[Paper Review] Quantum Entanglement and Detection of Topological Order in Numerics

Tarun Grover|arXiv (Cornell University)|Dec 9, 2011

Quantum many-body systems1 references3 citations

TL;DR

This paper proposes a method to detect topological order in numerical simulations by exploiting the entanglement structure of degenerate ground states in topologically ordered phases. By analyzing the Schmidt decomposition of these states, the non-universal area-law term in entanglement entropy cancels out, allowing direct extraction of the universal topological entanglement entropy $γ_{\text{topo}}$ via a simple formula, validated numerically on a 18-site chiral spin-liquid state with excellent agreement to the theoretical prediction of $\gamma_{\text{topo}} = \ln(2) \approx 0.69$. The method is compatible with DMRG and exact diagonalization and works regardless of the presence of edge states.

ABSTRACT

"Topological ordered" phases such as gapped quantum spin-liquids and fractional quantum Hall states possess ground state degeneracy on a torus. We show that the topological nature of this degeneracy has interesting consequences for the entanglement structure of the degenerate ground states. This leads to a simple method for detecting topological order, which is ready-made for numerical schemes such as density matrix renormalization group. The method is valid for phases that do or do not possess edge states alike. We demonstrate it by calculating the entanglement spectrum and the topological entanglement entropy for a chiral spin-liquid state on a 18 site system.

Motivation & Objective

• To develop a practical numerical method for detecting topological order in gapped quantum phases, especially where quantum Monte Carlo fails due to the fermionic sign problem.
• To overcome the difficulty of extracting the universal topological entanglement entropy $\gamma_{\text{topo}}$ from standard DMRG simulations, which are limited by finite-size effects and lack of tripartite entanglement calculations.
• To provide a scheme that cancels the non-universal area-law term in entanglement entropy using degenerate ground states, enabling robust extraction of $\gamma_{\text{topo}}$.
• To validate the method on a small system (18 sites) with a chiral spin-liquid state, demonstrating agreement with the theoretical $\gamma_{\text{topo}} = \ln(2)$.
• To extend the applicability of entanglement-based diagnostics to quantum-ordered phases with emergent gauge bosons, distinguishing topological degeneracy from gapless photon-like modes.

Proposed method

• The method relies on the fact that all degenerate ground states of a topologically ordered phase share the same non-universal area-law term in their entanglement entropy, enabling cancellation when comparing Schmidt spectra of different ground states.
• It uses the Schmidt decomposition of two or more degenerate ground states $|\psi_1\rangle$ and $|\psi_2\rangle$ obtained via DMRG or exact diagonalization, with the same reduced density matrix structure.
• The key equation is $S_2(\phi) = \cos^2(\phi) S_2(|\psi_1\rangle) + \sin^2(\phi) S_2(|\psi_2\rangle)$, where the difference $S_{2,\text{max}} - S_{2,\text{min}} = 2\gamma_{\text{topo}}$ is independent of the superposition angle $\phi$.
• The method is applied to a chiral spin-liquid wavefunction parameterized by a mean-field d-wave BCS gap $\Delta$, allowing systematic variation of the entanglement entropy across superpositions of degenerate states.
• The topological entanglement entropy is extracted from the maximum and minimum Renyi entropy $S_2$ over the superposition parameter $\phi$, with the difference yielding $2\gamma_{\text{topo}}$.
• The approach avoids complex tripartite entanglement measures (e.g., Eq. 3) and remains accurate even for small systems with finite-size effects, as long as the correlation length $\xi \ll L$.

Experimental results

Research questions

• RQ1 Can the topological entanglement entropy $\gamma_{\text{topo}}$ be reliably extracted from degenerate ground states in small systems using standard DMRG or exact diagonalization?
• RQ2 Does the entanglement spectrum of degenerate ground states in a topologically ordered phase remain identical, even when the states are only approximately degenerate?
• RQ3 Can the difference in Renyi entropy $S_2$ between superpositions of degenerate states be used to extract $\gamma_{\text{topo}}$ without requiring tripartite entanglement measures?
• RQ4 Is the method robust to finite-size effects and approximate degeneracy, as seen in realistic numerical simulations of quantum spin liquids?
• RQ5 Can this entanglement-based method distinguish topological degeneracy from gapless gauge modes in quantum-ordered phases?

Key findings

• The method successfully extracts $\gamma_{\text{topo}}$ from a 18-site chiral spin-liquid state, yielding $S_{2,\text{max}} - S_{2,\text{min}} \approx 0.67$, in close agreement with the theoretical value of $\ln(2) \approx 0.69$.
• The entanglement spectrum of two nearly degenerate ground states ($|0,0\rangle$ and $|0,\pi\rangle$) is identical, confirming that the Schmidt eigenvalues $\lambda_\alpha$ are the same across degenerate states.
• The Schmidt states corresponding to the same $\lambda_\alpha$ are orthogonal across different degenerate ground states, as predicted by Eq. (5), validating the theoretical framework.
• The difference in $S_2$ between superpositions of degenerate states is independent of the superposition angle $\phi$, confirming the theoretical prediction that $S_2(\phi)$ depends only on the individual $S_2$ values of the states.
• The method is numerically feasible for systems up to $N=30-40$ sites for spin-$1/2$ systems, as long as the correlation length $\xi \ll L$.
• The approach is applicable to both topological order with and without edge states, and can distinguish topological degeneracy from gapless gauge modes in quantum-ordered phases via entanglement structure.