Abstract We study the SYK model with complex fermions, in the presence of an all-to-all q-body interaction, with a non-vanishing chemical potential. We find that, in the large q limit, this model can be solved exactly and the corresponding Lyapunov exponent can be obtained semi-analytically. The resulting Lyapunov exponent is a sensitive function of the chemical potential μ. Even when the coupling J, which corresponds to the disorder averaged values of the all to all fermion interaction, is large, values of μ which are exponentially small compared to J lead to suppression of the Lyapunov exponent.
We show that the integrable state of charged SYK model does not thermalize even though the state has non-zero Lyapunov exponent. The chaotic state always thermalizes which is expected. So whether the system thermalizes or not depends on the initial state. We also show that the introduction of random mass deformation (q=2 SYK term) slows down thermalization but the system thermalizes exponentially fast. This is observed despite the fact that large (q=2) SYK interaction forces spectral statistics to obey Poisson statistics. This means that a precise measure of chaos and its prediction of thermalization is still lacking. In the chaotic state, the effective temperature is non-monotonic. It has a bump at relatively long time before settling down to the final value. With non-zero chemical potential, the effective temperature oscillates noticeably before settling down to the final value. The spectral asymmetry frequency does not change during quantum quenches.
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.
The Sachdev-Ye-Kitaev (SYK) model is a concrete model for a non-Fermi liquid with maximally chaotic behavior in
(0+1) dimensions. In order to gain some insights into real materials in higher dimensions where fermions could hop between different sites, here we consider coupling a SYK lattice by constant hopping. We call this the dispersive SYK model. Focusing on
(1+1)-dimensional homogeneous hopping, by either tuning the temperature or the relative strength of the random interaction (hopping) and constant hopping, we find a crossover between a dispersive metal to an incoherent metal, where the dynamic exponent
z changes from 1 to $\ensuremath{\infty}$. We study the crossover by calculating the spectral function, charge density correlator, and the Lyapunov exponent. We further find the Lyapunov exponent becomes larger when the chemical potential is tuned to approach a van Hove singularity because of the large density of states near the Fermi surface. The effect of the topological nontrivial bands is also discussed.
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=A T, 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.
Quantum chaos is one of the distinctive features of the Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions in 0+1 dimensions with infinite-range two-body interactions, which is attracting a lot of interest as a toy model for holography. Here we show analytically and numerically that a generalized SYK model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic and retains most of its holographic features for a fixed value of the perturbation and sufficiently high temperature. However, a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. We speculate about the gravity dual of this transition.