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Related Concept Videos

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It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Related Experiment Video

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Many-Body Localization Implies that Eigenvectors are Matrix-Product States.

M Friesdorf1, A H Werner1, W Brown2

  • 1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany.

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Many-body localization prevents thermalization in quantum systems. This study links vanishing transport to eigenvector entanglement, proving strong localization implies clustering correlations and enabling efficient approximations in one dimension.

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Area of Science:

  • Condensed-matter physics
  • Quantum statistical mechanics
  • Many-body localization

Background:

  • Many-body localization (MBL) is crucial for understanding insulators at nonzero temperatures and the absence of thermalization in quantum systems.
  • MBL provides insights into the foundations of quantum statistical mechanics and out-of-equilibrium dynamics.
  • Recent attention highlights MBL's role in systems exhibiting non-thermalizing behavior.

Purpose of the Study:

  • To establish a novel link between dynamical properties and entanglement properties of individual eigenvectors in many-body localized systems.
  • To investigate the implications of strong dynamical localization on eigenvector correlations.
  • To explore the existence of a mobility edge and its relation to spectral properties.

Main Methods:

  • Establishing a theoretical link between dynamical properties (vanishing group velocity, absence of transport) and entanglement properties of individual eigenvectors.
  • Proving that strong dynamical localization implies clustering correlations for many-body eigenvectors in generic spectra.
  • Analyzing spectral properties to identify conditions for the existence of a mobility edge.

Main Results:

  • A direct correlation is proven between strong dynamical localization and clustering correlations in all many-body eigenvectors.
  • These findings extend to specific parts of the spectrum, indicating the possibility of a mobility edge.
  • In one-dimensional systems, these results lead to an entanglement area law, allowing efficient eigenvector approximation via matrix-product states.

Conclusions:

  • Strong dynamical localization is intrinsically linked to specific entanglement properties of eigenvectors.
  • The presence of a mobility edge is theoretically supported, distinguishing localized and conducting spectral regions.
  • The findings offer a pathway for efficient numerical simulations of one-dimensional many-body localized systems.