gftool.density

gftool.density(gf_iw, potential, beta, return_err=True, matrix=False, total=False)

Calculate the number density of the Green’s function gf_iw at finite temperature beta.

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.

Deprecated since version 0.8.0: Mostly superseded by more flexible density_iw, thus this function will likely be discontinued. Currently density is a little more accurate for matrix=True, compared to density_iw without using fitting.

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.