Lehmann Representations and Spectral Functions in Matter

The Lehmann representation resolves a correlator into exact transitions between many-body eigenstates. For a canonical fermion operator it separates addition and removal weight, proves positivity and normalization of the diagonal spectral function, and shows whether a feature is a discrete finite-volume line, an isolated pole, or part of a continuum. The spectral resolution and positivity argument originate in Lehmann 1954, pp. 342–346.

Required background. Quantum-Matter Correlators and Observable Conventions fixes the functions being represented. Spectral Decomposition of Two-Point Functions, Thermal Propagators and Spectral Representations, and The Källén–Lehmann Representation supply the general spectral framework and relativistic contrast.

Grand-canonical eigenstates

Let $K=H-\mu N$ and $K|m\rangle=K_m|m\rangle$ . For a fermionic annihilation operator $c_\alpha$ , the diagonal spectral function is

A_\alpha(\omega) =\frac{2\pi}{Z}\sum_{mn} (e^{-\beta K_m}+e^{-\beta K_n}) |\langle m|c_\alpha|n\rangle|^2 \delta(\omega-K_n+K_m).

Every coefficient is nonnegative. Integrating and using completeness gives

\int\frac{\mathrm d\omega}{2\pi}A_\alpha(\omega) =\langle\{c_\alpha,c_\alpha^\dagger\}\rangle=1.

For a matrix of orbitals, $u^\dagger A(\omega)u\ge0$ for every vector $u$ . Off-diagonal entries need not be individually positive.

At zero temperature, choose a ground state $|0,N\rangle$ . Writing frequency relative to $\mu$ separates

\begin{aligned} A^+(\omega)&=2\pi\sum_n |\langle n,N+1|c^\dagger|0,N\rangle|^2 \delta[\omega-(E_n^{N+1}-E_0^N-\mu)],\\ A^-(\omega)&=2\pi\sum_n |\langle n,N-1|c|0,N\rangle|^2 \delta[\omega+(E_n^{N-1}-E_0^N+\mu)]. \end{aligned}

Positive and negative frequency correspond to addition and removal only after this frequency origin is declared.

Resolvent and analytic structure

The retarded function is

G^R(z)=\int\frac{\mathrm d\omega'}{2\pi} \frac{A(\omega')}{z-\omega'}, \qquad \operatorname{Im}z>0.

In finite volume, $A$ is a sum of delta functions. The thermodynamic limit can make transition energies dense, producing branch cuts and continua. An isolated delta function with nonzero weight becomes a real-axis pole in a stable system; coupling to a continuum generally moves a resonance pole off the real axis on an analytically continued sheet.

The large- $z$ expansion,

G(z)=\frac{\mu_0}{z}+\frac{\mu_1}{z^2}+\cdots, \qquad \mu_n=\int\frac{\mathrm d\omega}{2\pi}\omega^nA(\omega),

connects exact spectral moments to equal-time commutators. The zeroth moment is fixed even when most weight is incoherent.

An interacting atomic example

For the single-site Hubbard operator

K=U n_\uparrow n_\downarrow-\mu(n_\uparrow+n_\downarrow),

the spin- $\uparrow$ Green function sees two possible local environments. If the down-spin occupation probability is $\langle n_\downarrow\rangle$ , then

A_\uparrow(\omega) =2\pi[1-\langle n_\downarrow\rangle]\delta(\omega+\mu) +2\pi\langle n_\downarrow\rangle\delta(\omega+\mu-U).

The two weights sum to one. They are not two quasiparticles generated by a weak self-energy; they are exact addition/removal transitions conditioned on local occupation. This example warns against reading every multi-peak spectrum as a set of independent particles.

Composite operators and detailed balance

For a Hermitian operator $O$ , define the unsymmetrized structure factor

S_O(\omega)=\frac{2\pi}{Z}\sum_{mn} e^{-\beta K_m}|\langle n|O|m\rangle|^2 \delta(\omega-K_n+K_m).

It is nonnegative and obeys $S_O(-\omega)=e^{-\beta\omega}S_O(\omega)$ . The retarded commutator spectral density is

\rho_O(\omega)=[1-e^{-\beta\omega}]S_O(\omega),

which is odd for a time-reversal-symmetric scalar channel under the appropriate momentum reversal and changes sign across zero frequency. Positivity statements must name which of $A$ , $S$ , or $\rho$ is meant.

Finite volume, degeneracy, and normalization

Degenerate states require a complete trace or a declared symmetry-broken density matrix; selecting one vector can change matrix elements. Delta functions in finite volume acquire continuum densities of states only after the volume limit, with state-normalization factors transformed consistently. Broadening a finite spectrum for plotting is not a physical lifetime unless the broadening survives a controlled thermodynamic and resolution analysis.

Common pitfalls

Using $H$ energies with a frequency measured from $\mu$ without translation. Grand-canonical differences involve $K=H-\mu N$ . Addition and removal thresholds shift accordingly.

Demanding positivity of a commutator spectrum at negative frequency. The dynamic structure factor is nonnegative; the retarded commutator density includes detailed-balance signs.

Calling plotted broadening a decay rate. Numerical or instrumental convolution is not an imaginary self-energy. Vary it and take the relevant limits.

Exercises

Prove the zeroth moment

Integrate the finite-temperature Lehmann sum and recover the canonical anticommutator.

Solution

The first Boltzmann term sums to $\langle cc^\dagger\rangle$ . Relabeling $m\leftrightarrow n$ in the second gives $\langle c^\dagger c\rangle$ . Their sum is $\langle\{c,c^\dagger\}\rangle=1$ .

Check the atomic spectrum

Compute the first moment of the atomic spectral function.

Solution

Using the two delta functions,

\mu_1=(-\mu)[1-\langle n_\downarrow\rangle] +(U-\mu)\langle n_\downarrow\rangle =-\mu+U\langle n_\downarrow\rangle.

This agrees with $\langle\{[c_\uparrow,K],c_\uparrow^\dagger\}\rangle$ .

Continue

Dyson Equations and Self-Energies separates pole and incoherent contributions. Quasiparticle Poles, Residues, and Lifetimes supplies the interpretation criteria. Spectral Moments and Many-Body Sum Rules derives higher moments from commutators.

References

Lehmann, Harry. “On the Properties of Propagation Functions and Renormalization Constants of Quantized Fields.” Il Nuovo Cimento 11 (1954): 342–357. DOI.