Spectral Functions

A spectral function is an operator-, state-, and convention-resolved distribution of transition strength over energy and, when available, momentum or other quantum numbers. It records not only where the Hamiltonian has allowed energy differences, but also how strongly a specified insertion prepares and detects the corresponding exact states.

That qualification is essential. A many-body spectrum is not merely a list of eigenvalues. A spectral function also contains:

• matrix-element selection rules;
• thermal populations;
• addition, removal, or number-preserving channel information;
• coherent pole residues and incoherent weight;
• continuum thresholds and line shapes;
• the normalization attached to the chosen operators.

A measured intensity also includes probe matrix elements, occupation factors, backgrounds, final-state effects, and instrumental resolution. A peak in measured counts therefore needs a measurement model before it can be interpreted as an intrinsic excitation.

Helpful background. Single-particle Green functions explains addition and removal weights; structure factors supplies number-preserving scattering spectra.

Spectral variables and normalization

Energy rather than angular frequency

This page uses the target energy transfer

$$E = \hbar\omega$$

as the spectral variable. Positive $E$ means that the target absorbs energy. Negative $E$ means that an initially excited target can release energy, or, for a number-changing fermionic spectrum, that the feature lies on the removal side of the chemical potential.

The energy-resolved Fourier convention is

$$\mathcal S_{AB}(E) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dt\, e^{iEt/\hbar} \left\langle A(t)B(0) \right\rangle.$$

With this normalization,

$$\int_{-\infty}^{\infty} dE\, \mathcal S_{AB}(E) = \langle AB\rangle.$$

An angular-frequency spectrum differs by the Jacobian

$$\mathcal S_E(E) = \frac{1}{\hbar} \mathcal S_\omega \left( \frac{E}{\hbar} \right).$$

Factors of $\hbar$, $2\pi$, and $\pi$ are therefore part of the definition, not cosmetic notation.

Generator and energy zero

For number-preserving observables, time evolution normally uses the physical Hamiltonian $H$. For single-particle addition and removal in equilibrium, it is often convenient to use

$$\mathcal K = H-\mu\hat N.$$

The corresponding spectral energy is measured relative to the chemical potential. The same physical process can be described using $H$, but every frequency must then be shifted consistently. A plot labeled only by “energy” is incomplete unless its zero is stated.

Intrinsic and displayed widths

The notation

$$i0^+$$

selects a retarded boundary value. It is not a finite decay rate. A numerical replacement

$$0^+ \longrightarrow \eta\gt0$$

creates a displayed width set by $\eta$. That width becomes physical only when a controlled self-energy, bath, disorder average, or measurement model supplies the same scale.

The Generic Transition Spectrum

Let

$$H|n\rangle = E_n|n\rangle, \qquad \rho = \sum_n p_n |n\rangle\langle n|,$$

with $[\rho,H]=0$. For an operator $O$, define

$$\mathcal S_{O O^\dagger}(E) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dt\, e^{iEt/\hbar} \left\langle O(t)O^\dagger(0) \right\rangle.$$

Inserting a complete set gives the Lehmann form

$$\mathcal S_{O O^\dagger}(E) ={} \sum_{n,m} p_n \left| \left\langle m \middle| O^\dagger \middle| n \right\rangle \right|^2 \times \delta \left( E-E_m+E_n \right).$$

This compact expression already contains most of the physics:

1. Location. A line occurs at an exact transition energy $E_m-E_n$.
2. Preparation. The initial state appears with probability $p_n$.
3. Visibility. The operator matrix element determines whether the transition is bright or dark.
4. Weight. The squared overlap, not degeneracy alone, sets the line strength.
5. Support. At zero temperature, a ground-state target has no inelastic weight at $E\lt0$.

The zeroth moment is

$$\int dE\, \mathcal S_{O O^\dagger}(E) = \left\langle OO^\dagger \right\rangle.$$

Thus the total weight depends on the operator normalization and the reference state. It is not generally one.

Matrix-valued spectra

For a family of operators $O_a$,

$$\mathcal S_{ab}(E) ={} \sum_{n,m} p_n \langle n|O_a|m\rangle \langle m|O_b^\dagger|n\rangle \times \delta \left( E-E_m+E_n \right).$$

At each energy this matrix is Hermitian and positive semidefinite as a distribution. For every complex channel vector $v$,

$$v^\dagger \boldsymbol{\mathcal S}(E) v \geq 0.$$

Individual off-diagonal entries may be complex or change sign. Positivity applies to physical quadratic forms, not component by component.

Operator selectivity

Two operators acting on the same Hamiltonian can produce completely different spectra. If

$$\langle m|O^\dagger|n\rangle = 0,$$

then the transition is absent from this channel even though $|m\rangle$ is an exact eigenstate. Symmetry, momentum, polarization, particle number, and locality all create such selection rules.

Conversely, one bare operator may overlap many exact eigenstates. Its unit or finite total weight can then split among:

• a coherent pole;
• satellite poles;
• a multiparticle continuum;
• a threshold singularity;
• high-energy incoherent weight.

The spectral function is therefore an operator-resolved projection of the spectrum.

One Name, Several Objects

The phrase “spectral function” is used for several related but inequivalent objects.

Object	Representative definition	Positivity	Physical use
ordered transition spectrum	$\mathcal S_{O O^\dagger}(E)$	nonnegative in a conjugate channel	fluctuations and transition strength
fermionic single-particle function	$A=(i/2\pi)(G^{\mathrm R}-G^{\mathrm A})$	positive semidefinite	particle addition and removal
response spectral density	$\rho=(\chi^{\mathrm R}-\chi^{\mathrm A})/(2\pi i)$	generally signed	absorption minus emission
dynamic structure factor	$S_O(\mathbf q,E)$	nonnegative in a conjugate channel	momentum- and energy-resolved scattering
density of states	trace or momentum sum of $A$	nonnegative in standard fermionic use	state counting weighted by orbital content
bath spectral density	$J(\omega)=\sum_k\lvert g_k\rvert^2\delta(\omega-\omega_k)$	conventionally nonnegative	coupling-weighted reservoir modes
observed intensity	probe-dependent forward model	detector counts are nonnegative	experimental data

No formula should be moved from one row to another without checking operator order, normalization, statistics, and the energy variable.

Fermionic single-particle spectral function

For canonical fermion operators,

$$A(E) = \frac{i}{2\pi} \left[ G^{\mathrm R}(E) - G^{\mathrm A}(E) \right].$$

For a diagonal channel,

$$A_{aa}(E) = - \frac{1}{\pi} \operatorname{Im} G_{aa}^{\mathrm R}(E).$$

At zero temperature its addition and removal content is schematically

$$A_{aa}(E) ={} \sum_m |X_{am}^{+}|^2 \delta(E-\Delta_m^+) + \sum_n |X_{an}^{-}|^2 \delta(E+\Delta_n^-).$$

The positive-energy side adds a particle and the negative-energy side removes one, using energies relative to $\mu$. Canonical anticommutation gives

$$\int_{-\infty}^{\infty} dE\, A_{aa}(E) = 1.$$

The full derivation, finite-temperature formula, matrix positivity, and occupation relations belong to Green Functions in Many-Body QM.

Response spectral density

An observable response uses a commutator rather than one ordered product. In a compatible energy normalization,

$$\rho_{AB}(E) = \mathcal S_{AB}(E) - \mathcal S_{BA}(-E).$$

For a Hermitian stationary channel, define $\chi_E^R(E)=\int dt\,e^{iEt/\hbar}\chi^R(t)=\chi_\omega^R(E/\hbar)$. Unlike a spectral density, this response transform acquires no extra Jacobian merely from relabeling its argument. Then

$$\rho_{AA}(E) = \frac{1}{\pi} \operatorname{Im} \chi_{AA}^{\mathrm R}(E),$$

under the sign convention used in Retarded and Advanced Response. It is odd in energy and therefore cannot be nonnegative on both sides.

At thermal equilibrium, detailed balance supplies the preview

$$\rho_{AA}(E) = \left( 1-e^{-\beta E} \right) \mathcal S_{AA}(E)$$

for a conjugate Hermitian channel. The full set of Bose factors, symmetrized spectra, zero-frequency limits, and convention checks belongs to Fluctuation–Dissipation Theorem.

Dynamic structure factor

A density or spin structure factor is an ordered spectrum with a declared momentum transfer:

$$S_O(\mathbf q,E) \propto \int dt\, e^{iEt/\hbar} \left\langle O_{\mathbf q}(t) O_{-\mathbf q}(0) \right\rangle.$$

It describes a number-preserving excitation channel. It is not the same as a one-particle addition spectrum

$$A(\mathbf k,E).$$

A collective mode may be sharp in $S(\mathbf q,E)$ while carrying little or no weight in $A(\mathbf k,E)$, and the reverse can also occur.

Density of states

For a translation-invariant fermionic system, a common density of states is

$$\nu(E) = \frac{1}{\mathcal V} \sum_{\mathbf k} \operatorname{Tr} A(\mathbf k,E),$$

with the momentum measure and internal trace declared. Momentum integration discards dispersion information. Two systems can have similar $\nu(E)$ but very different $A(\mathbf k,E)$.

A local spectral density

$$\nu_i(E) = \operatorname{Tr} A_{ii}(E)$$

retains orbital or spatial selectivity but still depends on the chosen local basis.

Bath spectral density

The open-system quantity

$$J(\omega) = \sum_k |g_k|^2 \delta(\omega-\omega_k)$$

weights environmental modes by their coupling to the system. It is not the target’s single-particle spectral function, even though both can have peaks, gaps, and continua. Their operators, dimensions, and roles in a calculation differ.

An interacting atom: lines, weights, and removal intensity

Consider one isolated spin-degenerate orbital, with grand-canonical generator

$$\mathcal K=\xi(n_\uparrow+n_\downarrow)+Un_\uparrow n_\downarrow, \qquad \xi=\epsilon_d-\mu,\qquad U>0.$$

The atomic impurity model gives the four states and their energies. We use its exact atomic Green function to follow the same spectrum from transitions to an ideal measurement. There is no bath hybridization or hopping here.

For spin $\sigma$, adding an electron costs $\xi$ when the opposite spin is absent and $\xi+U$ when it is present. Since $n_{\bar\sigma}$ is conserved, these are separate exact transitions. With $\bar n=\langle n_{\bar\sigma}\rangle$,

$$\begin{aligned} G_\sigma(z)&=\frac{1-\bar n}{z-\xi}+\frac{\bar n}{z-\xi-U}, \qquad \operatorname{Im}z>0,\\ A_\sigma(E)&=(1-\bar n)\delta(E-\xi)+\bar n\delta(E-\xi-U). \end{aligned}$$

This is the atomic spectrum in Kollar’s §3.2, equation (28), with the orbital energy included in $\xi$. The two residues are probabilities for the opposite-spin occupation, so they are nonnegative and sum to one. The interaction splits the weight without giving either exact line a lifetime.

A finite-temperature calculation

Take an energy scale $J>0$ and choose $\xi=-J$, $U=4J$, and $\beta J=\ln2$. In the ordered basis $(|0\rangle,|\uparrow\rangle,|\downarrow\rangle,|\uparrow\downarrow\rangle)$, the grand energies and normalized Gibbs probabilities are

$$K_r=(0,-J,-J,2J), \qquad p_r=\frac1{21}(4,8,8,1).$$

Thus $\bar n=3/7$ and

$$A_\sigma(E)=\frac47\delta(E+J)+\frac37\delta(E-3J).$$

The spectral sum rules give

$$\int dE\,A_\sigma(E)=1, \qquad \int dE\,E A_\sigma(E)=\frac{5J}{7}.$$

The first moment can be checked without integrating the spectrum: $[c_\sigma,\mathcal K]=(\xi+Un_{\bar\sigma})c_\sigma$ gives

$$\left\langle\left\{[c_\sigma,\mathcal K],c_\sigma^\dagger\right\}\right\rangle =\xi+U\bar n=\frac{5J}{7}.$$

The centroid $5J/7$ lies between the lines; there is no excitation at that energy. Likewise, the full spectral variance will reflect their separation, not a decay width.

For an ideal removal probe with unit matrix element, no background, and perfect resolution, the intrinsic signal is $I_{\rm rem}=fA_\sigma$, where $f(E)=(e^{\beta E}+1)^{-1}$. Here $f(-J)=2/3$ and $f(3J)=1/9$, giving

$$I_{\rm rem}(E)=\frac{8}{21}\delta(E+J)+\frac1{21}\delta(E-3J), \qquad \int dE\,I_{\rm rem}(E)=\frac37=\langle n_\sigma\rangle.$$

The upper line’s removal weight comes from the thermally occupied doubly occupied state. At nonzero temperature, removal is selected by $fA$, not by cutting off $A$ at $E=0$. The raw line-area ratio is $1/8$, whereas the ratio in $A_\sigma$ is $3/4$.

A real detector broadens these already weighted lines. Its normalized energy-resolution kernel $R$ predicts

$$I_{\rm obs}(E)=\frac{8}{21}R(E+J)+\frac1{21}R(E-3J).$$

This full-axis integral remains $3/7$; integrating a finite window or dividing the convolved signal by $f(E)$ need not recover the intrinsic weights. The exercises below quantify these distinctions.

In the noninteracting limit $U=0$, the two poles coincide and recover $A_\sigma(E)=\delta(E-\xi)$ and $G_\sigma^{\rm R}(E)=(E-\xi+i0^+)^{-1}$. At $\xi=-U/2$, a spin-symmetric thermal atom instead has equal residues at $\pm U/2$. Neither two peaks nor a depleted interval alone establishes a bulk Mott phase: this isolated model has no spatial transport, and orbital hybridization can also produce two peaks without interactions.

What a Spectral Plot Can Establish

A well-defined spectral plot can support several distinct claims.

Excitation energy

A peak or threshold identifies an energy at which the chosen operator has appreciable transition weight. It does not by itself identify the microscopic composition of the excitation.

Dispersion

Tracking a feature across momentum gives a dispersion

$$E_\star(\mathbf k)$$

or

$$E_\star(\mathbf q).$$

The momentum label belongs to the operator insertion. A single-particle momentum $\mathbf k$ and a transferred momentum $\mathbf q$ are not interchangeable.

Spectral weight

The integrated area under a resolved component measures an overlap or transition strength after normalization and probe factors are removed. Peak height alone is not spectral weight.

Lifetime or damping

An intrinsic linewidth can imply a decay or dephasing scale under a stated pole model. A displayed or measured width does not establish that lifetime until numerical, instrumental, inhomogeneous, and final-state broadening have been separated.

Selection rules

Absence of weight can indicate a symmetry or matrix-element zero. It need not mean that no state exists at that energy.

Exact Lines, Poles, and Delta Functions

For a finite, closed system with discrete spectrum, an exact Lehmann representation contains delta functions:

$$\mathcal S(E) = \sum_j W_j \delta(E-E_j).$$

The coefficient $W_j$ is an area. A delta function has no finite height and no intrinsic full width at half maximum.

The corresponding retarded function has a boundary-value pole,

$$G^{\mathrm R}(E) \supset \frac{Z_j}{ E-E_j+i0^+ }.$$

Using

$$\frac{1}{ x+i0^+ } = \operatorname{PV}\frac1x - i\pi\delta(x),$$

the pole produces

$$A(E) \supset Z_j\delta(E-E_j).$$

The pole and delta function are two descriptions of the same pure-point spectral contribution. The pole lives in the analytic Green function; the delta function lives in its real-axis spectral measure.

Stable excitation

An isolated real-axis pole with nonzero residue identifies a stable exact excitation in the chosen channel. “Stable” here means that no energetically and symmetry-allowed continuum gives it a decay width in the idealized closed problem.

Operator-dark state

An exact eigenstate with zero matrix element has no pole in this Green function. The Hamiltonian spectrum can therefore be richer than any one operator’s pole inventory.

Degenerate lines

Several states at the same energy contribute a summed projector or summed weight. One observed delta line does not prove a nondegenerate state.

Continua and Thresholds

Continuous spectral weight appears when a limiting or averaging procedure creates a continuum of accessible energies. Common mechanisms include:

• the thermodynamic limit;
• multiparticle phase space;
• coupling to a reservoir;
• quenched-disorder averaging;
• ensemble averaging;
• unresolved dense levels;
• instrumental convolution.

The thermodynamic, multiparticle, and reservoir mechanisms can produce intrinsic continuous support. Disorder and ensemble averages define declared averaged spectra whose smoothness need not describe any one realization. Unresolved levels and instrumental convolution smooth the representation or observation without changing the exact target spectrum.

Branch cuts

If an analytic Green function has continuous real-axis discontinuity, its continuation is commonly represented by a branch cut. The cut is not an extra excitation. It is the analytic encoding of continuous spectral support.

Threshold energy

A continuum often begins at a kinematic threshold

$$E_{\mathrm{th}}(\mathbf q).$$

Near the edge,

$$\mathcal S(\mathbf q,E) \propto \Theta \left( E-E_{\mathrm{th}} \right) \left( E-E_{\mathrm{th}} \right)^\alpha$$

only if the density of states and matrix element have the corresponding regular behavior. Interactions can change the exponent, bind a state below threshold, or create a resonance near the edge.

Two-particle threshold benchmark

Take two gapped particles with

$$\epsilon(\mathbf p) = \Delta + \frac{\mathbf p^2}{2m}.$$

At total physical momentum $\hbar\mathbf q$, with $\mathbf q$ a wavevector and $\mathbf p$ a relative momentum, write

$$\mathbf p_1 = \frac{\hbar\mathbf q}{2} + \mathbf p, \qquad \mathbf p_2 = \frac{\hbar\mathbf q}{2} - \mathbf p.$$

Their total energy is

$$E = 2\Delta + \frac{\hbar^2\mathbf q^2}{4m} + \frac{\mathbf p^2}{m}.$$

The threshold is therefore

$$E_{\mathrm{th}}(\mathbf q) = 2\Delta + \frac{\hbar^2\mathbf q^2}{4m}.$$

For a smooth nonvanishing matrix element, the near-threshold joint density of states scales as

$$\mathcal S(\mathbf q,E) \propto \left( E-E_{\mathrm{th}} \right)^{d/2-1}.$$

Thus the same two-particle kinematics gives a divergent edge in one dimension, a step in two dimensions, and a square-root onset in three dimensions. A symmetry-forced matrix-element zero can soften each of these onsets.

Quasiparticle Peaks

A quasiparticle is not merely a visible maximum. It is an excitation that, over a controlled energy and momentum range, behaves like a weakly damped pole with a sufficiently isolated dispersion and nonzero overlap with a simple operator.

For one fermionic band,

$$G^{\mathrm R}(\mathbf k,E) = \frac1{ E-\xi_{\mathbf k} - \Sigma^{\mathrm R}(\mathbf k,E) }.$$

Write

$$\Sigma^{\mathrm R} = \Sigma' + i\Sigma'', \qquad \Sigma''\leq0$$

for a passive fermionic retarded channel. Away from a zero-width pole,

$$\begin{aligned} A(\mathbf k,E) ={}& \frac1\pi \frac{ -\Sigma''(\mathbf k,E) }{ D(\mathbf k,E) }, \\ D(\mathbf k,E) ={}& \left[ E-\xi_{\mathbf k} -\Sigma'(\mathbf k,E) \right]^2 + \left[ \Sigma''(\mathbf k,E) \right]^2. \end{aligned}$$

The distributional $i0^+$ contribution must be retained when $\Sigma''$ vanishes exactly.

Renormalized energy

A candidate quasiparticle energy solves

$$E_{\mathbf k}^\star - \xi_{\mathbf k} - \Sigma' \left( \mathbf k,E_{\mathbf k}^\star \right) = 0.$$

This equation can have several roots or no clean isolated root. Matrix-valued problems require an eigenchannel or determinant analysis rather than applying the scalar equation to every diagonal entry.

Residue

When the self-energy is smooth near the root,

$$Z_{\mathbf k} = \left[ 1 - \left. \partial_E \Sigma'(\mathbf k,E) \right|_{E=E_{\mathbf k}^\star} \right]^{-1}.$$

$Z_{\mathbf k}$ is the coherent pole weight in the stated single-particle channel. It is not:

• a permanent probability that a microscopic particle survives;
• the peak height;
• automatically equal to an effective-mass ratio;
• invariant under every change of orbital basis;
• guaranteed to remain finite at a non-Fermi-liquid or critical point.

In a generic interacting one-dimensional liquid, Luttinger Liquid Preview shows how the pole is replaced by power-law thresholds and the thermodynamic quasiparticle residue vanishes.

Intrinsic half-width

Define

$$\gamma_{\mathbf k} = - Z_{\mathbf k} \Sigma'' \left( \mathbf k,E_{\mathbf k}^\star \right).$$

If $\gamma_{\mathbf k}\geq0$ is small and the background varies slowly,

$$A(\mathbf k,E) \simeq{} \frac{Z_{\mathbf k}}{\pi} \frac{ \gamma_{\mathbf k} }{ \left( E-E_{\mathbf k}^\star \right)^2 + \gamma_{\mathbf k}^2 } + A_{\mathrm{inc}}(\mathbf k,E).$$

The coherent Lorentzian has area $Z_{\mathbf k}$, half width at half maximum

$$\operatorname{HWHM} = \gamma_{\mathbf k},$$

and full width at half maximum

$$\Gamma_E = 2\gamma_{\mathbf k}.$$

The remaining canonical weight belongs to $A_{\mathrm{inc}}$ after all coherent contributions are included.

When the Lorentzian approximation is controlled

A quasiparticle fit requires more than a narrow-looking feature. Check that:

1. $\Sigma'$ and $\Sigma''$ vary slowly across the fitted width;
2. the peak is separated from nearby poles and thresholds;
3. the background is not carrying comparable curvature;
4. the extracted width is larger than numerical uncertainty but smaller than the relevant dispersion scale;
5. the same feature follows a continuous dispersion across neighboring momenta;
6. the residue and width are stable under fit-window and resolution changes.

Near a threshold, avoided crossing, matrix-valued degeneracy, Green-function zero, or critical continuum, a Lorentzian fit can return precise numbers with no quasiparticle meaning.

Linewidth and Lifetime

Consider a retarded pole written as

$$G^{\mathrm R}(E) \simeq \frac{Z}{ E-E_\star+i\gamma }.$$

Its time-domain amplitude contains

$$e^{-iE_\star t/\hbar} e^{-\gamma t/\hbar}, \qquad t\gt0.$$

The spectral FWHM is

$$\Gamma_E = 2\gamma.$$

If this is a survival amplitude whose squared magnitude describes the population of the prepared excitation, with no additional pure dephasing, that population decays as

$$e^{-t/\tau},$$

then this pole convention gives

$$\tau = \frac{\hbar}{\Gamma_E} = \frac{\hbar}{2\gamma}.$$

The amplitude decay time is twice this value. A spectral coherence can also decay by pure dephasing without comparable population relaxation; in that case the linewidth alone does not determine a population lifetime. Exact spectra bounded below can produce short-time and late-time deviations from a purely exponential pole model. Other communities define the quoted “lifetime” from a coherence, population, transport, or escape rate. A lifetime–linewidth statement is incomplete until the decaying quantity and width convention are named.

Width dictionary

Width source	Origin	Does it imply an intrinsic lifetime?
self-energy damping	coupling to allowed decay channels	often, within a controlled pole approximation
open-system decay	relaxation or dephasing through external modes	identifies the decaying coherence; population lifetime needs a specified model
finite plotting $\eta$	chosen numerical kernel	no
finite observation time	Fourier resolution and window	no
detector resolution	instrumental response	no
inhomogeneous broadening	average over static environments	not a single homogeneous lifetime
thermal broadening	occupation factors or thermally varying self-energy	only after the mechanism is separated
unresolved level splitting	overlapping sharp transitions	no
final-state broadening	probe-specific outgoing state	not necessarily the target excitation lifetime

Lorentzian, Gaussian, and Voigt profiles

A normalized Lorentzian is

$$L_\gamma(E) = \frac1\pi \frac{\gamma}{ E^2+\gamma^2 }.$$

Two Lorentzian convolutions add their HWHM values. Two Gaussian convolutions add variances. A Lorentzian intrinsic line convolved with Gaussian instrumental resolution gives a Voigt profile, not another exact Lorentzian.

Asymmetric peaks

An asymmetric line can arise from:

• an energy-dependent self-energy;
• a nearby threshold;
• overlapping unresolved modes;
• interference between a discrete path and a continuum;
• an energy-dependent matrix element;
• background subtraction.

A Fano profile,

$$I(\epsilon) \propto \frac{ (q+\epsilon)^2 }{ 1+\epsilon^2 },$$

is an interference intensity. It should not automatically be identified with a positive intrinsic spectral density.

A Figure for the Three Distinctions

Panel A illustrates discrete lines. Panels B and C are a separate normalized continuous example: $A(E)=0.65L_{0.12}(E-0.5)+0.35\Theta(E-2)(E-2)e^{-(E-2)}$ in dimensionless energy units. Panel C is the numerical convolution of that same $A$ with a normalized Gaussian of width $\sigma=0.25$, with unit probe factor and no background. Convolution lowers and broadens the peak while preserving the full-axis integral. Only a finite energy window is shown; actual probes can additionally weight the spectrum by matrix elements and occupations.

Spectral Weight Is Area

For the normalized Lorentzian,

$$\int_{-\infty}^{\infty} dE\, L_\gamma(E-E_\star) = 1.$$

Therefore the coherent contribution

$$ZL_\gamma(E-E_\star)$$

has integrated area $Z$ for every $\gamma\gt0$. Its height,

$$\frac{Z}{\pi\gamma},$$

changes when the width changes. Comparing peak heights without comparing widths can therefore reverse the conclusion about spectral weight.

Coherent and incoherent decomposition

A useful local decomposition is

$$A(\mathbf k,E) = A_{\mathrm{coh}}(\mathbf k,E) + A_{\mathrm{inc}}(\mathbf k,E).$$

For one canonical orbital,

$$\int dE\, A(\mathbf k,E) = 1.$$

If one coherent pole has weight $Z_{\mathbf k}$, then

$$\int dE\, A_{\mathrm{inc}}(\mathbf k,E) = 1-Z_{\mathbf k}$$

only when this is the sole coherent contribution. If several coherent branches are separated out, the remaining incoherent integral is $1-\sum_jZ_j$. A finite-width Lorentzian is a local approximation; extending it over all energies is justified for area estimates only when omitted background and tail corrections are controlled.

Partial windows

Experiments and computations integrate over finite windows:

$$W(E_1,E_2) = \int_{E_1}^{E_2} dE\, A(E).$$

For the intrinsic $A(E)$, a change in this partial integral means redistribution across the window boundaries, including a moving or broadening peak. In measured intensity, occupation factors, probe matrix elements, resolution leakage, and background subtraction can additionally change the apparent area even when intrinsic $A$ is unchanged. Neither observation automatically violates a full-domain sum rule.

Moment constraints

Zeroth moments test normalization. Higher moments test commutators, kinetic energy, interactions, and asymptotic expansions. Sum Rules develops those exact constraints. Here the practical rule is:

When comparing integrated weights, specify the measure, window, operator normalization, and which other energies or channels were actually measured. Weight outside the accessible range can remain unknown.

Worked Benchmark: BCS Coherence Peaks

For a translationally invariant BCS saddle, the electron spectral function is

$$A(\mathbf k,E) ={} u_{\mathbf k}^2 \delta(E-E_{\mathbf k}) + v_{\mathbf k}^2 \delta(E+E_{\mathbf k}),$$

where

$$E_{\mathbf k} = \sqrt{ \xi_{\mathbf k}^2 + |\Delta_{\mathbf k}|^2 }.$$

The coherence factors satisfy

$$u_{\mathbf k}^2 + v_{\mathbf k}^2 = 1.$$

The two branches therefore split one unit of canonical electron weight. At zero temperature,

$$n_{\mathbf k} = \int_{-\infty}^{0} dE\, A(\mathbf k,E) = v_{\mathbf k}^2.$$

The negative-energy peak is not a negative-norm state. It is particle-removal weight created by particle–hole mixing. BCS Mean-Field Theory derives the Bogoliubov transformation and coherence factors.

Bogoliubov Quasiparticles explains why these pole weights are canonical coherence factors, how they differ from an interaction-renormalized residue, and why bosonic commutator spectra instead carry signed negative-frequency weight.

Interactions beyond the saddle can broaden the peaks, shift their energies, create satellites, and transfer weight into continua. A broadened two-peak fit alone does not establish weak-coupling BCS physics.

Energy and Momentum Cuts

A two-dimensional spectral map

$$A(\mathbf k,E)$$

is commonly analyzed through two kinds of cuts.

Energy-distribution curve

An energy-distribution curve holds momentum fixed and scans $E$. Its line shape directly samples the energy dependence of the self-energy, matrix element, occupation factor, and background at that momentum.

Momentum-distribution curve

A momentum-distribution curve holds $E$ fixed and scans momentum. If $\mathbf k$ is a wavevector and the group velocity is $\mathbf v^\star=\hbar^{-1}\nabla_{\mathbf k}E^\star$, local linearization gives

$$E-E_{\mathbf k}^\star \simeq - \hbar\mathbf v^\star \cdot (\mathbf k-\mathbf k_\star),$$

For a cut along a unit direction $\hat{\mathbf n}$, if the self-energy and matrix element vary weakly with momentum, the wavevector HWHM is approximately

$$\Delta k_{\mathrm{HWHM}} \simeq \frac{\gamma}{ \hbar|\mathbf v^\star\cdot\hat{\mathbf n}| }.$$

This conversion fails near a band extremum, flat band, strong momentum-dependent self-energy, overlapping band, or matrix-element zero. An energy width and momentum width are not interchangeable without a local dispersion model.

From Intrinsic Spectrum to Measured Intensity

A useful schematic forward model is

$$I_{\mathrm{obs}}(x,E) ={} \int dx'\,dE'\, R_x(x-x') R_E(E-E') \times |M(x',E')|^2 \mathcal F(E') \mathcal S(x',E') + B(x,E).$$

Here:

• $x$ denotes momentum, position, angle, polarization, or another resolved variable;
• $\mathcal S$ is the intrinsic operator spectrum;
• $M$ is a probe matrix element;
• $\mathcal F$ is an occupation or detailed-balance factor;
• $R_x$ and $R_E$ are normalized resolution kernels;
• $B$ is background.

The appropriate $\mathcal S$ and $\mathcal F$ depend on the experiment.

Removal spectroscopy

Photoemission, radio-frequency ejection, and related removal probes can approach

$$I_{\mathrm{rem}}(\mathbf k,E) \propto |M(\mathbf k,E)|^2 f(E) A(\mathbf k,E),$$

before final-state effects, escape depth, kinematic factors, and resolution are included.

At zero temperature, $f(E)$ suppresses positive-energy addition weight. Dividing noisy data by a small Fermi factor is not a model-independent reconstruction of the unoccupied spectrum.

Addition spectroscopy

Inverse photoemission or injection probes can emphasize

$$I_{\mathrm{add}}(\mathbf k,E) \propto |M(\mathbf k,E)|^2 \left[ 1-f(E) \right] A(\mathbf k,E).$$

Addition and removal experiments need not have the same matrix elements, backgrounds, or final-state broadening.

Tunneling spectroscopy

In a weak-tunneling limit, current involves a convolution of tip and sample spectral densities:

$$I(V) \propto \int dE\, \nu_{\mathrm{tip}}(E-eV) \nu_{\mathrm{sample}}(E) \times \left[ f(E-eV)-f(E) \right] |T(E,V)|^2.$$

The often-used approximation

$$\frac{dI}{dV} \propto \nu_{\mathrm{sample}}(eV)$$

requires a sufficiently featureless tip, weak tunneling, controlled voltage drop, low enough temperature, and slowly varying matrix element. It is not a universal identity.

Scanning Tunneling Microscopy and Spectroscopy explains junction operation, feedback and setpoint effects, conductance maps, and quasiparticle interference.

Scattering spectroscopy

Neutron, X-ray, electron, and cold-atom Bragg probes transfer momentum and energy without adding the same constituent whose single-particle propagator defines $A$. Their intrinsic target object is commonly a dynamic structure factor or susceptibility:

$$S_O(\mathbf q,E)$$

or

$$\chi_{OO}^{\mathrm R}(\mathbf q,E).$$

Polarization factors, form factors, kinematics, and detailed balance multiply the target spectrum. A mode seen in scattering and a peak seen in photoemission may arise from related physics while remaining different operator channels.

Matrix-element zeros

If

$$M(x,E) = 0,$$

the measured intensity vanishes even when the intrinsic spectrum is nonzero. Changing photon energy, polarization, probe geometry, tip orbital, or transferred momentum can therefore make a feature appear or disappear without changing the target Hamiltonian.

Resolution convolution

Even normalized resolution kernels preserve total weight only when the full domain is integrated:

$$\int dE\, R_E(E-E') = 1.$$

Finite detector windows, nonlinear calibration, clipping, and background subtraction can spoil apparent conservation. Deconvolution amplifies noise and is not unique without regularization or prior information.

Numerical Spectra

This section records the interpretation of common numerical outputs. Dynamical Correlation Functions Numerically gives the full comparative workflow, common benchmark and ways to distinguish finite-size, finite-time, and continuation resolution.

Exact diagonalization

Finite exact diagonalization produces weighted delta functions. A common display replaces

$$\delta(E-E_j)$$

by

$$L_\eta(E-E_j) = \frac1\pi \frac{\eta}{ (E-E_j)^2+\eta^2 }.$$

The integrated weight is preserved, but peak height, apparent overlap, and gap visibility depend on $\eta$. A convergence study should vary:

• system size;
• boundary conditions;
• symmetry sector;
• broadening kernel;
• broadening width;
• frequency window.

For a sparse Hamiltonian, Lanczos Method Preview shows how the operator-generated seed $A|0\rangle$ yields a tridiagonal continued fraction, why the response run can occupy a different symmetry sector, and which residual and moment checks belong in the numerical record.

Real-time calculation

A record of duration $T$ has characteristic energy resolution

$$\Delta E \sim \frac{2\pi\hbar}{T}.$$

A rectangular time window produces sinc-like ringing. Smooth windows reduce sidelobes but broaden the main lobe and convolve the true spectrum with the window transform. Zero padding interpolates the plotted frequency grid; it does not increase physical resolution.

Imaginary-time continuation

Imaginary-time or Matsubara data constrain a smoothed integral transform of the real-frequency spectrum. Recovering narrow lines, nearby peaks, or sharp thresholds from finite noisy data is ill conditioned.

A continuation result should report:

1. the data covariance and fitted residuals;
2. the positivity assumption and whether it is valid for the channel;
3. exact normalization and known moments;
4. prior or regularizer dependence;
5. synthetic-data resolution tests;
6. stability under removing imaginary-time or Matsubara points;
7. uncertainty on integrated features, not only a smooth central curve.

Two reconstructions that fit the imaginary-axis data equally well can disagree on linewidths smaller than the continuation resolution.

Analytic Continuation explains covariance whitening, singular-value resolution, and synthetic tests that reveal which spectral features are actually resolved.

Correction-vector and resolvent methods

After shifting by a declared reference energy $E_{\mathrm{ref}}$, methods that evaluate

$$\langle\psi| O \frac1{ E+i\eta-(H-E_{\mathrm{ref}}) } O^\dagger |\psi\rangle$$

compute an $\eta$-broadened object directly. The linear-solver tolerance, variational truncation, finite size, and $\eta$ all contribute errors. Extrapolating $\eta\to0^+$ is a separate limiting problem.

Comparing Theory and Experiment

A useful comparison matches more than peak position.

Band Structure Workflows explains converged one-electron band calculations. Their eigenvalues are a useful starting point, but an interacting spectral function can redistribute weight into satellites and continua. DMFT for Quantum Materials describes one approach to calculating that redistribution, within its local-self-energy approximation.

Forward comparison

Prefer applying the experimental forward model to the theoretical spectrum:

$$\mathcal S_{\mathrm{theory}} \longrightarrow I_{\mathrm{predicted}},$$

then compare $I_{\mathrm{predicted}}$ with raw or minimally processed data. This keeps resolution, occupation, matrix elements, and background assumptions explicit.

Quantities to compare

At minimum compare:

• peak or threshold energy;
• integrated weight over declared windows;
• intrinsic and observed widths separately;
• momentum dispersion;
• temperature and control-parameter dependence;
• polarization or channel dependence;
• asymmetry and continuum shape;
• exact moments available to both sides.

Agreement in one peak position does not establish agreement in spectral function.

Uncertainty propagation

Fit uncertainties should include correlations among:

• energy calibration;
• resolution width;
• background parameters;
• matrix-element model;
• occupation temperature;
• overlapping peaks;
• finite-window tails.

Subtracting a flexible background can trade continuum weight for peak area. That covariance should be reported rather than hidden by a single best-fit curve.

Common pitfalls

A maximum need not be a quasiparticle. Thresholds, overlapping modes, and probe matrix elements can all create maxima. A quasiparticle interpretation needs a locally controlled pole and a consistent dispersion.

Height is not residue. For a Lorentzian of coherent area $Z$, the height is $Z/(\pi\gamma)$. A lower peak can carry the same weight after broadening.

A linewidth may describe coherence rather than population. Pure dephasing and static inhomogeneity broaden spectra without providing the same population decay time as irreversible escape from an excitation.

A finite window is not a complete sum rule. Unobserved tails, other branches, and probe-dependent suppression can explain missing measured area. Keep the intrinsic spectral integral separate from an occupation-weighted intensity.

Exercises

Exercise 1: Lorentzian weight and delta limit

For

$$L_\gamma(E-E_0) = \frac1\pi \frac{\gamma}{ (E-E_0)^2+\gamma^2 }, \qquad \gamma\gt0,$$

show that its integral is one, identify its HWHM and FWHM, and explain the sense in which it approaches $\delta(E-E_0)$ as $\gamma\to0^+$.

Solution

Set

$$x = \frac{E-E_0}{\gamma}.$$

Then

$$\begin{aligned} \int_{-\infty}^{\infty} dE\, L_\gamma(E-E_0) &= \frac1\pi \int_{-\infty}^{\infty} \frac{dx}{1+x^2} \\ &= 1. \end{aligned}$$

The maximum is

$$L_\gamma(0) = \frac1{\pi\gamma}.$$

Half the maximum occurs at

$$|E-E_0| = \gamma.$$

Thus

$$\operatorname{HWHM} = \gamma, \qquad \operatorname{FWHM} = 2\gamma.$$

For every smooth compactly supported test function $f$,

$$\lim_{\gamma\to0^+} \int dE\, L_\gamma(E-E_0)f(E) = f(E_0).$$

That distributional statement, rather than pointwise convergence at $E_0$, defines the delta limit.

Exercise 2: The same atom in a density channel

In the atomic benchmark, replace electron removal by a source $H'=-f(t)n_\sigma$. Find the connected ordered spectrum of $\delta n_\sigma$, its isolated retarded self-response, and the isothermal derivative after re-equilibration. Why do the two finite-energy lines of $A_\sigma$ not appear?

Solution

The atom conserves $n_\sigma$, so $\delta n_\sigma(t)=\delta n_\sigma$. Since $n_\sigma^2=n_\sigma$ and $\langle n_\sigma\rangle=3/7$,

$$\mathcal S_{\delta n_\sigma\delta n_\sigma}(E) =\left(\frac37-\frac9{49}\right)\delta(E) =\frac{12}{49}\delta(E).$$

Its commutator vanishes at every time, hence $\chi^{\rm R}_{n_\sigma n_\sigma}=0$. An equilibrating bath can instead reweight the occupation sectors, giving the isothermal derivative

$$\left.\frac{\partial\langle n_\sigma\rangle_f}{\partial f}\right|_{f=0} =\beta\,\operatorname{Var}(n_\sigma) =\frac{12\ln2}{49J}.$$

The electron operator changes particle number and connects different atomic energies; the density operator is diagonal in these states. The missing finite-energy lines are a channel selection rule. The remaining connected zero-energy fluctuation is invisible to the commutator, exactly as the zero-frequency FDT discussion requires.

Exercise 3: A spectral variance without a lifetime

Compute the second moment and variance of the atomic $A_\sigma$. Check the second moment using $\langle\{[[c_\sigma,\mathcal K],\mathcal K],c_\sigma^\dagger\}\rangle$. Then convolve $A_\sigma$ with a centered Gaussian of standard deviation $s=J/4$. Which moments change, and does the resulting width measure a lifetime?

Solution

The line weights give

$$M_2=\frac47J^2+\frac37(3J)^2=\frac{31J^2}{7}, \qquad \operatorname{Var}_A(E)=M_2-M_1^2=\frac{192J^2}{49}.$$

Because $n_{\bar\sigma}$ commutes with both $\mathcal K$ and $c_\sigma$,

$$[[c_\sigma,\mathcal K],\mathcal K]=(\xi+Un_{\bar\sigma})^2c_\sigma.$$

The anticommutator expectation is therefore $\xi^2+(2\xi U+U^2)\bar n=31J^2/7$, independently reproducing the spectral result.

A centered normalized Gaussian preserves $M_0=1$ and $M_1=5J/7$. It adds $s^2M_0$ to $M_2$, so

$$M_2^{\rm obs}=\frac{503J^2}{112}, \qquad \operatorname{Var}_{\rm obs}(E)=\frac{192J^2}{49}+\frac{J^2}{16}.$$

The original variance describes two sharp transitions; the extra term describes detector resolution. Neither establishes decay in the isolated atom. A Lorentzian display kernel would instead make the ordinary full-axis second moment diverge, so the same variance subtraction would be invalid.

Exercise 4: Linearized self-energy

Consider

$$\Sigma^{\mathrm R}(E) = \Sigma_0 + \left( 1-Z_0^{-1} \right)E - i\frac{\gamma_0}{Z_0},$$

with real $\Sigma_0$, $0\lt Z_0\leq1$, and $\gamma_0\gt0$. For a bare level $\xi$, find the renormalized energy, residue, and Lorentzian HWHM.

Solution

The pole equation is

$$E^\star - \xi - \Sigma_0 - \left( 1-Z_0^{-1} \right)E^\star = 0.$$

Therefore

$$E^\star = Z_0 \left( \xi+\Sigma_0 \right).$$

The derivative formula gives

$$\begin{aligned} Z &= \left[ 1- \left( 1-Z_0^{-1} \right) \right]^{-1} \\ &= Z_0. \end{aligned}$$

Finally,

$$\gamma = - Z\operatorname{Im}\Sigma^{\mathrm R} = - Z_0 \left( -\frac{\gamma_0}{Z_0} \right) = \gamma_0.$$

The spectral FWHM is $2\gamma_0$, provided the linearized form remains valid across that width. This local model integrates to $Z_0$ if extended over the whole line; a canonical fermionic spectrum needs additional weight $1-Z_0$ outside the modeled pole region.

Exercise 5: Two-particle threshold exponent

For two particles in $d$ dimensions with

$$\epsilon(\mathbf p) = \Delta + \frac{\mathbf p^2}{2m},$$

derive the threshold at fixed total physical momentum $\hbar\mathbf q$ and show that a constant matrix element gives near-threshold weight proportional to $(E-E_{\mathrm{th}})^{d/2-1}$.

Solution

Use relative momentum $\mathbf p$:

$$\mathbf p_{1,2} = \frac{\hbar\mathbf q}{2} \pm \mathbf p.$$

Then

$$\epsilon(\mathbf p_1) + \epsilon(\mathbf p_2) = 2\Delta + \frac{\hbar^2\mathbf q^2}{4m} + \frac{\mathbf p^2}{m}.$$

The minimum occurs at $\mathbf p=\mathbf0$, so

$$E_{\mathrm{th}}(\mathbf q) = 2\Delta + \frac{\hbar^2\mathbf q^2}{4m}.$$

Let

$$\varepsilon = E-E_{\mathrm{th}}.$$

The phase-space integral is

$$\int d^dp\, \delta \left( \varepsilon-\frac{\mathbf p^2}{m} \right).$$

In spherical coordinates it scales as

$$p^{d-1} \left| \frac{d(p^2/m)}{dp} \right|^{-1} \propto p^{d-2}.$$

Since $p\propto\sqrt\varepsilon$,

$$\mathcal S(E) \propto \varepsilon^{d/2-1} \Theta(\varepsilon).$$

A matrix element that vanishes as $p^\ell$ contributes an additional power $\varepsilon^\ell$ after squaring the amplitude.

Exercise 6: Fermi weighting must precede resolution

For the atomic benchmark, take a centered Gaussian detector kernel $R_J(E)=e^{-E^2/(2J^2)}/(\sqrt{2\pi}J)$. Compare the correct $I_{\rm obs}=R_J*(fA_\sigma)$ with $I_{\rm wrong}=f(E)(R_J*A_\sigma)(E)$ at $E=0$. Also find the fraction of the correct removal signal recorded in $-2J<E<2J$.

Solution

The Fermi factor weights the exact transition energies before the detector mixes neighboring energies. Thus

$$\begin{aligned} I_{\rm obs}(0)&=\frac{8e^{-1/2}+e^{-9/2}}{21\sqrt{2\pi}J} \simeq\frac{0.09239}{J},\\ I_{\rm wrong}(0)&=\frac{4e^{-1/2}+3e^{-9/2}}{14\sqrt{2\pi}J} \simeq\frac{0.07008}{J}. \end{aligned}$$

The latter uses $f(0)=1/2$ for contributions originating at both $-J$ and $3J$, where the actual occupations differ.

Let $\Phi(x)$ be the standard-normal cumulative distribution. Integrating the correct two Gaussian lines over the stated window gives

$$W_{\rm rem} =\frac8{21}\bigl[\Phi(3)-\Phi(-1)\bigr] +\frac1{21}\bigl[\Phi(-1)-\Phi(-5)\bigr] \simeq0.32755.$$

The full removal weight is $3/7$, so the recorded fraction is $7W_{\rm rem}/3\simeq0.7643$. The missing area is detector-window leakage, even though the underlying spectrum obeys its exact normalization. Dividing the convolved curve by a pointwise Fermi factor cannot undo this mixing.

Exercise 7: Finite time and artificial linewidth

A real-time calculation reaches $T=200\,\hbar/J$, where $J$ is an energy scale. Estimate the characteristic energy resolution. Explain why zero padding cannot resolve two lines separated by $0.01J$.

Solution

The characteristic resolution is

$$\Delta E \sim \frac{2\pi\hbar}{T} = \frac{2\pi}{200}J \approx 0.031J.$$

Two lines separated by $0.01J$ lie within this main resolution scale. Zero padding inserts additional frequency-grid points between the original discrete transform samples, making the curve look smoother. It does not add late-time information and therefore cannot create the missing resolving power.

A different window can alter ringing and numerical peak width, but resolving the pair requires a longer reliable time record or independent prior information.

Exercise 8: Choose the spectral object

For each task, identify the most direct intrinsic object and one forward-model factor that must still be considered:

1. remove an electron with angle-resolved photoemission;
2. scatter neutrons from spin fluctuations;
3. infer local electronic states with a tunneling tip;
4. model a qubit coupled to oscillator bath modes.

Solution

1. Use the fermionic removal part of $A(\mathbf k,E)$. The Fermi factor, photoemission matrix element, final state, and instrumental resolution intervene.
2. Use the spin dynamic structure factor $S^{\alpha\beta}(\mathbf q,E)$ or the related susceptibility. Magnetic form factors and polarization projectors intervene.
3. Use local sample and tip spectral densities in the tunneling-current convolution. The tunneling matrix element, tip density of states, temperature, and voltage drop intervene.
4. Use the bath coupling spectral density $J(\omega)$. The system coupling operator and thermal occupation factors intervene.

The objects can share similar line shapes, but their operators and normalizations are different.

Related reading

• Angle-Resolved Photoemission Spectroscopy explains how occupied electron spectra become measured momentum maps.
• Unconventional Superconductivity compares gap and coherence information with crystal symmetry and other probes.
• Lifetime and Spectral Weight relates decay rates to propagation and quasiparticle criteria.
• Dynamical Correlation Functions Numerically compares Lehmann, resolvent, and real-time methods.
• Spectral Densities explains coupling-weighted bath spectra.

References

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