Spread of entanglement in a Sachdev-Ye-Kitaev chain

We study the spread of R\'enyi entropy between two halves of a Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions, prepared in a thermofield double (TFD) state. The SYK chain model is a model of chaotic many-body systems, which describes a one-dimensional lattice of Majorana fermions, with spatially local random quartic interaction. We find that for integer R\'enyi index $n>1$, the R\'enyi entanglement entropy saturates at a parametrically smaller value than expected. This implies that the TFD state of the SYK chain does not rapidly thermalize, despite being maximally chaotic: instead, it rapidly approaches a prethermal state. We compare our results to the signatures of thermalization observed in other quenches in the SYK model, and to intuition from nearly-$\mathrm{AdS}_2$ gravity.Comment: 1+46 pages, 11 figure

Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

The Sachdev-Ye-Kitaev model is a $(0+1)$-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and quantum chaos. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with $N$ Majorana fermions at each site and random interactions between them. Our model can be defined on arbitrary lattices in arbitrary spatial dimensions. In the large $N$ limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the out-of-time-ordered correlation functions. In addition, we obtain new properties unique to higher dimensions such as diffusive energy transport and a "butterfly velocity" describing the propagation of chaos in space. We mainly present results for a $(1+1)$-dimensional example, and discuss the general case near the end.Comment: 1+37 pages, published versio

Population dynamics under demographic and environmental stochasticity

The present paper is devoted to the study of the long term dynamics of diffusion processes modelling a single species that experiences both demographic and environmental stochasticity. In our setting, the long term dynamics of the diffusion process in the absence of demographic stochasticity is determined by the sign of $\Lambda_0$, the external Lyapunov exponent, as follows: $\Lambda_00$ implies convergence to a unique positive stationary distribution $\mu_0$. If the system is of size $\frac{1}{\epsilon^{2}}$ for small $\epsilon>0$ (the intensity of demographic stochasticity), demographic effects will make the extinction time finite almost surely. This suggests that to understand the dynamics one should analyze the quasi-stationary distribution (QSD) $\mu_\epsilon$ of the system. The existence and uniqueness of the QSD is well-known under mild assumptions. We look at what happens when the population size is sent to infinity, i.e., when $\epsilon\to 0$. We show that the external Lyapunov exponent still plays a key role: 1) If $\Lambda_0<0$, then $\mu_\epsilon\to \delta_0$, the mean extinction time is of order $|\ln \epsilon|$ and the extinction rate associated with the QSD $\mu_{\epsilon}$ has a lower bound of order $\frac{1}{|\ln\epsilon|}$; 2) If $\Lambda_0>0$, then $\mu_\epsilon\to \mu_0$, the mean extinction time is polynomial in $\frac{1}{\epsilon^{2}}$ and the extinction rate is polynomial in $\epsilon^{2}$. Furthermore, when $\Lambda_0>0$ we are able to show that the system exhibits multiscale dynamics: at first the process quickly approaches the QSD $\mu_\epsilon$ and then, after spending a polynomially long time there, it relaxes to the extinction state. We give sharp asymptotics in $\epsilon$ for the time spent close to $\mu_\epsilon$.Comment: 59 page

Holographic duality between (2+1)(2+1)-d quantum anomalous Hall state and (3+1)(3+1)-d topological insulators

In this paper, we study $(2+1)$-dimensional quantum anomalous Hall states, i.e. band insulators with quantized Hall conductance, using the exact holographic mapping. The exact holographic mapping is an approach to holographic duality which maps the quantum anomalous Hall state to a different state living in $(3+1)$-dimensional hyperbolic space. By studying topological response properties and the entanglement spectrum, we demonstrate that the holographic dual theory of a quantum anomalous Hall state is a $(3+1)$-dimensional topological insulator. The dual description enables a new characterization of topological properties of a system by the quantum entanglement between degrees of freedom at different length scales.Comment: 10 pages, 9 figure

ObjSim: Lightweight Automatic Patch Prioritization via Object Similarity

In the context of test case based automatic program repair (APR), patches that pass all the test cases but fail to fix the bug are called overfitted patches. Currently, patches generated by APR tools get inspected manually by the users to find and adopt genuine fixes. Being a laborious activity hindering widespread adoption of APR, automatic identification of overfitted patches has lately been the topic of active research. This paper presents engineering details of ObjSim: a fully automatic, lightweight similarity-based patch prioritization tool for JVM-based languages. The tool works by comparing the system state at the exit point(s) of patched method before and after patching and prioritizing patches that result in state that is more similar to that of original, unpatched version on passing tests while less similar on failing ones. Our experiments with patches generated by the recent APR tool PraPR for fixable bugs from Defects4J v1.4.0 show that ObjSim prioritizes 16.67% more genuine fixes in top-1 place. A demo video of the tool is located at https://bit.ly/2K8gnYV.Comment: Proceedings of the 29th ACM SIGSOFT International Symposium on Software Testing and Analysis (ISSTA '20), July 18--22, 2020, Virtual Event, US