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Last edited by Gabriel Wlazłowski
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Reference scales

General info

During computation, W-SLDA codes exploit information about the typical scales present in the problem. Precisely, reference scales define typical orders of magnitude for computed quantities. The most important reference scale is the Fermi momentum. For a uniform system, it is defined as

  • k_F^{(1D)}=\frac{\pi n}{2}
  • k_F^{(2D)}=\sqrt{2\pi n}
  • k_F^{(3D)}=(3\pi^2 n)^{1/3}

Other reference scales computed automatically from k_F are:

  • \varepsilon_F=\frac{1}{2}k_F^2 - Fermi energy,
  • E_{\textrm{ffg}}=c_E N\varepsilon_F - energy of free Fermi gas, where the coefficient in front depends on dimensionality:
    • c_E^{(1D)}=\frac{1}{3},
    • c_E^{(2D)}=\frac{1}{2},
    • c_E^{(3D)}=\frac{3}{5}.

Finally, chemical potentials also serve as reference scales for static problems:

  • \mu_{a} - chemical potential is spin-up particles (particles of type a),
  • \mu_{b} - chemical potential is spin-down particles (particles of type b).

Defining reference scales for static calculation

Fermi momentum kF

There are the following methods of defining the k_F reference scale:

  • via input file: k_F is provided by user in input file. To activate this mode, you need to uncomment the tag referencekF:
referencekF             1.0    # hard set for reference value of kF
  • via problem-definition.h file: (VERSION>=2022.02.21) by editing function:
/**
 * This function computes the Fermi momentum, which is used as the reference value. 
 * Other reference scales are set automatically to: eF=kF^2/2, Effg=(3/5)*N*eF (N-total number of particles)
 * For more details, see: Wiki -> Setting reference scales.
 * NOTE units are: hbar=m=k_b=1
 * @param it iteration number
 * @param h_densities structure with densities, see (wiki) documentation for the list of fields
 * @param params array of input parameters, before the call of this routine, the params array is processed by process_params() routine
 * @param extra_data_size size of extra_data in bytes, if extra_data size=0 the optional data is not uploaded
 * @param extra_data optional set of data uploaded by load_extra_data()
 * @return value of Fermi momentum for your problem
 * */
double referencekF(int it, wslda_density h_densities, double *params, size_t extra_data_size, void *extra_data)
{
    if(input->referencekF>0.0) return input->referencekF; // take it from input file
    
    // define here your prescription for computing kF
    // ...
    // default: extract max density and use it for definition of kF
    double max_dens=0.0, kF;
    int ixyz;
    for(ixyz=0; ixyz<h_densities.nx*h_densities.ny*h_densities.nz; ixyz++) 
        if(h_densities.rho_a[ixyz]+h_densities.rho_b[ixyz]>max_dens) max_dens=h_densities.rho_a[ixyz]+h_densities.rho_b[ixyz];
    
    // depending on the dimensionality of the problem
    if(NY==1 && NZ==1) kF = 0.5*M_PI*max_dens;                // 1D
    else if(NZ==1)     kF = pow(2.0*M_PI*max_dens,1./2.);     // 2D
    else               kF = pow(3.*M_PI*M_PI*max_dens,1./3.); // 3D
    
    return kF;
}

Fermi energy eF

Over the entire code, it is defined as \varepsilon_F=\frac{1}{2}k_F^2.

Free Fermi gas energy Effg

API_VERSION>=20221120
The quantity is used only for reporting energy values. The definition can be controlled via the logger.h file, by changing the body of the function energy_unit(...). Default values are computed as:

/**
 * This function defines the unit in which energies are printed in stdout.
 * @param kF typical Fermi momentum scale of the problem, value returned by referencekF() function.
 * @param mu array with chemical potentials: mu[SPINA], mu[SPINB].
 * @param npart array with computed particle numbers: npart[SPINA] and npart[SPINB].
 * @param params array of input parameters, before the call of this routine, the params array is processed by process_params() routine
 * @param extra_data_size size of extra_data in bytes, if extra_data size=0 the optional data is not uploaded
 * @param extra_data optional set of data uploaded by load_extra_data()
 * */
double energy_unit(double kF, double *mu, double *npart,
           double *params, size_t extra_data_size, void *extra_data)
{
    double Effg;
    double eF = kF*kF/2.0; // Fermi energy
    double N = npart[SPINA]+npart[SPINB]; // total number of particles

    // depending on the dimensionality of the problem
    if(NY==1 && NZ==1) Effg=(1./3.)*N*eF;   // 1D
    else if(NZ==1)     Effg=(1./2.)*N*eF;   // 2D
    else               Effg=(3./5.)*N*eF;   // 3D

    return Effg;
}

Chemical potentials mu

Chemical potentials are automatically adjusted when the mode with fixed particle number is used. For the mode with fixed chemical potential, see here.

Examples

Fermi momentum is fixed by density in the box center

double referencekF(int it, wslda_density h_densities, double *params, size_t extra_data_size, void *extra_data)
{
    if(input->referencekF>0.0) return input->referencekF; // take it from input file
    
    // Fermi momentum is fixed by density in the box center
    // DETERMINE LOCAL SIZES OF ARRAYS (CODE DIMENSIONALITY DEPENDENT)
    int lNX=h_densities.nx, lNY=h_densities.ny, lNZ=h_densities.nz; // local sizes

    // take value of density in box center and save it to extra_data
    int ixyz = lNZ/2 + lNZ*lNY/2 + lNZ*lNY*lNX/2;
    double dens = h_densities.rho_a[ixyz]+h_densities.rho_b[ixyz];

    
    // depending on the dimensionality of the problem
    double kF;
    if(NY==1 && NZ==1) kF = 0.5*M_PI*dens;                // 1D
    else if(NZ==1)     kF = pow(2.0*M_PI*dens,1./2.);     // 2D
    else               kF = pow(3.*M_PI*M_PI*dens,1./3.); // 3D
    
    return kF;
}

Defining reference scales for time-dependent calculations

All reference scales are provided along with an initial state; i.e., the binary files produced by static codes contain this information. Presently, there is no option to change values for reference scales in time-dependent codes.