A,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'RASC') THEN !
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!
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CALL RASC_LP_3D(X,XC,U0,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'SBOH') THEN !
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!
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CALL SBOH_LP_3D(U0,W,D,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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END IF !
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!
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END SUBROUTINE LANDAU_PARAMETERS_3D
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!
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!=======================================================================
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!
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SUBROUTINE LANDAU_PARAMETERS_2D(RS,LANDAU,X,IX, &
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F0S,F0A,F1S,F1A,F2S,F2A)
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!
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! This subroutine computes the standard dimensionless Landau parameters
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! in the 2D case.
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!
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!
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! Input parameters:
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!
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * LANDAU : model chosen for the calculation of the parameters
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! LANDAU = 'ERZA' Engelbrecht-Randeria-Zhang approach
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! LANDAU = 'GVYO' Giuliani-Vignale parametrization of
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! Yasuhara-Ousaka approach
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! LANDAU = 'KCMP' Kwoon-Ceperley-Martin parametrization
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! * X : either MU or EG (for 'ERZA' only)
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! * IX : switch for input value X (for 'ERZA' only)
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! IX = 1 --> X = chemical potential MU in SI
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! IX = 2 --> X = ground state energy EG in SI
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!
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!
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! Output parameters:
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!
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! * F0S,F0A,F1S,F1A,F2S,F2A
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Jun 2020
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!
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IMPLICIT NONE
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!
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CHARACTER (LEN = 4) :: LANDAU
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!
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REAL (WP) :: RS,X,G
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REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
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!
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INTEGER :: IX
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!
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IF(LANDAU == 'ERZA') THEN !
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!
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CALL ERZA_LP_2D(X,IX,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'GVYO') THEN !
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!
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CALL GVYO_LP_2D(RS,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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ELSE IF(LANDAU == 'KCMP') THEN !
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!
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CALL KCMP_LP_2D(RS,F0S,F0A,F1S,F1A,F2S,F2A) !
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!
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END IF !
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!
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END SUBROUTINE LANDAU_PARAMETERS_2D
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!
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!=======================================================================
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!
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SUBROUTINE ANBR_LP_3D(U0,F0S,F0A,F1S,F1A,F2S,F2A)
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!
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! This subroutine computes the standard dimensionless Landau parameters
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! for the Anderson-Brinkman model in the 3D case
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!
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! References: (1) D. Volhardt, Rev. Mod. Phys. 58, 99-120 (1984)
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!
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!
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! Input parameters:
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!
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! * U0 : bare interaction constant
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!
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!
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! Output parameters:
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!
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! * F0S,F0A,F1S,F1A,F2S,F2A
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,EIGHT,FOURTH
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USE CONSTANTS_P1, ONLY : H_BAR,M_E
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USE FERMI_SI, ONLY : EF_SI,KF_SI
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USE PI_ETC, ONLY : PI2
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!
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IMPLICIT NONE
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!
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REAL (WP) :: U0,UC,XU
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REAL (WP) :: NF_SI
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REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
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!
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! Initialization
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!
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F0S=ZERO !
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F0A=ZERO !
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F1S=ZERO !
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F1A=ZERO !
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F2S=ZERO !
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F2A=ZERO !
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!
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NF_SI=M_E*KF_SI/(PI2*H_BAR*H_BAR) ! n(E_F) in SI
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!
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! Cut-off interaction
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!
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UC=EIGHT*EF_SI !
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XU=U0/UC !
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!
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F1S=THREE*(ONE/(ONE-XU*XU)) - ONE ! ref. (1) eq. (24)
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F0A=-FOURTH*NF_SI*U0*(TWO+XU)/((ONE*XU)*(ONE*XU)) ! ref. (41) eq. (25)
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!
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END SUBROUTINE ANBR_LP_3D
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!
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!=======================================================================
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!
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SUBROUTINE CHEN_LP_3D(U0,F0S,F0A,F1S,F1A,F2S,F2A)
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!
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! This subroutine computes the standard dimensionless Landau parameters
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! in Chen's approach, in the 3D case
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!
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! References: (1) V. A. Belyakov, Soviet Phys. JETP 13, 850-851 (1961)
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! (2) J.-S. Chen, J. Stat. Mech. L08002 (2009)
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!
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!
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!
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! Input parameters:
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!
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! * U0 : bare interaction constant
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!
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!
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! Output parameters:
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!
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! * F0S,F0A,F1S,F1A,F2S,F2A
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,FOUR,EIGHT, &
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NINE,THIRD,FOURTH
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USE CONSTANTS_P1, ONLY : H_BAR,M_E
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USE FERMI_SI, ONLY : KF_SI
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USE PI_ETC, ONLY : PI,PI_INV
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!
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IMPLICIT NONE
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!
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REAL (WP) :: U0
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REAL (WP) :: A,AKF,P2AKF
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REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
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!
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! Initialization
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!
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F0S=ZERO !
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F0A=ZERO !
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F1S=ZERO !
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F1A=ZERO !
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F2S=ZERO !
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F2A=ZERO !
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!
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! Scattering amplitude from Fermi pseudopotential
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!
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A= FOURTH*PI_INV*U0 * (M_E/(H_BAR*H_BAR)) ! ref. 1 eq. (3)
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!
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AKF=A*KF_SI !
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P2AKF=PI-TWO * AKF !
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!
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F1S=FOUR*AKF*AKF/(THREE * P2AKF**2) ! ref. (2) eq. (28)
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F0S=(ONE+THIRD*F1S)*(ONE + TWO*AKF/P2AKF + & !
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20.0E0_WP*AKF*AKF/(NINE*P2AKF*P2AKF) + & ! ref. (2) eq. (25)
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EIGHT*AKF*AKF*AKF/(NINE*(P2AKF**3))) !
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!
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END SUBROUTINE CHEN_LP_3D
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!
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!=======================================================================
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!
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SUBROUTINE ERZA_LP_2D(X,IX,F0S,F0A,F1S,F1A,F2S,F2A)
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!
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! This subroutine computes the standard dimensionless Landau parameters
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! in the 2D case.
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!
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! References: (1) J. R. Engelbrecht, M. Randeria and L. Zhang,
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! Phys. Rev. B 45, 10135-10138 (1992)
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!
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!
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! Input parameters:
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!
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! * X : either MU or EG
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! * IX : switch for input value X
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! IX = 1 --> X = chemical potential MU in SI
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! IX = 2 --> X = ground state energy EG in SI
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!
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! Intermediate parameters:
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!
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! * G : low-density expansion parameter (ref. 1)
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!
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!
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! Output parameters:
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!
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! * F0S,F0A,F1S,F1A,F2S,F2A
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ZERO,TWO,FOUR
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!
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IMPLICIT NONE
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!
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INTEGER :: IX
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!
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REAL (WP) :: X,G
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REAL (WP) :: LN2
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REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
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!
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LN2=DLOG(TWO) !
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!
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! Initialization
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!
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F0S=ZERO !
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F0A=ZERO !
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F1S=ZERO !
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F1A=ZERO !
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F2S=ZERO !
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F2A=ZERO !
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!
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! Computing the expansion parameter G
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!
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CALL MU_EG_TO_G(X,IX,G) !
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!
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F0S=TWO*G+FOUR*G*G*(TWO-LN2) !
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F0A=-FOUR*G*G*LN2 !
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F1S= TWO*G*G !
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F1A=- TWO*G*G !
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!
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END SUBROUTINE ERZA_LP_2D
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!
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!=======================================================================
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!
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SUBROUTINE MU_EG_TO_G(X,IX,G)
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!
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! This subroutine computes the low-density expansion parameter g
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! from the knowledge of either the chemical potential mu or
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! from the ground state energy EG
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!
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! References: (1) J. R. Engelbrecht, M. Randeria and L. Zhang,
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! Phys. Rev. B 45, 10135-10138 (1992)
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!
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!
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! Input parameters:
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!
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! * X : either MU or EG
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! * IX : switch for input value X
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! IX = 1 --> X = chemical potential MU in SI
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! IX = 2 --> X = ground state energy EG in SI
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!
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!
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! Output parameters:
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!
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! * G : low-density expansion parameter (ref. 1)
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 5 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,TWO,THREE,FOUR
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USE FERMI_SI, ONLY : EF_SI
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USE POLYNOMIAL_EQ, ONLY : QUADRATIC_EQUATION,CHECK_ROOTS2
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!
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IMPLICIT NONE
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!
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INTEGER :: IX
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!
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REAL (WP) :: X,G
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REAL (WP) :: MU,EG
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REAL (WP) :: LN2
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!
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COMPLEX (WP) :: AA,BB,CC
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COMPLEX (WP) :: X1,X2
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!
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LN2=DLOG(TWO) !
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!
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IF(IX == 1) THEN !
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!
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MU=X !
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!
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AA=DCMPLX(FOUR*(ONE-LN2)) !
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BB=DCMPLX(TWO) !
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CC=DCMPLX(ONE-MU/EF_SI) !
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!
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CALL QUADRATIC_EQUATION(AA,BB,CC,X1,X2) !
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!
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CALL CHECK_ROOTS2(X1,X2,G) !
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!
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ELSE IF(IX == 2) THEN !
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!
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EG=X !
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!
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AA=DCMPLX(THREE-FOUR*LN2) !
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BB=DCMPLX(TWO) !
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CC=DCMPLX(ONE-TWO*EG/EF_SI) !
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!
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CALL QUADRATIC_EQUATION(AA,BB,CC,X1,X2) !
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!
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CALL CHECK_ROOTS2(X1,X2,G) !
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!
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END IF !
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!
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END SUBROUTINE MU_EG_TO_G
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!
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!=======================================================================
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!
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SUBROUTINE GUTZ_LP_3D(U0,F0S,F0A,F1S,F1A,F2S,F2A)
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!
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|
! This subroutine computes the standard dimensionless Landau parameters
|
|
! for the Gutzwiller approach, in the 3D case
|
|
!
|
|
! References: (1) D. Volhardt, Rev. Mod. Phys. 58, 99-120 (1984)
|
|
!
|
|
!
|
|
!
|
|
! Input parameters:
|
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!
|
|
! * U0 : bare interaction constant
|
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!
|
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!
|
|
! Output parameters:
|
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!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
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!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,FOUR,EIGHT, &
|
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THIRD,FOURTH
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USE CONSTANTS_P1, ONLY : H_BAR,M_E
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USE FERMI_SI, ONLY : EF_SI,KF_SI
|
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USE PI_ETC, ONLY : PI2
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!
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IMPLICIT NONE
|
|
!
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|
REAL (WP) :: U0,UC,XU
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REAL (WP) :: NF_SI,P,I,AA,BB
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REAL (WP) :: NUM,DEN
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REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
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! Initialization
|
|
!
|
|
F0S=ZERO !
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F0A=ZERO !
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F1S=ZERO !
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F1A=ZERO !
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F2S=ZERO !
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F2A=ZERO !
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!
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NF_SI=M_E*KF_SI/(PI2*H_BAR*H_BAR) ! n(E_F) in SI
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!
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! Cut-off interaction
|
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!
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UC=EIGHT*EF_SI !
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XU=U0/UC !
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!
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P=FOURTH*UC*NF_SI !
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I=XU !
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AA=-ONE + (FOUR - I*I)*(ONE - P) !
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BB=( ONE + I*I*(ONE-P) )**2 !
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NUM=BB + TWO*THIRD*P*AA !
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DEN=BB*(ONE+I*I) + TWO*THIRD*P*AA !
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!
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F0A=-P*(ONE - ONE/((ONE+I)*(ONE+I))) ! ref. (1) eq. (50)
|
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F0S= P*(ONE/((ONE+I)*(ONE+I) - ONE) ) ! ref. (1) eq. (54)
|
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F1A=-THREE*I*I * NUM/DEN ! ref. (1) eq. (71)
|
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!
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END SUBROUTINE GUTZ_LP_3D
|
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!
|
|
!=======================================================================
|
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!
|
|
SUBROUTINE GVYO_LP_3D(RS,F0S,F0A,F1S,F1A,F2S,F2A)
|
|
!
|
|
! This subroutine computes the Yasuhara-Ousaka
|
|
! parametrization of Landau parameters F0s,F0a,
|
|
! F1s,F1a,F2s,F2a.
|
|
!
|
|
! We use a 4-degree polynomial to fit the data of
|
|
! table 8.1 and table 8.6 of reference (1)
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * RS : Wigner-Seitz radius (in units of a_0)
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
!
|
|
! Reference: (1) G. F. Giuliani and G. Vignale,
|
|
! "Quantum Theory of the Electron Liquid",
|
|
! (Cambridge University Press 2005)
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL (WP) :: RS,F0S,F0A,F1S,F1A,F2S,F2A
|
|
REAL (WP) :: Y,Y2,Y3,Y4
|
|
REAL (WP) :: A1(0:4),A2(0:4),A3(0:4),A4(0:4),A5(0:4),A6(0:4)
|
|
!
|
|
DATA A1 / -0.066667E0_WP , -0.13392E0_WP , -0.0094444E0_WP ,&! F0s
|
|
0.00018519E0_WP, 6.8305E-18_WP / !
|
|
DATA A2 / -0.10833E0_WP , -0.039854E0_WP , -0.029514E0_WP ,&! F0a
|
|
0.008287E0_WP , -0.000625E0_WP / !
|
|
DATA A3 / -0.07E0_WP , 0.049167E0_WP , -0.024583E0_WP ,&! F1s
|
|
0.0058333E0_WP , -0.00041667E0_WP / !
|
|
DATA A4 / -0.0242E0_WP , 0.0066333E0_WP, -0.00068333E0_WP,&! F1a
|
|
0.00016667E0_WP, -1.6667E-05_WP / !
|
|
DATA A5 / -0.0221E0_WP , -0.00265E0_WP , 0.00395E0_WP ,&! F2s
|
|
-0.00075E0_WP , 5.0E-05_WP / !
|
|
DATA A6 / -0.0242E0_WP , 0.0066333E0_WP, -0.00068333E0_WP,&! F2a
|
|
0.00016667E0_WP, -1.6667E-05_WP / !
|
|
!
|
|
! Powers of RS
|
|
!
|
|
Y = RS !
|
|
Y2 = Y*Y !
|
|
Y3 = Y2*Y !
|
|
Y4 = Y3*Y !
|
|
!
|
|
! Computing the Landau parameters
|
|
!
|
|
F0S=A1(0) + A1(1)*Y + A1(2)*Y2 + A1(3)*Y3 + A1(4)*Y4 ! F0s
|
|
F0A=A2(0) + A2(1)*Y + A2(2)*Y2 + A2(3)*Y3 + A2(4)*Y4 ! F0a
|
|
F1S=A3(0) + A3(1)*Y + A3(2)*Y2 + A3(3)*Y3 + A3(4)*Y4 ! F1s
|
|
F1A=A4(0) + A4(1)*Y + A4(2)*Y2 + A4(3)*Y3 + A4(4)*Y4 ! F1a
|
|
F2S=A5(0) + A5(1)*Y + A5(2)*Y2 + A5(3)*Y3 + A5(4)*Y4 ! F2s
|
|
F2A=A6(0) + A6(1)*Y + A6(2)*Y2 + A6(3)*Y3 + A6(4)*Y4 ! F2a
|
|
!
|
|
END SUBROUTINE GVYO_LP_3D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE GVYO_LP_2D(RS,F0S,F0A,F1S,F1A,F2S,F2A)
|
|
!
|
|
! This subroutine computes the Yasuhara-Ousaka
|
|
! parametrization of Landau parameters F0s,F0a,
|
|
! F1s.
|
|
!
|
|
! We use a 4-degree polynomial to fit the data of
|
|
! table 8.5 of reference (1)
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * RS : Wigner-Seitz radius (in units of a_0)
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
!
|
|
! Reference: (1) G. F. Giuliani and G. Vignale,
|
|
! "Quantum Theory of the Electron Liquid",
|
|
! (Cambridge University Press 2005)
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ZERO
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL (WP) :: RS,F0S,F0A,F1S,F1A,F2S,F2A
|
|
REAL (WP) :: Y,Y2,Y3,Y4
|
|
REAL (WP) :: A1(0:4),A2(0:4),A3(0:4)
|
|
!
|
|
DATA A1 / -0.13E0_WP , -0.19917E0_WP , -0.12958E0_WP ,& ! F0s
|
|
0.0091667E0_WP, -0.00041667E0_WP / !
|
|
DATA A2 / -0.1019E0_WP , -0.24296E0_WP , 0.063813E0_WP,& ! F0a
|
|
-0.0094683E0_WP, 0.00051984E0_WP / !
|
|
DATA A3 / -0.087143E0_WP , 0.10986E0_WP , 0.010738E0_WP,& ! F1s
|
|
-0.0037143E0_WP, 0.0002619E0_WP / !
|
|
!
|
|
! Initialization
|
|
!
|
|
F0S=ZERO !
|
|
F0A=ZERO !
|
|
F1S=ZERO !
|
|
F1A=ZERO !
|
|
F2S=ZERO !
|
|
F2A=ZERO !
|
|
!
|
|
! Powers of RS
|
|
!
|
|
Y = RS !
|
|
Y2 = Y*Y !
|
|
Y3 = Y2*Y !
|
|
Y4 = Y3*Y !
|
|
!
|
|
! Computing the Landau parameters
|
|
!
|
|
F0S=A1(0) + A1(1)*Y + A1(2)*Y2 + A1(3)*Y3 + A1(4)*Y4 ! F0s
|
|
F0A=A2(0) + A2(1)*Y + A2(2)*Y2 + A2(3)*Y3 + A2(4)*Y4 ! F0a
|
|
F1S=A3(0) + A3(1)*Y + A3(2)*Y2 + A3(3)*Y3 + A3(4)*Y4 ! F1s
|
|
!
|
|
END SUBROUTINE GVYO_LP_2D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE IPWI_LP_3D(A,F0S,F0A,F1S,F1A,F2S,F2A)
|
|
!
|
|
! This subroutine computes the standard dimensionless Landau parameters
|
|
! for the Iwamoto-Pines hard-sphere model, in the 3D case
|
|
!
|
|
! References: (1) N. Iwamoto and D. Pines, Phys. Rev. B 29, 3924 (1984)
|
|
!
|
|
!
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * A : hard sphere radius (in SI)
|
|
!
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,FOUR,SEVEN,THIRD
|
|
USE CONSTANTS_P1, ONLY : H_BAR,M_E
|
|
USE FERMI_SI, ONLY : KF_SI
|
|
USE PI_ETC, ONLY : PI_INV
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL*8 A,AKF
|
|
REAL*8 LN2
|
|
REAL*8 F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
LN2=DLOG(TWO) !
|
|
!
|
|
! Initialization
|
|
!
|
|
F0S=ZERO !
|
|
F0A=ZERO !
|
|
F1S=ZERO !
|
|
F1A=ZERO !
|
|
F2S=ZERO !
|
|
F2A=ZERO !
|
|
!
|
|
AKF=A*KF_SI !
|
|
!
|
|
F0S= TWO*PI_INV*AKF*( & !
|
|
ONE+FOUR*THIRD*PI_INV* & !
|
|
(TWO+LN2)*AKF & !
|
|
) ! ref. (1) eq. (B1)
|
|
F0A=-TWO*PI_INV*AKF*( & !
|
|
ONE+FOUR*THIRD*PI_INV* & !
|
|
(ONE-LN2)*AKF & !
|
|
) ! ref. (1) eq. (B2)
|
|
F1A= 1.6E0_WP*PI_INV*PI_INV*(SEVEN*LN2-ONE)*AKF*AKF ! ref. (1) eq. (B3)
|
|
F1S=-1.6E0_WP*PI_INV*PI_INV*(TWO+LN2)*AKF*AKF ! ref. (1) eq. (B4)
|
|
!
|
|
END SUBROUTINE IPWI_LP_3D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE KCMP_LP_2D(RS,F0S,F0A,F1S,F1A,F2S,F2A)
|
|
!
|
|
! This subroutine computes the Kwoon-Ceperley-Martin
|
|
! parametrization of Landau parameters F0s,F0a,
|
|
! F1s,F1a,F2s and F2a.
|
|
!
|
|
! We use a 4-point Lagrange interpolation to fit the data of
|
|
! table VIII of reference (1)
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * RS : Wigner-Seitz radius (in units of a_0)
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
! Reference: (1) Y. Kwoon, D. M. Ceperley and R. M. Martin,
|
|
! Phys. Rev. B 50, 1684-1694 (1994)
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ONE,TWO,THREE,FIVE
|
|
USE INTERPOLATION, ONLY : LAG_4P_INTERP
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL (WP) :: RS,F0S,F0A,F1S,F1A,F2S,F2A
|
|
REAL (WP) :: A1(4),A2(4),A3(4),A4(4),A5(4),A6(4)
|
|
REAL (WP) :: X(4)
|
|
!
|
|
! Data of table VIII
|
|
!
|
|
DATA A1 / -0.60E0_WP, -0.99E0_WP, -1.63E0_WP, -3.70E0_WP / ! F0s
|
|
DATA A2 / -0.34E0_WP, -0.41E0_WP, -0.49E0_WP, -0.51E0_WP / ! F0a
|
|
DATA A3 / -0.14E0_WP, -0.10E0_WP, -0.03E0_WP, 0.12E0_WP / ! F1s
|
|
DATA A4 / -0.19E0_WP, -0.24E0_WP, -0.26E0_WP, -0.27E0_WP / ! F1a
|
|
DATA A5 / -0.07E0_WP, -0.16E0_WP, -0.27E0_WP, -0.50E0_WP/ ! F2s
|
|
DATA A6 / 0.01E0_WP, 0.07E0_WP, 0.14E0_WP, 0.32E0_WP/ ! F2a
|
|
!
|
|
! RS values un table VIII
|
|
!
|
|
X(1)=ONE !
|
|
X(2)=TWO !
|
|
X(3)=THREE !
|
|
X(4)=FIVE !
|
|
!
|
|
! F0s --> A1 data
|
|
!
|
|
F0S=LAG_4P_INTERP(X,A1,RS) !
|
|
!
|
|
! F0a --> A2 data
|
|
!
|
|
F0A=LAG_4P_INTERP(X,A2,RS) !
|
|
!
|
|
! F1s --> A3 data
|
|
!
|
|
F1S=LAG_4P_INTERP(X,A3,RS) !
|
|
!
|
|
! F1a --> A4 data
|
|
!
|
|
F1A=LAG_4P_INTERP(X,A4,RS) !
|
|
!
|
|
! F2s --> A5 data
|
|
!
|
|
F2S=LAG_4P_INTERP(X,A5,RS) !
|
|
!
|
|
! F2a --> A6 data
|
|
!
|
|
F2A=LAG_4P_INTERP(X,A6,RS) !
|
|
!
|
|
END SUBROUTINE KCMP_LP_2D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE RASC_LP_3D(X,XC,U0,F0S,F0A,F1S,F1A,F2S,F2A)
|
|
!
|
|
! This subroutine computes the standard dimensionless Landau parameters
|
|
! using a Rayleigh-Schrödinger expansion in the 3D case
|
|
!
|
|
! References: (1) Slides
|
|
!
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * X : dimensionless factor --> X = q / (2 * k_F)
|
|
! * XC : dimensionless cut-off --> XC = q_c / (2 * k_F)
|
|
! * U0 : bare interaction constant
|
|
!
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,SEVEN, &
|
|
HALF,THIRD,FOURTH
|
|
USE CONSTANTS_P1, ONLY : M_E
|
|
USE FERMI_SI, ONLY : KF_SI
|
|
USE PI_ETC, ONLY : PI2
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL (WP) :: X,XC,U0,UB
|
|
REAL (WP) :: Q_SI,Q_CO
|
|
REAL (WP) :: LN2
|
|
REAL (WP) :: F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
LN2=DLOG(TWO) !
|
|
!
|
|
! Initialization
|
|
!
|
|
F0S=ZERO !
|
|
F0A=ZERO !
|
|
F1S=ZERO !
|
|
F1A=ZERO !
|
|
F2S=ZERO !
|
|
F2A=ZERO !
|
|
!
|
|
Q_SI=TWO*X*KF_SI ! q in SI
|
|
Q_CO=TWO*XC*KF_SI ! q_c in SI
|
|
!
|
|
! Screened interaction
|
|
!
|
|
UB=U0*(ONE - U0*FOURTH*M_E/PI2 * HALF*( Q_CO + &!
|
|
(Q_SI*Q_SI-Q_CO*Q_CO)* &! ref. 1 p.
|
|
DLOG(DABS(Q_CO-Q_SI)/(Q_CO+Q_SI))/ &!
|
|
Q_SI)) !
|
|
!
|
|
F0S= UB * (ONE + UB*(ONE+HALF*THIRD*(TWO+LN2))) !
|
|
F0A=-UB * (ONE + UB*(ONE-TWO*THIRD*(ONE-LN2))) ! ref. (1) p. 22
|
|
F1S= UB*UB * TWO*(SEVEN*LN2-ONE)/15.0E0_WP !
|
|
!
|
|
END SUBROUTINE RASC_LP_3D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE SBOH_LP_3D(U0,W,D,F0S,F0A,F1S,F1A,F2S,F2A)
|
|
!
|
|
! This subroutine computes the standard dimensionless Landau parameters
|
|
! of the slave-boson one-band Hubbard model in the 3D case
|
|
!
|
|
! References: (1) T. Li and P. Bénard, Phys. Rev. B 50, 17837 (1994)
|
|
! (2) Slides
|
|
!
|
|
!
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * U0 : bare interaction constant
|
|
! * W : half bandwidth for bare particle (ref. 7)
|
|
! * D : filling (dopant concentration) in ref. 7
|
|
!
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 5 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,FOUR,FIVE, &
|
|
SIX,EIGHT,NINE,TEN
|
|
USE COMPLEX_NUMBERS, ONLY : ZEROC,ONEC
|
|
USE CONSTANTS_P1, ONLY : M_E
|
|
USE FERMI_SI, ONLY : KF_SI
|
|
USE PI_ETC, ONLY : PI2
|
|
USE POLYNOMIAL_EQ, ONLY : CUBIC_EQUATION,CHECK_ROOTS3
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL*8 U0
|
|
REAL*8 W,D,U
|
|
REAL*8 D2,D4,D6
|
|
REAL*8 Y,Y2,Y3,Y4,Y5,Y6,Y7
|
|
REAL*8 NUM,DEN
|
|
REAL*8 F0S,F0A,F1S,F1A,F2S,F2A
|
|
!
|
|
COMPLEX*16 AC,BC,CC,DC
|
|
COMPLEX*16 X1,X2,X3
|
|
!
|
|
! Initialization
|
|
!
|
|
F0S=ZERO !
|
|
F0A=ZERO !
|
|
F1S=ZERO !
|
|
F1A=ZERO !
|
|
F2S=ZERO !
|
|
F2A=ZERO !
|
|
!
|
|
! Scaled interaction
|
|
!
|
|
U=U0/(FOUR*W) ! ref. 7 eq. (59c)
|
|
!
|
|
!
|
|
! Computing Y, solution of:
|
|
!
|
|
! (1-Y)*Y^2
|
|
! --------- = U ref. (3) eq. (11)
|
|
! Y^2-D2
|
|
!
|
|
AC=ONEC !
|
|
BC=DCMPLX(U-ONE) !
|
|
CC=ZEROC !
|
|
DC=DCMPLX(-U*D*D) !
|
|
!
|
|
CALL CUBIC_EQUATION(AC,BC,CC,DC,X1,X2,X3) !
|
|
!
|
|
! Looking for a real and positive solution Y = x^2
|
|
!
|
|
CALL CHECK_ROOTS3(X1,X2,X3,Y) !
|
|
!
|
|
D2=D*D !
|
|
D4=D2*D2 !
|
|
D6=D4*D2 !
|
|
Y2=Y*Y !
|
|
Y3=Y2*Y !
|
|
Y4=Y3*Y !
|
|
Y5=Y4*Y !
|
|
Y6=Y5*Y !
|
|
Y7=Y6*Y !
|
|
!
|
|
NUM=(ONE-Y)*(TWO*D2-FIVE*D2*Y+TWO*Y2+D2*Y2+Y3-Y4) !
|
|
DEN=(D2-TWO*Y+Y2)*(-TWO*D2+THREE*D2*Y-Y3) !
|
|
F0S=NUM/DEN ! ref. (1) eq. (61)
|
|
!
|
|
NUM=(ONE-D2)*(Y2-D2) !
|
|
DEN=(TWO*Y-Y2-D2)**2 !
|
|
F0A=-ONE + NUM/DEN ! ref. (1) eq. (60)
|
|
!
|
|
NUM=THREE*(Y-ONE)**2 * ( -TEN*D4 + 19.0E0_WP*D4*Y - &!
|
|
EIGHT*D2*Y2 - &!
|
|
TWO*D2*Y3+TWO*Y4-Y5 &!
|
|
) !
|
|
DEN=( 16.0E0_WP*D4 - SIX*D6 - 48.0E0_WP*D4*Y + &!
|
|
NINE*D6*Y + EIGHT*D2*Y2 + 48.0E0_WP*D4*Y2 - &!
|
|
EIGHT*D2*Y3 - 25.0E0_WP*D4*Y3 - EIGHT*Y4 + &!
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TEN*D2*Y4 + EIGHT*Y5-D2*Y5 - FOUR*Y6 + &!
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Y7 &!
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) !
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F1A=NUM/DEN ! ref. (1) eq. (65)
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!
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NUM=THREE*(ONE-Y2)**2 !
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DEN=(TWO*Y-Y2-D2) !
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F1S=NUM/DEN ! ref. (1) eq. (23)
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!
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END SUBROUTINE SBOH_LP_3D
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!
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END MODULE LANDAU_PARAM
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