gftool.density
- gftool.density(gf_iw, potential, beta, return_err=True, matrix=False, total=False)[source]
Calculate the number density of the Green’s function gf_iw at finite temperature beta.
Deprecated since version 0.8.0: Mostly superseded by more flexible
density_iw
, thus this function will likely be discontinued. Currentlydensity
is a little more accurate for matrix=True, compared todensity_iw
without using fitting.As Green’s functions decay only as \(1/ω\), the known part of the form \(1/(iω_n + μ - ϵ - ℜΣ_{\text{static}})\) will be calculated analytically. \(Σ_{\text{static}}\) is the ω-independent mean-field part of the self-energy.
- Parameters:
- gf_iwcomplex ndarray
The Matsubara frequency Green’s function for positive frequencies \(iω_n\). The last axis corresponds to the Matsubara frequencies.
- potentialfloat ndarray or float
The static potential for the large-ω behavior of the Green’s function. It is the real constant \(μ - ϵ - ℜΣ_{\text{static}}\). The shape must agree with gf_iw without the last axis. If
matrix
, then potential needs to be a (N, N) matrix. It is the negative of the Hamiltonian matrix and thus needs to be hermitian.- betafloat
The inverse temperature beta = 1/T.
- return_errbool or float, optional
If True (default), the error estimate will be returned along with the density. If return_err is a float, a warning will Warning will be issued if the error estimate is larger than return_err. If False, no error estimate is calculated. See
density_error
for description of the error estimate.- matrixbool, optional
Whether the given potential is a matrix (default: False).
- totalbool or tuple
If total the total density (summed over all dimensions) is returned. Also a tuple can be given, over which axes the sums is taken.
- Returns:
- xfloat
The number density of the given Green’s function gf_iw.
- errfloat
An estimate for the density error. Only returned if return_err is True.
Notes
The number density can be obtained from the Matsubara frequency Green’s function using
\[⟨n⟩ = \lim_{ϵ↗0} G(τ=-ϵ) = 1/β ∑_{n=-∞}^{∞} G(iω_n)\]As Green’s functions decay only as \(O(1/ω)\), truncation of the summation yields a non-vanishing contribution of the tail. For the analytic structure of the Green’s function see [2], [3]. To take this into consideration the known part of the form \(1/(iω_n + μ - ϵ - ℜΣ_{\text{static}})\) will be calculated analytically. This yields [1]
\[\begin{split}⟨n⟩ = 1/β ∑_{n=-∞}^{∞} [G(iω_n) - 1/(iω_n + μ - ϵ - ℜΣ_{\text{static}})] \\ + 1/2 + 1/2 \tanh[1/2 β(μ - ϵ - ℜΣ_{\text{static}})].\end{split}\]We can use the symmetry \(G(z*) = G^*(z)\) do reduce the sum only over positive Matsubara frequencies
\[\begin{split}∑_{n=-∞}^{∞} G(iω_n) &= ∑_{n=-∞}^{-1} G(iω_n) + ∑_{n=0}^{n=∞} G(iω_n) \\ &= ∑_{n=0}^{∞} [G(-iω_n) + G(iω_n)] \\ &= 2 ∑_{n=0}^{∞} ℜG(iω_n).\end{split}\]Thus we get the final expression
\[\begin{split}⟨n⟩ = 2/β ∑_{n≥0} ℜ[G(iω_n) - 1/(iω_n + μ - ϵ - ℜΣ_{\text{static}})] \\ + 1/2 + 1/2 \tanh[1/2 β(μ - ϵ - ℜΣ_{\text{static}})].\end{split}\]References
[1]Hale, S. T. F., and J. K. Freericks. “Many-Body Effects on the Capacitance of Multilayers Made from Strongly Correlated Materials.” Physical Review B 85, no. 20 (May 24, 2012). https://doi.org/10.1103/PhysRevB.85.205444.
[2]Eder, Robert. “Introduction to the Hubbard Mode.” In The Physics of Correlated Insulators, Metals and Superconductors, edited by Eva Pavarini, Erik Koch, Richard Scalettar, and Richard Martin. Schriften Des Forschungszentrums Jülich Reihe Modeling and Simulation 7. Jülich: Forschungszentrum Jülich, 2017. https://www.cond-mat.de/events/correl17/manuscripts/eder.pdf.
[3]Luttinger, J. M. “Analytic Properties of Single-Particle Propagators for Many-Fermion Systems.” Physical Review 121, no. 4 (February 15, 1961): 942–49. https://doi.org/10.1103/PhysRev.121.942.