⚛️ high-energy theory

Quasiparticle tunnelling in two coupled chiral SYK model

This paper demonstrates that weakly coupling two chiral SYK models in 1+1 dimensions via a relevant interaction preserves the gapless nature and entropy of the individual systems, leading to massless collective modes and quasiparticle tunnelling without a mass gap or thermal phase transition, thereby highlighting a sharp qualitative distinction from the behavior of coupled SYK models in 0+1 dimensions.

Original authors: Avik Chakraborty, Manavendra Mahato

Published 2026-06-26✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Avik Chakraborty, Manavendra Mahato

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of quantum physics as a giant, chaotic dance floor. For a long time, physicists have been studying a specific type of dance called the SYK model. Think of this as a dance where thousands of partners (particles) are randomly paired up to spin and interact. This dance is famous because it's mathematically solvable, meaning we can predict exactly how the partners move, and it behaves strangely like a black hole.

Usually, this dance happens in a single line (one dimension). But in this paper, the researchers asked: What happens if we move this dance to a two-dimensional strip, like a long hallway, and make the dancers "chiral"?

"Chiral" is a fancy word for "handedness." Imagine all the dancers in this hallway are forced to spin and move only to the right. They cannot move left. This creates a one-way street for quantum energy.

The Experiment: Connecting Two Hallways

The researchers took two of these "right-moving only" hallways (let's call them Hallway A and Hallway B) and built a small, weak bridge between them. This bridge allows a dancer from Hallway A to occasionally step over into Hallway B and vice versa.

In the world of zero-dimensional physics (the original SYK model), connecting two systems like this usually creates a "gap." Imagine the dance floor suddenly freezing up; the dancers get stuck in a specific pattern, and the energy levels separate into distinct steps, like a ladder. This is often compared to a "traversable wormhole," a tunnel connecting two distant points in space.

The Big Surprise: No Freezing, Just Flowing

The authors of this paper discovered that in their 1+1 dimensional hallway, nothing freezes.

Even though they built the bridge and broke some of the symmetry rules (like time-reversal), the dancers kept moving freely.

No Gap: Unlike the zero-dimensional version, the energy levels did not separate into a ladder. The system remained "gapless," meaning the dancers could move at any speed without hitting a wall.
No Phase Change: Usually, when you connect two systems, you expect a dramatic shift in how they behave (a phase transition). Here, the system stayed smooth and continuous.
The Entropy Stays the Same: The "messiness" or disorder of the system (entropy) didn't change just because they added the bridge. It remained exactly the same as if the two hallways were completely separate.

The New Discovery: Ghostly Waves

While the dancers didn't freeze, something new appeared. The bridge allowed for massless collective waves to travel between the two hallways.

Think of it like this: If you shout in Hallway A, the sound doesn't just stay there or disappear. It creates a ripple that travels perfectly through the bridge into Hallway B, and back again, without losing energy or getting stuck. These ripples are "quasiparticles"—ghostly waves of information that move between the two systems.

Why This Matters (According to the Paper)

The paper highlights a sharp difference between the "single line" version of this physics and the "hallway" version:

In the single line (0+1D): Connecting two systems creates a gap and a "wormhole" effect where things get stuck or locked in.
In the hallway (1+1D): Connecting two systems keeps everything flowing. The "chiral" nature (the one-way street) is so robust that even a bridge between two systems cannot stop the flow or create a gap.

The Bottom Line

The researchers found that these "chiral" quantum systems are incredibly tough. You can try to break their rules or connect them to other systems, but they refuse to develop an energy gap. Instead, they simply develop a new way for information to tunnel between them, behaving like massless waves. This suggests that the "wormhole" intuition we have from simpler models doesn't work the same way in these higher-dimensional, one-way quantum systems. The edge of these systems remains a free-flowing, gapless highway for quantum information.

Technical Summary: Quasiparticle Tunnelling in Two Coupled Chiral SYK Models

Problem Statement
The Sachdev-Ye-Kitaev (SYK) model is a paradigmatic solvable quantum mechanical system (0+1 dimensions) exhibiting maximal chaos, conformal symmetry in the infrared (IR), and a connection to holographic duals of near-extremal black holes. A relevant quadratic deformation in the 0+1-dimensional coupled SYK model (two copies interacting via a bilinear term) is known to open a mass gap, induce a thermal phase transition, and realize physics analogous to traversable wormholes.

This paper investigates the consequences of introducing an identical relevant quadratic interaction between two coupled 1+1-dimensional chiral SYK models. The chiral SYK model is a generalization involving $N$ right-moving chiral Majorana fermions with homogeneous random 4-fermion interactions. Unlike its 0+1-dimensional counterpart, the 1+1-dimensional chiral SYK model possesses exact scaling symmetry and exhibits gapless, topological edge-like dynamics with quantized thermal Hall conductance. The central question is whether the relevant quadratic coupling, which breaks scaling and time-reversal symmetry, induces a mass gap and a phase transition in the 1+1-dimensional chiral system, or if the gapless nature of the chiral edge modes is robust against such deformations.

Methodology
The authors analyze the system in the large- $N$ limit using the disorder-averaged path integral formalism.

Model Construction: Two chiral SYK systems ( $A$ and $B$ ) are coupled via a Hamiltonian $H_I = i\mu \sum_j \psi^j_A \psi^j_B$ , where $\mu$ is a dimensionful coupling constant. This term breaks time-reversal and scaling symmetries.
Dyson-Schwinger (DS) Equations: The authors derive the large- $N$ DS equations for the two-point functions (Green's functions) $G_{ab}$ and self-energies $\Sigma_{ab}$ (where $a,b \in \{A, B\}$ ).
Perturbative Solution: Working in the regime of weak intersystem coupling ( $\mu^2 \ll J$ ), the authors solve the DS equations perturbatively. The leading correction to the inter-system correlator $G_{AB}$ is computed analytically at finite temperature.
Regularization: A consistent point-splitting regularization scheme (splitting in the spatial $x$ direction) is employed to handle UV divergences in real-space integrals and to define the two-point functions rigorously. This is crucial for computing physical quantities like free energy.
Analytic Continuation: The Euclidean correlators are analytically continued to real time to compute retarded correlators and spectral densities.

Key Contributions and Results

Analytic Solution for Coupled Correlators: The authors obtain an exact analytic expression for the leading-order correction to the inter-system two-point function $G_{AB}(\tau, x)$ at finite temperature. The solution is expressed in terms of complete elliptic integrals of the first kind, $K(c)$ , where the arguments depend on the temperature $\beta^{-1}$ and the modified velocities $u_\pm = 1 \pm J/2\pi$ .
Absence of Mass Gap: Unlike the 0+1-dimensional coupled SYK model, the 1+1-dimensional coupled system does not develop a mass gap. The two-point functions do not exhibit exponential decay at large Euclidean times. This indicates that the system remains in a gapless phase at all temperatures.
Thermodynamics: The leading correction to the thermodynamic free energy density is calculated. Crucially, this correction is temperature-independent. Consequently, the entropy density of the coupled system remains identical to that of two uncoupled chiral SYK systems. There is no enhancement of ground-state degeneracy or thermal phase transition.
Quasiparticle Tunnelling and Collective Modes:
- Real-time analysis of the retarded correlator at zero temperature reveals a logarithmic behavior, characteristic of massless free scalar fields in 1+1 dimensions.
- The spectral function confirms the existence of massless collective bosonic modes propagating between the two subsystems with velocities $u_\pm$ .
- These modes are identified as quasiparticle tunnelling events between the two chiral systems. The tunnelling occurs without generating a gap.
Symmetry Preservation: The solution respects the $Z_4$ symmetry of the Lagrangian, which constrains the correlators to be antisymmetric around $\tau = \beta/2$ . The symmetry remains unbroken in the effective theory of bilinears to leading order in $\mu$ .

Significance and Claims
The paper claims a sharp qualitative distinction between relevant deformations of SYK models in 0+1 and 1+1 spatial dimensions:

0+1 Dimensions: A relevant quadratic deformation leads to a gapped phase, a thermal phase transition, and wormhole-like physics (traversable wormholes).
1+1 Dimensions (Chiral): The same deformation preserves the gapless nature of the system. It gives rise to propagating collective modes (quasiparticle tunnelling) rather than a gapped ground state.

The authors conclude that the intuition derived from traversable wormholes in 0+1 dimensions does not straightforwardly extend to higher-dimensional chiral systems. The results highlight the robustness of gapless chiral edge dynamics against explicit scaling symmetry-breaking interactions. This supports the view that interacting chiral fermions of this type can exist as stable edge modes of 2+1-dimensional gapped topological phases, where the gaplessness is protected topologically as long as the bulk gap does not close. The work underscores that the topological nature of the chiral SYK model, evidenced by its quantized thermal Hall conductance, persists even in the presence of the symmetry-breaking coupling.