Seeing the Quantum Butterfly Effect

Bin Y, Nikolai Sinitsyn

Lei Wang, Matthias Troyer

We propose a way to detect the fractal energy spectrum of the Hofstadter model from the density distributions of ultracold fermions in an external trap. At low temperature, the local compressibility is proportional to the density of states of the system, which reveals the fractal energy spectrum. However, thermal broadening and noises in the real experimental situation inevitably smear out fine features in the density distribution. To overcome this difficulty, we use the maximum entropy method to extract the density of states directly from the noisy thermal density distributions. Simulations show that one is able to restore the core feature of the Hofstadter's butterfly spectrum with current experimental techniques. By further reducing the noise or the temperature, one can refine the resolution and observe fine structures of the butterfly spectrum.

David Deutsch, ARTUR K. EKERT

Daniel A. Roberts, Brian G. Swingle

As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound-the Lieb-Robinson bound-and the butterfly effect in strongly coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the "butterfly" velocity v_{B}. Similarly, the Lieb-Robinson velocity places a state-independent ballistic upper bound on the size of time evolved operators in nonrelativistic lattice models. Here, we argue that v_{B} is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with free-particle computations to understand the role of strong coupling. We find that v_{B} remains constant or decreases with decreasing temperature. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling.

Yi Ling, Zhuo-Yu Xian

We investigate the butterfly effect and charge diffusion near the quantum phase transition in holographic approach. We argue that their criticality is controlled by the holographic scaling geometry with deformations induced by a relevant operator at finite temperature. Specifically, in the quantum critical region controlled by a single fixed point, the butterfly velocity decreases when deviating from the critical point. While, in the non-critical region, the behavior of the butterfly velocity depends on the specific phase at low temperature. Moreover, in the holographic Berezinskii-Kosterlitz-Thouless transition, the universal behavior of the butterfly velocity is absent. Finally, the tendency of our holographic results matches with the numerical results of Bose-Hubbard model. A comparison between our result and that in the O(N ) nonlinear sigma model is also given.

Don Lincoln