Momentum-Resolved Relaxation Time Framework

Updated 15 August 2025

Momentum-resolved relaxation time frameworks are theoretical models that capture how the return to equilibrium in excited electronic and quasiparticle systems varies with momentum.
They integrate experimental techniques like trARPES and momentum microscopy with advanced simulations to quantify anisotropic scattering and gap-dependent decay rates.
These frameworks unify microscopic scattering processes with macroscopic conservation laws, offering insights for superconductivity, ultrafast optoelectronics, and strongly correlated systems.

A momentum-resolved relaxation time framework is a class of theoretical and phenomenological models wherein the time scale governing the return to equilibrium of excited electronic, spin, or quasiparticle populations depends explicitly on momentum—either as a continuous variable $\mathbf{k}$ (in crystalline systems) or via quantum numbers associated with momentum-like degrees of freedom. This formalism has become crucial for understanding ultrafast dynamics, transport, and dissipation in systems ranging from superconductors, graphene, and Mott insulators to strongly correlated quantum fluids, where relaxation processes exhibit pronounced momentum dependence due to symmetry, anisotropy, or collective effects.

1. Theoretical Foundation and Two-Temperature Model Approaches

The foundational concept is that, after an ultrafast excitation (e.g., by a femtosecond laser pulse), the non-equilibrium distributions of electrons (or quasiparticles) do not relax uniformly throughout the Brillouin zone. Instead, local dynamics are governed by the interplay between microscopic scattering processes (such as electron–electron and electron–phonon collisions) and macroscopic conservation laws, with the momentum dependence traced to features such as the superconducting gap symmetry, Fermi surface topology, or coupling to collective excitations.

In high- $T_c$ d-wave superconductors, a canonical example is the two-temperature model (Wang et al., 2013). Post-pump, the electronic subsystem rapidly thermalizes among itself (electron–electron scattering, $\tau_{ee}\sim$ few fs), establishing a quasi-equilibrium characterized by $T_e$ (electronic temperature), whereas the lattice remains at $T_l$ . The recombination of quasiparticles into Cooper pairs governs relaxation:

$\frac{d(T_e^2)}{dt} = -r (T_e^4 - T_l^4)$

with $r$ capturing the recombination kinetics. Momentum dependence emerges via the nodal d-wave gap, $\Delta_k = \frac{\Delta(T_e)}{2}(\cos k_x - \cos k_y)$ , so that the population decay rate is strongly angle-dependent: near the node ( $\varphi = 45^\circ$ ), the gap vanishes and relaxation is governed by thermal broadening; away from the node, gap reduction dominates, producing disparate momentum-resolved decay times.

2. Experimental Frameworks and Momentum-Resolved Spectroscopies

Ultrafast experimental probes able to resolve momentum-dependent relaxation include time- and angle-resolved photoemission spectroscopy (trARPES) (Wang et al., 2013, Nicholson et al., 2018), time-resolved momentum microscopy (Reutzel et al., 22 Feb 2024), and time/momentum-resolved tunneling spectroscopy (Zawadzki et al., 2019, Yoo et al., 2023). These techniques project the evolution of the excited electronic distribution onto both $(E, \mathbf{k})$ space and time, enabling direct measurement of differential relaxation rates, energy transfer mechanisms, and band-dependent decay.

A prototypical workflow in trARPES involves using a pump to excite the electronic population and a time-delayed probe to photoemit electrons, measuring the energy and momentum-dependent population as a function of delay $\Delta t$ . Population decay at a given $\mathbf{k}$ (or along a momentum cut) is analyzed via integration of the spectral function above the Fermi level, allowing extraction of momentum-resolved lifetimes or recombination rates. The time evolution is then fit using exponential or multi-exponential models, and often compared with spectral function-based simulations:

$A(E, k, \Delta t) = \epsilon(k, \Delta t) f(E, \Delta t)$

where $\epsilon(k, \Delta t)$ is the time-dependent band dispersion and $f(E, \Delta t)$ is the Fermi–Dirac function at a transient electronic temperature (Nicholson et al., 2018).

Time-resolved momentum microscopy further extends this paradigm to direct measurement of excitonic dynamics, allowing real-space reconstruction of exciton wavefunctions via orbital tomography (Reutzel et al., 22 Feb 2024).

3. Microscopic Origins of Momentum Dependence

Momentum-resolved relaxation arises from several sources, including but not limited to:

Superconducting gap anisotropy: As in d-wave superconductors where $\Delta_k$ is highly anisotropic, the density of states, electronic heat capacity, and available relaxation channels vary strongly with $\mathbf{k}$ , leading to direction-dependent QP recombination and lifetime (Wang et al., 2013).
Spin–orbit interactions and disorder: In graphene, the Bychkov–Rashba term and disorder produce momentum-locked spin precession, yielding a Dyakonov–Perel type relation for the spin relaxation time, $\tau_s \propto 1/\tau$ (with $\tau$ the momentum lifetime) (Offidani et al., 2018).
Electron–phonon and phonon–phonon coupling: In surface nanowires and semiconductors, energy transfer proceeds from hot electrons to high-energy optical phonons, with subsequent slower coupling to the lattice, producing mode- and $k$ -dependent relaxation times (Nicholson et al., 2018, Wörle et al., 2021).
Degree of integrability: In one-dimensional Bose gases and quantum spin chains, integrability-breaking perturbations introduce momentum-resolved damping rates via a generalized relaxation time approximation (GRTA), transitioning from quasi-ballistic GHD to diffusive hydrodynamics (Lopez-Piqueres et al., 2020).

4. Generalized and Extended Relaxation Time Approximations

The relaxation time approximation (RTA) for the Boltzmann kinetic equation is adapted to momentum-resolved frameworks in multiple ways:

Momentum-dependent relaxation time: The collision term is generalized as $C[f] = -(p^\mu u_\mu/\tau_R(x, p)) (f - f_0)$ , with $\tau_R(x, p)$ parameterized (often as a power law, e.g., $\tau_R(x,p) = \tau_0 (u \cdot p/T)^\ell$ ) to capture microscopic scattering (Mitra, 2020, Singh et al., 19 Mar 2024, Singh et al., 29 Jan 2025, Gangadharan et al., 31 Mar 2025).
Elastic/inelastic separation: Distinct relaxation times $\tau_{el}$ and $\tau_{in}$ are assigned to conserve particle number (elastic) or allow for particle production/annihilation (inelastic), leading to a combined collision kernel with separate momentum-dependent components (Florkowski et al., 2016, Bhadury et al., 2020).
Multi-species generalization: For mixtures (e.g., quark-gluon plasma, hadron resonance gas), counter-terms in the collision operator enforce conservation of energy and momentum, even with species- and momentum-dependent relaxation times (Rocha et al., 20 May 2025).
Spin dependence: In relativistic spin hydrodynamics, the relaxation time is allowed to depend on both momentum and spin, $\tau_R(x,p,s) = \tau_{eq}(x,p) (u\cdot s)^{2\ell}$ , modifying all transport coefficients and allowing for spin-resolved dissipative corrections (Bhadury, 26 Aug 2024).

These scattering kernels are solved, typically via Chapman–Enskog gradient expansions or moment methods, resulting in evolution equations for dissipative currents (shear and bulk stress, diffusion, spin current) whose coefficients directly reflect the underlying momentum dependence and enable frame-invariant hydrodynamic formulations (Mitra, 2020, Singh et al., 19 Mar 2024, Singh et al., 29 Jan 2025, Bhadury, 26 Aug 2024, Rocha et al., 20 May 2025).

5. Phenomenological and Numerical Implications

Explicit momentum dependence in $\tau_R$ leads to substantial modifications in hydrodynamic evolution and transport:

Distinct relaxation times for different modes: Diffusion, shear, and bulk dissipative channels may relax on parametrically different timescales, as seen in the ratio $\tau_n/\tau_\pi$ for number diffusion versus shear stress, which departs from unity when $\ell \neq 0$ and can significantly impact dissipative evolution (Singh et al., 29 Jan 2025).
Extended matching conditions: The energy density and other local equilibrium properties acquire correction terms (“extended Landau matching”) tied to the nontrivial momentum structure of $\tau_R$ , affecting both the hydrodynamic closure and the stability of simulations (Mitra, 2020).
Modification of transport coefficients: Shear viscosity, bulk viscosity, and diffusion constants become explicit functions of the momentum dependence parameter $\ell$ , leading to measurable effects in the anisotropy and relaxation profiles—especially pertinent in strongly magnetized QCD matter (Singh et al., 19 Mar 2024).
Anomalous behavior in the presence of strong correlations: In 1D Mott insulators and quantum Hall systems, momentum-resolved relaxation can be suppressed (e.g., due to spin-charge separation, topological spin textures), resulting in persistent non-thermal populations and long-lived prethermal states (Zawadzki et al., 2019, Yoo et al., 2023).

Numerical solutions typically require handling infinite hierarchies of coupled moment equations, especially under Bjorken flow or in far-from-equilibrium conditions, where momentum-dependent $\tau_R$ introduces nontrivial algebraic structure and necessitates resummation or regularization strategies (Gangadharan et al., 31 Mar 2025).

6. Applications and Impact in Materials and Strongly Correlated Systems

Momentum-resolved relaxation time frameworks have facilitated advances in several domains:

High- $T_c$ and unconventional superconductivity: trARPES, analyzed within two-temperature and momentum-resolved frameworks, has elucidated nodal/antinodal discrepancies in QP decay and the fluence dependence of pairing recovery (Wang et al., 2013).
Ultrafast optoelectronics in semiconductors: Phase-resolved transient absorption and momentum microscopy reveal how hot carrier relaxation, energy transfer, and exciton thermalization depend on $k$ -space location, providing blueprints for device optimization (Wörle et al., 2021, Reutzel et al., 22 Feb 2024).
Quantum transport and spintronics: The link between momentum and spin relaxation times in Dirac and Rashba materials clarifies the mechanisms underpinning spintronic device operation (Offidani et al., 2018, Vollmar et al., 2022).
Hydrodynamic modeling in heavy-ion collisions: Incorporating moment and momentum dependence in relaxation times enables realistic matching to exact field-theory results, crucial for interpreting observables sensitive to non-equilibrium effects and for constraining transport coefficients from data (Singh et al., 29 Jan 2025, Rocha et al., 20 May 2025).
Correlated 2DES and quantum Hall states: Pump–probe tunneling spectroscopy has established momentum-resolved decay timescales for collective spin textures and energy relaxation in systems where transport is governed by topological constraints (Yoo et al., 2023).

These frameworks collectively provide the quantitative tools and conceptual basis required to interpret ultrafast and out-of-equilibrium experiments, facilitate first-principles computation of dissipative coefficients, and unify kinetic theory, hydrodynamics, and advanced spectroscopies under a common momentum-resolved perspective.

7. Future Directions and Challenges

Despite substantial progress, several challenges and frontiers remain:

Microscopic–macroscopic bridging: Accurately parameterizing the momentum and species dependence of $\tau_R$ from full scattering matrices, especially in presence of strong correlations or when deviations from quasi-particle pictures are significant.
Nonlinear and quantum extensions: Inclusion of higher-order corrections, quantum fluctuations, and non-Gaussian initial conditions, particularly important in transient, highly non-equilibrium states.
Integration with machine-learned and/or ab-initio scattering rates: For realistic multi-band and multi-species materials, developing efficient schemes to input momentum-resolved $\tau_R$ directly from first-principles electronic structure or experimental data.
Application to topological and strongly entangled phases: Extending current frameworks to accommodate the peculiar momentum dependence that arises from topological textures, emergent gauge fields, or non-Fermi liquid behavior.
Comparison and matching to strongly coupled theories: Systematic exploration of how weakly coupled kinetic-theory-based (branch-cut) structures connect with the quasinormal mode spectra (isolated poles) found in holographic and strong-coupling approaches (Bajec et al., 26 Mar 2024).

A plausible implication is that as experimental time, momentum, and energy resolution continue to improve, the requirement for theoretically robust, momentum-resolved relaxation frameworks will intensify, with kinetic and hydrodynamic models needing to keep pace with the anisotropic and mode-selective relaxation dynamics measurable in modern ultrafast experiments.

Table: Selected Features of Momentum-Resolved Relaxation Time Frameworks

System/Technique	Origin of $\tau$ Momentum Dependence	Key Observable
d-wave superconductor/trARPES	Gap anisotropy ( $\Delta_k$ )	Angle-dependent QP decay rates
Graphene/Dirac-Rashba	Spin–orbit, disorder	Dyakonov–Perel spin/momentum lifetimes
In/Si(111) nanowires/trARPES	Bandstructure, electron–phonon coupling	$k$ -resolved thermalization
QGP kinetic theory/ERTA	Microphysics, species/mass, magnetic fields	Mode-dependent transport coefficients