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Entropy and exact matrix-product representation of the Laughlin wave function.

S Iblisdir1, J I Latorre, R Orús

  • 1Departament d'Estructura i Constituents de la Matèria, Universitat de Barcelona, 647 Diagonal, 08028 Barcelona, Spain.

Physical Review Letters
|March 16, 2007
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Summary

Researchers derived an analytical expression for von Neumann entropy in Laughlin wave functions. This provides bounds for matrix-product state representations, crucial for understanding quantum systems.

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Area of Science:

  • Quantum Hall Effect
  • Condensed Matter Physics
  • Many-Body Quantum Systems

Background:

  • The Laughlin wave function is a key model for understanding the fractional quantum Hall effect.
  • Matrix-product states (MPS) are essential for numerically simulating one-dimensional quantum systems.
  • Understanding the entanglement properties of quantum states is crucial for characterizing their complexity.

Purpose of the Study:

  • To derive an analytical expression for the von Neumann entropy of the Laughlin wave function for any bipartition.
  • To establish an upper bound for the von Neumann entropy for specific filling fractions.
  • To determine bounds on the size of matrices required for exact MPS representations of the Laughlin state.

Main Methods:

  • Analytical calculation of von Neumann entropy for the Laughlin wave function.
  • Derivation of an upper bound for entropy at nu=1/m (m odd).
  • Development of an analytical MPS representation using Clifford algebra.

Main Results:

  • An exact analytical expression for von Neumann entropy is obtained for nu=1.
  • An upper bound on the von Neumann entropy is found for nu=1/m (m odd).
  • The study provides a bound on the minimum matrix size for exact MPS representations.
  • An analytical MPS representation of the Laughlin state is proposed.

Conclusions:

  • The derived entropy expressions and bounds are vital for efficient numerical simulations of quantum Hall states.
  • The proposed MPS representation offers a pathway to study large systems with controlled accuracy.
  • The findings contribute to a deeper understanding of entanglement in topological phases of matter.