som.spectral_stats: Statistical analysis of noisy spectral functions

Functions in this module implement the statistical analysis technique for ensembles of spectral functions described in Sections I-II of [GMPPS2017].

som.spectral_stats.spectral_integral()

Evaluate spectral integral

\[i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)\]

for a single energy interval.

Parameters:

Z_m:float, Center of the energy interval. Delta_m:float, Length of the energy interval. C:som.Configuration, Spectral function \(A^{(j)}(z)\). R_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: float, Value of the integral.

Signature(float z_m, float delta_m, configuration c, resolution_function r_func) -> float

Evaluate spectral integral

\[i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)\]

for a single energy interval.

Parameters:

z_m:float, Center of the energy interval. delta_m:float, Length of the energy interval. c:som.Configuration, Spectral function \(A^{(j)}(z)\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: float, Value of the integral.

Signature(triqs::mesh::refreq mesh, configuration c, resolution_function r_func) -> vector<double>

Evaluate spectral integrals

\[i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. c:som.Configuration, Spectral function \(A^{(j)}(z)\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(i_m^{(j)}\).

Signature(list[std::pair<double,double>] intervals, configuration c, resolution_function r_func) -> vector<double>

Evaluate spectral integrals

\[i_m^{(j)} = \int_{-\infty}^\infty dz \bar K(m, z) A^{(j)}(z)\]

for a list of real energy intervals.

Parameters:

intervals:list [float, float], List of pairs (left interval boundary, right interval boundary). c:som.Configuration, Spectral function \(A^{(j)}(z)\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(i_m^{(j)}\).

som.spectral_stats.spectral_avg()

Compute spectral averages over a set of accumulated particular solutions

\[i_m = \frac{1}{J} \sum_{j=1}^J i_m^{(j)}\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

Cont:Som, Analytic continuation object. I:int, Index of the diagonal matrix element of the observable used to construct cont. Mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. R_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(i_m\).

Signature(som_core cont, int i, triqs::mesh::refreq mesh, resolution_function r_func) -> vector<double>

Compute spectral averages over a set of accumulated particular solutions

\[i_m = \frac{1}{J} \sum_{j=1}^J i_m^{(j)}\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

cont:Som, Analytic continuation object. i:int, Index of the diagonal matrix element of the observable used to construct cont. mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(i_m\).

Signature(som_core cont, int i, list[std::pair<double,double>] intervals, resolution_function r_func) -> vector<double>

Compute spectral averages over a set of accumulated particular solutions

\[i_m = \frac{1}{J} \sum_{j=1}^J i_m^{(j)}\]

for a list of real energy intervals.

Parameters:

cont:Som, Analytic continuation object. i:int, Index of the diagonal matrix element of the observable used to construct cont. intervals:list [float, float], List of pairs (left interval boundary, right interval boundary). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(i_m\).

som.spectral_stats.spectral_disp()

Compute spectral dispersions of a set of accumulated particular solutions

\[\sigma^2_m = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m)^2\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

Cont:Som, Analytic continuation object. I:int, Index of the diagonal matrix element of the observable used to construct cont. Mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. Avg:Real 1D NumPy array of precomputed averages \(i_m\). R_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(\sigma_m\).

Signature(som_core cont, int i, triqs::mesh::refreq mesh, vector<double> avg, resolution_function r_func) -> vector<double>

Compute spectral dispersions of a set of accumulated particular solutions

\[\sigma^2_m = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m)^2\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

cont:Som, Analytic continuation object. i:int, Index of the diagonal matrix element of the observable used to construct cont. mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. avg:Real 1D NumPy array of precomputed averages \(i_m\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(\sigma_m\).

Signature(som_core cont, int i, list[std::pair<double,double>] intervals, vector<double> avg, resolution_function r_func) -> vector<double>

Compute spectral dispersions of a set of accumulated particular solutions

\[\sigma^2_m = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m)^2\]

for a list of real energy intervals.

Parameters:

cont:Som, Analytic continuation object. i:int, Index of the diagonal matrix element of the observable used to construct cont. intervals:list [float, float], List of pairs (left interval boundary, right interval boundary). avg:Real 1D NumPy array of precomputed averages \(i_m\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 1D NumPy array of values \(\sigma_m\).

som.spectral_stats.spectral_corr()

Compute spectral two-point correlators of a set of accumulated particular solutions

\[\sigma_{mm'} = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m) (i_{m'}^{(j)} - i_{m'})\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

Cont:Som, Analytic continuation object. I:int, Index of the diagonal matrix element of the observable used to construct cont. Mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. Avg:Real 1D NumPy array of precomputed averages \(i_m\). R_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 2D NumPy array of values \(\sigma_{mm'}\).

Signature(som_core cont, int i, triqs::mesh::refreq mesh, vector<double> avg, resolution_function r_func) -> matrix<double>

Compute spectral two-point correlators of a set of accumulated particular solutions

\[\sigma_{mm'} = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m) (i_{m'}^{(j)} - i_{m'})\]

for energy intervals centered around points of a regular energy mesh.

Parameters:

cont:Som, Analytic continuation object. i:int, Index of the diagonal matrix element of the observable used to construct cont. mesh:triqs.gf.meshes.MeshReFreq, Real energy mesh. avg:Real 1D NumPy array of precomputed averages \(i_m\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 2D NumPy array of values \(\sigma_{mm'}\).

Signature(som_core cont, int i, list[std::pair<double,double>] intervals, vector<double> avg, resolution_function r_func) -> matrix<double>

Compute spectral two-point correlators of a set of accumulated particular solutions

\[\sigma_{mm'} = \frac{1}{J} \sum_{j=1}^J (i_m^{(j)} - i_m) (i_{m'}^{(j)} - i_{m'})\]

for a list of real energy intervals.

Parameters:

cont:Som, Analytic continuation object. i:int, Index of the diagonal matrix element of the observable used to construct cont. intervals:list [float, float], List of pairs (left interval boundary, right interval boundary). avg:Real 1D NumPy array of precomputed averages \(i_m\). r_func:str, Name of the resolution function \(\bar K(m, z)\), one of rectangle, lorentzian, gaussian.

Returns: Real 2D NumPy array of values \(\sigma_{mm'}\).