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

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.