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 ofrectangle
,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 ofrectangle
,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 ofrectangle
,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 ofrectangle
,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 constructcont
.Mesh: triqs.gf.meshes.MeshReFreq
, Real energy mesh.R_func: str
, Name of the resolution function \(\bar K(m, z)\), one ofrectangle
,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 constructcont
.mesh: triqs.gf.meshes.MeshReFreq
, Real energy mesh.r_func: str
, Name of the resolution function \(\bar K(m, z)\), one ofrectangle
,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 constructcont
.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 ofrectangle
,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 constructcont
.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 ofrectangle
,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 constructcont
.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 ofrectangle
,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 constructcont
.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 ofrectangle
,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 constructcont
.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 ofrectangle
,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 constructcont
.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 ofrectangle
,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 constructcont
.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 ofrectangle
,lorentzian
,gaussian
.Returns: Real 2D NumPy array of values \(\sigma_{mm'}\).