Green’s function and Lehmann representation

\[ \newcommand{\iv}{{\mathrm{i}\nu}} \newcommand{\wmax}{{\omega_\mathrm{max}}} \newcommand{\dd}{{\mathrm{d}}} \]

One-particle Green’s function

We introduce a Green’s function with imaginary arguments in time and frequency. This has no physical meaning but a mere mathematical trick to make calculations easier (to give another example of this: in Minkowski spacetime we take advantage of a similiar substitution).

The so-called imaginary-frequency (Matsubara) Green’s functions are defined as follows:

\[ G_{ij}(\tau-\tau') = -\langle T_\tau [c_i(\tau){c}^\dagger_j(\tau')]\rangle, \]

where \(i\) and \(j\) denote spin/orbital/band and \(T_\tau\) is the time-ordering operator. Here, \(\tau\) represents a imaginary time unit \(\mathrm{i}t\), while \(c_i\)/\(c_j\) is a fermionic or bosonic annihilation/creation operator.

The Fourier Transformation of \(G_{ij}(\tau)\) (defined for \(0 < \tau < \beta\)) reads

\[ G_{ij}(\iv) = \int_0^{\beta} \dd \tau e^{\iv\tau} G_{ij}(\tau), \]

where \(\nu = n\pi/\beta\) is a Matsubara frequency. The integer \(n\) is the reduced frequency, which the libraries take as input: \(n \equiv \zeta \pmod 2\), i.e., \(n\) is odd for fermions and even for bosons, where \(\zeta\) is the parity of the statistics, \(\zeta = 1\) (fermion) and \(\zeta = 0\) (boson). In terms of the ordinary Matsubara index \(m\), \(n = 2m + \zeta\), i.e., \(\nu = (2m+1)\pi/\beta\) (fermion) and \(\nu = 2m\pi/\beta\) (boson). The inverse temperature is denoted by \(\beta\) (We take \(\hbar=1\)). The inverse transformation is given by

\[ G_{ij}(\tau) = \frac{1}{\beta}\sum_{\nu} e^{-\iv\tau}G_{ij}(\iv), \]

where the sum runs over all fermionic (bosonic) Matsubara frequencies.

Continuing \(G_{ij}(\iv)\) to a holomorphic function in the upper half of the complex plane, the imaginary-frequency (Matsubara) Green’s function can be related to the “ordinary” retarded Green’s function as

\[ G_{ij}^\mathrm{R}(\omega)=G_{ij}(z \rightarrow \omega+\mathrm{i}0^{+}). \]

In the following, we omit the indices \(i\) and \(j\) unless there is confusion. The notation is summarized in Notation and conventions.

Lehmann representation

In the imaginary-frequency domain, the Lehmann representation reads

\[ \begin{align} G(\iv) &= \int_{-\infty}^\infty \dd\omega \underbrace{\frac{1}{\iv - \omega}}_{\equiv K(\iv, \omega)} A(\omega), \end{align} \]

where \(A(\omega)\) is a spectral function. In terms of retarded and advanced Green’s functions in real frequency, the spectral function is related to them as

\[ A_{ii}(\omega) = -\frac{1}{\pi} \operatorname{Im} G^R_{ii}(\omega) \]

for the diagonal (local) components, and more generally

\[ \boxed{A_{ij}(\omega) = \frac{\mathrm{i}}{2\pi}\left(G^R_{ij}(\omega) - G^A_{ij}(\omega)\right)}. \]

Here \(G^R\) and \(G^A\) denote the retarded and advanced Green’s functions, respectively. \(K(\iv, \omega)\) is the so-called analytic continuation kernel. The Lehmann representation can be transformed to the imaginary-time domain as

(1)\[ \begin{align} G(\tau) &= - \int_{-\infty}^\infty \dd\omega K(\tau, \omega) A(\omega), \end{align} \]

where the primary domain is the open interval \(0 < \tau < \beta\) and

\[\begin{split} \begin{align} K(\tau, \omega) &\equiv - \frac{1}{\beta} \sum_{\nu} e^{-\iv \tau} K(\iv, \omega) = \begin{cases} \frac{e^{-\tau\omega}}{1+e^{-\beta\omega}} & (\mathrm{fermion}),\\ \frac{e^{-\tau\omega}}{1-e^{-\beta\omega}} & (\mathrm{boson}) \end{cases}. \end{align} \end{split}\]

The minus sign in (1) is a convention that makes the fermionic kernel positive, \(K(\tau, \omega) > 0\); this kernel is the logistic kernel \(K^\mathrm{L}\) used below. The bosonic kernel is negative for \(\omega < 0\).

Imaginary-time domain, (anti-)periodicity, and special points

The (anti-)periodicity in imaginary time is a symmetry dictated by statistics and should be shared consistently by the Green’s function \(G(\tau)\) and any basis functions used to represent it (e.g. IR/DLR basis functions in \(\tau\)). With the parity \(\zeta=1\) (fermion) and \(\zeta=0\) (boson), the rule is

\[ f(\tau+\beta)=(-1)^\zeta f(\tau), \]

for \(\tau\) away from boundary/special points, where \(f\) may stand for \(G\) itself or a basis function.

In practice, \(G(\tau)\) (and likewise the basis functions) are smooth on \((0,\beta)\), while the boundary points \(\tau\in\{-\beta,\pm 0,\beta\}\) require one-sided interpretations:

• \(\tau=0\) and \(\tau=\beta\) are understood as limits, \(0^+\) and \(\beta^-\); the floating-point input \(-0.0\) is read as \(0^-\) and \(\tau=-\beta\) as \((-\beta)^+\). The periodicity relates the endpoint values as \(G(0^-)=(-1)^\zeta G(\beta^-)\).

• \(G(\tau)\) is discontinuous at \(\tau=0\): for the Green’s function of an elementary operator \(c\), \(G(\tau) = -\langle T_\tau c(\tau) c^\dagger(0) \rangle\), the jump is \(G(0^+)-G(0^-)=-1\) for both statistics, so that \(G(0^+)-(-1)^\zeta G(\beta^-)=-1\).

• Values at negative \(\tau\), \(-\beta \le \tau < 0\), follow from \(f(\tau)=(-1)^\zeta f(\tau+\beta)\). The libraries evaluate functions of \(\tau\) on \([-\beta,\beta]\) and raise an error outside this range.

For implementation details (including the distinction between \(+0\) and \(-0\) in floating-point arithmetic), see Periodicity of Green’s functions in imaginary time.

Regularization of the bosonic kernel

The bosonic kernel \(K(\tau, \omega)\) above diverges at \(\omega = 0\):

\[ K(\tau, \omega) = \frac{e^{-\tau\omega}}{1-e^{-\beta\omega}} \sim \frac{1}{\beta\omega} \quad (\omega \to 0). \]

To perform the singular value expansion numerically, this divergence must be regularized. There are two common approaches:

Method 1: Logistic kernel with modified spectral function

This approach, introduced in [Kaye et al., 2022], uses the logistic kernel for both fermions and bosons:

(2)\[ K^\mathrm{L}(\tau, \omega) = \frac{e^{-\tau\omega}}{1+e^{-\beta\omega}}. \]

The Lehmann representation is reformulated as

\[ G(\tau)= - \int_{-\infty}^\infty\dd{\omega} K^\mathrm{L}(\tau,\omega) \rho(\omega), \]

where \(\rho(\omega)\) is the modified spectral function:

\[\begin{split} \begin{align} \rho(\omega) &\equiv \begin{cases} A(\omega) & (\mathrm{fermion}),\\ \displaystyle\frac{A(\omega)}{\tanh(\beta \omega/2)} & (\mathrm{boson}). \end{cases} \end{align} \end{split}\]

Advantage: The same kernel \(K^\mathrm{L}\) can be used for both fermions and bosons, simplifying the implementation. It is the default kernel of the libraries for both statistics.

Note: For bosons, the spectral function \(A(\omega)\) must vanish at least linearly at \(\omega = 0\) to compensate for the \(1/\tanh(\beta\omega/2) \sim 2/(\beta\omega)\) factor, so that \(\rho(\omega)\) stays finite.

Method 2: Regularized Bose kernel (deprecated)

This approach, used in [Shinaoka et al., 2017], introduces a regularized bosonic kernel:

(3)\[ K^\mathrm{B}(\tau, \omega) = \omega \cdot \frac{e^{-\tau\omega}}{1-e^{-\beta\omega}}. \]

This kernel is deprecated; use the logistic kernel (Method 1) instead. Its definition here, together with the dimensionless form given below, is the reference definition of the irbasis paper [Chikano et al., 2019].

The factor \(\omega\) cancels the \(1/\omega\) divergence, making the kernel well-behaved at \(\omega = 0\). The Lehmann representation becomes

\[ G(\tau)= - \int_{-\infty}^\infty\dd{\omega} K^\mathrm{B}(\tau,\omega) \rho'(\omega), \]

where \(\rho'(\omega) = A(\omega)/\omega\) is the modified spectral function.

Basis size: As shown in [Shinaoka et al., 2017], the number of basis functions grows only logarithmically with \(\Lambda = \beta\wmax\). The same holds for the logistic kernel of Method 1, so this is not an advantage of this kernel.

Note: The physical spectral function \(A(\omega)\) must vanish at least linearly at \(\omega = 0\) for the integral to converge.

Comparison

Property	Logistic kernel	Regularized Bose kernel
Status	Default	Deprecated
Fermion support	Yes	No
Boson support	Yes (with modified \(\rho\))	Yes
Kernel form	\(K^\mathrm{L} = \frac{e^{-\tau\omega}}{1+e^{-\beta\omega}}\)	\(K^\mathrm{B} = \omega \cdot \frac{e^{-\tau\omega}}{1-e^{-\beta\omega}}\)
Modified spectral function	\(\rho = A/\tanh(\beta\omega/2)\)	\(\rho' = A/\omega\)
Implementation	Unified for F/B	Separate for B

Non-dimensionalization of kernels

For numerical work it is convenient to introduce dimensionless variables. We define the dimensionless parameters

\[ \Lambda \equiv \beta \wmax, \qquad x \equiv 2\tau/\beta - 1 \in [-1,1], \qquad y \equiv \omega/\wmax \in [-1,1]. \]

In terms of \((x,y)\), the logistic kernel becomes

\[ K^\mathrm{L}(x, y) = \frac{\exp[-\Lambda y (x+1)/2]}{1 + \exp[-\Lambda y]}, \]

which is the form used internally in the IR and DLR implementations. The physical kernel in \((\tau,\omega)\) is obtained by the change of variables above, with the integration range \(\omega \in [-\wmax, \wmax]\).

For the (deprecated) regularized Bose kernel, the dimensionless form is

\[ K^\mathrm{B}(x, y) = y \frac{\exp[-\Lambda y (x+1)/2]}{1 - \exp[-\Lambda y]}, \]

and the dimensional kernel is recovered via

\[ K^\mathrm{B}(\tau, \omega) = \wmax \; K^\mathrm{B}(x, y), \]

with the same definitions of \(x\), \(y\), and \(\Lambda\) as above. This convention is consistent with the implementation in the Rust backend (see kernel.rs), and allows us to tabulate and manipulate kernels on the compact domain \(x,y \in [-1,1]\) while keeping the dependence on \(\beta\) and \(\wmax\) only through \(\Lambda\). With this definition, the singular values of \(K^\mathrm{B}(\tau, \omega)\) are \(S_l = \wmax^{+1} \sqrt{\beta\wmax/2}\, s_l\), where \(s_l\) are those of \(K^\mathrm{B}(x, y)\) (see Notation and conventions). libsparseir releases without the fix of sparse-ir-rs#273 scale the singular values of bases built from this kernel by \(\wmax^{-1}\) instead of \(\wmax^{+1}\).