We investigate the many-body quantum chaos of non-Fermi liquid states with Fermi surfaces in two spatial dimensions by computing their out-of-time-order correlation functions. Using a recently proposed large N theory for the critical Fermi surface, and the ladder identity of Gu and Kitaev, we show that the chaos Lyapunov exponent takes the maximal value of 2πk_{B}T/ℏ, where T is the absolute temperature. We also examine a phenomenological model that can be continuously tuned between a non-Fermi liquid without quasiparticles and a Fermi liquid with quasiparticles. We find that the Lyapunov exponent becomes smaller than the maximal value precisely when quasiparticles are restored.
We investigate the many-body quantum chaos of non-Fermi liquid states with Fermi surfaces in two spatial dimensions by computing their out-of-time-order correlation functions. Using a recently proposed large N theory for the critical Fermi surface, and the ladder identity of Gu and Kitaev, we show that the chaos Lyapunov exponent takes the maximal value of 2πk_{B}T/ℏ, where T is the absolute temperature. We also examine a phenomenological model that can be continuously tuned between a non-Fermi liquid without quasiparticles and a Fermi liquid with quasiparticles. We find that the Lyapunov exponent becomes smaller than the maximal value precisely when quasiparticles are restored.
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 N complex fermions and M 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.
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.
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-N 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 increases, and it can reach an order of ${\ensuremath{\lambda}}_{L}\ensuremath{\approx}T$ as JK approaches the multichannel Kondo fixed point (MCK). Furthermore, we also demonstrate that ${\ensuremath{\lambda}}_{L}$ decreases monotonically as the impurity and bosonic bath coupling g 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 compute the Lyapunov exponent characterizing quantum scrambling in a family of generalized Sachdev-Ye-Kitaev models, which can be tuned between different low temperature states from Fermi liquids, through non-Fermi liquids, to fast scramblers. The analytic calculation, controlled by a small coupling constant and large N, allows us to clarify the relations between the quasiparticle relaxation rate $1/\ensuremath{\tau}$ and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ characterizing scrambling. In the Fermi liquid states we find that the quasiparticle relaxation rate dictates the Lyapunov exponent. In non-Fermi liquids, where $1/\ensuremath{\tau}\ensuremath{\gg}T$, we find that ${\ensuremath{\lambda}}_{L}$ is always T linear with a prefactor that is independent of the coupling constant in the limit of weak coupling. Instead it is determined by a scaling exponent that characterizes the relaxation rate. ${\ensuremath{\lambda}}_{L}$ approaches the general upper bound $2\ensuremath{\pi}T$ at the transition to a fast scrambling state. Finally, in a marginal Fermi liquid state the exponent is linear in temperature with a prefactor that vanishes as a nonanalytic function $\ensuremath{\sim}gln(1/g)$ of the coupling constant g.
Quantum chaos in many-body systems may be characterized by the Lyapunov exponent defined as the exponential growth rate of out-of-time-order correlators (OTOC). So far Lyaponov exponents around various quantum critical points (QCP) remain largely unexplored. Here, we investigate the Lyapunov exponent around QCPs of the Gross-Neveu (GN) model with N flavors of Dirac fermions in (2+1) dimensions. Around the GN quantum phase transition between a Dirac semimetal and a gapped insulator breaking Z2 symmetry (e.g., inversion symmetry of the honeycomb lattice), we find that the Lyaponov exponent λL≈3.5T/N at temperature T and to the leading order of 1/N in the large-N expansion. We also obtain the quantum scattering rate of an excitation with energy ε, which is proportional to εT/N at low energy. We further discuss possible experimental relevances of the GN model in many-body systems.
The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent λL=2πT at temperature T), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, g≡J0/δJ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing g yields the following effects: (i) The system remains a fast scrambler with λL=AT, but with a suppressed coefficient A<2π. (ii) In the limit g→∞, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with λL≃0. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of g. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all g values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.