[Paper Review] Angular Superdiffusion and Directional Memory in Two-Dimensional Electron Fluids

Patrick J. Ledwith, Haoyu Guo+1 more|arXiv (Cornell University)|Aug 6, 2017

Cold Atom Physics and Bose-Einstein Condensates19 citations

TL;DR

This paper proposes that two-dimensional electron fluids exhibit a novel 'super-Fermi-liquid' regime characterized by angular superdiffusion and directional memory, driven by correlated, lock-step angular momentum transfers in collinear two-particle collisions. Unlike conventional Fermi liquids, these excitations evade $T^2$ dissipation and instead relax via superdiffusive angular dynamics on the Fermi surface, leading to anomalously long-lived states that dominate at times $t \gg \tau_*$. The key result is a fourth-order Fokker-Planck equation describing non-Brownian angular diffusion, with long-lived odd-parity modes persisting well beyond the standard Fermi-liquid relaxation time.

ABSTRACT

We demonstrate that 2D Fermi liquids can support peculiar excitations that are not subject to Landau's $T^2$ dissipation. The long-lived excitations relax through correlated angular dynamics involving "lock-step" angular displacements along the Fermi surface occurring in collinear two-particle collisions, a surprising behavior that is unique to 2D systems. We develop a microscopic picture of the non-Brownian random walk, describing the angular dynamics as anomalous diffusion ("superdiffusion") on the Fermi surface. Strongly-correlated dynamics with directional memory, mediated by novel undamped excitations, dominates at moderately long times, pushing the onset of conventional hydrodynamics to abnormally large timescales. This exotic behavior can be directly probed by momentum-resolved tunneling techniques.

Motivation & Objective

• To understand the non-Fermi-liquid dynamics in two-dimensional electron systems beyond conventional hydrodynamics.
• To identify the origin of anomalously long-lived excitations in 2D Fermi liquids that evade standard $T^2$ relaxation.
• To develop a microscopic theory of angular superdiffusion on the Fermi surface driven by kinematically constrained, collinear two-particle collisions.
• To demonstrate that directional memory and correlated angular dynamics dominate at times $t \gg \tau_*$, forming a distinct transport regime.

Proposed method

• Modeling the angular dynamics of quasiparticles on the Fermi surface using a Fokker-Planck approach with angular diffusion.
• Deriving a fourth-order partial differential equation $\partial_t \tilde{f}(\theta) = -D \partial_\theta^4 \tilde{f}(\theta)$ to describe superdiffusive angular relaxation of the odd-parity momentum distribution $\tilde{f}(\theta)$.
• Identifying that only collinear, soft, and head-on two-particle collisions—enforced by Fermi exclusion and momentum conservation—generate lock-step angular displacements.
• Using a Fourier decomposition of the momentum distribution to separate even- and odd-parity modes, with distinct relaxation rates $\gamma_m$.
• Simulating time evolution of the angular distribution using a model with $\gamma_{m>1} \propto m^4$ for odd $m$, and $\gamma_{m \neq 0} = \gamma_0$ for even $m$, to illustrate the hierarchy of relaxation timescales.
• Proposing momentum-resolved tunneling in double-layer 2D electron systems as a direct experimental probe, where magnetic fields tune the angular separation of tunneling hot spots.

Experimental results

Research questions

• RQ1 What causes the breakdown of conventional Fermi-liquid hydrodynamics in two-dimensional electron systems at times $t \gg \tau_*$?
• RQ2 How do kinematic constraints and Pauli exclusion in 2D Fermi liquids lead to correlated angular momentum transfers in two-particle collisions?
• RQ3 Why do odd-parity momentum distribution modes relax significantly slower than even-parity modes, and what is the origin of the $\tau_{\text{odd}} \sim (T_F/T)^2 \tau_{\text{even}}$ scaling?
• RQ4 Can angular superdiffusion on the Fermi surface be described by a non-Brownian, fourth-order Fokker-Planck equation, and what are its implications for transport?
• RQ5 How can the super-Fermi-liquid regime with directional memory be experimentally detected using momentum-resolved tunneling?

Key findings

• The odd-parity part of the momentum distribution $\tilde{f}(\theta)$ evolves via superdiffusive angular dynamics governed by $\partial_t \tilde{f} = -D \partial_\theta^4 \tilde{f}$, indicating anomalous, non-Brownian diffusion.
• The relaxation time for odd-parity modes scales as $\tau_{\text{odd}} \sim (T_F/T)^2 \tau_{\text{even}}$, with $\tau_{\text{even}} \sim \tau_* \sim 1/T^2$, leading to a wide regime $\tau_{\text{even}} \ll t \ll \tau_{\text{odd}}$ where directional memory persists.
• Directional memory is preserved at times $t \gg \tau_*$ because angular relaxation is dominated by correlated, lock-step momentum transfers in collinear two-particle collisions.
• The system transitions to conventional hydrodynamics only at $t \sim \tau_{\text{odd}}$, when only the $m=1$ Fourier harmonic of the distribution remains.
• Momentum-resolved tunneling in double-layer 2D electron systems with tunable magnetic fields can directly probe the superdiffusive angular dynamics and detect the formation of bump/antibump structures on the Fermi surface.
• The experimental setup allows time-resolved detection of the dynamical phases via voltage-pulse reversal, enabling mapping of the super-Fermi-liquid regime between ballistic and hydrodynamic regimes.