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This study numerically tests a hypothesis on operator complexity growth in many-body systems. The hypothesis is largely supported for Ising models, but not fully observed in Heisenberg models within current numerical limits.

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Area of Science:

  • Quantum mechanics
  • Condensed matter physics
  • Statistical mechanics

Background:

  • A recent hypothesis proposes linear growth of Lanczos coefficients in generic, nonintegrable many-body systems.
  • This complexity growth is expected to be slower in integrable or free models.
  • The hypothesis includes a logarithmic correction for one-dimensional systems.

Purpose of the Study:

  • To numerically investigate the hypothesis on operator complexity growth.
  • To test the hypothesis across diverse systems like Ising and Heisenberg models.
  • To explore the relationship between operator growth and geometric bounds.

Main Methods:

  • Numerical simulations of one-dimensional and two-dimensional Ising models.
  • Numerical simulations of one-dimensional Heisenberg models.
  • Derivation and analysis of a related geometric bound on operator growth.

Main Results:

  • The hypothesis on complexity growth is practically fulfilled for all simulated Ising models.
  • The universal behavior predicted by the hypothesis was not observed in the Heisenberg model data.
  • The derived geometric bound was not sharply achieved, but the hypothesis held in most cases.

Conclusions:

  • The study provides numerical evidence supporting the hypothesis for certain nonintegrable systems.
  • Limitations in numerical data acquisition affect observations in integrable systems like the Heisenberg model.
  • Operator complexity growth appears linked to geometric properties of the Hamiltonian and lattice structure.