[Paper Review] Topological squashed entanglement: nonlocal order parameter for one-dimensional topological superconductors

Alfonso Maiellaro, Antonio Marino+1 more|arXiv (Cornell University)|Jan 28, 2022

Topological Materials and PhenomenaPhysics and Astronomy127 references16 citations

TL;DR

This paper introduces topological squashed entanglement (TSE), a nonlocal entanglement measure based on edge-to-edge quantum conditional mutual information, as a robust, quantized order parameter for one-dimensional topological superconductors. It demonstrates that TSE is quantized at log(2)/2 in the Kitaev chain's topological phase and vanishes in the trivial phase, remaining stable under interactions, disorder, and geometric frustration.

ABSTRACT

Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to log(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi 1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.

Motivation & Objective

• To identify a nonlocal, entanglement-based order parameter that unambiguously characterizes topological superconductors beyond conventional local order parameters.
• To overcome the limitations of bipartite block entanglement and topological entanglement entropy in detecting nonlocal correlations between edge modes.
• To establish a measure that remains quantized and stable under interactions, disorder, and in systems with suppressed bulk-edge correspondence.
• To distinguish topological order from symmetry-breaking order using a nonlocal entanglement observable.
• To generalize entanglement-based diagnostics to multipartite partitions and apply them to complex quasi-1D topological systems.

Proposed method

• Proposes edge squashed entanglement (TSE) as a nonlocal bipartite entanglement measure defined via edge-to-edge quantum conditional mutual information (QCMI).
• Defines two forms of TSE: one using tripartitions (edge-bulk-edge) and another using quadripartitions (edge-bulk-bulk-edge), both based on upper bounds of squashed entanglement.
• Applies the TSE formalism to model Hamiltonians of 1D and quasi-1D topological superconductors, including the Kitaev chain and Kitaev ladder.
• Uses the quantum conditional mutual information to detect long-distance, nonlocal correlations between Majorana edge modes.
• Compares TSE behavior in the fermionic Kitaev chain with the spin-1/2 Ising model to distinguish topological order from symmetry-breaking order.
• Extends the analysis to systems with long-range hopping (e.g., Kitaev tie) to test robustness under geometric frustration and broken bulk-edge correspondence.

Experimental results

Research questions

• RQ1 Can a nonlocal entanglement measure detect and quantify topological order in one-dimensional superconductors where standard entanglement measures fail?
• RQ2 Is the edge squashed entanglement quantized in the topological phase and vanishing in the trivial phase, and does it scale correctly at quantum phase transitions?
• RQ3 Can TSE distinguish topological superconductors from symmetry-broken systems like the transverse-field Ising model?
• RQ4 How does TSE behave under interactions and local disorder, and is it robust in systems with suppressed bulk-edge correspondence?
• RQ5 Can TSE count the number of Majorana zero modes and distinguish multiple topological phases in complex geometries like the Kitaev ladder?

Key findings

• The edge squashed entanglement is quantized at log(2)/2 in the entire topological phase of the Kitaev chain, corresponding to half the maximal Bell-state entanglement.
• TSE vanishes in the trivial phase, confirming its role as a nonlocal order parameter for topological superconductivity.
• TSE exhibits correct scaling at the quantum phase transition and remains stable under interactions and local perturbations.
• TSE successfully distinguishes topological superconductors from symmetry-breaking magnets, such as the spin-1/2 Ising chain, by detecting nonlocal edge correlations.
• In the two-leg Kitaev ladder, TSE counts the number of Majorana edge modes and distinguishes different topological phases.
• Even in the Kitaev tie with long-range hopping and geometric frustration, TSE remains a valid and robust detector of topological features, demonstrating resilience to suppressed bulk-edge correspondence.