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Complexity in the Lipkin-Meshkov-Glick model.

Kunal Pal1, Kuntal Pal1, Tapobrata Sarkar1

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Summary
This summary is machine-generated.

We explored quantum complexity in the Lipkin-Meshkov-Glick (LMG) model. Nielsen complexity shows logarithmic divergence near phase transitions, but a finite discontinuity in time-dependent scenarios.

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Area of Science:

  • Quantum Information Theory
  • Condensed Matter Physics
  • Statistical Mechanics

Background:

  • Quantum complexity quantifies the resources needed to implement quantum operations.
  • The Lipkin-Meshkov-Glick (LMG) model is a fundamental model for studying many-body quantum systems with infinite-range interactions.
  • Understanding complexity in different quantum models reveals universal and system-specific behaviors.

Purpose of the Study:

  • To derive exact expressions for Nielsen complexity (NC) and Fubini-Study complexity (FSC) in the LMG model.
  • To investigate the behavior of quantum complexity near phase transitions and in time-dependent settings.
  • To compare complexity measures in the LMG model with those in other spin systems.

Main Methods:

  • Exact derivation of Nielsen complexity (NC) and Fubini-Study complexity (FSC) for the LMG model.
  • Application of the Lewis-Riesenfeld theory for time-dependent invariant operators.
  • Numerical analysis of geodesic behavior in relation to complexity.

Main Results:

  • In a time-independent LMG model, NC exhibits logarithmic divergence near phase transitions, similar to entanglement entropy.
  • In time-dependent scenarios, this logarithmic divergence transitions to a finite discontinuity.
  • FSC in a variant LMG model diverges logarithmically near the separatrix, a behavior distinct from quasifree models.

Conclusions:

  • The LMG model displays unique complexity characteristics, particularly in its response to time dependence and proximity to the separatrix.
  • Complexity measures like NC and FSC offer insights into the quantum dynamics and phase transitions of many-body systems.
  • The study highlights the nuanced relationship between quantum complexity, phase transitions, and system dynamics.