Single Particle Excitations

• Single-particle excitations are defined as the addition or removal of a particle from a quantum many-body system, marked by characteristic poles in the Green’s function.
• They quantify quasiparticle energies, lifetimes, and momentum-dependent spectral weight, offering insights into Fermi-liquid behavior, localization, and metal-insulator transitions.
• Advanced experimental and computational methods, including RPA and diagrammatic Monte Carlo, enable precise mapping of these excitations across diverse quantum systems.

Single-particle excitations characterize the addition or removal of a particle from a quantum many-body system and thus encode essential information about its spectral, transport, and response properties. In condensed matter, atomic, and mesoscopic physics, these excitations are defined via poles (or peaks) in the single-particle Green’s function, and their spectral function provides direct access to quasiparticle energies, lifetimes, coherence factors, and momentum-dependent weight. Single-particle excitations reveal the interplay between interactions, symmetry, disorder, topology, and external perturbations, and their universal properties underpin a wide range of quantum phenomena from Fermi-liquid theory to many-body localization, metal-insulator transitions, superfluidity, and quantum criticality.

1. Formal Definition and Theoretical Frameworks

The single-particle Green’s function $G(\mathbf{k},\omega)$ describes the amplitude for adding or removing a particle of momentum $\mathbf{k}$ and energy $\omega$: $$G(\mathbf{k},\omega) = \langle\langle\,c_{\mathbf{k}\sigma}\,;\,c_{\mathbf{k}\sigma}^\dagger\rangle\rangle_\omega = \int_0^\infty dt\,e^{i(\omega+i0^+)t} \langle\{c_{\mathbf{k}\sigma}(t),\,c_{\mathbf{k}\sigma}^\dagger(0)\}\rangle$$ The spectral function,

$$A(\mathbf{k},\omega) = -\frac{1}{\pi}\mathrm{Im}\,G(\mathbf{k},\omega+i0^+)$$

quantifies the energy- and momentum-resolved density of single-particle excitations, including bare bands and interaction effects. Quasiparticles are sharply defined when $A(\mathbf{k},\omega)$ exhibits narrow Lorentzian peaks; their width and shift encode the interaction-renormalized dispersion and finite lifetime through the self-energy $\Sigma(\mathbf{k},\omega)$: $$G^R(\mathbf{k},\omega) = \frac{1}{\omega + \mu - \epsilon_{\mathbf{k}} - \Sigma^R(\mathbf{k},\omega)}$$ In Fermi liquids, the quasiparticle residue

$$Z = \left[1 - \partial_\omega \mathrm{Re} \Sigma^R(\mathbf{k}_F,\omega)\rvert_{\omega=0} \right]^{-1}$$

and effective mass $m^*/m$, as well as Landau parameters ($F_0^s$, $F_0^a$), provide a complete phenomenology; diagrammatic Monte Carlo techniques now enable their high-precision extraction for archetypal models such as the 3D uniform electron gas (Haule et al., 2020).

2. Disorder, Localization, and Many-Body Localization

In disordered systems, single-particle excitations discriminate between extended (delocalized) and localized states. The local density of states (LDOS),

$$\rho(i,\omega) = -\frac{1}{\pi} \mathrm{Im} G(i,i,\omega^+)$$

where $\omega^+ = \omega + i\eta$, quantifies the spatial inhomogeneity of spectral weight. Universal features near localization and many-body localization (MBL) transitions are captured by the ratio of typical to average LDOS or scattering rate: $$R_X(\omega) = \frac{X_{\rm typ}(\omega)}{X_{\rm avg}(\omega)}, \quad X \in \{\rho, \Gamma\}$$ Typical values vanish in the localized phase ($R_X \to 0$), while remaining $O(1)$ in the metallic phase. Finite-size scaling of $R_X(W,L)$ reveals continuous transitions—critical exponents obeying the Chayes–Chayes–Fisher–Spencer (CCFS) bound $\nu \ge 2/d$ in random disorder systems, with clean data collapses indicating scale invariance (Jana et al., 2022). For quasi-periodic potentials, the critical exponent $\nu \ge 1$, distinct universality classes arise compared to random disorder (Prasad et al., 2023). Intriguingly, level-spacing diagnostics (which probe the Fock-space connectivity) can yield anomalously small $\nu$, demonstrating observable-dependent scaling and the need for real-space diagnostics.

3. Strong Correlation: Mott, Kondo, and Pseudogap Physics

Single-particle dispersions in correlated lattice models set the stage for emergent collective excitations. In the half-filled Hubbard model, the Mott transition links single-particle dispersion to spin excitations: by forming particle-hole composites from the single-particle band, the spinon continuum in 1D or magnon mode in 2D is reconstructed (Kohno, 2014). Doping-induced states above the Mott gap evolve continuously from charged quasiparticles to purely spin-carrying excitations as the spectral weight vanishes while the dispersion survives. In Kondo insulators, the mean-field hybridization opens a gap,

$$G_c(\mathbf{k},\omega) = \frac{\omega+\lambda}{(\omega-E_{k,+})(\omega-E_{k,-})}$$

with a momentum-dependent direct gap $$\Delta(\mathbf{k}) = \sqrt{(\epsilon_{\mathbf{k}}+\lambda)^2 + 4b^2}$$ smallest on the locus of the small Fermi surface. The momentum distribution $n_c(\mathbf{k})$ retains rapid crossovers at this locus, indicating "incipient" Fermi-surface jumps at the quantum critical point (Pixley et al., 2015). Exact-diagonalization Monte Carlo approaches have established pseudogap-like dips and pronounced momentum/pseudo-gap anisotropies in the spectral function $A(\mathbf{k},\omega)$ of the Hubbard model, including particle-hole asymmetric dispersions along high-symmetry cuts and Fermi surfaces (Rashid et al., 2023).

4. Superfluid, Bose–Einstein Condensate, and Polariton Contexts

In bosonic systems, the single-particle Green’s function becomes a $2\times2$ matrix in Nambu space, and excitations are hybridized with collective density modes. The correspondence is exact in the low-energy, low-momentum limit: $$\det G^{-1}(\mathbf{p},\omega) = 0 \implies \omega = c_T |\mathbf{p}|$$ with $c_T^2 = n/(m \,\partial n/\partial \mu)$ the isothermal sound speed. The matrix formalism by Gavoret-Nozières and Nepomnyashchii–Nepomnyashchii ensures that both single-particle and density response share this phonon pole; explicit identities such as $\Sigma_{12}(0,0)=0$ and the Hugenholtz–Pines relation $\Sigma_{11}(0,0)-\Sigma_{12}(0,0)=\mu$ ensure consistency (Watabe, 2020). At $T>0$, RPA calculations show satellite peaks and broadening in the spectral function and structure factor, with many-body effects emerging. In 2D dipolar Bose gases, path integral Monte Carlo and stochastic optimization methods reveal sharp quasiparticle resonances hybridized with collective density modes (phonons, rotons, multi-roton branches), which collapse in the normal phase into broad free-particle-like branches (Filinov et al., 2012). Bogoliubov–de Gennes theory underpins the crossover from single-particle to collective sound-wave-like excitations in driven polariton gases and box-trapped BECs, experimentally confirmed by angle-resolved FWM and center-of-mass oscillation measurements (Kohnle et al., 2011, Garratt et al., 2018).

5. Lifetimes, Damping, and Non-equilibrium Broadening

The single-particle excitation lifetime $\tau_{\mathbf{p}}$ is determined by the imaginary part of the self-energy. In dilute Bose–Einstein condensates at zero temperature,

$$\delta\omega_{\mathbf{p}} = \text{Im} \Sigma(\omega = E_{\mathbf{p}}^B) \propto (U n_0) \propto a$$

generating a finite lifetime scaling as $\tau_\mathbf{p} \sim a^{-1}$, contrasting with $a^{-2}$ scaling in normal states. The momentum-dependence exhibits undamped phonons at long wavelengths, maximal damping at intermediate $|\mathbf{p}|$, and recovery of infinite lifetime (free-particle) at large $|\mathbf{p}|$ (Tsutsui et al., 2014). In dissipative contexts, such as polariton gases and cavity-coupled Fermi systems, dissipation and filling factors can cause severe broadening, selectively breaking down the quasiparticle picture via resonance-induced coupling to polariton modes. The broadening $\Gamma$ is maximized when dissipation matches recoil energy, and is spin- and filling-selective (Feng et al., 2017). Nonlinear excitation, quench dynamics, and Goldstone-mode oscillations can be visualized directly in population inversion and oscillatory single-particle occupations in ultracold atomic gases (Kettmann et al., 2015).

6. Single-Electron/Single-Hole Generation and Quantum Optics

Single-particle excitations in noninteracting electron gases can be engineered via voltage pulses. Lorentzian voltage pulses of unit flux generate a pure electron excitation above the Fermi level and a correlated hole below; the final state factorizes into the untouched Fermi sea and a single wave-packet. This protocol is unique for producing minimal-excitation single-electron sources for electron optics, with the total transmitted charge quantized and independent of pulse shape (Hassler et al., 2011).

7. Core-Level and Semiconductor Quantum Structures

The binding energy of a core-state single-particle excitation is computed via the $\Delta$SCF approach as the difference in total energies between ground and core-hole configurations, correctly capturing final-state screening (renormalized core-hole density) and spin-orbit splitting. This yields core-level binding energies in close agreement with experiment when compared to Voigt-profile fits, resolving discrepancies with Kohn–Sham DOS-based interpretations in systems such as silicene (Lee et al., 2016). In vertically-stacked quantum-dot wires, single-particle continua are analytically derived via RPA: intrasubband and intersubband transitions delimit the energy-momentum region in which $\mathrm{Im}\,\epsilon^{-1}$ is nonzero. Adjusting barrier and well widths or applying a magnetic field tunes the miniband dispersion, density of states, and thereby the SPE continuum and its boundaries (Kushwaha, 2011, Kushwaha, 2021).

8. Collective Coupling, Screening, and Crossover

In graphene, single-particle interband excitations (π and σ transitions) dominate the loss function at small momentum transfer $q$, but evolve into collective modes (plasmons) as screening builds up for finite $q$: $$S(q,\omega) = -\text{Im}[\mathcal{E}_M^{-1}(q,\omega)] = \frac{\mathcal{E}_2(q,\omega)}{\mathcal{E}_1^2(q,\omega) + \mathcal{E}_2^2(q,\omega)}$$ A screening factor $D(q,\omega)$ sharpens and blue-shifts peaks into broad Landau-damped plasmons for $q \sim 0.03$–$0.3$ a.u., showing a continuous crossover from unscreened single-particle excitations ($\omega \sim q^2$) to collective plasmonic modes ($\omega \sim \sqrt{q}$), confirmed quantitatively by DFT–RPA and EELS measurements (Novko et al., 2015).

---

In all contexts, single-particle excitation spectra unify dynamical response, universality across quantum transitions, the interplay of locality and collective behavior, and the spectral signatures of quasiparticles and their coherent or incoherent decay. Their rigorous characterization and scaling analysis provide a foundational diagnostic for quantum matter in equilibrium, non-equilibrium, and strongly correlated settings.