Nonequilibrium Relaxation and Odd-Even Effect in Finite-Temperature Electron Gases

TL;DR

This work develops a numerically exact framework to solve the linearized Fermi-liquid collision integral for a two-dimensional electron gas with screened Coulomb interactions, enabling the full spectrum of relaxational eigenmodes to be computed across temperatures up to $T_F$. It reveals an isolated band of long-lived odd-parity modes whose decay rates scale as $\\gamma_{m,\\text{odd}} \\sim (T/T_F)^4 m^4$ at low $T$, while higher modes in all sectors follow conventional Fermi-liquid scaling $\\gamma_m \\sim T^2/(\\hbar T_F)$. The odd-parity lifetimes are largely insensitive to the interaction strength $r_s$, whereas even-parity modes are strongly $r_s$-dependent, allowing tunability of the odd–even separation by Coulomb screening. These results provide a comprehensive, quantitative description of nonequilibrium relaxation in 2D electron gases and establish a foundation for extended hydrodynamic (tomographic) descriptions that incorporate long-lived odd-parity modes. The methodology and findings have direct relevance for interpreting transport experiments in clean 2D metals and for developing beyond-hydrodynamic theories that capture parity-selective relaxation channels.

Abstract

Pauli blocking in Fermi liquids imposes strong phase-space constraints on quasiparticle lifetimes, leading to a well-known quadratic-in-temperature decay rate of quasiparticle modes at low temperatures. In two-dimensional systems, however, even longer-lived modes are predicted (dubbed ``odd-parity'' modes) that involve a collective deformation of the Fermi distribution. Here, we present an efficient method to evaluate the full spectrum of relaxational eigenmodes of a Fermi liquid within kinetic theory. We employ this method to study the experimentally relevant case of a Fermi liquid with screened Coulomb interactions and map out the decay rates of quasiparticle modes beyond the asymptotic low-temperature limit up to the Fermi temperature, thus covering the entire temperature range of typical experiments. We confirm the existence of anomalously long-lived odd-parity modes and provide a comprehensive classification and detailed analysis of the relaxation spectrum. In particular, we find that (i) the odd-parity effect in the decay rates extends to temperatures as large as $T=0.15T_F$, (ii) there is only a small number of long-lived odd-parity modes, with an infinite number of remaining modes that show standard Fermi-liquid scaling, and (iii) the ratio between the odd- and even-parity lifetimes is tunable with the Coulomb interaction strength, in addition to temperature, which reflects a difference in the microscopic relaxation mechanism of the modes. Our findings provide a comprehensive description of the nonequilibrium relaxation behavior of two-dimensional electron gases and bridge a significant gap in our understanding of these systems.

Figures (10)

• Figure 1: Microscopic origin of the odd-even effect in the relaxation of the quasiparticle distribution. Sketched are two nonequilibrium distributions in momentum space with a deformed Fermi surface and a small broadening of the Fermi edge of width ${\it O}(T)$, where the first distribution (blue) has even and the second (orange) odd parity. At low temperatures, relaxation corresponds to angular dynamics on the Fermi surface (black circle). Even-parity deformations of the quasiparticle distribution (blue) relax by head-on scattering, which gives the canonical Fermi-liquid dependence $\gamma_{\text{even}} \sim (T/T_F)^2$. Odd-parity deformations (orange), by contrast, relax by small-angle scattering, which leads to a much smaller relaxation rate at low temperatures $\gamma_{\text{odd}} \sim (T/T_F)^4$.
• Figure 2: Symmetry of Fermi surface deformations ordered by the angular harmonic index $m$ [cf. \ref{['eq:psi_m']}]. The black circle indicates the Fermi surface. Note that the full relaxational eigenmodes of the collision integral, \ref{['eq:collision_def']}, are labeled by the angular symmetry, but they are in general not sharp deformations of the Fermi distribution as sketched here [i.e., \ref{['eq:f_pert_def']} with $\psi = {\rm const}$]. Instead, they have a thermal and interaction-induced broadening, where higher eigenmodes have an increasing number of radial nodes in the function $\psi_m(p)$ [cf. Eqs. \ref{['eq:basis_expansion']} and \ref{['eq:lowT_exact_polys']} for the low-temperature case]. Deformations with $m=0$ symmetry determine the density response and bulk viscosity, $m=1$ to charge and heat currents, and $m=2$ to the shear viscosity, for example, and higher modes contribute to a finite wave vector response.
• Figure 3: Spectra of decay rates $\gamma_m$ as a function of angular mode number $m$ for $r_s = 1$ at three different temperatures (a) $T/T_F=10^{-4}$, (b) $T/T_F = 10^{-2}$, and (c) $T/T_F=0.5$. We show the lowest eight relaxational eigenmodes for each angular channel, where compressional modes ($m=0$) are black, current modes ($m=1$) gray, and remaining even-parity modes blue and odd-parity modes orange, respectively, and we indicate higher modes by the gray shaded area. We use the same range in each plot to compare spectra at different temperatures. As the temperature decreases, Pauli blocking becomes increasingly important with a phase-space constraint on scattering that increases the lifetime, where the magnitude of most relaxation rates asymptotically follows a Fermi-liquid scaling with $\gamma_m \sim T^2/(\hbar T_F)$. Crucially, however, there is an odd-even effect in the relaxation rates at low temperatures, with a band of isolated odd-parity modes (the number of which increases at low temperature as $\sim \sqrt{T_F/T}$) that decouple from the remaining spectrum with anomalously long lifetimes and an asymptotic scaling $\gamma_m \sim m^4 T^4/(\hbar T_F^3)$ [highlighted by the blue shaded area in (a) and (b)].
• Figure 4: Relaxation rates as a function of temperature for the (a) lowest, (b) second-lowest, and (c) third-lowest relaxational eigenmodes (left to right) in each angular sector up to $m \leq 20$. We show compressional modes [$m=0$, (c)] in black, current modes [$m=1$, (b) and (c)] in gray, and remaining even-parity modes in blue and odd-parity modes in orange, respectively. The insets show the local temperature scaling exponent $\alpha$ for each mode, $\gamma_m \sim (T/T_F)^\alpha$, as determined from a logarithmic derivative of the numerical data in the main figures. The relaxation rate of all modes decreases with decreasing temperature, reflecting the enhanced lifetime due to Pauli blocking. For blue even-parity modes in (a) and all higher eigenmodes [(b) and (c)], the relaxation rates are suppressed at low temperatures with standard Fermi-liquid scaling, $\gamma_m \sim T^2/(\hbar T_F)$, corresponding to $\alpha=2$. The odd-even effect is present only in the lowest eigenmode of each angular momentum sector (a): Here, with decreasing temperature, a successively increasing number of order ${\it O}(\sqrt{T_F/T})$ of odd-parity modes decouple from the Fermi-liquid scaling with an asymptotic temperature dependence $\gamma_m \sim T^4$, corresponding to $\alpha=4$.
• Figure 5: Lowest $m = 2$ decay rate (blue points) and $m=3$ decay rate (orange points) at $T/T_F = 10^{-4}$ as a function of the Coulomb interaction strength $r_s$. The continuous blue line is the analytical even-parity result \ref{['eq:gamma_m2_exact']}, and the dashed black line includes only the direct contribution in \ref{['eq:direct_exchange_m2']}. The latter expression becomes exact only at asymptotically small $r_s$ as illustrated in the inset, which shows the ratio between the exchange and direct scattering contributions. The continuous orange line is the exact odd-parity result \ref{['eq:lowT_odd_gamma']} with a momentum-dependent Coulomb scattering amplitude, and the dashed black line is the decay rate for a constant matrix element at large $r_s$ for comparison. Overall, there is a strong odd-even effect for all $r_s$, but the separation in the decay rates is strongly dependent on the interaction strength $r_s$.