🔬 condensed matter

Entanglement spectrum of gapless topological phases: a case study with topological superconductors

This paper demonstrates that the subsystem correlation spectrum effectively characterizes the topology of gapless topological superconductors in both 1D and 2D by capturing signatures of both edge and bulk modes, thereby generalizing the Li-Haldane entanglement spectrum framework to systems with gapless bulk excitations.

Original authors: Archi Banerjee, Meng Zeng

Published 2026-07-08

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CC BY 4.0

Original authors: Archi Banerjee, Meng Zeng

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 you have a giant, complex machine made of tiny, invisible gears (quantum particles). Sometimes, this machine is perfectly smooth and silent (gapped). Other times, it has a few loose gears rattling around inside, making it "gapless" or noisy. Physicists want to know: Is this machine built with a special, hidden blueprint that makes it "topological" (robust and unique), even when it's noisy?

Traditionally, to check the blueprint, scientists would look at the edges of the machine. If the machine is special, it usually hums with a specific song at its edges. But what if the machine is so noisy inside that you can't hear the edge song clearly?

This paper introduces a clever new way to listen to the machine's "internal echo" to figure out its blueprint, even when it's noisy.

The "Shadow" vs. The "Machine"

Think of the whole machine as a Shadow Puppet Theater.

The Machine: The actual quantum system (the superconductor).
The Shadow: The "Entanglement Spectrum." This is a mathematical shadow cast by the machine. It doesn't show you the gears directly, but it reveals the shape of the connections between them.

In the past, scientists knew that if the machine was smooth (gapped), the shadow would show a perfect, clear silhouette of the edge songs. This paper asks: What does the shadow look like when the machine is noisy (gapless)?

The Experiment: Two Types of Noisy Machines

The authors built two digital models of these "noisy" machines (Topological Superconductors) to test their theory:

1. The "Steady Stream" Machine ( $d_{x^2-y^2}$ pairing)

The Setup: Imagine a river flowing smoothly down a channel. Inside the river, there are some whirlpools (bulk nodes), but the water at the very edges flows in a special, one-way current (chiral edge modes).
The Shadow Result: When they looked at the shadow (the correlation spectrum), they saw a clear, bright line representing that special edge current. Even though the river inside was swirling, the shadow clearly showed the edge was still there and unique.
The Metaphor: It's like looking at a noisy crowd in a stadium. Even though the crowd is loud, if you look at the shadow on the wall, you can still clearly see the distinct shape of the person standing on the stage.

2. The "Merged Stream" Machine ( $d_{xy}$ pairing)

The Setup: Imagine a river where the special edge current crashes directly into the whirlpools in the middle. The edge and the noise mix together.
The Shadow Result: Here, the shadow was trickier. The special edge shape disappeared because it had merged with the noise. However, the shadow still showed a different kind of pattern: it showed that the "noise" itself had a special structure.
The Metaphor: It's like mixing red and blue paint. You can't see the red line anymore, but the resulting purple color tells you that red was there, even if it's now blended in.

The "Magic Ruler" (Trace Index)

The authors also used a special "Magic Ruler" called the Trace Index.

Think of this as counting how many particles are "hanging out" in a specific section of the machine.
In the first machine, the ruler showed a sudden, sharp jump (a "half-step") right where the special edge current was. This was a clear signal: "Yes, a special edge mode exists here!"
In the second machine, the ruler didn't jump at the edge because the edge had merged with the noise. But the ruler still helped confirm the machine's overall topological nature.

The Big Discovery

The paper proves that even when a quantum machine is "noisy" inside (gapless), its shadow (the correlation spectrum) still holds the secrets of its blueprint.

For smooth machines: The shadow shows the edge songs perfectly.
For noisy machines: The shadow shows both the edge songs and the internal noise, but they look different. The edge songs stand out clearly in some cases, while in others, the noise itself reveals the topological nature.

What They Didn't Say

The authors are careful to note that this works perfectly for their specific "free fermion" models (machines where the gears don't push or pull on each other). They explicitly state they do not know yet if this "shadow" trick works for machines where the gears interact strongly (interacting systems). They leave that mystery for future explorers.

In short: You don't need to silence the noise to find the blueprint. If you know how to read the shadow, you can still tell if the machine is special, even when it's making a racket.

Problem Statement
Quantum entanglement has become a central tool for characterizing phases of matter, particularly through the entanglement spectrum (ES). While the correspondence between the low-energy ES and topological edge modes is well-established for bulk-gapped topological systems (e.g., fractional quantum Hall states and symmetry-protected topological phases), its applicability to systems with a gapless bulk remains an open question. In gapless topological phases, robust symmetry-protected edge modes can coexist with a gapless bulk spectrum. Theoretical frameworks relying on "spectral flattening" (as used by Fidkowski for gapped free fermions) are ill-defined in the thermodynamic limit for gapless systems, making it unclear how to extract topological signatures from the ES or the subsystem correlation spectrum (CS). This paper addresses the challenge of characterizing the topology of non-trivial ground states in gapless topological superconductors using the CS.

Methodology
The authors utilize exactly solvable free-fermion models to calculate the subsystem correlation spectrum (CS), defined as the spectrum of the subsystem two-point correlation matrix. For free fermions, the CS contains the same information as the many-body ES. The study focuses on:

1D Critical Majorana Chains: Reproducing known results for the $\alpha$ -chain to analyze the scaling of degeneracy splitting in the CS at critical points compared to gapped phases.
2D Nodal Topological Superconductors (TSCs): Investigating $d$ $d$ -wave nodal TSCs with a 4-by-4 Bogoliubov-de Gennes (BdG) Hamiltonian. Two specific pairing symmetries are examined:
- $d_{x^2-y^2}$ pairing: Features chiral Majorana edge modes separated in momentum space from bulk nodal points.
- $d_{xy}$ pairing: Features bulk nodal points that merge with edge modes at the Fermi level, resulting in a topologically non-trivial phase without protected chiral edge modes.
Geometric Setup: Calculations are performed on a cylindrical geometry with periodic boundary conditions along the $y$ -direction and open (or virtual cut) boundary conditions along the $x$ -direction.
Complementary Metrics: The authors calculate the trace index (average subsystem particle number) to quantify spectral flow and compare results between gapless bulk cases and artificially gapped bulk cases (achieved by adding $C_4$ -breaking terms).

Key Results

1D Critical Scaling: In bulk-gapped 1D chains, the degeneracy splitting of the CS decays exponentially with subsystem size, consistent with the spectral flattening argument. In contrast, for the critical (gapless) chain, the splitting follows a power law ( $\sim 1/L_A^\alpha$ ). This provides numerical evidence that the subsystem correlation matrix in gapless systems is fundamentally distinct from the physical subsystem Hamiltonian, rendering the spectral flattening proof inapplicable.
2D $d_{x^2-y^2}$ TSC: The CS exhibits a clear chiral spectral flow near $k_y=0$ , corresponding to the chiral Majorana edge modes. At the virtual cut, a mode sits exactly at eigenvalue $\xi = 1/2$ . The trace index shows a half-quantized jump ( $1/2$ ) at $k_y=0$ , confirming the Majorana nature of the edge mode. Crucially, the CS also reveals signatures of the bulk gapless nodes near $k_y \approx \pi/2$ . While high-energy bulk modes remain near $\xi=0$ or $1$, low-energy bulk modes near the nodes deviate from these limits and approach $\xi=1/2$ , contributing significantly to the entanglement entropy.
2D $d_{xy}$ TSC: In this case, the edge modes merge with the bulk gapless states at $k_y=0$ . Consequently, the CS does not show a robust $1/2$ -mode at $k_y=0$ , nor does the trace index exhibit a half-quantized jump. The in-gap eigenvalues show oscillatory behavior dependent on the cut position, attributed to the oscillatory nature of correlations in gapless liquids rather than topological edge modes.
Bulk Gapping: When $C_4$ -breaking terms are added to gap out the bulk nodes, the CS in both pairing cases predominantly displays the edge contribution, with bulk modes pushed cleanly to $\xi=0$ or $1$, recovering the behavior typical of gapped topological systems.

Significance and Claims
The paper claims to generalize the Li-Haldane characterization of topological edge states via the ES to 2D topological systems with a gapless bulk. The primary contribution is demonstrating that the CS effectively distinguishes between topological edge modes and bulk gapless modes, even when both appear in the low-energy spectrum.

The CS acts as a "magnifier," separating the robust, cut-independent signatures of topological edge modes (e.g., the $1/2$ mode and half-quantized trace index jump) from the cut-dependent or power-law scaling behaviors of bulk gapless modes.
The work establishes that while spectral flattening fails in the thermodynamic limit for gapless systems, the CS still retains sufficient information to identify non-trivial topology and the nature of edge states.
The authors explicitly note that the relation between the CS and physical edge modes in the presence of interactions remains an open question, as the CS is defined solely by two-point correlations, whereas interactions may require higher-point correlations to fully capture topological information.

The "Shadow" vs. The "Machine"

The Experiment: Two Types of Noisy Machines

The "Magic Ruler" (Trace Index)

The Big Discovery

What They Didn't Say

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