Spectral Function

⚡ Key Takeaways

The spectral function $A(\mathbf{k},\omega)$ is the imaginary part of the retarded single-particle Green's function; it measures the probability density that adding or removing a particle of momentum $\mathbf{k}$ costs energy $\omega$ . Key result: sharp delta-peaks for free/quasiparticle excitations broaden into Lorentzians of width $\sim 1/\tau$ (inverse lifetime) once interactions are switched on, while satisfying an exact sum rule. Where it applies: many-body quantum theory, condensed matter (ARPES, tunneling), quantum field theory (Källén–Lehmann), and operator/spectral theory in functional analysis. Who/when: Källén and Lehmann (1954), with roots in Lebesgue–Stieltjes spectral measures and the spectral theorem.

💡 The Core Insight

A resolvent $(\hat H - z)^{-1}$ is analytic off the real axis, and its discontinuity across the real axis is a positive measure — the spectral measure. The spectral function is the density of that measure. Equivalently: $\dfrac{1}{x - \omega - i0^+} = \mathcal{P}\dfrac{1}{x-\omega} + i\pi\,\delta(x-\omega)$ converts the analytic structure of a Green's function into a real, positive, normalized distribution telling you exactly which energies the system can absorb.

DefinitionRetarded Green's function and spectral function

For a many-body system with field operators $c_{\mathbf{k}}$ , the retarded single-particle Green's function at temperature $T=1/\beta$ is

1

\quad G^R(\mathbf{k},t) = -i\,\theta(t)\,\big\langle \{ c_{\mathbf{k}}(t),\, c^\dagger_{\mathbf{k}}(0) \}_{\pm} \big\rangle,

Derivation hint: the anticommutator $\{,\}_+$ is for fermions, commutator $[,]_-$ for bosons; $\theta(t)$ enforces causality (retardation), and $\langle\cdot\rangle$ is the grand-canonical thermal average. Its Fourier transform $G^R(\mathbf{k},\omega) = \int dt\, e^{i\omega t} G^R(\mathbf{k},t)$ defines the spectral function:

2

\quad A(\mathbf{k},\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,G^R(\mathbf{k},\omega).

Derivation hint: $\mathrm{Im}$ extracts the discontinuity across the real- $\omega$ axis; the sign is fixed so that $A \ge 0$ , since $\mathrm{Im}\,G^R \le 0$ for $\omega+i0^+$ analytic continuation. Here $\mathbf{k}$ is momentum, $\omega$ is the (real) excitation energy measured relative to the chemical potential $\mu$ , and the symbols are otherwise the operator algebra of the model.

DefinitionLehmann (spectral) representation

Inserting a complete set of exact eigenstates $|n\rangle$ with energies $E_n$ gives the Lehmann representation

3

\quad A(\mathbf{k},\omega) = \frac{1}{Z}\sum_{m,n}\Big( e^{-\beta E_m} + e^{-\beta E_n}\Big)\,\big|\langle n| c^\dagger_{\mathbf{k}}|m\rangle\big|^2\,\delta\!\big(\omega - (E_n - E_m)\big),

Derivation hint: insert $\mathbb{1}=\sum_n |n\rangle\langle n|$ between operators in [1], Fourier transform $\theta(t)e^{-i\Omega t}$ to $1/(\omega-\Omega+i0^+)$ , and apply the Sokhotski–Plemelj identity to take $\mathrm{Im}$ . The Green's function itself is recovered by a Hilbert (Kramers–Kronig) transform:

4

\quad G^R(\mathbf{k},\omega) = \int_{-\infty}^{\infty} d\omega'\,\frac{A(\mathbf{k},\omega')}{\omega - \omega' + i0^+}.

Derivation hint: substitute the delta-function form [3] back into the Fourier integral; the $i0^+$ prescription keeps the pole in the lower half-plane, guaranteeing retardation.

TheoremSum rule and positivity

The spectral function is non-negative and, for canonical (anti)commutation relations, normalized to unity:

5

\quad A(\mathbf{k},\omega) \ge 0 \quad\text{and}\quad \int_{-\infty}^{\infty} d\omega\, A(\mathbf{k},\omega) = \big\langle \{c_{\mathbf{k}}, c^\dagger_{\mathbf{k}}\}_{\pm}\big\rangle = 1.

Derivation hint: positivity follows term-by-term from the modulus-squared matrix elements in [3]; the integral kills the delta-functions and reassembles the equal-time (anti)commutator, which equals 1 by the CAR/CCR algebra.

ProofFrom resolvent to positive measure

Step 1 — Spectral theorem. Any self-adjoint $\hat H$ on a Hilbert space admits a projection-valued measure $E_\lambda$ with $\hat H = \int \lambda\, dE_\lambda$ . For a fixed vector $|\psi\rangle = c^\dagger_{\mathbf{k}}|m\rangle$ , the scalar measure $d\mu_\psi(\lambda) = \langle\psi|dE_\lambda|\psi\rangle$ is a positive Borel measure on $\mathbb{R}$ .

Step 2 — Resolvent. The diagonal resolvent matrix element is the Stieltjes transform of $\mu_\psi$ :

6

\quad \big\langle\psi\big|(\hat H - z)^{-1}\big|\psi\big\rangle = \int \frac{d\mu_\psi(\lambda)}{\lambda - z}, \qquad z\in\mathbb{C}\setminus\mathbb{R}.

Derivation hint: apply the spectral theorem to the function $f(\hat H)=(\hat H-z)^{-1}$ , $f(\lambda)=(\lambda-z)^{-1}$ , then pair with $|\psi\rangle$ .

Step 3 — Stieltjes inversion. Approaching the real axis with $z = \omega + i\eta$ , $\eta\to 0^+$ , the Sokhotski–Plemelj theorem gives

7

\quad d\mu_\psi(\omega) = -\frac{1}{\pi}\,\mathrm{Im}\,\big\langle\psi\big|(\hat H - \omega - i0^+)^{-1}\big|\psi\big\rangle\, d\omega.

Derivation hint: $\frac{1}{\lambda-\omega-i\eta}=\frac{\lambda-\omega}{(\lambda-\omega)^2+\eta^2}+i\frac{\eta}{(\lambda-\omega)^2+\eta^2}$ ; the second term is a nascent delta of weight $\pi$ , recovering the measure density.

Step 4 — Identification. Summing $\mu_\psi$ over the thermal ensemble of source vectors $|\psi\rangle$ reproduces exactly equation [3]. Thus $A(\mathbf{k},\omega)\,d\omega$ is a thermally weighted spectral measure: positive because each $\mu_\psi$ is positive, normalized because the total weight equals $\|\psi\|^2$ summed with Boltzmann factors. $\blacksquare$

∎

HL

Harry Lehmann & Gunnar Källén

1924–1998 / 1926–1968

Working independently around 1954, Lehmann and Källén derived the model-independent spectral representation of the two-point function in quantum field theory. By inserting a complete set of physical states between field operators, they expressed the propagator as a weighted integral over a positive spectral density $\rho(s)$ supported on the mass spectrum. The Källén–Lehmann representation proved that the exact propagator's pole structure (one-particle states) and continuous part (multi-particle thresholds) are constrained by positivity and Lorentz invariance alone — independent of perturbation theory. Källén, a Swedish theorist of fierce rigor, and Lehmann, a pillar of the Hamburg axiomatic-QFT school, thereby gave the spectral function its enduring nonperturbative foundation.

🧠 Intuition

Think of injecting an electron of momentum $\mathbf{k}$ into the system and asking: "what energy did that cost?" For a free particle the answer is sharp — energy $\varepsilon_{\mathbf{k}}$ exactly — so $A$ is a single delta-spike. Turn on interactions and the injected electron is no longer an eigenstate; it decays. The energy answer fuzzes out into a Lorentzian of width equal to the decay rate $1/\tau$ , centered on the renormalized energy. The area under the peak (the quasiparticle residue $Z$ ) tells you what fraction of the original electron survives as a coherent quasiparticle; the rest of the unit sum-rule weight is smeared into an "incoherent background." In a Mott insulator or strongly correlated metal, $Z\to 0$ and the peak dissolves entirely — there is no quasiparticle, only a broad continuum. The spectral function is thus the system's honest answer-sheet for "which excitations actually exist."

Figure

Free-particle delta-peak (gold) broadening into an interacting Lorentzian as the inverse lifetime grows

ExampleFree fermion gas

Given: Non-interacting fermions with dispersion $\varepsilon_{\mathbf{k}} = \frac{\hbar^2 k^2}{2m} - \mu$ . The retarded Green's function is $G^R(\mathbf{k},\omega) = \dfrac{1}{\omega - \varepsilon_{\mathbf{k}} + i0^+}$ .

Steps: 1. Apply Sokhotski–Plemelj: $\mathrm{Im}\,\dfrac{1}{\omega-\varepsilon_{\mathbf{k}}+i0^+} = -\pi\,\delta(\omega-\varepsilon_{\mathbf{k}})$ . 2. Use definition [2]: $A(\mathbf{k},\omega) = -\frac{1}{\pi}(-\pi)\,\delta(\omega-\varepsilon_{\mathbf{k}})$ . 3. Check the sum rule: $\int d\omega\,\delta(\omega-\varepsilon_{\mathbf{k}}) = 1$ . ✓

Boxed answer:

\boxed{A(\mathbf{k},\omega) = \delta(\omega - \varepsilon_{\mathbf{k}}) \quad [\text{units: } (\text{energy})^{-1}]}

A single sharp line: a free particle is an exact eigenstate, all weight $Z=1$ .

ExampleQuasiparticle with finite lifetime

Given: An interacting metal where the self-energy near the Fermi surface is $\Sigma(\mathbf{k},\omega) = \Delta_{\mathbf{k}} - i\Gamma_{\mathbf{k}}$ , with renormalized energy $\tilde\varepsilon_{\mathbf{k}} = \varepsilon_{\mathbf{k}} + \Delta_{\mathbf{k}}$ . Take $\tilde\varepsilon_{\mathbf{k}} = 0.10\ \text{eV}$ and $\Gamma_{\mathbf{k}} = 0.02\ \text{eV}$ .

Steps: 1. Dyson form: $G^R = \dfrac{1}{\omega - \varepsilon_{\mathbf{k}} - \Sigma} = \dfrac{1}{\omega - \tilde\varepsilon_{\mathbf{k}} + i\Gamma_{\mathbf{k}}}$ . 2. Spectral function: $A = -\frac{1}{\pi}\,\mathrm{Im}\,G^R = \dfrac{1}{\pi}\dfrac{\Gamma_{\mathbf{k}}}{(\omega-\tilde\varepsilon_{\mathbf{k}})^2 + \Gamma_{\mathbf{k}}^2}$ — a Lorentzian. 3. Peak height at $\omega=\tilde\varepsilon_{\mathbf{k}}$ : $A_{\max} = \frac{1}{\pi\Gamma_{\mathbf{k}}} = \frac{1}{\pi(0.02)} \approx 15.9\ \text{eV}^{-1}$ . 4. FWHM $= 2\Gamma_{\mathbf{k}} = 0.04\ \text{eV}$ ; lifetime $\tau = \hbar/(2\Gamma_{\mathbf{k}}) = \dfrac{6.58\times10^{-16}\,\text{eV}\,\cdot\,\text{s}}{0.04\,\text{eV}} \approx 1.6\times10^{-14}\ \text{s}$ .

Boxed answer:

\boxed{A(\mathbf{k},\omega)=\frac{1}{\pi}\frac{0.02}{(\omega-0.10)^2+0.02^2}\ \text{eV}^{-1},\quad \tau \approx 16\ \text{fs}}

⚠️ Common mistake

The spectral function is not a wavefunction and not generally a probability over a single particle's states. It is a spectral density — a measure. Its non-negativity and unit normalization make it look like a probability distribution over $\omega$ at fixed $\mathbf{k}$ , and indeed it is the probability density for the energy cost of adding/removing a particle. But $A$ can be a sum of delta-functions plus a continuum; it has dimensions of inverse energy; and at finite $T$ it weights both particle addition and removal. Do not confuse $\int d\omega\,A=1$ (operator algebra) with $\int d\omega\, f(\omega)A=n_{\mathbf{k}}$ (the occupation, which uses the Fermi function $f$ ).

Counterexample

"A peak in $A$ always means a long-lived quasiparticle." False. In a Luttinger liquid (1D interacting fermions) or at a Mott transition, the spectral function can show power-law singularities or split into Hubbard bands with no Lorentzian quasiparticle pole at all — the residue $Z=0$ . The spectral function is broad and structured, yet there is no underlying coherent particle: $\lim_{\omega\to E_F} \mathrm{Im}\,\Sigma$ does not vanish like a Fermi liquid's $\omega^2$ , so no sharp pole forms. A feature in $A(\mathbf{k},\omega)$ proves the system can absorb energy there, not that a particle-like excitation exists.

connection

The spectral function sits at the crossroads of several frameworks. In functional analysis it is the density of the Spectral Measure supplied by the Spectral Theorem for self-adjoint operators, recovered via Stieltjes Transform inversion. In field theory it is the Källén–Lehmann density constrained by Analyticity and Causality through Kramers–Kronig Relations (equation [4] is precisely a Hilbert transform). The bridge between $A$ and dynamics is the Dyson Equation via the self-energy $\Sigma$ , and the bridge to experiment is the Fluctuation-Dissipation Theorem, which relates $A$ to the imaginary part of response functions. The thermal occupation follows from convolving $A$ with the Fermi-Dirac Distribution.

🔬 Where this appears

⏳ History

- 1929–1932: John von Neumann formalizes the spectral theorem for unbounded self-adjoint operators, supplying projection-valued (spectral) measures — the mathematical ancestor of the spectral function.

- 1932: Thomas Stieltjes' inversion formula (1894) is wedded to the resolvent, establishing that the discontinuity of

(\hat H - z)^{-1}

across

\mathbb{R}

yields a positive measure.

- 1947–1949: Matsubara, Schwinger, and others develop Green's-function methods; the propagator's analytic structure becomes central.

- 1954: Gunnar Källén and, independently, Harry Lehmann derive the spectral (Källén–Lehmann) representation of the field-theoretic propagator with a positive spectral density.

- 1957: Landau formulates Fermi-liquid theory; the quasiparticle pole and residue

Z

acquire spectral-function meaning.

- 1959–1961: Abrikosov, Gor'kov, Dzyaloshinski and Kadanoff–Baym systematize the spectral representation in condensed-matter many-body theory; the sum rule [5] becomes standard.

- 1963: Hubbard introduces his model; the splitting of the spectral function into Hubbard bands is born.

- 1989–1996: Metzner, Vollhardt, Georges, and Kotliar develop DMFT, making

A(\omega)

computable for correlated electrons.

- 1990s–present: High-resolution ARPES turns

A(\mathbf{k},\omega)

into a directly imaged quantity, validating quasiparticle theory and exposing its breakdown in cuprates.

open-problem

The reconstruction of $A(\omega)$ from imaginary-time (Matsubara) data is a numerically ill-posed inverse problem: the kernel $K(\tau,\omega)=e^{-\tau\omega}/(1+e^{-\beta\omega})$ has exponentially small singular values, so tiny statistical noise in Quantum Monte Carlo data permits wildly different spectral functions. Maximum Entropy and stochastic analytic-continuation methods regularize this, but no method provably recovers sharp high-energy features without bias. Open questions remain over (i) the precise nature of spectral functions at quantum critical points, where scaling forms replace Lorentzians and the very notion of a quasiparticle dissolves; (ii) marginal and non-Fermi-liquid spectral lineshapes in the cuprate "strange metal," whose $A(\mathbf{k},\omega)$ resists any controlled microscopic derivation; and (iii) the rigorous existence and analyticity of spectral densities for interacting relativistic QFTs beyond perturbation theory — part of the broader constructive field theory program.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also