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In a recent comment to the paper Chaotic Integrable transition in the SYK model, it was claimed that, in a certain region of parameters, the Lyapunov exponent of the N Majoranas Sachdev-Ye-Kitaev model with a quadratic perturbation, is always positive. This implies that the model is quantum chaotic. In this reply, we show that the employed perturbative formalism breaks down precisely in the range of parameters investigated in the comment due to a lack of separation of time scales. Moreover, based on recent analytical results, we show that for any large and fixed N, the model has indeed a chaotic-integrable transition that invalidate the results of the comment.
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We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $ SU(M) $ complex fermion representation is that it need no local constraints, so a $1/M $ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $ J_{ij} $ instead of four indices $ J_{ijkl} $, so it may fit the bulk string theory better. By performing a $1/M $ expansion at $ N=infty $, we show that a quantum spin liquid (QSL) state remains stable at a finite $ M $. The $ 1/M $ corrections are exactly marginal, so the system remains conformably invariant at any finite $ M $. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $ N=infty $ at any finite $ M $, nor at $ M=infty $ at any finite $ N $. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $ M $. We work out a criterion for the two large numbers $ N $ and $ M $ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $ N $, the quantum chaos appears at the order of $ 1/M^{0} $, which is the subleading order in the $ 1/M $ expansion. When the $ 1/N $ quantum fluctuations at any finite $ M $ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.
Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground state properties of complex quantum systems and gives relations between correlation functions. In this work, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes, and are thus fundamentally linked to the models Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics, and has consequences away from the ground state, including in $q$-body dynamical correlation functions.
We investigate chaotic to integrable transition in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or $(q=2)^2$ SYK with an interaction $ sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex fermion and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10 fold way by Random Matrix Theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study Chaotic to Integrable transitions (CIT) in both types of hybrid SYK models. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $ N=infty $, $ 1/N $ expansions at a finite $ N $ and the CIT characterized by the RMT at a finite $ N $. The corresponding distinctions between the edge states and bulk states in Fock spaces are studied. Some future perspectives, especially the failure of the Zamoloddchikovs c-theorem in 1d CFT are outlined.
We introduce a spinful variant of the Sachdev-Ye-Kitaev model with an effective time reversal symmetry, which can be solved exactly in the limit of a large number $N$ of degrees of freedom. At low temperature, its phase diagram includes a compressible non-Fermi liquid and a strongly-correlated spin singlet superconductor that shows a tunable enhancement of the gap ratio predicted by BCS theory. These two phases are separated by a first-order transition, in the vicinity of which a gapless superconducting phase, characterized by a non-zero magnetization, is stabilized upon applying a Zeeman field. We study equilibrium transport properties of such superconductors using a lattice construction, and propose a physical platform based on topological insulator flakes where they may arise from repulsive electronic interactions.
Periodically driven quantum matter can realize exotic dynamical phases. In order to understand how ubiquitous and robust these phases are, it is pertinent to investigate the heating dynamics of generic interacting quantum systems. Here we study the thermalization in a periodically-driven generalized Sachdev-Ye-Kitaev (SYK)-model, which realizes a crossover from a heavy Fermi liquid (FL) to a non-Fermi liquid (NFL) at a tunable energy scale. Developing an exact field theoretic approach, we determine two distinct regimes in the heating dynamics. While the NFL heats exponentially and thermalizes rapidly, we report that the presence of quasi-particles in the heavy FL obstructs heating and thermalization over comparatively long time scales. Prethermal high-frequency dynamics and possible experimental realizations of non-equilibrium SYK physics are discussed as well.
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