Green's functions & LKAG

The single-particle Green’s function is the central object of interacting electronic structure: it is the amplitude for adding a particle (or hole) to the interacting ground state, propagating it, and removing it again. Its poles are the exact charged excitation energies, the weights of those poles are how “particle-like” the excitations remain once interactions are switched on, and everything the rest of the electrons do to a given particle is packaged into one self-energy $\Sigma$. I keep two uses of $G$ in view here. The first is spectroscopic — the spectral function $A(\omega )=-\frac{1}{\pi }\mathrm{Im}G$ is what photoemission measures, and resumming $\Sigma =iGW$ gives the GW quasiparticle bands. The second is magnetic — the LKAG / magnetic force theorem writes the Heisenberg couplings $J_{ij}$ as an energy integral of products of intersite Green’s functions, the machinery behind TB2J and the Green’s-function route to exchange tensors from DFT.

Definition and the spectral (Lehmann) representation

Work in the grand-canonical ensemble at chemical potential $\mu$, with field operators $\hat{\psi }(x)$ in the Heisenberg picture, $x=(\mathbf{r},\sigma ,t)$. The time-ordered single-particle Green’s function is

$$G(x,x^{\prime })=-i⟨\mathcal{T}\hat{\psi }(x){\hat{\psi }}^{†}(x^{\prime })⟩,$$

with $\mathcal{T}$ the time-ordering operator (a relative minus sign on fermionic interchange) and $⟨\cdots ⟩$ the ground-state (or thermal) average. For a time-independent Hamiltonian $G$ depends on $t-t^{\prime }$; Fourier transforming and inserting the complete set of exact $(N\pm 1)$-particle eigenstates $|m⟩$ of $\hat{H}-\mu \hat{N}$ yields the Lehmann representation

$$G(\mathbf{r},{\mathbf{r}}^{\prime };\omega )=\sum _m\left[\frac{⟨0|\hat{\psi }(\mathbf{r})|m⟩⟨m|{\hat{\psi }}^{†}({\mathbf{r}}^{\prime })|0⟩}{\omega -(E_m^{N+1}-E_0^N)+i\eta }+\frac{⟨0|{\hat{\psi }}^{†}({\mathbf{r}}^{\prime })|m⟩⟨m|\hat{\psi }(\mathbf{r})|0⟩}{\omega -(E_0^N-E_m^{N-1})-i\eta }\right],$$

( $\eta \to 0^{+}$ ). The first term has poles at the electron-addition energies, the second at the removal energies; measured from $\mu$, all addition poles sit at $\omega >0$ and all removal poles at $\omega <0$. In an eigenbasis of a single quantum number (e.g. momentum/band $\nu$), $G_{\nu }(\omega )={\sum }_m|⟨m|{\hat{c}}_{\nu }^{†}|0⟩{|}^2/(\omega -{\epsilon }_m+i\eta \mathrm{sgn}{\epsilon }_m)$. The exact excitation spectrum is thus encoded in the analytic structure of $G$ — a fact that no finite-order perturbation theory changes.

The spectral function and sum rules

The spectral function is the (positive) discontinuity of $G$ across the real axis,

$$A(\mathbf{r},{\mathbf{r}}^{\prime };\omega )=-\frac{1}{\pi }\mathrm{Im}{G}^R(\mathbf{r},{\mathbf{r}}^{\prime };\omega )=\sum _m[⟨0|\hat{\psi }(\mathbf{r})|m⟩⟨m|{\hat{\psi }}^{†}({\mathbf{r}}^{\prime })|0⟩\delta (\omega -E_m^{+})+(\text{removal})],$$

using the retarded $G^R$ (poles pushed into the lower half-plane, the correct analytic continuation for response). Diagonal in $\mathbf{r}$ and integrated, $A(\omega )={\sum }_{\nu }{A}_{\nu }(\omega )\ge 0$ is the density of charged single-particle excitations — exactly the (angle-resolved) photoemission / inverse-photoemission lineshape. Two exact constraints anchor it:

$${\int }_{-\infty }^{\infty }{A}_{\nu }(\omega )d\omega =1(\text{spectral-weight sum rule, from the CAR}),G_{\nu }(\omega )={\int }_{-\infty }^{\infty }\frac{A_{\nu }({\omega }^{\prime })}{\omega -{\omega }^{\prime }+i\eta \mathrm{sgn}({\omega }^{\prime }-\mu )}d{\omega }^{\prime }.$$

The first is the statement that the total weight available to one mode is one particle; the second (a Kramers–Kronig / Hilbert transform) says $A$ determines $G$ everywhere off the real axis. The occupied part, ${\int }_{-\infty }^{\mu }{A}_{\nu }d\omega =n_{\nu }$, returns the momentum distribution, whose discontinuity at the Fermi surface is the quasiparticle weight $Z$ below.

Dyson equation and the self-energy

Split $\hat{H}={\hat{H}}_0+\hat{V}$ with ${\hat{H}}_0$ solvable (Green’s function $G_0$) and $\hat{V}$ the interaction. Summing the perturbation series for $G$ defines the self-energy $\Sigma$ as the sum of all one-particle-irreducible insertions; the resummation is the Dyson equation

$$G(\omega )=G_0(\omega )+G_0(\omega )\Sigma (\omega )G(\omega )⟺G(\omega )=[\omega -H_0-\Sigma (\omega ){]}^{-1},$$

(operator/matrix form in any single-particle basis). $\Sigma (\omega )$ is in general non-Hermitian, nonlocal in $\mathbf{r}$, and energy-dependent: it is the complete container for the effect of all other particles on the one being propagated. Its real part shifts levels (and, in DFT-based many-body theory, replaces the exchange–correlation potential: $\Sigma$ is the nonlocal, frequency-dependent generalization of $v_{\mathrm{xc}}$); its imaginary part gives excitations a finite lifetime. The Hedin equations close the set $\{G,\Sigma ,W,P,\Gamma \}$ self-consistently; truncating the vertex at $\Gamma =1$ gives $\Sigma =iGW$, the bridge this note builds toward in GW.

Quasiparticle pole versus incoherent weight

Near a solution of the quasiparticle equation $\omega ={\epsilon }_{\nu }^0+\mathrm{Re}{\Sigma }_{\nu }(\omega )$ at $\omega =E_{\nu }$, expand $\Sigma$ to first order. The spectral function splits into a sharp pole plus a smooth background,

$$A_{\nu }(\omega )=\underset{\text{coherent quasiparticle}}{\underbrace{Z_{\nu }\frac{1}{\pi }\frac{|\mathrm{Im}{\Sigma }_{\nu }(E_{\nu })|}{(\omega -E_{\nu }{)}^2+(\mathrm{Im}{\Sigma }_{\nu }(E_{\nu }){)}^2}}}+A_{\nu }^{\mathrm{inc}}(\omega ),Z_{\nu }={\left[1-\frac{\partial \mathrm{Re}{\Sigma }_{\nu }}{\partial \omega }{|}_{E_{\nu }}\right]}^{-1}\in (0,1].$$

The quasiparticle: a Lorentzian of width ${\Gamma }_{\nu }=Z_{\nu }|\mathrm{Im}{\Sigma }_{\nu }(E_{\nu })|$ (inverse lifetime) centred at the renormalized energy $E_{\nu }$, carrying only a fraction $Z_{\nu }$ of the unit spectral weight. The missing $1-Z_{\nu }$ is the incoherent weight $A^{\mathrm{inc}}$ — satellites (e.g. plasmon sidebands), Hubbard bands, the lower/upper bands of a Mott insulator — the part of the electron that is not a long-lived single-particle excitation. $Z_{\nu }$ is simultaneously the quasiparticle residue, the strength of the Fermi-surface step in $n_{\nu }$, and (one factor in) the mass renormalization

$$\frac{m^{\ast }}{m}=\frac{1-{\partial }_{\omega }\mathrm{Re}\Sigma }{1+v_{F0}^{-1}{\partial }_k\mathrm{Re}\Sigma }=\frac{Z^{-1}}{1+{\partial }_{{\epsilon }_k}\mathrm{Re}\Sigma },v_{F0}={\partial }_k{\epsilon }_k^0,$$

along the band — where $Z^{-1}=1-{\partial }_{\omega }\mathrm{Re}\Sigma$ is the frequency derivative at fixed $\mathbf{k}$ (the residue), and the momentum derivative enters only through the dimensionless ratio $v_{F0}^{-1}{\partial }_k\mathrm{Re}\Sigma ={\partial }_{{\epsilon }_k}\mathrm{Re}\Sigma$ in the denominator, normalized by the bare Fermi velocity $v_{F0}={\partial }_k{\epsilon }_k^0$ — ${\partial }_k\mathrm{Re}\Sigma$ on its own carries units of velocity and cannot sit inside a dimensionless $1-(\cdots )$. $Z\to 0$ signals the breakdown of the quasiparticle picture (Mott physics). Keeping $Z$ explicit is what distinguishes a genuine many-body spectrum from a renormalized band structure.

Matsubara (finite-temperature) formalism

At temperature $T=1/\beta$ the cleanest route is imaginary time $\tau \in [0,\beta )$ and the Matsubara Green’s function

$${\mathcal{G}}_{\nu }(\tau )=-⟨{\mathcal{T}}_{\tau }{\hat{c}}_{\nu }(\tau ){\hat{c}}_{\nu }^{†}(0)⟩,{\mathcal{G}}_{\nu }(i{\omega }_n)=[i{\omega }_n-{\epsilon }_{\nu }^0-{\Sigma }_{\nu }(i{\omega }_n){]}^{-1},$$

which (anti)periodicity $\mathcal{G}(\tau +\beta )=\mp \mathcal{G}(\tau )$ restricts to the discrete Matsubara frequencies ${\omega }_n=(2n+1)\pi /\beta$ (fermions) or $2n\pi /\beta$ (bosons). Its advantages are practical: the perturbation expansion has the same diagram rules as $T=0$ but with discrete sums $\frac{1}{\beta }{\sum }_n$ replacing $\int \frac{d\omega }{2\pi }$, and thermal weights enter automatically. The spectral function recovers everything via the same Lehmann kernel, ${\mathcal{G}}_{\nu }(i{\omega }_n)=\int d\omega A_{\nu }(\omega )/(i{\omega }_n-\omega )$, and the retarded function is the analytic continuation $i{\omega }_n\to \omega +i\eta$. The bridge to occupied-state integrals that the magnetic force theorem needs is the contour identity

$$\frac{1}{\beta }\sum _{n}F(i{\omega }_n)e^{i{\omega }_n{0}^{+}}=-\oint \frac{dz}{2\pi i}f(z)F(z)=\frac{1}{\pi }\mathrm{Im}{\int }_{-\infty }^{\infty }d\omega f(\omega )F^R(\omega ),$$

with $f$ the Fermi function — a Matsubara sum collapses to an energy integral over occupied states weighted by $f$, exactly the form $J_{ij}$ takes below (at $T=0$, $f$ is a step and the integral runs to $E_F$, with the contour usually deformed onto the imaginary axis for smoothness).

The LKAG magnetic force theorem

Now the magnetic application. Take a converged collinear spin-polarized electronic structure (DFT, in a localized/Wannier or LMTO basis), with site- and spin-resolved one-electron Green’s functions $G_{ij}^{\uparrow }$, $G_{ij}^{\downarrow }$ (orbital indices suppressed within a site; $i,j$ label magnetic atoms). The local exchange splitting is the on-site potential difference between the two spin channels,

$${\Delta }_i={\hat{H}}_{ii}^{\uparrow }-{\hat{H}}_{ii}^{\downarrow },$$

an on-site matrix in orbital space. The magnetic force theorem (Liechtenstein–Katsnelson– Antropov–Gubanov, LKAG) treats an infinitesimal rotation of spin $i$ relative to spin $j$ as a perturbation and identifies the second variation of the total (band) energy with a Heisenberg coupling. Because the first variation vanishes at self-consistency, only the change in the band energy at fixed potential is needed — that is the force-theorem simplification — and the result is an energy integral over occupied states of a product of intersite Green’s functions sandwiching the on-site splittings,

$$\boxed{J_{ij}=\frac{1}{2\pi }\mathrm{Im}{\int }_{-\infty }^{E_F}d\epsilon {\mathrm{Tr}}_L[{\Delta }_i{G}_{ij}^{\uparrow }(\epsilon ){\Delta }_j{G}_{ji}^{\downarrow }(\epsilon )]}$$

(${\mathrm{Tr}}_L$ over orbitals; the spin-rotation Hamiltonian is mapped onto ${\hat{H}}_{\mathrm{spin}}=-{\sum }_{ij}{J}_{ij}{\mathbf{e}}_i\cdot {\mathbf{e}}_j$, conventions for the prefactor and the sign of $J_{ij}$ — ferromagnetic $>0$ — vary between implementations). The structure is transparent: ${\Delta }_i$ turns the propagating electron’s spin, $G_{ij}^{\uparrow }$ carries it from $i$ to $j$, ${\Delta }_j$ turns it back, $G_{ji}^{\downarrow }$ returns it — a closed loop whose occupied-state energy integral is the coupling. The same construction generalizes:

$$J_{ij}^{\alpha \beta }=\frac{1}{2\pi }\mathrm{Im}{\int }^{E_F}d\epsilon {\mathrm{Tr}}_L[{\Delta }_i^{\alpha }{G}_{ij}(\epsilon ){\Delta }_j^{\beta }{G}_{ji}(\epsilon )]$$

with the spin-rotation generators ${\Delta }^{\alpha }$ now $2\times 2$ in spin (built from converged spinor Green’s functions including spin–orbit coupling). Decomposing $J_{ij}^{\alpha \beta }$ into its symmetric-traceless, isotropic, and antisymmetric parts yields the full exchange tensor — the isotropic Heisenberg $J_{ij}$, the symmetric anisotropy, and the antisymmetric Dzyaloshinskii–Moriya vector ${\mathbf{D}}_{ij}$ — all from one electronic-structure calculation and its Green’s functions. This is precisely the algorithm TB2J implements on top of a Wannier or SIESTA/LCAO Hamiltonian, and it is the Green’s-function front end of exchange tensors from DFT. Two caveats I always note: the force theorem assumes the magnetic configuration is a (meta)stable self-consistent state (so the linear term truly vanishes), and the mapping onto a bilinear Heisenberg model is an expansion in small relative rotations — strong itinerancy or large noncollinearity can make higher (biquadratic, ring) terms non-negligible, in which case $J_{ij}$ acquires a configuration dependence.

Prerequisites

second quantization · perturbation theory (the diagrammatic/ Dyson series and analytic continuation)

Builds toward: GW · exchange tensors from DFT

Key references

• The LKAG magnetic force theorem — A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov & V. A. Gubanov, J. Magn. Magn. Mater. 67, 65 (1987).
• TB2J implementation — X. He, N. Helbig, M. J. Verstraete & E. Bousquet, Comput. Phys. Commun. 264, 107938 (2021).
• Many-body texts — A. L. Fetter & J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971); A. A. Abrikosov, L. P. Gor’kov & I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, 1963); G. D. Mahan, Many-Particle Physics, 3rd ed. (Kluwer/Plenum, 2000).
• GW / self-energy — L. Hedin, Phys. Rev. 139, A796 (1965).