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Generic Local Hamiltonians are Gapless.

Ramis Movassagh1

  • 1Department of Mathematics, IBM TJ Watson Research Center, Yorktown Heights, NY, 10598.

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|December 30, 2017
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This summary is machine-generated.

Generic quantum local Hamiltonians are proven to be gapless, exhibiting a continuous density of states above the ground state. This finding applies to various lattice and graph structures, impacting models like many-body localization.

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Area of Science:

  • Quantum mechanics
  • Condensed matter physics
  • Statistical mechanics

Background:

  • Understanding the spectral properties of quantum many-body systems is crucial for predicting their behavior.
  • Local Hamiltonians, common in condensed matter models, often exhibit complex energy spectra.
  • The presence or absence of an energy gap significantly influences a system's thermalization and topological properties.

Purpose of the Study:

  • To determine the spectral properties of generic quantum local Hamiltonians.
  • To investigate the existence of energy gaps and the nature of the ground state.
  • To provide constructive proofs for these properties in diverse physical settings.

Main Methods:

  • Analysis of quantum local Hamiltonians on lattices and graphs.
  • Incorporation of translational invariance in a disordered sense.
  • Calculation of gap scaling using Gaussian β orthogonal random matrix ensembles.
  • Examination of discrete local eigenvalue distributions.

Main Results:

  • Proof that generic quantum local Hamiltonians are gapless.
  • Demonstration of a continuous density of states above the ground state.
  • Identification of conditions leading to finite-size ground state degeneracies for discrete eigenvalue distributions.
  • Corollary: existence of finite-size partitions where the ground state approaches a product state.

Conclusions:

  • Generic quantum local Hamiltonians lack an energy gap, implying unique low-energy physics.
  • The ground state properties, including potential degeneracies, are sensitive to the nature of local eigenvalue distributions.
  • The results offer fundamental insights into the spectral landscape of disordered quantum systems.