Tyler Guglielmo

We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order of N , the standard result for the q-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter, k. We present an algorithm to sum the diagrams at any given order of 1/(kq)n. We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finite N becomes more significant for larger values of N .

We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order of N, the standard result for the q-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter, k. We present an algorithm to sum the diagrams at any given order of 1/(kq)ⁿ. We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finite N becomes more significant for larger values of N.

In order to study the chaotic behavior of a system with non-local interactions, we will consider weakly coupled non-commutative field theories. We compute the Lyapunov exponent of this exponential growth in the large Moyal-scale limit to leading order in the t’Hooft coupling and 1/N. We found that in this limit, the Lyapunov exponent remains comparable in magnitude to (and somewhat smaller than) the exponent in the commutative case. This can possibly be explained by the infrared sensitivity of the Lyapunov exponent. Another possible explanation is that in examples of weakly coupled non-commutative field theories, non-local contributions to various thermodynamic quantities are sub-dominant.

We check formally that the Hubeny-Rangamani-Takayanagi prescription for holographic entanglement entropy—when applied to a static black brane spacetime and to a wide class of subregions that do not lie on a constant time slice—gives rise to volume-law entropy in the limit of large subregion. By volume-law entropy, we mean that the entanglement entropy scales with the volume of the projection of the boundary subregion onto a static time slice with respect to the boundary thermal state (with the same coefficient as in the volume law on the static time slice). Our result applies to subregions that have reflection symmetry as well as strips, and we also present field-theoretic arguments in support of our holographic findings.

We explore the effect of introducing mild nonlocality into otherwise local, chaotic quantum systems, on the rate of information spreading and associated rates of entanglement generation and operator growth. We consider various forms of nonlocality, both in 1-dimensional spin chain models and in holographic gauge theories, comparing the phenomenology of each. Generically, increasing the level of nonlocality increases the rate of information spreading, but in lattice models we find instances where these rates are slightly suppressed.