Spectral Moments and Many-Body Sum Rules

Spectral moments are exact short-time data. For a canonical single-particle field, nested commutators with $K=H-\mu N$ fix the coefficients of the large-frequency Green function. For density response, a double commutator gives the $f$ -sum rule. These constraints test normalization and redistributed spectral weight even when individual peaks are unresolved; the commutator and high-frequency derivations are collected in Fetter and Walecka 2003, ch. 7.

Required background. Lehmann Representations and Spectral Functions in Matter supplies the spectral sum, and Densities, Currents, and Nonrelativistic Ward Identities supplies equal-time current algebra.

Helpful background. Transport Sum Rules and Ultraviolet Constraints develops the broader response setting.

One-particle moments

For a fermionic operator $c$ , define

\mu_n=\int_{-\infty}^{\infty}\frac{\mathrm d\omega}{2\pi} \omega^nA(\omega).

Let $L O=[O,K]$ . The Lehmann representation gives

\mu_n=\langle\{L^n c,c^\dagger\}\rangle,

provided the moment exists. The first two are

\mu_0=1, \qquad \mu_1=\langle\{[c,K],c^\dagger\}\rangle.

They appear in

G(z)=\frac{\mu_0}{z}+\frac{\mu_1}{z^2} +\frac{\mu_2}{z^3}+\cdots.

For a free level $K=\xi c^\dagger c$ , $[c,K]=\xi c$ , so $\mu_n=\xi^n$ . For the Hubbard atom,

\mu_1=-\mu+U\langle n_{\downarrow}\rangle,

which checks both the Hartree shift and total weight without assuming a quasiparticle.

Higher moments contain increasingly local composite operators. They may diverge for idealized zero-range theories unless ultraviolet tails and subtractions are treated consistently. A divergent raw moment is not a failed sum rule; it signals that the moment requires renormalization or does not exist.

The density $f$ -sum rule

For $\rho_{\mathbf q}=\sum_{j=1}^Ne^{-i\mathbf q\cdot\mathbf r_j}$ , define the total dynamic structure factor at $T=0$ by

S(\mathbf q,\omega) =2\pi\sum_n |\langle n|\rho_{\mathbf q}|0\rangle|^2 \delta(\omega-E_n+E_0), \qquad \omega\ge0.

Completeness gives

\int_0^\infty\frac{\mathrm d\omega}{2\pi} \omega S(\mathbf q,\omega) =\frac12\langle[\rho_{-\mathbf q},[H,\rho_{\mathbf q}]]\rangle.

For particles of mass $m$ with translation-invariant position-dependent interactions, the interaction commutes with density and

\int_0^\infty\frac{\mathrm d\omega}{2\pi} \omega S(\mathbf q,\omega) =\frac{N\mathbf q^2}{2m}.

Dividing $S$ by $N$ changes the right-hand side accordingly. On a lattice, the double commutator yields band-curvature or kinetic-energy expectation values rather than $Nq^2/(2m)$ . The continuum formula must not be transplanted unchanged.

Compressibility and limit order

The inverse-frequency density sum is related to the corresponding total static susceptibility,

\chi^R(\mathbf q,0) =-2\int_0^\infty\frac{\mathrm d\omega}{2\pi} \frac{S(\mathbf q,\omega)}{\omega}

at zero temperature in the present convention. Because both $\rho_{\mathbf q}$ and $S$ above are extensive, taking $\mathbf q\to0$ in the static order connects this $\chi^R$ to $-\partial N/\partial\mu$ . Equivalently, after division by volume, $\chi^R/V=-\partial n/\partial\mu$ . The $f$ -sum weights high frequencies, while the compressibility sum weights low frequencies. Satisfying one does not guarantee the other.

In a charged fluid, the plasmon can exhaust much of the long-wavelength $f$ -sum while the static compressibility involves screening and background conventions. This is another reason to keep static and dynamic limits distinct.

Ultraviolet tails and subtracted moments

Short-range interactions generate power-law spectral and momentum tails. A nominal high moment can then depend on contact operators or diverge. Before using a moment:

determine the asymptotic spectral power;
check convergence of the weighted integral;
include required contact or diamagnetic terms; and
compare the same regulated quantity on both sides.

Frequency cutoffs in experiment or numerics leave a missing-tail contribution. Estimate it from a controlled asymptotic form rather than silently treating the measured window as complete.

Using moments to test approximations

Given an approximate spectrum $A_{\rm app}$ , calculate moments directly and from equal-time operators. Disagreement locates missing or misplaced weight even if the low-energy peak looks plausible. A finite broadening kernel preserves the zeroth moment only if normalized; truncating the plotted frequency window generally does not.

For response, verify the commutator sum, contact term, volume convention, and per-particle normalization. A self-energy approximation can yield a normalized $G$ while an inconsistent bare response vertex violates the $f$ -sum rule.

Common pitfalls

Using a moment that does not converge. Zero-range tails can invalidate unsubtracted high moments. Establish the ultraviolet behavior first.

Forgetting the lattice modification. The continuum $Nq^2/(2m)$ result assumes quadratic kinetic energy and continuous translations.

Checking only total weight. Zeroth normalization cannot detect weight shifted to the wrong energy; higher moments and independent response sums are needed.

Exercises

Derive the continuum double commutator

For $H=\sum_j\mathbf p_j^2/(2m)+V(\{\mathbf r\})$ , evaluate $[\rho_{-\mathbf q},[H,\rho_{\mathbf q}]]$ .

Solution

$V$ commutes with every position-dependent density. For each particle, $[\mathbf p^2,e^{-i\mathbf q\cdot\mathbf r}]=e^{-i\mathbf q\cdot\mathbf r}(-2\mathbf q\cdot\mathbf p+q^2)$ in the chosen phase convention. The second commutator cancels the momentum term and gives $q^2/m$ per particle. The prefactor $1/2$ in the spectral identity yields $Nq^2/(2m)$ .

Diagnose missing weight

An approximate positive spectrum has a single pole $A=2\pi Z\delta(\omega-E)$ with $Z<1$ . Can it satisfy $\mu_0=1$ ?

Solution

No. Its zeroth moment is $Z$ . A positive incoherent contribution of total weight $1-Z$ is required. Renormalizing the pole to unit weight would change its physical residue and generally spoil higher moments.

Continue

Polarization, the Lindhard Function, and Particle–Hole Continua supplies an exact free response for sum-rule checks. Current Vertices and Ward-Consistent Response explains how compatible vertices preserve them. Universal Relations and Tan Contact treats short-distance tails controlled by contact.

References

Fetter, Alexander L., and John Dirk Walecka. Quantum Theory of Many-Particle Systems. Mineola, NY: Dover, 2003; originally published 1971. Publisher record.