Soliton-antisoliton pairs in the supersymmetric gapped phase of an interacting Majorana chain

This paper demonstrates that in the supersymmetric gapped phase of an interacting Majorana chain, supersymmetry persists as indicated by a diverging-then-decaying diagnostic, and the lowest excitations consist of soliton-antisoliton pairs that bind emergent localized Majorana modes to form nonlocal Dirac fermions distinguishing even and odd fermion parity.

The Gist

Imagine a long chain of tiny, magical beads. In the world of quantum physics, these beads are called Majorana fermions. They are special because they act like their own antiparticles, and when they interact in a specific way, they create a hidden "super-symmetry" (SUSY). Think of this symmetry like a perfect dance where every move by a particle has a matching, mirrored move by its partner.

This paper investigates what happens to this perfect dance when the chain is pushed into a specific state called the "gapped phase." It's like asking: "If we turn up the volume on the music, does the dance break, or does it just change its steps?"

Here is a breakdown of the paper's findings using everyday analogies:

1. The Perfect Dance vs. The Broken Dance

At a very specific setting (the "tricritical point"), the system is perfectly balanced. The "dance" (supersymmetry) is visible and well-understood.

The researchers wanted to know: What happens if we move slightly away from this perfect balance?

• On one side (The "Ising" side): The moment they step away from the perfect balance, the dance breaks immediately. It's like a tightrope walker who loses balance the instant they step off the center line. The mathematical tools used to detect the symmetry suddenly go wild (diverge), signaling that the symmetry is gone.
• On the other side (The "Gapped" side): Here, the story is different. Even though they moved away from the perfect balance, the dance doesn't stop immediately. Instead, it slowly fades away. The symmetry survives for a while, lingering in the system before eventually disappearing deep inside the "gapped" zone. It's like a spinning top that keeps wobbling and spinning for a long time even after you stop pushing it, before finally falling over.

2. The Two Patterns of the Chain

In this "gapped" zone, the chain of beads settles into one of two possible patterns, like a zipper that can be closed from the top or the bottom.

• Pattern A: The beads pair up in one specific way.
• Pattern B: The beads pair up in the opposite way.

Usually, the chain picks one pattern and sticks to it. However, because the chain is quantum, it can exist in a state where it is both patterns at once, depending on how you look at it. The researchers found that these two patterns are actually distinguished by a property called "fermion parity" (think of it as the chain being either "even" or "odd" in a specific quantum sense).

3. The Excited State: A Traveling Fault Line

When the chain is in its lowest energy state (the ground state), it is uniform—it's all Pattern A or all Pattern B. But what happens when you give it a little energy (an "excitation")?

The researchers discovered that the lowest energy excitation isn't a single bead jumping up. Instead, it looks like a fault line or a kink traveling through the chain.

• Imagine a long carpet that is rolled up one way on the left and the other way on the right. The place where the rolling direction changes is the "fault line."
• In this quantum chain, this fault line is a Soliton-Antisoliton (SA) pair. It's a pair of "defects" that separate a region of Pattern A from a region of Pattern B.
• These defects aren't stuck in one spot; they are fuzzy and can be anywhere along the chain, existing as a superposition (a quantum mix) of all possible locations.

4. The Hidden Ghosts (Emergent Majoranas)

Here is the most magical part. At the exact spot where the pattern changes (the fault line), something new appears.

• When the pairing of the beads switches from Pattern A to Pattern B, two beads get "left behind." They don't fit into the new pattern.
• These two leftover beads become localized Majorana modes. Think of them as "ghosts" that get trapped at the fault line.
• One ghost lives at the start of the fault, and the other lives at the end. Even though they are far apart, they are connected. Together, they form a single, invisible "Dirac fermion" (a standard particle made of two halves).

5. The Key to the Mystery

The paper explains that the difference between the "even" and "odd" states of the chain comes down to this invisible Dirac fermion.

• If the "ghost" pair is empty, the chain is in one state (even parity).
• If the "ghost" pair is occupied, the chain is in the other state (odd parity).

So, the entire quantum nature of the excited state is determined by whether these two trapped ghosts are holding hands or not.

Summary

The paper shows that in a specific quantum chain:

1. Symmetry survives for a while after the perfect balance is broken, unlike on the other side where it breaks instantly.
2. Excitations are not just random jitters; they are organized pairs of defects (solitons) that separate two different ordering patterns.
3. New particles (Majorana modes) appear trapped at these defects, acting as the "switch" that determines the quantum state of the whole system.

The researchers used powerful computer simulations (DMRG) to prove that this picture holds true, even though the defects are fuzzy and moving, and the "ghosts" are hidden deep within the quantum math.

Technical Summary

Technical Summary: Soliton-antisoliton pairs in the supersymmetric gapped phase of an interacting Majorana chain

Problem Statement
While the realization of supersymmetry (SUSY) at the tricritical Ising (TCI) point in one-dimensional interacting Majorana fermion chains is well-established, the behavior of SUSY in the adjacent gapped, symmetry-broken phase remains less understood. Specifically, it is unclear how SUSY manifests away from the critical point and what constitutes the nature of low-energy excitations in this gapped regime. This paper investigates these questions within the O'Brien–Fendley (OF) model, which realizes the TCI point at an interaction strength of order one.

Methodology
The authors employ numerical simulations, primarily using the Density Matrix Renormalization Group (DMRG), to analyze the OF model Hamiltonian:
$$H_0 = \sum_{j=0}^{N-1} it\gamma_j\gamma_{j+1} + g\gamma_{j-2}\gamma_{j-1}\gamma_{j+1}\gamma_{j+2}$$
The study focuses on the Ramond sector (periodic boundary conditions on Majorana fermions). The methodology involves three primary approaches:

1. SUSY Diagnostics: The authors utilize a ratio $R$, defined via matrix elements of the lattice supercharge operator $Q$ between ground and excited states in even and odd fermion parity sectors. This ratio serves as a diagnostic for the survival of SUSY away from the superconformal point.
2. Order Parameter Analysis: A dimerization order parameter is constructed based on two inequivalent pairings of neighboring Majorana fermions into Dirac fermions ($c_j$ and $d_j$). The authors apply uniform and staggered symmetry-breaking fields ($\Delta$ and $\Delta_{SA}$) to lift ground-state degeneracies and probe the system's response.
3. Excitation Characterization: To detect soliton-antisoliton (SA) pairs, which are delocalized in translation-invariant eigenstates, the authors introduce a "pinning field" that energetically favors opposite dimerization patterns in different halves of the chain. This allows for the spatial resolution of domain walls and the analysis of emergent Majorana modes via Green's functions.

Key Contributions and Results

• Persistence of SUSY in the Gapped Phase:
  The study reveals a striking asymmetry in the behavior of the SUSY diagnostic $R$ across the TCI point. On the Ising (critical) side, $R$ diverges immediately upon crossing the transition, signaling spontaneous SUSY breaking. Conversely, on the gapped (ordered) side, $R$ changes continuously with a finite derivative at the TCI point and decays to zero only deep within the gapped phase. This indicates that SUSY survives over a finite region of the ordered phase adjacent to the TCI point.

• Nature of Low-Energy Excitations:
  The first excited states are identified as superpositions of configurations containing a single soliton-antisoliton (SA) pair separating the two distinct dimerization orders. Because translation symmetry forces expectation values to be uniform, the authors use the susceptibility to the SA-pinning field to confirm that the first excited state is dominated by these pairs, whereas the ground state remains uniformly ordered.

• Emergent Localized Majorana Modes:
  The analysis of Majorana Green's functions ($G_{n,m} = i\langle \gamma_n \gamma_m \rangle$) reveals that the difference between even and odd fermion-parity sectors is localized near the soliton and antisoliton positions. This supports the theoretical picture that each soliton and antisoliton binds an emergent localized Majorana mode ($\Gamma_1$ and $\Gamma_2$). These two modes form a nonlocal Dirac fermion.

• Fermion Parity and Occupation:
  The paper establishes that the distinction between excited states with even and odd total fermion parity arises solely from the occupation of the nonlocal Dirac fermion formed by the bound Majorana modes. In the absence of the pinning field, the two degenerate ordered ground states are distinguished by their global fermion parity, which protects them from mixing.

Significance
The authors claim that these results provide numerical evidence that important signatures of emergent SUSY persist beyond the critical point, organizing the low-energy physics throughout a finite region of the ordered phase. The work offers a microscopic picture of the excitation spectrum in the supersymmetric gapped phase, demonstrating that the excitation energy arises from SA pairs and that the associated fermion parity is determined by emergent degrees of freedom bound to these defects rather than the bulk ordering itself. This clarifies the fate of SUSY in the gapped phase and characterizes the nature of its excitations as soliton-bound Majorana modes forming nonlocal Dirac fermions.