[Paper Review] Entanglement Spectroscopy and its Application to the Quantum Hall Effects

N. Regnault|arXiv (Cornell University)|Oct 26, 2015

Quantum and electron transport phenomena10 citations

TL;DR

This paper introduces entanglement spectroscopy as a powerful tool to diagnose topological order in quantum Hall systems by analyzing the spectrum of the reduced density matrix upon system partitioning. It demonstrates that the entanglement spectrum reveals edge excitations and topological invariants, providing deep insights into fractional quantum Hall states and fractional Chern insulators with high numerical efficiency.

ABSTRACT

The entanglement spectroscopy, initially introduced by Li and Haldane in the context of the fractional quantum Hall effects, has stimulated an extensive range of studies. The entanglement spectrum is the spectrum of the reduced density matrix, when we partition the system into two. For many quantum systems, it unveils a unique feature: Computed from the bulk ground state wave function, the entanglement spectrum give access to the physics of edge excitations. Using this property, the entanglement spectroscopy has proved to be a highly valuable tool to diagnose topological ordering. These lectures intend to provide an overview of the entanglement spectroscopy, mainly in the context of the fractional quantum Hall effect. We introduce the basic concepts through the case of the quantum spin chains. We discuss the connection with the entanglement entropy and the matrix product state representation. We show how the entanglement spectrum can be computed for non-interacting topological phases and how it reveals the edge excitation from the ground state. We then present an extensive review of the entanglement spectra applied to the fractional quantum Hall phases, showing how much information is encoded within the ground state and how different partitions probe different type of excitations. Finally, we discuss the application of this tool to study the fractional Chern insulators.

Motivation & Objective

• To establish entanglement spectroscopy as a diagnostic tool for topological order in quantum Hall systems.
• To demonstrate how the entanglement spectrum encodes information about edge excitations in topological phases.
• To extend the method to fractional Chern insulators and non-interacting topological systems.
• To provide a practical framework for identifying topological order from ground state wave functions alone.

Proposed method

• Uses bipartite partitioning of the system to compute the reduced density matrix ρA = TrB |Ψ⟩⟨Ψ|.
• Analyzes the spectrum of ρA to extract entanglement levels, which mirror edge state excitations.
• Applies the method to various systems: spin chains, integer and fractional quantum Hall states, Chern insulators, and fractional Chern insulators.
• Employs matrix product state (MPS) representations to efficiently compute entanglement spectra in one-dimensional systems.
• Uses orbital, particle, and real-space partitioning to probe different types of excitations.
• Leverages the connection between entanglement entropy and topological order, especially in systems with gapless edge modes.

Experimental results

Research questions

• RQ1 How does the entanglement spectrum reflect the presence of topological order in quantum Hall systems?
• RQ2 Can the entanglement spectrum reveal edge excitations even when no local order parameter exists?
• RQ3 To what extent can the entanglement spectrum distinguish between different topological phases, such as fractional quantum Hall states and fractional Chern insulators?
• RQ4 How does the choice of partition (orbital, particle, real space) affect the information extracted from the entanglement spectrum?
• RQ5 Can entanglement spectroscopy be used to characterize non-interacting topological phases and their edge modes?

Key findings

• The entanglement spectrum of the fractional quantum Hall effect closely mimics the spectrum of edge excitations, providing a direct signature of topological order.
• For the integer quantum Hall effect, the entanglement spectrum reveals Landau level-like levels that reflect the chiral edge modes.
• In fractional Chern insulators, the entanglement spectrum exhibits a clear gap and level counting consistent with anyonic statistics.
• The orbital entanglement spectrum for model wave functions (e.g., Laughlin states) reproduces the expected topological degeneracy and anyon statistics.
• The method successfully identifies topological order in non-interacting systems, such as Chern insulators, via the entanglement spectrum's characteristic structure.
• Entanglement spectroscopy provides a robust, numerically efficient alternative to traditional order parameters for diagnosing topological phases from the ground state alone.