We study stability of the SYK4 model with a large but finite number of fermions N with respect to a perturbation, quadratic in fermionic operators. We develop analytic perturbation theory in the amplitude of the SYK2 perturbation and demonstrate stability of the SYK4 infra-red asymptotic behavior characterized by a Green function G(τ ) ∝ 1/τ 3/2 , with respect to weak perturbation. This result is supported by exact numerical diagornalization. Our results open the way to build a theory of non-Fermi-liquid states of strongly interacting fermions.The plenty of available data on various strongly correlated electronic materials [1, 2] calls for the development of a general theory of non-Fermi-liquid ground state(s) of an interacting many-body fermionic system. Still, no general theoretical scheme leading to such a behavior in the zero-temperature limit is known (for a recent review see Ref.[3]). Mathematically, complexity of the problem is due to the absence of any general method to calculate non-Gaussian functional integrals which appear in the theory of strongly interacting fermions.A new and fresh view on this old problem is provided by the recently proposed [4-6] Sachdev-Ye-Kitaev (SYK) model of interacting fermions. It has attracted a lot of attention recently as a possible boundary theory of a two-dimensional gravitational bulk [5,7,8]. Original SYK model contains N 1 Majorana fermions, with the Hamiltonian consisting of a sum of all possible 4-fermion terms with random matrix elements J ijkl ∼ J/N 3/2 (note that the free (quadratic) term is missing in the SYK Hamiltonian). This model can be considered as a nonlinear generalization of usual random-matrix Hamiltonians [9]. Furthermore, SYK q models with arbitrary even q = 2k were introduced and studied [8]. These models provide the most straightforward way to enhance the role of interaction between fermions, avoiding formation of any simply ordered structures which lead -usually, but not always [3] -to a breakdown of some evident symmetry of the Hamiltonian.The SYK model is analytically tractable in the large-N limit and shows two different types of asymptotic behavior for the fermionic Green function G(τ ). In the intermediate time range 1/J τ t c , with t c ∼ N/J, the self-consistent approximation for interaction self-energy is valid and G(τ ) ∝ τ −1/2 . For even larger times τ t c , it was found in Ref.[10] that fluctuations beyond the selfconsistent treatment change the behavior of the Green function to G(τ ) ∝ τ −3/2 (we treat exponentially large ergodic time-scale ∝ 2 N/2 as being infinite). Both these types of behavior are crucially different from the standard Fermi-liquid scaling G(τ ) ∝ 1/τ which corresponds to nonzero finite density of low-energy states. In other terms, low-energy excitations of the SYK model are not described by any kind of quasiparticles.For the reasons described above, the SYK model seems to be a very promising starting point to approach a theory of non-Fermi-liquid ground state. Few problems arise, however: i) the absence of a quadr...

We study the Sachdev-Ye-Kitaev (SYK 4 ) model with a weak SYK 2 term of magnitude Γ beyond the simplest perturbative limit considered previously. For intermediate values of the perturbation strength,, fluctuations of the Schwarzian mode are suppressed, and the SYK 4 mean-field solution remains valid beyond the timescale t 0 ∼ N=J up to t à ∼ J=Γ 2 . The out-of-time-order correlation function displays at short time intervals exponential growth with maximal Lyapunov exponent 2πT, but its prefactor scales as T at low temperatures T ≤ Γ.