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The phase rule describes the relationship between the variance (degrees of freedom), the number of components, and the number of phases in a system at equilibrium.Variance is a concept that denotes the number of independent intensive properties (properties are those that do not depend on the amount of material in the system), such as temperature, pressure, and composition, that can be altered without impacting the number of phases in equilibrium.In a single-component system, such as pure water,...
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Nonergodic Phases in Strongly Disordered Random Regular Graphs.

B L Altshuler1, E Cuevas2, L B Ioffe3,4

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|October 22, 2016
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Summary
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We prove a new nonergodic, delocalized phase in the Anderson model on disordered hierarchical lattices. This finding, supported by population dynamics and random regular graph analysis, reveals critical transitions in quantum systems.

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Area of Science:

  • Condensed Matter Physics
  • Disordered Systems
  • Quantum Chaos

Background:

  • The Anderson model describes electron localization in disordered materials.
  • Understanding delocalized states in disordered systems is crucial for quantum and many-body physics.
  • Hierarchical lattices and random regular graphs (RRG) offer simplified yet relevant models for studying complex phenomena.

Purpose of the Study:

  • To prove the existence of an intermediate nonergodic but delocalized phase.
  • To develop new methods for detecting ergodicity violations in delocalized states.
  • To investigate phase transitions and critical exponents in disordered systems.

Main Methods:

  • Combining numerical diagonalization with semianalytical calculations.
  • Introducing a generalized population dynamics method.
  • Analyzing statistics of random wave functions on disordered random regular graphs (RRG) with N sites and connectivity K=2.

Main Results:

  • Existence of a nonergodic, delocalized phase in the Anderson model on hierarchical lattices.
  • Detection of ergodicity violation in delocalized states using generalized population dynamics.
  • Calculation of fractal dimensions D(W) and D(W) and population dynamics exponent D(W) for disorder strength W.
  • Evidence for a first-order phase transition between delocalized phases on RRG at W ≈ 10.0.

Conclusions:

  • The Anderson model exhibits a novel nonergodic, delocalized phase.
  • Generalized population dynamics effectively detects ergodicity violations.
  • The findings have implications for understanding quantum and classical nonintegrable and many-body systems.