Entanglement entropy as a probe of topological phase transitions

This paper introduces an exact entanglement entropy-based framework that robustly identifies topological phase transitions in disordered Su-Schrieffer-Heeger models by leveraging the vanishing difference in entropy between half-filled and near-half-filled states to distinguish genuine topological edge states from trivial localized states.

The Gist

Imagine you are trying to figure out if a mysterious, long chain of beads is made of a special "magic" material or just ordinary plastic. You can't see the inside, and the chain might be covered in dust or have some beads missing (disorder). How do you tell the difference?

This paper introduces a clever new way to answer that question using a concept from quantum physics called Entanglement Entropy. Think of this not as a physical tool, but as a way of measuring how "connected" different parts of the chain are to each other.

Here is the breakdown of their discovery using simple analogies:

1. The Setup: The SSH Chain

The scientists are studying a specific type of chain called the Su-Schrieffer-Heeger (SSH) model. Imagine a necklace where the beads come in pairs (A and B).

• The "Magic" (Topological) Phase: The beads are linked tightly between the pairs, but loosely within the pairs. This creates a special property: if you cut the necklace, two loose, "ghostly" beads appear at the very ends. These are edge states. They are protected by the chain's structure and won't disappear easily.
• The "Ordinary" (Trivial) Phase: The beads are linked tightly within the pairs, but loosely between them. If you cut this chain, nothing special happens at the ends.

2. The Problem: Disorder

In the real world, things aren't perfect. The chain might have random dust (random disorder) or a weird, repeating pattern of dust (quasiperiodic disorder).

• Old Tools Fail: Traditional ways of identifying the "magic" phase rely on looking at the chain's perfect, repeating pattern. If the chain is messy or disordered, these old tools break down and give confusing answers.
• The New Tool: The authors propose using Entanglement Entropy (EE). Think of EE as a measure of how much information one part of the chain shares with the rest.

3. The Magic Trick: The "Guest" Particle

The researchers developed a specific test:

1. Take the chain at a "half-full" state (every seat on the bus is taken).
2. Add one extra guest (one extra particle) to the chain.
3. Measure the "connectedness" (Entanglement) of a small section in the middle of the chain (the bulk).

The Result:

• In the Magic Phase (Topological): The extra guest doesn't want to sit in the middle. Because of the special "ghostly" edge states, the guest immediately runs to the very ends of the chain and hides there. Since the guest is far away from the middle section, the middle section doesn't notice the guest arrived. The "connectedness" measurement stays exactly the same (zero difference).
• In the Ordinary Phase (Trivial): The guest has nowhere special to hide. They wander around and sit in the middle of the chain. The middle section notices the guest. The "connectedness" measurement changes (a non-zero difference).

The Analogy: Imagine a crowded theater.

• Topological: If you add one more person, they immediately run to the exit doors (the edges) and stand there. The people sitting in the middle rows don't even know a new person arrived.
• Trivial: The new person just sits in an empty seat in the middle row. The people around them notice immediately.

4. The Twist: Fake Ghosts (Domain Walls)

The authors realized there's a trick. Sometimes, if you join two different chains together, you can create a "fake ghost" in the middle (a domain wall) that looks like it's hiding at an edge. This could fool the test.

The Solution: The "Shrink" Test
To catch the fakes, they proposed a second step: Shrink the middle section.

• Real Magic: The real edge states are so robust that even if you shrink the middle section, the "ghost" stays at the very end, and the middle section still doesn't notice. The measurement stays zero.
• Fake Ghosts: The "fake" ghosts are fragile. If you shrink the middle section, the fake ghost gets caught in the middle, and the measurement suddenly changes.

5. Why This Matters

• Robustness: This method works perfectly even when the chain is messy, dusty, or disordered.
• Better than the Old Way: The paper shows that their new method is actually better than the standard "Topological Quantum Number" (Q) used by physicists, which often gets confused and gives wrong answers when disorder is present.
• Bridging Worlds: It connects two big fields of physics: Quantum Information (how things are connected) and Condensed Matter (how materials behave). It suggests that the way particles are "entangled" can tell us deep secrets about the material's shape and structure.

Summary

The paper says: "Don't just look at the pattern of the chain. Instead, add a tiny guest and see if they hide at the edges. If they do, and they stay hidden even when you poke the chain, you have found a topological phase. This works even if the chain is messy!"

This is a powerful new diagnostic tool that helps scientists identify exotic materials in the real, messy world, not just in perfect theoretical models.

Technical Summary

1. Problem Statement

Topological phase transitions (TPTs) are traditionally identified using momentum-space invariants (e.g., winding numbers, $\mathbb{Z}_2$ invariants). However, these methods rely on translational symmetry and fail in disordered systems where momentum is not a good quantum number. While real-space alternatives exist (e.g., real-space winding numbers, scattering coefficients), the role of Entanglement Entropy (EE) in identifying TPTs in disordered systems remains underexplored. Specifically, there is a lack of a systematic framework to distinguish genuine topological edge states from trivial localized states (which can also appear at edges due to disorder or domain walls) using many-body EE.

2. Methodology

The authors propose a comprehensive framework combining analytical and numerical approaches to study the Su-Schrieffer-Heeger (SSH) model and its disordered variants.

• Models Studied:

  1. Clean SSH Model: A 1D dimerized chain with intracell ($t_1$) and intercell ($t_2$) hopping.
  2. Quasiperiodic SSH: Hopping terms modulated by a quasiperiodic potential (Aubry-André type).
  3. Binary Disordered SSH: Hopping terms subject to random binary disorder.
     All models preserve chiral symmetry.

• Analytical Approach (Transfer Matrix & Lyapunov Exponent):

  • The authors derive exact phase boundaries by analyzing the localization properties of zero-energy edge modes.
  • They employ the Transfer Matrix Method to calculate the Lyapunov Exponent (LE), $\gamma_0$, for edge states.
  • Criterion: $\gamma_0 > 0$ indicates localized edge states (Topological Phase), while $\gamma_0 \to 0$ indicates delocalization (Trivial Phase). This provides exact analytical expressions for phase boundaries in all three models.

• Numerical Approach (Entanglement Entropy Diagnostic):

  • They define a specific diagnostic quantity: $\Delta S_A = |S_A^{\text{hf}} - S_A^{\text{hf}+1}|$, where $S_A$ is the entanglement entropy of a subsystem $A$ (located deep in the bulk), calculated for the ground state at half-filling ($N/2$ particles) and half-filling plus one particle ($N/2 + 1$).
  • Logic: In the topological phase, the extra particle occupies a zero-energy edge state localized at the boundary. Since subsystem $A$ is in the bulk, the extra particle does not contribute to $S_A$, making $\Delta S_A \approx 0$. In the trivial phase, the extra particle occupies bulk states, contributing to $S_A$, resulting in $\Delta S_A > 0$.
  • Domain Wall Analysis: To distinguish topological states from accidental localized states, they analyze domain-wall configurations (e.g., joining two SSH chains) and systematically tune the size of subsystem $A$.

• Comparison Metric:

  • They compare their EE-based diagnostic against the Topological Quantum Number ($Q$), a scattering-based invariant defined via the reflection matrix, to test robustness in disordered regimes.

3. Key Contributions & Results

A. Exact Phase Boundaries

Using the transfer matrix method, the authors derived exact analytical phase boundaries for:

• Clean SSH: $|t_1| = |t_2|$ (i.e., $\lambda = 0$).
• Quasiperiodic SSH: $\delta = t + \lambda$ (where $\delta$ is disorder strength).
• Binary Disordered SSH: A generalized expression involving the disorder strength $W$, dimerization $\lambda$, and probability $P$.
  These analytical boundaries were found to match numerical results perfectly.

B. The $\Delta S_A$ Diagnostic

• Robustness: The quantity $\Delta S_A$ successfully distinguishes topological ($\Delta S_A \approx 0$) from trivial ($\Delta S_A > 0$) phases in clean, quasiperiodic, and binary disordered systems.
• Physical Origin: Analysis of site occupation numbers ($\Delta \Omega$) confirms that in the topological phase, the added particle is strictly localized at the edges, leaving the bulk subsystem $A$ unaffected.

C. Distinguishing Topological vs. Trivial Localization

A critical finding is that $\Delta S_A \approx 0$ can occur for non-topological localized states (e.g., at domain walls between two trivial chains or in the Aubry-André-Harper model).

• Solution: The authors propose Subsystem Tuning.
  • If a state is genuinely topological (zero-energy, chiral symmetry protected), $\Delta S_A$ remains zero even when the subsystem $A$ is shrunk to partially overlap the domain wall.
  • If the state is a trivial finite-energy localized state, shrinking the subsystem causes the added particle to occupy both $A$ and $B$, causing $\Delta S_A$ to become finite.
• This establishes a protocol to filter out "accidental" edge states.

D. Superiority over Topological Invariant $Q$

• In the quasiperiodic case, the topological invariant $Q$ exhibits ambiguity. In the trivial regime, $Q$ fluctuates between 0 and 1 (averaging to ~0.5) due to sign changes in the underlying scattering parameter, leading to false positives in single disorder realizations.
• In contrast, $\Delta S_A$ provides a sharp, unambiguous distinction between phases even for a single disorder realization, making it a more reliable diagnostic for disordered systems.

4. Significance

• Disorder Robustness: The work demonstrates that entanglement entropy is a robust probe for topological phases in systems lacking translational symmetry, where traditional momentum-space invariants fail.
• Bridge Between Fields: It unifies concepts from quantum information (entanglement) and condensed matter physics (topological order), offering a practical tool for identifying topological matter in experimental setups (e.g., cold atoms, photonic systems) where disorder is inevitable.
• Diagnostic Protocol: The proposed "Subsystem Tuning" method provides a rigorous way to differentiate between topological edge modes and trivial localized states, addressing a long-standing challenge in the field.
• Analytical-Numerical Consistency: The paper validates the EE-based numerical approach against exact analytical results derived from Lyapunov exponents, confirming the reliability of the method.

Conclusion

The paper establishes $\Delta S_A$ as a powerful, exact, and disorder-robust diagnostic for topological phase transitions in 1D systems. By combining entanglement entropy analysis with systematic subsystem tuning, the authors provide a complete framework to identify topological phases and distinguish them from trivial localized states, outperforming traditional topological invariants in disordered regimes.