Quantum chaos, unlike its classical counterpart, lacks a universally accepted definition due to the absence of well-defined trajectories in Hilbert space. Quantum systems can show chaotic behavior through quantities like the spectral form factor (SFF) and out-of-time-ordered correlators (OTOCs), which characterize energy level statistics and the dynamical sensitivity to perturbations. These diagnostics have attracted considerable attention in fields ranging from condensed matter to high-energy physics, especially in the study of quantum many-body systems and holographic dualities.
In this context, the Sachdev-Ye-Kitaev (SYK) model—a (0+1)-dimensional model of randomly interacting Majorana fermions—has emerged as a key platform for studying strong coupling, maximal chaos, and emergent conformal symmetry. Building on this foundation, my research aims to understand how randomness, integrability, and disorder affect the chaotic signatures in both simple and extended quantum systems. Specifically, I have investigated the relationship between the SFF and OTOC, the effect of disorder correlations in the SYK$_2$ model, and the spectral diagnostics in qubit systems and simplified variants of the SYK model.
We prove that in bosonic quantum mechanics the two-point spectral form factor can be obtained as an average of the two-point out-of-time ordered correlation function, with the average taken over the Heisenberg group. In quantum field theory, there is an analogous result with the average taken over the tensor product of many copies of the Heisenberg group, one copy for each field mode. The resulting formula is expressed as a path integral over two fields, providing a promising approach to the computation of the spectral form factor. We develop the formula that we have obtained using a coherent state description from the JC model and also in the context of the large-$N$ limit of CFT and Yang-Mills theory from the large-$N$ matrix quantum mechanics.
We introduce randomness into a class of integrable models and study the spectral form factor as a diagnostic to distinguish between randomness and chaos. Spectral form factors exhibit a characteristic dip-ramp-plateau behavior in the $N>2$ SYK$_2$ model at high temperatures that is absent in the $N=2$ SYK$_2$ model. Our results suggest that this dip-ramp-plateau behavior implies the existence of random eigenvectors in a quantum many-body system. To further support this observation, we examine the Gaussian random transverse Ising model and obtain consistent results without suffering from small $N$ issues. Finally, we demonstrate numerically that expectation values of observables computed in a random quantum state at late times are equivalent to the expectation values computed in the thermal ensemble in a Gaussian random one-qubit model.
We study the SYK$_2$ model of Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling the model to two- and four-point sources. In particular, we define the generalized spectral form factor (SFF) and level spacing distribution function by generalizing from the partition function to the generating function. For $N=2$, we obtain an exact solution of the generalized SFF. It exhibits qualitatively similar behavior to the higher $N$ case with a source term. The exact solution helps understand the behavior of the generalized SFF. We calculate the generalized level spacing distribution function and the mean value of the adjacent gap ratio defined by the generating function. For the SYK$_2$ model with a four-point source term, we find a Gaussian unitary ensemble behavior in the near-integrable region of the theory, which indicates a transition to chaos. This behavior is confirmed by the connected part of the generalized SFF with an unfolded spectrum. The departure from this Gaussian random matrix behavior as the relative strength of the source term is increased is consistent with the observation that the four-point source term alone, without the SYK$_2$ couplings turned on, exhibits an integrable spectrum from the SFF and level spacing distribution function in the large $N$ limit.
Entanglement entropy gives quantitative understanding to the entanglement. We use the decomposition of the Hilbert space to discuss the properties of Quantum Entanglement. Therefore, it is hard to define the reduced density matrix from a partial trace operation with different centers. The center commutes with all elements in the Hilbert space. The choice of centers corresponds to observation on an entangling surface. We discuss the decomposition of the Hilbert space for the strong subadditivity and other related inequalities. In the Hamiltonian formulation, it is easier to obtain symmetry structure. We consider massless p-form theory as an example. The massless p-form theory in ($2p+2$)-dimensions has global symmetry, similar to the electric-magnetic duality, connecting centers in the ground state. We find a duality structure in centers. It is hard to compute the entanglement entropy from the partial trace operation in Quantum Field Theory. We propose the Lagrangian formulation from the Hamiltonian formulation to compute the entanglement entropy with centers. From the Lagrangian method, the codimension two surface term in the Einstein gravity theory should correspond to a non-tensor product decomposition. Finally, we use the strong-coupling expansion to compute the entanglement entropy of the SU($N$) Yang-Mills lattice gauge theory in the fundamental representation from the extended lattice model. Our result shows a spatial area term when total dimensions are higher than two and $N>1$.