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Near-frozen high energy state in a chiral channel driven out of equilibrium

S. Fischer, Yigal Meir, Yuval Gefen, Bernd Rosenow

Published Aug 2, 2021• 0 citations
OpenAlex

Abstract

Ergodic many-body systems are expected to reach quasi-thermal equilibrium. Here we demonstrate that, surprisingly, high-energy electrons, which are injected into an interacting one-dimensional quantum Hall edge mode, stabilize at a far-from-thermalized state over a long-time scale. To detect this non-equilibrium state, one positions an energy-resolved detector downstream of the point of injection. Previous works have shown that electron distributions, which undergo short-ranged interactions, generically relax to near-thermal asymptotic states. Here, we consider screened interactions of finite range. The thus-obtained many-body state comprises fast-decaying transient components, followed by a nearly frozen distribution with a peak near the injection energy.

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Ergodic many-body systems are expected to reach quasi-thermal equilibrium. Here we demonstrate that, surprisingly, high-energy electrons, which are injected into an interacting one-dimensional quantum Hall edge mode, stabilize at a far-from-thermalized state over a long-time scale. To detect this non-equilibrium state, one positions an energy-resolved detector downstream of the point of injection. Previous works have shown that electron distributions, which undergo short-ranged interactions, generically relax to near-thermal asymptotic states. Here, we consider screened interactions of finite range. The thus-obtained many-body state comprises fast-decaying transient components, followed by a nearly frozen distribution with a peak near the injection energy.
Ergodic many-body systems are expected to reach thermal equilibrium. Here, we demonstrate that, surprisingly, high-energy electrons, which are injected into a quantum Hall edge mode with finite-range interactions, stabilize at a far-from-thermalized state over a long timescale. To detect this nonequilibrium state, one positions an energy-resolved detector downstream of the injection point. So far, nonequilibrium distributions in integrable systems were either found not to display relaxation at all, or generically relax to near-thermal asymptotic states. In stark contrast, the here-obtained many-body state comprises fast-decaying transient components, followed by a nearly frozen distribution with a peak near the injection energy.
We study equilibration of quantum Hall edge states at integer filling factors, motivated by experiments involving point contacts at finite bias. Idealizing the experimental situation and extending the notion of a quantum quench, we consider time evolution from an initial nonequilibrium state in a translationally invariant system. We show that electron interactions bring the system into a steady state at long times. Strikingly, this state is not a thermal one: Its properties depend on the full functional form of the initial electron distribution and not simply on the initial energy density. Further, we demonstrate that measurements of the tunneling density of states at long times can yield either an overestimate or an underestimate of the energy density, depending on details of the analysis, and discuss this finding in connection with an apparent energy loss observed experimentally. More specifically, we treat several separate cases: for filling factor $\ensuremath{\nu}=1$ we discuss relaxation due to finite-range or Coulomb interactions between electrons in the same channel, and for filling factor $\ensuremath{\nu}=2$ we examine relaxation due to contact interactions between electrons in different channels. In both instances we calculate analytically the long-time asymptotics of the single-particle correlation function. These results are supported by an exact solution at arbitrary time for the problem of relaxation at $\ensuremath{\nu}=2$ from an initial state in which the two channels have electron distributions that are both thermal but with unequal temperatures, for which we also examine the tunneling density of states.
This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.
Cascade is known as the transfer of conserved quantities across scales through nonlinear mode coupling. Originally studied in classical fluids, it has recently been realized in quantum systems. However, whether a stable cascade can emerge in a strongly interacting quantum system during the driving process remains an open question, where heating competes with the cascade. In this work, we investigate the far-from-equilibrium dynamics of a one-dimensional Bose–Hubbard system at half-filling, driven periodically by a spatially modulated chemical potential that injects energy at a single intermediate wavevector. By performing large-scale simulations, we uncover that the momentum distribution follows a quasi-steady power-law scaling in the region [π/4,π/2]. Notably, this particle cascade in momentum space is stable only under weak driving in the weakly interacting regime against thermalization, while it persists up to substantially larger amplitudes in the strongly interacting regime. Our results reveal an interacting many-body quantum counterpart of periodically driven turbulent wave cascades and indicate a mechanism by which interactions stabilize universal distributions under far-from-equilibrium conditions.
We theoretically study a Luttinger liquid (LL) driven out of equilibrium by injection of high-energy electrons. The electrons are injected into the LL locally at a fixed energy and their spectral properties are detected at another spatial point some distance away by evaluating the average tunneling current from the LL into a resonant level with tunable energy. For energies slightly below the injection energy, the dependence of the detected current on the difference between injection, and detection energies is described by a power law whose exponent depends continuously on the Luttinger parameter. In contrast, for tunneling into a chiral LL edge of a fractional quantum-Hall state from the Laughlin sequence, we find that the detected current grows linearly with energy difference, despite a decreasing density of states.
A Tomonaga-Luttinger (TL) liquid is known as an integrable system, in which a non-equilibrium many-body state survives without relaxing to a thermalized state. This intriguing characteristic is tested experimentally in copropagating quantum Hall edge channels at bulk filling factor nu = 2. The unidirectional transport allows us to investigate the time evolution by measuring the spatial evolution of the electronic states. The initial state is prepared with a biased quantum point contact, and its spatial evolution is measured with a quantum-dot energy spectrometer. We find strong evidences for a non-thermal metastable state in agreement with the TL theory before the system relaxes to thermal equilibrium with coupling to the environment.