Abstract
Understanding the fluctuations of observables is one of the main goals in science, be it theoretical or experimental, quantum or classical. We investigate such fluctuations in a subregion of the full system, focusing on geometries with sharp corners. We report that the angle dependence is super-universal: up to a numerical prefactor, this function does not depend on anything, provided the system under study is uniform, isotropic, and correlations do not decay too slowly. The prefactor contains important physical information: we show in particular that it gives access to the long-wavelength limit of the structure factor. We exemplify our findings with fractional quantum Hall states, topological insulators, scale invariant quantum critical theories, and metals. We suggest experimental tests, and anticipate that our findings can be generalized to other spatial dimensions or geometries. In addition, we highlight the similarities of the fluctuation shape dependence with findings relating to quantum entanglement measures.
Highlights
Understanding the fluctuations of observables is one of the main goals in science, be it theoretical or experimental, quantum or classical
We show that there exists a large, and experimentally relevant, set of states and observables that share the same universal shape dependence for their fluctuations
We will focus on a uniform and isotropic systems, for which the above correlation function only depends on the distance separating the two positions hρðrÞρðr0Þic 1⁄4 f ðjr À r0jÞ, yielding ðΔOAÞ2 1⁄4 dr dr[0] f ðjr À r0jÞ : ð2Þ
Summary
Understanding the fluctuations of observables is one of the main goals in science, be it theoretical or experimental, quantum or classical. We investigate such fluctuations in a subregion of the full system, focusing on geometries with sharp corners. We highlight the similarities of the fluctuation shape dependence with findings relating to quantum entanglement measures. 1234567890():,; In quantum mechanics, measurements on identically prepared systems of an observable O will generally yield different outcomes. This is a consequence of the fact that the state of the system is in a quantum superposition of states having welldefined values of O. Fluctuations of most physical systems behave for large regions A as[4]
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