Entanglement in a second order topological insulator on a square lattice

In a $d$-dimensional topological insulator of order $d$, there are zero energy states on its corners which have close relationship with its entanglement behaviors. We studied the bipartite entanglement spectra for different subsystem shapes and found that only when the entanglement boundary has corners matching the lattice, exact zero modes exist in the entanglement spectrum corresponding to the zero energy states caused by the same physical corners. We then considered finite size systems in which case these corner states are coupled together by long range hybridizations to form a multipartite entangled state. We proposed a scheme to calculate the quadripartite entanglement entropy on the square lattice, which is well described by a four-sites toy model and thus provides another way to identify the higher order topological insulators from the multipartite entanglement point of view.Comment: 5 pages, 3 figure

Electronic structure near an impurity and terrace on the surface of a 3-dimensional topological insulator

Motivated by recent scanning tunneling microscopy experiments on surfaces of Bi$_{1-x}$Sb$_{x'}$\cite{yazdanistm,gomesstm} and Bi$_2$Te$_3$,\cite{kaptunikstm,xuestm} we theoretically study the electronic structure of a 3-dimensional (3D) topological insulator in the presence of a local impurity or a domain wall on its surface using a 3D lattice model. While the local density of states (LDOS) oscillates significantly in space at energies above the bulk gap, the oscillation due to the in-gap surface Dirac fermions are very weak. The extracted modulation wave number as a function of energy satisfies the Dirac dispersion for in-gap energies and follows the border of the bulk continuum above the bulk gap. We have also examined analytically the effects of the defects by using a pure Dirac fermion model for the surface states and found that the LDOS decays asymptotically faster at least by a factor of 1/r than that in normal metals, consistent with the results obtained from our lattice model.Comment: 7 pages, 5 figure

Heisenberg Limit Superradiant Superresolving Metrology

We propose a superradiant metrology technique to achieve the Heisenberg limit super-resolving displacement measurement by encoding multiple light momenta into a three-level atomic ensemble. We use $2N$ coherent pulses to prepare a single excitation superradiant state in a superposition of two timed Dicke states that are $4N$ light momenta apart in momentum space. The phase difference between these two states induced by a uniform displacement of the atomic ensemble has $1/4N$ sensitivity. Experiments are proposed in crystals and in ultracold atoms

Efficient Task Replication for Fast Response Times in Parallel Computation

One typical use case of large-scale distributed computing in data centers is to decompose a computation job into many independent tasks and run them in parallel on different machines, sometimes known as the "embarrassingly parallel" computation. For this type of computation, one challenge is that the time to execute a task for each machine is inherently variable, and the overall response time is constrained by the execution time of the slowest machine. To address this issue, system designers introduce task replication, which sends the same task to multiple machines, and obtains result from the machine that finishes first. While task replication reduces response time, it usually increases resource usage. In this work, we propose a theoretical framework to analyze the trade-off between response time and resource usage. We show that, while in general, there is a tension between response time and resource usage, there exist scenarios where replicating tasks judiciously reduces completion time and resource usage simultaneously. Given the execution time distribution for machines, we investigate the conditions for a scheduling policy to achieve optimal performance trade-off, and propose efficient algorithms to search for optimal or near-optimal scheduling policies. Our analysis gives insights on when and why replication helps, which can be used to guide scheduler design in large-scale distributed computing systems.Comment: Extended version of the 2-page paper accepted to ACM SIGMETRICS 201