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arXiv:physics/9810046

Quantum response of finite Fermi systems and the relation of Lyapunov exponent to transport coefficients

Klaus Morawetz

Published Oct 26, 1998• 0 citations
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Abstract

Within the frame of kinetic theory a response function is derived for finite Fermi systems which includes dissipation in relaxation time approximation and a contribution from additional chaotic processes characterized by the largest Lyapunov exponent. A generalized local density approximation is presented including the effect of many particle relaxation and the additional chaotic scattering. For small Lyapunov exponents relative to the product of wave vector and Fermi velocity in the system, the largest Lyapunov exponent modifies the response in the same way as the relaxation time. Therefore the transport coefficients can be connected with the largest positive Lyapunov exponent in the same way as known from the transport theory in relaxation time approximation.

DOI
10.48550/arxiv.physics/9810046

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Within the frame of kinetic theory a response function is derived for finite Fermi systems which includes dissipation in relaxation time approximation and a contribution from additional chaotic processes characterized by the largest Lyapunov exponent. A generalized local density approximation is presented including the effect of many particle relaxation and the additional chaotic scattering. For small Lyapunov exponents relative to the product of wave vector and Fermi velocity in the system, the largest Lyapunov exponent modifies the response in the same way as the relaxation time. Therefore the transport coefficients can be connected with the largest positive Lyapunov exponent in the same way as known from the transport theory in relaxation time approximation.
Abstract In this paper, we study the chaotic behavior of the unitary Fermi gas in both high- and low-temperature limits by calculating the quantum Lyapunov exponent defined in terms of the out-of-time-order correlator. We take the method of generalized Boltzmann equations, derived from the augmented Keldysh approach (Aleiner et al 2016 Ann. Phys. 375 , 378). At high temperatures, the system is described by weakly interacting fermions with two spin components and the Lyapunov exponent is found to be λ L = 21 n T 1 / 2 . Here n is the density of fermions for a single-spin component. In the low-temperature limit, the system is a superfluid and can be described by phonon modes. Using the effective action derived in (Son and Wingate 2006 Ann. Phys. 321 , 197), we find λ L = 9 × 10 3 T T F 4 T where T F is the Fermi energy. By comparing these to existing results of heat conductivity, we find that D E ≪ v 2 / λ L where D E is the energy diffusion constant and v is some typical velocity. We argue that this is related to the conservation law for such systems with quasi-particles.
For perturbative scalar field theories, the late-time-limit of the out-of-time-ordered correlation function that measures (quantum) chaos is shown to be equal to a Boltzmann-type kinetic equation that measures the total gross (instead of net) particle exchange between phase-space cells, weighted by a function of energy. This derivation gives a concrete form to numerous attempts to derive chaotic many-body dynamics from ad hoc kinetic equations. A period of exponential growth in the total gross exchange determines the Lyapunov exponent of the chaotic system. Physically, the exponential growth is a front propagating into an unstable state in phase space. As in conventional Boltzmann transport, which follows from the dynamics of the net particle number density exchange, the kernel of this kinetic integral equation for chaos is also set by the 2-to-2 scattering rate. This provides a mathematically precise statement of the known fact that in dilute weakly coupled gases, transport and scrambling (or ergodicity) are controlled by the same physics.
Linearizing the appropriate kinetic equation we derive general response functions including self-consistent mean fields or density functionals and collisional dissipative contributions. The latter ones are considered in relaxation time approximation conserving successively different balance equations. The effect of collisions is represented by correlation functions which are possible to calculate with the help of the finite temperature Lindhard random-phase approximation expression. The presented results are applicable to the finite temperature response of interacting quantum systems if the quasiparticle or mean-field energy is parametrized within Skyrme-type functionals including density, current, and energy dependencies which can be considered alternatively as density functionals. In this way we allow to share correlations between density functional and collisional dissipative contributions appropriate for special treatment. We present results for collective modes such as the plasmon in plasma systems and the giant resonance in nuclei. The collisions lead in general to an enhanced damping of collective modes. If the collision frequency is close to the frequency of the collective mode, resonance occurs and the collective mode is enhanced showing a collisional narrowing.
A single-time quantum transport equation, which includes effects beyond the quasiparticle approximation, is derived for Fermi systems in the framework of nonequilibrium real-time Green's functions theory. Ternary correlations are incorporated in the kinetic description via a cluster expansion for the self-energies (e.g., the transport vertex and the width) truncated at the level of three-body scattering amplitudes. A finite temperature/density formulation of the three-body problem is given. Corresponding three-body equations reduce to the well-known Faddeev equations in the vacuum limit. In equilibrium the equation of state contains virial corrections proportional to the third quantum virial coefficient.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
We reanalyzed the recently derived response function for interacting systems in relaxation-time approximation respecting density, momentum, and energy conservation. We find that momentum conservation leads exactly to the local-field corrections for both cases respecting only density conservation and respecting density and energy conservation. This rewriting simplifies the former formulas dramatically. We discuss the small wave vector expansion and find that the response function shows a high-frequency dependence of omega(-5), which allows to fulfill higher-order sum rules. The momentum conservation also resolves a puzzle about the conductivity, which should only be finite in multicomponent systems.