🔬 mesoscale physics

Entanglement signatures of topological phase transitions in a dirty Weyl semimetal

This paper investigates how entanglement spectrum and entropy signatures evolve in a disordered Weyl semimetal, demonstrating that disorder broadens the characteristic $\xi=1/2$ eigenvalue locus associated with Fermi arcs, which subsequently vanishes either through gradual spectral weight loss during node annihilation or by merging into the background at the transition to a diffusive metallic state.

Original authors: Vinícius Mohr, Shunji Tsuchiya, Andras Szabo

Published 2026-07-30

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CC BY 4.0

Original authors: Vinícius Mohr, Shunji Tsuchiya, Andras Szabo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of materials as a vast, bustling city where electrons are the commuters. In most cities, traffic flows smoothly, but in some exotic neighborhoods, the roads are built with strange, invisible rules that make the traffic behave in ways that seem to defy common sense. This is the world of topological phases, a branch of physics that studies how the "shape" of these electron paths protects them from getting messed up by small bumps or potholes (disorder). Think of a topological material like a perfect, unbreakable knot; you can wiggle it or stretch it, but you can't untie it unless you cut the string completely.

One of the most fascinating neighborhoods in this city is the Weyl semimetal. Here, the electrons don't just flow; they zip around like massless particles, creating "traffic hubs" called Weyl nodes. These hubs are connected by special, one-way streets on the surface of the material known as Fermi arcs. What makes this exciting is that these structures are incredibly robust. They can survive a little bit of dirt or chaos (disorder) without falling apart. However, if the chaos gets too loud, or if the city planners decide to close the hubs, the whole special structure can collapse into a boring, ordinary metal. Scientists are eager to understand exactly how this collapse happens, because knowing the difference between a "messy but still special" state and a "completely ruined" state could help us build better, more resilient quantum computers in the future.

In this new study, researchers Vinícius Mohr, Shunji Tsuchiya, and András L. Szabó decided to peek behind the curtain of these exotic materials to see what happens when things go wrong. They didn't just look at the traffic flow; they looked at how "entangled" the electrons are with each other. Imagine two groups of friends at a party; if they are "entangled," their conversations are so deeply linked that you can't understand one group without knowing the other. The researchers used a mathematical tool called the entanglement spectrum to map these connections. They found that in a perfect Weyl semimetal, there is a very specific, sharp signal in this map: a bunch of connections that are exactly halfway between "totally linked" and "totally separate" (a value of 0.5). This signal is the fingerprint of the Fermi arcs.

The team then simulated what happens when they start throwing "dirt" (disorder) into the system. They discovered that a little bit of dirt doesn't destroy the fingerprint; instead, it blurs it. That sharp, single line of 0.5 connections turns into a fuzzy, wide cloud of values around 0.5. As they added more and more dirt, this cloud got wider and wider until, at a critical point (around a disorder strength of $W \approx 3.75t$ ), the cloud melted completely into the background noise, and the special signal vanished. This marked the moment the material lost its topological superpowers and became a normal, messy metal.

But the researchers also asked a different question: What if we didn't add dirt, but instead changed the city's layout to close the traffic hubs? They simulated this by adjusting a parameter called $m$ until the Weyl nodes crashed into each other and disappeared. In this scenario, the fuzzy cloud didn't just melt; it slowly shrank and lost its strength until it was gone.

The key takeaway from their simulations is that the "fingerprint" of the material changes in two very different ways depending on how you break it. If you break it with too much disorder, the signal dissolves into chaos. If you break it by closing the nodes, the signal simply fades away. By tracking these subtle changes in the entanglement map, the team showed that we can tell exactly how a topological material is dying, even when it's covered in dirt. Their work suggests that looking at these quantum connections is a powerful way to spot the difference between a material that is just messy and one that has lost its magic entirely.

Technical Summary: Entanglement Signatures of Topological Phase Transitions in a Dirty Weyl Semimetal

Problem Statement
Three-dimensional Weyl semimetals (WSMs) are paradigmatic gapless topological phases characterized by nodal points in momentum space and topological surface states known as Fermi arcs. While these phases are robust against weak perturbations, their topology can be destroyed via two distinct mechanisms: (1) the pairwise annihilation of Weyl nodes in momentum space (driven by tuning band parameters), or (2) the destruction of the quasiparticle pole by sufficiently strong disorder, which drives the system into a non-Fermi-liquid diffusive-metallic (DM) state.

Probing these transitions in disordered systems is challenging because the loss of translational invariance precludes standard momentum-space formulations of topological invariants. While entanglement entropy (EE) and the entanglement spectrum (ES) have proven effective for studying topological transitions in one- and two-dimensional disordered systems, their behavior in three-dimensional gapless systems undergoing these specific transitions remains to be fully characterized. This work investigates how entanglement measures evolve as a WSM undergoes both node annihilation and the disorder-induced transition to a diffusive metal.

Methodology
The authors employ a minimal two-orbital lattice model constructed by stacking two-dimensional topological insulators, described by a tight-binding Hamiltonian with a tunable mass parameter $m$ and interlayer hopping $t_z$ . Disorder is introduced as an orbital-independent on-site potential drawn from a uniform random distribution with strength $W$ .

The study utilizes the following computational approaches:

Phase Diagram Reconstruction: The average density of states (DOS) is computed using the kernel polynomial method on a cubic system ( $L=100$ ) to map the phase boundaries in the $(m, W)$ plane, identifying the WSM, normal insulator (NI), and diffusive metal (DM) phases.
Entanglement Spectrum (ES) Analysis: The system is bipartitioned in real space into two equal subsystems. For the clean limit, exact diagonalization is performed on $L=10$ systems to resolve the ES eigenvalues $\xi_n$ of the reduced correlation matrix $C_{ij}$ .
Disorder Averaging: To handle the lack of translational invariance in disordered systems, the ES is computed for 10,000 disorder realizations at fixed $W$ . The distribution of these eigenvalues is reconstructed to form the Entanglement Density of States (EDOS), denoted as $\nu_S(\xi)$ .
Entanglement Entropy Scaling: The authors compute disorder-averaged Rényi entropies ( $S_i$ for $i=1, 2, 3$ ) and analyze their scaling with disorder strength $W$ via the local power-law exponent $\alpha_i$ .

Key Contributions and Results

Clean Limit Signature: In the clean WSM, the ES exhibits a distinct locus of eigenvalues pinned exactly at $\xi = 1/2$ . These eigenvalues correspond to the Fermi arc surface states projected onto the virtual boundary, connecting the Weyl nodes in the $(k_y, k_z)$ plane. This feature mirrors the topological obstruction to the atomic limit, yielding maximal entanglement ( $\log 2$ ).
Effect of Weak Disorder: Upon introducing weak disorder ( $W > 0$ ), the sharp pinning of eigenvalues at $\xi = 1/2$ is lost. Instead, the $\delta$ -like peak broadens into a finite-width distribution centered around $\xi = 1/2$ . This broadened peak remains distinct from the continuum of the rest of the spectrum, serving as a signature of the surviving topological band structure despite the loss of translational invariance.
Disorder-Driven Transition (WSM to DM): As disorder strength increases toward the critical value $W_c \approx 3.75t$ , the broadened peak in the EDOS gradually loses coherence and "melts" into the background spectrum. This dissolution coincides with the collapse of the quasiparticle pole and the transition to the diffusive metal phase. The Fermi arcs lose their surface localization and merge with the bulk states.
Band-Structure-Driven Transition (WSM to NI): When tuning the mass parameter $m$ to annihilate the Weyl nodes (at finite disorder $W=2.0t$ ), the EDOS peak at $\xi = 1/2$ behaves differently. Rather than broadening into the background, the peak gradually loses spectral weight as the nodes approach and annihilate, eventually vanishing as the system becomes a trivial normal insulator.
Scaling of Rényi Entropies: The disorder-averaged Rényi entropies ( $S_1, S_2, S_3$ ) increase monotonically throughout the WSM phase, peaking deep in the DM phase before decreasing. Crucially, the entropies evolve smoothly across the critical disorder strength $W_c$ without a sharp anomaly. However, the deviation from the clean limit, $\Delta S_i$ , exhibits a quadratic scaling ( $\sim W^2$ ) for weak disorder, consistent with perturbative corrections to the correlation matrix. This power-law scaling breaks down as the system approaches $W_c$ , signaling the increasing importance of higher-order disorder corrections and the approach to the dirty quantum critical point.

Significance
The paper demonstrates that entanglement measures, specifically the structure of the entanglement spectrum and the scaling of Rényi entropies, remain sensitive probes of topological transitions in gapless three-dimensional systems even in the presence of strong disorder.

The authors establish that the EDOS provides a robust diagnostic tool capable of distinguishing between two fundamentally different mechanisms of topological destruction:

Disorder-induced transition: Characterized by the broadening and eventual melting of the $\xi=1/2$ peak into the background.
Band-structure-induced transition (node annihilation): Characterized by the gradual suppression of the spectral weight of the $\xi=1/2$ peak.

Furthermore, the breakdown of the quadratic scaling of entanglement entropy near the critical disorder strength offers a signature of the dirty quantum critical point associated with the transition to a non-Fermi-liquid state. These findings confirm that while the pinning of eigenvalues to $\xi=1/2$ is fragile to disorder, the topological character of the WSM leaves a distinct, broadened imprint on the entanglement spectrum that persists until the quasiparticle description itself collapses.

Technical Summary: Entanglement Signatures of Topological Phase Transitions in a Dirty Weyl Semimetal

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