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arXiv:1809.01303 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum chaos for the unitary Fermi gas from the generalized Boltzmann equations

Pengfei Zhang

Published Feb 27, 2019• 6 citations
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Abstract

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.

DOI
10.1088/1361-6455/ab0af9

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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.
We calculate the quantum Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ and butterfly velocity vB{v}_{B}vB​ in the dilute Bose gas at temperature TTT deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time-ordered correlators, from which ${\ensuremath{\lambda}}_{L}$ and vB{v}_{B}vB​ are extracted. At very low temperature where elementary excitations are phonon-like, we find ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{T}^{5}$ and ${v}_{B}\ensuremath{\sim}c$, the sound velocity. At relatively high temperature, we have ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}T$ and ${v}_{B}\ensuremath{\sim}c{(T/{T}_{*})}^{0.23}$. We find that ${\ensuremath{\lambda}}_{L}$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on TTT. The chaos diffusion constant ${D}_{L}={v}_{B}^{2}/{\ensuremath{\lambda}}_{L}$, on the other hand, differs from the energy diffusion constant DE{D}_{E}DE​. We find ${D}_{E}\ensuremath{\ll}{D}_{L}$ at very low temperature and ${D}_{E}\ensuremath{\gg}{D}_{L}$ otherwise.
The growth rate of the out-of-time-ordered correlator in a NNN-flavor Fermi gas is investigated and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is calculated to the order of 1/N1/N1/N. We find that the Lyapunov exponent monotonically increases as the interaction strength increases from the BCS limit to the unitary region. At the unitarity, the Lyapunov exponent increases while the temperature drops and it can reach to the order of ${\ensuremath{\lambda}}_{L}\ensuremath{\sim}T$ around the critical temperature for the N=1N=1N=1 case. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the Lyapunov exponent reaches its maximum ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}4.35T/N$ at the critical temperature. The system scrambles faster for stronger pairing fluctuations. At the BCS limit, the Lyapunov exponent behaviors as ${\ensuremath{\lambda}}_{L}\ensuremath{\propto}{e}^{\ensuremath{\mu}/T}{a}_{s}^{2}{T}^{2}/N$.
We study the quantum chaos in the Bose-Fermi Kondo model in which the impurity spin interacts with conduction electrons and a bosonic bath at the intermediate temperature in the large-NNN limit. The out-of-time-ordered correlator is calculated based on the Bethe-Salpeter equation and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ is extracted. Our calculation shows that the Lyapunov exponent monotonically increases as the Kondo coupling JK{J}_{K}JK​ increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as JK{J}_{K}JK​ approaches the multichannel Kondo fixed point (MCKMCKMCK). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling ggg increases, which is contrary to the general expectation that the most chaotic property occurs at the quantum critical point with the non-Fermi-liquid nature.
We analyze the relation between quantum chaotic behavior and phase transitions of the Yukawa--Sachdev-Ye-Kitaev model as a function of filling and temperature, which describes random Yukawa interactions between NNN complex fermions and MMM bosons in zero spatial dimensions for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.
Fermionic matter is ubiquitous in nature, from the electrons in metals and semiconductors or the neutrons in the inner crust of neutron stars, to gases of fermionic atoms, like 40K or 6Li that can be created and studied under laboratory conditions. It is especially interesting to study these systems at very low temperatures, where we enter the world of quantum mechanical phenomena. Due to the Fermi-Dirac statistics, a dilute system of spin-polarised fermions exhibits no interactions and can be viewed as an ideal Fermi gas. However, interactions play a crucial role for fermions of several spin species. This thesis addresses several questions concerning interacting Fermi gases, in particular the transition between the normal and the superfluid phase and dynamical properties at higher temperatures. First we will look at the unitary Fermi gas: a two-component system of fermions interacting with divergent scattering length. This system is particularly interesting as it exhibits universal behaviour. Due to the strong interactions perturbation theory is inapplicable and no exact theoretical description is available. I will describe the Determinant Diagrammatic Monte Carlo algorithm with which the unitary Fermi gas can be studied from first principles. This algorithm fails in the presence of a spin imbalance (unequal number of particles in the two components) due to a sign problem. I will show how to apply reweighting techniques to generalise the algorithm to the imbalanced case, and present results for the critical temperature and other thermodynamic observables at the critical point, namely the chemical potential, the energy per particle and the contact density. These are the first numerical results for the imbalanced unitary Fermi gas at finite temperature. I will also show how temperatures beyond the critical point can be accessed and present results for the equation of state and the temperature dependence of the contact density. At sufficiently high temperatures a semiclassical description captures all relevant physical features of the system. The dynamics of an interacting Fermi gas can then be studied via a numerical simulation of the Boltzmann equation. I will describe such a numerical setup and apply it to study the collision of two spin-polarised fermionic clouds. When the two components are separated in an elongated harmonic trap and then released, they collide and for sufficiently strong interactions can bounce off each other several times. I will discuss the different types of the qualitative behaviour, show how they can be interpreted in terms of the equilibrium properties of the system, and explain how they relate to the coupling between different excitation modes. I will also demonstrate how transport coefficients, for instance the spin drag, can be extracted from the numerical data.
Many-body systems that saturate the quantum bound on chaos are attracting interest across a wide range of fields. Notable examples include the Sachdev-Ye-Kitaev model and its variations, all characterized by some form or randomness and all-to-all couplings. Here, we study many-body quantum chaos in a quantum impurity model showing non-Fermi-liquid physics, the overscreened multichannel S U ( N ) Kondo model. We exactly compute the low-temperature behavior of the out-of-time order correlator in the limit of large N and large number of channels, K , at a fixed ratio γ = K / N . Because of strong correlations at the impurity site, the spin fractionalizes in auxiliary fermions and bosons. We show that all the degrees of freedom of our theory acquire a Lyapunov exponent that is linear in temperature as T → 0 , with a prefactor that depends on γ . Remarkably, for N = K , the impurity spin displays maximal chaos, while bosons and fermions only reach half of the maximal Lyapunov exponent. Our results highlight two key features: a nondisordered model that is maximally chaotic due to strong correlations at its boundary, with the maximal chaos appearing in a composite gauge-invariant operator, the impurity spin, and not in the auxiliary single-particle degrees of freedom.