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arXiv:2609.05336

The Junction That Remembers: Infrared Response of Two Coupled SYK Majorana Systems

Ali Vahedi

Published Sep 4, 2026• 0 citations
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Abstract

Two strongly interacting Majorana systems of the Sachdev--Ye--Kitaev type, each with its own independent disorder, are joined by a link that is switched on in time. We ask how the infrared structure of the link is imprinted on the response and whether the system remembers how it was opened. At large $N$ the leading cross-correlation between the two systems is an exact product of two equilibrium propagators and the switching profile; because the disorder is independent, no ladder dresses it. The link operator has scaling dimension one half, which gives its susceptibility a universal logarithm whose coefficient and phase we fix and verify numerically, while the gap of the coupled equilibrium problem grows more slowly than the naive crossover estimate. Exact dynamics for up to twelve Majoranas per side show strong parity transfer and entanglement several times the free-fermion ceiling. The work done by the switch has a first-order part that does not depend on the switching time and a second-order part that grows in magnitude with it. Once the work is held fixed, the protocol dependence of the entanglement, the link expectation value and a scrambling diagnostic nearly vanishes, and the small residuals do not depend on whether the protocols differ at low or at high frequency. These small systems have no infrared window, so they neither confirm nor exclude memory at large $N$.

DOI
10.48550/arxiv.2609.05336

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Two strongly interacting Majorana systems of the Sachdev--Ye--Kitaev type, each with its own independent disorder, are joined by a link that is switched on in time. We ask how the infrared structure of the link is imprinted on the response and whether the system remembers how it was opened. At large NNN the leading cross-correlation between the two systems is an exact product of two equilibrium propagators and the switching profile; because the disorder is independent, no ladder dresses it. The link operator has scaling dimension one half, which gives its susceptibility a universal logarithm whose coefficient and phase we fix and verify numerically, while the gap of the coupled equilibrium problem grows more slowly than the naive crossover estimate. Exact dynamics for up to twelve Majoranas per side show strong parity transfer and entanglement several times the free-fermion ceiling. The work done by the switch has a first-order part that does not depend on the switching time and a second-order part that grows in magnitude with it. Once the work is held fixed, the protocol dependence of the entanglement, the link expectation value and a scrambling diagnostic nearly vanishes, and the small residuals do not depend on whether the protocols differ at low or at high frequency. These small systems have no infrared window, so they neither confirm nor exclude memory at large NNN.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of fermions interacting with qqq-body random couplings. For q=2q=2q=2, it describes free particles and is nonchaotic in the many-body sense, while for q>2q>2q>2 it is strongly interacting and exhibits many-body chaos. In this work we study the entanglement entropy (EE) of the SYKq\mathrm{SYK}qSYKq models for a bipartition of NNN real or complex fermions into subsystems containing 2m2m2m real/mmm complex fermions and $N\ensuremath{-}2m$/$N\ensuremath{-}m$ fermions in the remainder. For the free model SYK2\mathrm{SYK}2SYK2, we obtain an analytic expression for the EE, derived from the $\ensuremath{\beta}$-Jacobi random matrix ensemble. Furthermore, we use the replica trick and path-integral formalism to show that the EE is maximal when one subsystem is small, i.e., $m\ensuremath{\ll}N$, for arbitrary qqq. We also demonstrate that the EE for the SYK4 model is noticeably smaller than the Page value when the two subsystems are comparable in size, i.e., m/Nm/Nm/N is O(1)O(1)O(1). Finally, we explore the EE for a model with both SYK2 and SYK4 interactions and find a crossover from SYK2 (low-temperature) to SYK4 (high-temperature) behavior as we vary energy.
We study the Lindbladian dynamics of the Sachdev-Ye-Kitaev (SYK) model, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Here, the linear jump operators are nonrandom, while the quadratic jump operators are sampled from a Gaussian distribution. In the limit of large NNN, where NNN is the number of Majorana fermion operators, and also in the limit of large NNN and MMM, where MMM is the number of jump operators, the SYK Lindbladians are analytically tractable, and we obtain their stationary Green's functions, from which we can read off the decay rate. For finite NNN, we also study the distribution of the eigenvalues of the SYK Lindbladians.
The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated metal with all-to-all random interactions (average strength JJJ) between NNN fermions (complex Dirac fermions or real Majorana fermions). In the large-NNN limit a conformal symmetry emerges that renders the model exactly soluble. Here we study how the non-Fermi-liquid behavior of the closed system in equilibrium manifests itself in an open system out of equilibrium. We calculate the current-voltage characteristic of a quantum dot, described by the complex-valued SYK model, coupled to a voltage source via a single-channel metallic lead (coupling strength $\mathrm{\ensuremath{\Gamma}}$). A one-parameter scaling law appears in the large-NNN conformal regime, where the differential conductance G=dI/dVG=dI/dVG=dI/dV depends on the applied voltage only through the dimensionless combination $\ensuremath{\xi}=eVJ/{\mathrm{\ensuremath{\Gamma}}}^{2}$. Low and high voltages are related by the duality $G(\ensuremath{\xi})=G(\ensuremath{\pi}/\ensuremath{\xi})$. This provides for an unambiguous signature of the conformal symmetry in tunneling spectroscopy.
In chaotic quantum systems, scrambling and dissipation are governed by distinct timescales. We show that their relative ordering in the Sachdev-Ye-Kitaev model depends on the number of degrees of freedom NNN and undergoes a crossover in the large-NNN limit. In the conformal limit of SYKq\mathrm{SYK}_qSYKq​, the two-point function half-life tdiss∝βt_{\rm diss} \propto \betatdiss​∝β is NNN-independent, while the scrambling time grows as log⁡N\log NlogN. Equating them yields a crossover scale N∗(q)=exp⁡(2 acosh(2q/2))N^*(q) = \exp\left(2\,\mathrm{acosh}(2^{q/2})\right)N∗(q)=exp(2acosh(2q/2)), which grows exponentially with qqq: N∗≈62N^*\approx 62N∗≈62 for q=4q=4q=4, ≈254\approx 254≈254 for q=6q=6q=6, and ≈1022\approx 1022≈1022 for q=8q=8q=8. Below N∗N^*N∗, the OTOC decays before the two-point function; above N∗N^*N∗, the ordering reverses. Using exact diagonalization at N=8N=8N=8-222222, we confirm three qualitative predictions of this formula: (i) the ratio r=tdiss/tOTOCr = t_{\rm diss}/t_{\rm OTOC}r=tdiss​/tOTOC​ decreases with NNN for all qqq; (ii) at fixed NNN, rrr increases with qqq, confirming that N∗N^*N∗ grows with qqq; (iii) the ordering is robust across temperatures βJ=2\beta J = 2βJ=2-202020, consistent with the β\betaβ-independence of N∗N^*N∗ in the conformal limit. The crossover separates a few-body regime, where multi-point correlations are more fragile, from a many-body regime, where single-particle coherence is lost first.
A Majorana field cannot be split into two spatial pieces, but a propagating Majorana can be truncated by a time-dependent shutter. We study this truncation for the two chiral components of a massless Majorana field in one spatial dimension, modeling the shutter as a smooth time-dependent SO(2)SO(2)SO(2) rotation of the two components. Although orthogonal at the level of the real field algebra, this rotation does not preserve the positive- and negative-frequency splitting that defines the Fock vacuum, so the in-vacuum becomes a fermionic squeezed state in outgoing modes. At any finite infrared regulator, a single incident Majorana is then a single outgoing quasiparticle on top of this squeezed vacuum. The squeezed background carries a universal soft response: its anomalous Bogoliubov kernel falls as the inverse sum of the two frequencies at low energy, so the anomalous Hilbert--Schmidt norm grows logarithmically with the infrared cutoff, with a coefficient fixed by the asymptotic shutter mismatch. The vacuum overlap exhibits a Fermi-edge type orthogonality exponent which, for the rotation-angle lift ΔθΔθΔθ with ∣Δθ∣<π|Δθ|<π∣Δθ∣<π, is exactly (Δθ/2π)2(Δθ/2π)^2(Δθ/2π)2, in analogy with the Fermi-edge singularity and the Kondo and Luttinger-liquid problems. We prove local equivalence non-perturbatively for the squeezed-vacuum factor under a compact-support hypothesis, and give an exact local pullback identity for the full one-quasiparticle state on the even CAR algebra; combined with wavepacket-localization assumptions, these yield the expected local vacuum and one-Majorana descriptions. We verify profile independence using a smooth switching family and contrast the temporal benchmark with a number-conserving beam-splitter benchmark whose entropy is bounded by log⁡2\log 2log2.