Odd relaxation in three-dimensional Fermi liquids

TL;DR

The paper demonstrates that three-dimensional isotropic Fermi liquids host a tomographic-like hierarchy of long-lived, non-hydrodynamic modes, exhibiting a parity-based separation of relaxation rates between even- and odd-parity deformations. The authors develop a spherical-harmonic angular-basis framework and compute leading-order decay rates, showing a robust $T^2$ scaling with a significant odd-even prefactor difference that can reach ~$40\%$, enhanced by large-angle scattering. They connect these decay rates to transport observables by solving the Boltzmann equation, revealing signatures in the static transverse conductivity $\sigma_\perp(q)$ and in transverse collective modes, thereby providing experimentally accessible probes of the tomographic regime in 3D. The results extend tomographic transport beyond 2D, highlighting that parity-based relaxation phenomena are more generic and tunable by interaction specifics, with implications for identifying non-hydrodynamic modes in 3D Fermi liquids.

Abstract

Recent theoretical works predict a hierarchy of long-lived, non-hydrodynamic modes in two-dimensional Fermi liquids arising from the feature$-$supposedly unique to two dimensions$-$that relaxation by head-on scattering is not efficient in the presence of Pauli blocking. This leads to a parity-based separation of scattering rates, with odd-parity modes relaxing much more slowly than even-parity ones. In this work, we establish that a similar effect exists in isotropic three-dimensional (3D) Fermi liquids, even though relaxation does not proceed solely by head-on scattering. We show that while the relaxation rates of even and odd modes in 3D share the same leading-order $\sim T^2$ low-temperature scaling typical of Fermi liquids, their magnitudes differ, with odd-parity modes relaxing more slowly than even ones for a broad class of interactions. We find a relative difference between odd-parity and even-parity relaxation rates as large as $40\%$ just by Pauli blocking alone, with a strong additional dependence on the scattering potential, such that the odd-even staggering is further enhanced by interactions that favor large-angle scattering. We identify signatures of these odd-parity relaxation rates in the static transverse conductivity as well as the transverse collective mode structure. Our results establish the unexpected existence of a tomographic like regime in higher-dimensional Fermi liquids and suggest experimental probes via transport measurements.

Figures (6)

• Figure 1: Momentum space picture of the dominant two-particle collision processes for an isotropic Fermi liquid at low temperatures in (a) 2D and (b) 3D. Shaded regions indicate the occupied states in equilibrium, and ${\bf p}_1$ and ${\bf p}_2$ indicate initial and ${\bf p}_1'$ and ${\bf p}_2'$ final momenta, where ${\bf k} = {\bf p}_1 - {\bf p}_1'$ is the momentum transfer in the collision. In both cases, Eq. \ref{['eqn:angle_fixed']} constrains the scattering momenta to form a rectangle. In 2D, this enforces head-on scattering with only a single free parameter $\theta_1$, the angle between $\bf{k}$ and $\bf{p}_1$, which leads to a parity-based separation of relaxation rates. In 3D, in contrast, an additional parameter $\varphi_2$ enters that describes the angle between $\bf{p}_1+\bf{p}_1'$ and $\bf{p}_2+\bf{p}_2'$. Unless $\varphi_2 = \pi$, head-on scattering will not occur. Interactions that enhance large-angle scattering will prefer $\varphi_2\sim \pi$ and thus enhance the even-odd effect, while those suppressing it will reduce the even-odd effect.
• Figure 2: Decay rate coefficient $I_l$ of the longest-lived modes in a 3D Fermi liquid with constant interaction potential as a function of the angular index $l$, Eq. \ref{['eqn:constant']}. Blue points indicate decay rates for even $l$ and orange points for odd $l$. The smooth curves are drawn to guide the eye. The inset shows the relative difference $|\gamma_l-\gamma_{l-1}|/\gamma_l$ of odd and even relaxation rates for odd $l\geq 3$. The difference peaks at $40\%$ for $l=3$ and decreases at large $l$.
• Figure 3: Decay rate coefficient $I_l$ of the longest-lived modes in a 3D Fermi liquid for an interaction that prefers (a) large-angle scattering [Eq. \ref{['eqn:large_angle']} with $\sigma = 0.01$] and (b) small-angle scattering [Eq. \ref{['eqn:defn_screened_coulomb']} with $r_s = 0.01$], where $\langle|\overline{V}|^2\rangle$ is defined in Eq. \ref{['eqn:defn_ovVsq']}. Even (odd) mode scattering rates are shown in blue (orange). The insets display the relative difference between even and odd mode scattering for odd $l\geq 3$. (a) Large-angle scattering enhances the difference between even and odd rates. The inset shows a relative difference between odd and even rates as large as $40$. (b) Small-angle scattering suppresses the difference between even and odd scattering rates, but even here a relative difference of $10\%$ remains.
• Figure 4: Scaled static transverse conductivity $q^2 \sigma_\perp(q)$ as a function of the wave vector $q$. (a) Static conductivity for a constant interaction potential (blue curve), which displays the even-odd effect [cf. Eq. \ref{['eqn:constant']}], and a constant scattering rate $\gamma_l = \gamma$ for $l \geq 2$ (green curve). The wave numbers where corrections to hydrodynamic scaling appear are marked with a circle. For an even-odd staggering of decay rates, finite-wavelength corrections appear at significantly smaller wave numbers. (b) Static conductivity for with constant interactions [Eq. \ref{['eqn:constant']}] (blue curve), and large-angle interactions [Eq. \ref{['eqn:large_angle']}] (orange curve). The corresponding scattering rates are shown in Fig. \ref{['fig:large_small']}(a). Gray dashed lines indicate the values of $q$ for which the microscopic deformations of the Fermi surface are plotted in Fig. \ref{['fig:static_deformations']}. Note that compared to panel (a), the axes are rescaled by $\gamma_2$ so the different scaling behavior of the large-angle scattering can be cleanly observed.
• Figure 5: Fermi surface deformations plotted on the Fermi surface, with squares displaying the decomposition into spherical harmonics. The current $\bf{j}$ points to the right, while the wave vector $\bf{q}$ points out of the page. The first row assumes a constant interaction [Eq. \ref{['eqn:constant']}], and the second row assumes a large-angle interaction [Eq. \ref{['eqn:large_angle']}]. The columns correspond, from left to right, to the wave number indicated by gray dashed lines in Fig. \ref{['fig:static_cond']}. The hydrodynamic limit $v_Fq\ll \gamma_l$ is the first column, the 3D tomographic limit is the second column, and the collisionless limit $\gamma_l \ll v_Fq$ is the third column. In the hydrodynamic limit, only the conserved current is excited, while in the collisionless limit all modes are excited with a deformation that is strongly peaked about $\bf{q}$. In the tomographic like limit, the odd modes exhibit enhanced occupation, especially clear for the large-angle scattering in panel (e).