Entanglement and matrix elements of observables in interacting integrable systems

Area of Science:

• Quantum Information Theory
• Condensed Matter Physics
• Statistical Mechanics

Background:

• Understanding quantum entanglement is crucial for characterizing quantum systems.
• Distinguishing between quantum integrable and quantum chaotic systems is a fundamental challenge.
• The spin-1/2 XXZ chain serves as a paradigmatic model for studying interacting integrable systems.

Purpose of the Study:

• To investigate the behavior of entanglement entropy and local operator matrix elements in the eigenstates of an interacting integrable Hamiltonian.
• To contrast these properties with those of quantum chaotic systems.
• To establish entanglement entropy as a reliable measure for differentiating integrable models from generic ones.

Main Methods:

• Numerical study of the bipartite von Neumann entanglement entropy in the eigenstates of the spin-1/2 XXZ chain.
• Analysis of matrix elements of local operators in the same eigenstates.
• Comparison of results with those from quantum chaotic systems and translationally invariant quadratic fermionic models.

Main Results:

• The average eigenstate entanglement entropy in the XXZ chain exhibits a volume-law coefficient smaller than that of quantum chaotic systems.
• This coefficient is found to be close to, or the same as, that of translationally invariant quadratic fermionic models.
• Diagonal matrix elements of local operators show non-vanishing support, while their fluctuations vanish power-law-wise.
• Off-diagonal matrix elements display a near log-normal distribution with variance proportional to 1/D.

Conclusions:

• Entanglement entropy serves as a powerful tool to distinguish quantum integrable models from quantum chaotic ones.
• The spin-1/2 XXZ Hamiltonian's entanglement properties show remarkable similarity to quadratic fermionic models.
• The study provides detailed insights into the spectral properties and entanglement structure of integrable systems.