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Quasiparticle tunnelling in two coupled chiral SYK model

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Two weakly coupled chiral SYK systems do not develop a mass gap; instead, massless collective bosonic modes tunnel between them.

desk verdict A careful leading-order perturbative calculation of two coupled chiral SYK models, but the no-gap claim is not established beyond first order in the tunneling strength. read the letter →

arxiv 2507.18298 v3 pith:MBZMRPWJ submitted 2025-07-24 hep-th cond-mat.str-el

Avik Chakraborty , Manavendra Mahato This is my paper · ORCID
classification hep-thcond-mat.str-el
keywords chiralSYKmodelcoupledmodelslarge-NlimitDyson-Schwingerequationsgaplessspectrummasslessbosonicmodesquasiparticletunnellingtopologicaledgetheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies two copies of the 1+1-dimensional chiral SYK model coupled by a relevant bilinear term $\mu \sum_i \psi_A^i \psi_B^i$ that breaks scaling and time-reversal symmetry. Working at large $N$ and weak inter-system coupling, the authors solve the Dyson-Schwinger equations perturbatively and find that, in contrast to the 0+1-dimensional coupled SYK model, the coupled chiral system remains gapless at all temperatures. The inter-system correlator $G_{AB}(\tau,x)$ shows no exponential decay in Euclidean time, and the leading correction to the free energy is temperature independent, so the entropy density is unchanged. Analytic continuation of the retarded correlator reveals massless collective bosonic modes propagating between the two subsystems at zero temperature, with spectral peaks at $\omega = -u_{\pm} k$. This matters because it suggests that gapless chiral edge dynamics survive a relevant deformation, unlike the gapped wormhole-like phase of coupled SYK quantum mechanics.

What carries the argument

The load-bearing object is the finite-temperature chiral SYK two-point function $G^\beta(\tau,x) = (2\beta\sqrt{u_+u_-})^{-1}[\sin(\pi(\tau - i u_+^{-1}x)/\beta)\sin(\pi(\tau - i u_-^{-1}x)/\beta)]^{-1/2}$, taken as the exact unperturbed propagator for each subsystem. Convolving two copies of it through the Dyson-Schwinger equation gives $G^\beta_{AB}$ in Eq. (3.21) as a combination of complete elliptic integrals of the first kind, defined by $K(c)=\int_0^1 ds\,[(1-s^2)(1-c^2s^2)]^{-1/2}$. The elliptic-integral form is what makes the no-gap conclusion visible: its analytic structure has branch points but no exponential decay, and its analytic continuation produces the zero-temperature retarded correlator $G_{AB,\mathrm{ret}}(t,x)\propto \theta(t)\log|(t-u_+^{-1}x)/(t-u_-^{-1}x)|$. Point splitting in the $x$ direction supplies the UV regularisation that makes the convolution and the free-energy calculation finite.

What would settle it

Numerically solve the coupled large-$N$ Dyson-Schwinger equations at finite $\mu$ and inspect the inter-system correlator $G_{AB}(\tau,x)$ for large $\tau$: exponential decay would indicate a mass gap and falsify the no-gap claim. A second check is to compute the full temperature dependence of the free-energy correction beyond leading order in $\mu$; any $\beta$-dependent piece at order $\mu^2$ would contradict the temperature-independence asserted here.

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Extended reading notes

Core claim

The paper's central claim is that a relevant quadratic coupling between two chiral SYK systems does not open a mass gap and does not drive a thermal phase transition. To leading order in the inter-system coupling $\mu$, the Euclidean inter-system two-point function $G^\beta_{AB}(\tau,x)$ is built from the exact chiral SYK propagator and is expressed through complete elliptic integrals of the first kind; it falls off without exponential decay in $\tau$, which is the signature of a gapless spectrum. The leading free-energy correction is $\Delta F/L = -(\mu^2 N / 2J)\log(u_+/u_-)$, independent of temperature, so the entropy density remains that of $2N$ free chiral Majorana fermions. At zero temperature, the spectral function $\rho_{AB}(\omega,k)$ contains delta-function peaks at $\omega = -u_\pm k$ for $k<0$, signalling massless collective bosonic modes tunnelling between the subsystems. The result is framed as a sharp difference from the coupled SYK model in $0+1$ dimensions, where the same kind of deformation produces a gapped phase.

Load-bearing premise

The whole result depends on the finite-temperature two-point function of a single chiral SYK system, Eq. (2.33), being the exact large-$N$ propagator for all Euclidean times and momenta, and on the point-splitting regularisation used to convolve it giving the physical ultraviolet behaviour.

Editorial extensions

If this is right

  • At leading order in $\mu$, the coupled system has no thermal phase transition; the entropy density equals that of two decoupled chiral SYK systems.
  • The ground-state energy is lowered by the inter-system coupling, $\Delta F/L = -\mu^2N/(2J)\log(u_+/u_-)$, even though the spectrum remains gapless.
  • The zero-temperature spectral function $\rho_{AB}(\omega,k)$ has delta peaks at $\omega=-u_\pm k$, meaning quasiparticles can tunnel between the subsystems as massless collective bosonic modes.
  • The relevant deformation does not produce the Schwarzian/wormhole-like gapped physics familiar from 0+1-dimensional coupled SYK; chirality and spatial dimension change the fate of the deformation.
  • Gapless chiral edge dynamics survive an explicit scaling- and time-reversal-breaking interaction, supporting the interpretation of the chiral SYK system as a stable edge theory of a gapped 2+1-dimensional topological bulk.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A nonperturbative numerical solution of the coupled Dyson-Schwinger equations at $\mu$ of order $J$ would settle whether the gapless phase survives beyond leading order; the paper leaves this as open.
  • Because the zero-temperature retarded correlator is logarithmic, the effective low-energy theory is plausibly two pairs of chiral bosons with a mode-mixing interaction; if so, the coupled model may be exactly solvable by bosonisation for any $\mu$, not only for small $\mu$.
  • The inter-subsystem spectral weight at $\omega = -u_\pm k$ suggests that real-time tunnelling between the two edges is coherent and non-dissipative; an out-of-time-order correlator or conductance calculation could test whether this tunnelling carries information at the maximal chaos rate.
  • The absence of a gapped phase also removes the standard traversable-wormhole interpretation of the coupled-SYK deformation, implying any holographic bulk must be gapless or topological rather than a gapped wormhole geometry.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies two identical chiral SYK models in 1+1 dimensions coupled by the bilinear term i μ ψ_A^j ψ_B^j. Working at large N and weak coupling μ²≪J, the authors solve the Dyson–Schwinger equations to first order in μ and obtain a closed-form expression for the inter-system correlator G_AB in terms of complete elliptic integrals (Sec. 3.3). They compute the leading free-energy correction independently from the effective action and from a direct convolution (Sec. 3.4 and Appendix C), finding a temperature-independent shift, and analytically continue the Euclidean correlator to obtain the retarded function and spectral function (Sec. 3.5), whose zero-temperature delta peaks are interpreted as massless collective modes propagating between the subsystems. The paper's central claim is that, unlike the 0+1-dimensional coupled SYK model, the coupled chiral system remains gapless and exhibits no thermal phase transition.

Significance. The perturbative computation is carefully executed: the elliptic-integral expression for G_AB is cross-checked by two independent free-energy calculations with full agreement, and the computed G_AB respects the Z4/antiperiodicity constraint (Eq. (3.23)). If the leading-order picture extends nonperturbatively, the result is significant because it provides an analytic example of a relevant bilinear deformation preserving gapless chiral edge dynamics, with explicit massless mode velocities u_±, and it sharpens the distinction between SYK models in 0+1 and 1+1 dimensions. The paper also gives falsifiable spectral predictions, Eq. (3.44), that could be tested numerically. However, the significance is conditional: the no-gap conclusion is currently established only to first order in μ, and the input finite-temperature propagator is an ansatz that the convolution depends on.

major comments (2)
  1. [Sec. 3.3 / Abstract] The central claim that the coupled system 'does not develop a mass gap' and shows 'no thermal phase transition' is inferred from G_AB computed at first order in μ, Eqs. (3.20)–(3.21). The Dyson–Schwinger equations (3.15)–(3.18) are nonlinear, and O(μ²) feedback through G_AA→Σ_AA and through the J²G_AB³ term in Σ_AB could produce a nonperturbative gap that is invisible at any finite order in μ. This is not purely hypothetical: in the 0+1-dimensional coupled SYK model the same leading-order convolution of gapless conformal propagators would falsely suggest a gapless phase, whereas the exact solution is gapped. The manuscript itself states in Sec. 4 that 'it is important to perform a nonperturbative numerical analysis' beyond small μ, so the abstract's unconditional no-gap statement exceeds what the calculation establishes. The authors should either soften the claim to leading-order perturbative evidence or supply additional nonperturbative support.
  2. [Sec. 2.2 / Sec. 3.3] The input finite-temperature propagator G_β in Eq. (2.33) is introduced as a 'guess' and is verified only in momentum space with a specific UV regularization. Because the convolution in Eq. (3.20), the resulting G_AB in Eq. (3.21), and the temperature-independent free-energy correction in Eq. (3.33) all use this propagator as the exact unperturbed G_AA, any branch or regularization dependence of Eq. (2.33) propagates directly into the no-gap conclusion. The authors should either demonstrate that Eq. (3.21) is robust to the regularization choices discussed in Sec. 2.2, or state explicitly that the lack of exponential decay is conditional on the validity of the ansatz (2.33).
minor comments (4)
  1. [Sec. 2.2] The text says 'discreet Matsubara frequencies'; this should be 'discrete Matsubara frequencies'.
  2. [Sec. 2.2] There are several missing spaces in phrases such as 'Nfree chiral Majorana fermions' and 'O(N −2) [5]'; these should be corrected.
  3. [Sec. 3.1] The bosonized rewriting in Eq. (3.7) is heuristic for general N: the large-N chiral SYK model is not a free-boson theory, and the identification c_a^j = :e^{iφ_a^j}: is justified in the N=4 case. A clarifying sentence that this is illustrative would avoid overstating the bosonization.
  4. [Sec. 3.4] After Eq. (3.33), 'It remains same as the entropy density' should read 'It remains the same as the entropy density'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coupled-system calculation uses the chiral SYK propagator as an external input and derives G_AB from the Dyson-Schwinger equations; the later spectral and free-energy results follow from that derivation rather than being assumed.

full rationale

The paper's load-bearing input is the large-N finite-temperature chiral SYK propagator G_beta(τ,x) in Eq. (2.33), taken from Ref. [1] (Lian, Sondhi, Yang). This is an external prior result, not fitted here, and it is used as the unperturbed G_AA = G_BB in Eq. (3.19). The central new object, G_AB, is then obtained by the explicit first-order DS convolution in Eq. (3.20): G_AB(τ,x) = iµ ∫ G_AA G_BB. The subsequent expression in Eq. (3.21), the regularized coincident-point value in Eq. (3.30), the free-energy correction in Eq. (3.31), and the spectral function in Eq. (3.44) are all mathematical consequences of that convolution and of the stated analytic continuation of elliptic integrals. No parameter is adjusted to reproduce a target behavior, and the claimed consistency check between Eqs. (3.31) and (3.33) is an internal check of the same leading-order DS solution rather than a circular import. The velocities u± appearing in the massless modes are inherited from the input chiral SYK propagator; that is a derivation from stated inputs, not a renaming of a known result as a new prediction. The paper's own Conclusion explicitly limits the result to small µ and calls for nonperturbative numerical analysis, which is a caveat about the strength of the evidence but not evidence of circularity. Similarly, the concern that a first-order computation cannot see a nonperturbative gap is a correctness-risk objection, not a circularity objection. Overall, the derivation chain is self-contained given the external chiral SYK solution, and there are no fitted inputs or self-citations carrying the argument.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results rest on the finite-temperature chiral SYK propagator (an external input), the large-N melonic saddle, the perturbative expansion in mu, and the choice of point-splitting regularization. Two model parameters (mu, J) enter; neither is fitted to data. No new particles, forces, or conserved quantities are postulated; the massless modes are emergent from the computed correlators, not invented entities.

free parameters (2)
  • mu
    Inter-system bilinear coupling strength; chosen small (mu^2 much less than J) to justify the perturbative Dyson-Schwinger solution. Dimensionful coupling, not fitted to data.
  • J
    Chiral SYK random-coupling amplitude; restricted to 0 <= J < 2*pi for positive velocities. Dimensionless model parameter, not fitted.
assumptions (5)
  • domain assumption Finite-temperature two-point function of the chiral SYK model, Eq. (2.33), is exact in the large-N limit
    Used as the unperturbed propagator G_AA in the coupled Dyson-Schwinger equations (Sec. 3.3, Eq. 3.19). It is a guess reviewed from Ref [1] and verified in momentum space with a specific UV regularization.
  • domain assumption Large-N limit and melonic dominance
    Standard SYK assumption that only melonic diagrams contribute at leading order in 1/N, used in Sec. 3.2 to derive the effective action and Dyson-Schwinger equations.
  • ad hoc to paper Perturbative expansion in mu is valid and higher-order terms do not alter the gapless conclusion
    The no-mass-gap claim relies on the leading-order expression for G_AB; the paper acknowledges in the Conclusion that nonperturbative numerical analysis is needed to confirm the regime of validity and extend beyond small mu.
  • domain assumption Point-splitting regularization in the x direction is the consistent UV scheme
    Used throughout the real-space convolution integrals (Sec. 2.2 and Appendices B, C). Different schemes give different real-space correlators, though momentum-space results match.
  • domain assumption Z4 symmetry and antiperiodic boundary conditions constrain the correlators
    Used to impose antisymmetry of G_AB about tau = beta/2 (Sec. 3.3). The solution is shown to respect this constraint at leading order.

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Pith. "Pith review of Quasiparticle tunnelling in two coupled chiral SYK model." pith.science (2026). https://pith.science/paper/MBZMRPWJ

@misc{pith2026250718298,
  author       = {Pith},
  title        = {Pith review of: Quasiparticle tunnelling in two coupled chiral SYK model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBZMRPWJ}},
  note         = {Machine review of arXiv:2507.18298}
}
read the original abstract

The chiral SYK model is a 1+1 dimensional generalisation of the Sachdev-Ye-Kitaev model with chiral Majorana fermions and homogeneous random interactions. In the large-N limit, the model admits an exact solution of the two-point function due to its scaling symmetry and exhibits a quantised thermal Hall conductance consistent with that of a 2+1-dimensional gapped topological system. We study two chiral SYK systems coupled by a relevant quadratic interaction that explicitly breaks scaling and time-reversal symmetry. Working in the regime of weak intersystem coupling, we solve the Dyson-Schwinger equations perturbatively and obtain analytic expressions for two-point functions at finite temperature. Unlike the coupled SYK model in 0+1 dimensions, the 1+1-dimensional chiral system does not develop a mass gap, and no thermal phase transition is observed. We show that the leading correction to the thermodynamic free energy is temperature independent, implying that the entropy density remains identical to that of two uncoupled chiral SYK systems. A real-time analysis of the retarded correlator reveals the emergence of massless collective bosonic modes propagating between the two subsystems at zero temperature, signalling quasiparticle tunnelling without gap generation. Our results demonstrate a sharp qualitative distinction between relevant deformations of SYK models in zero and one spatial dimensions, and highlight the robustness of gapless chiral edge dynamics against explicit scaling symmetry-breaking interactions.

Figures

Figures reproduced from arXiv: 2507.18298 by the authors.

Figure 1
Figure 1. The spectral density of states of the chiral SYK model for fixed momentum [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. The real and imaginary parts of the two-point functions are shown for parameter values [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. (a) The circular contour |z˜| = 1 deformed to exclude the real ˜z axis where the poles are situated. (b) The square contour C˜ which reduces to the x integral when the length of the sides reach infinity. The two poles shown on the real x axis lie infinitesimally above for Im(t) = ϵ and infinitesimally below for Im(t) = −ϵ. the integrand is analytic inside the contour, the integral along the unit circle can be expres… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The unit circle contour is deformed to avoid the poles and the branch cuts. With no [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: Countably infinite branch cuts of the integrand of [PITH_FULL_IMAGE:figures/full_fig_p029_5.png]
Figure 6
Figure 6. Figure 6: (a) The circular contour |z ′ | = 1 deformed to exclude the poles situated on real ˜z axis. (b) The infinite square contour C ′ reduces to the x integral. The two poles shown on the real x axis lie infinitesimally above for Im(t) = ϵ and infinitesimally below for Im(t)…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.