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arXiv:2606.17161

The Helical SYK Model and Emergent Infrared Integrability

Gustavo Valdivia-Mera, Bhavay Tyagi, Eric R. Bittner, Pavan Hosur

Published Jun 15, 2026• 0 citations
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Abstract

We construct a helical generalization of the Sachdev-Ye-Kitaev (SYK) model in $1+1$ dimensions, built from left- and right-moving Majorana fermions with local quartic interactions and random couplings in flavor-chirality space. These interactions organize into a symmetry-controlled hierarchy of quartic chirality sectors. At the most restrictive end of this hierarchy, symmetry forces the quartic structure into a density-density form, which admits an exact solution using bosonization, rendering the theory integrable. Once the full quartic helical interaction space is allowed, including purely chiral, chirality-balanced, and chirality-imbalanced sectors, this symmetry-protected integrable structure is lost. Nevertheless, the large-$N$ infrared limit remains analytically tractable through short-distance selection rules and disorder averaging. Using conformal perturbation theory about the free fixed point, we show that the entire interaction space is marginally irrelevant, and the theory thus becomes free and integrable in the IR.

DOI
10.48550/arxiv.2606.17161

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Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.
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We study a 1+11+11+1 (space-time)-dimensional extension of the 0+10+10+1-dimensional Sachdev-Ye-Kitaev model for NNN Majorana fermions, with random all-to-all quartic interactions, averaged over disorder. At large interaction couplings and large NNN, the conformal symmetry of the effective action emerges, which is not broken spontaneously as in the original 0+10+10+1 SYK model. Two-point correlators are obtained from a coupling expansion of the Schwinger-Dyson equations. For N=4N=4N=4, the model can be mapped onto complex fermions and solved exactly via the bosonization technique, featuring two branches of excitations: a gapped ``pseudocharge'' mode and a gapless ``pseudospin'' mode. We give an approximate analytic form of the two-point correlators at large distances and zero temperature for N=4N=4N=4, which is adopted heuristically to evaluate an approximation to the large-NNN free energy in the zero temperature limit, numerically. The fact that this energy displays an absolute minimum at a finite value of the gap, though in a restricted range of parameters, suggests that a gapped branch of excitations is also present in our extension of the model at least in that range of parameters.
The SYK model of Sachdev, Ye and Kitaev is a recent model of N Majorana fermions coupled via quartic random interaction. It has the extraordinary properties of being exactly solvable at large N , having emergent conformal symmetry in the IR, and exhibiting maximal quantum chaos. The confluence of these properties make SYK a highly attractive candidate for a solvable AdS/CFT correspondence. What’s more, it was found to capture elements of non-Fermi-liquid behavior in strange metals. This thesis introduces the SYK model and discusses a recent variant, the SYK∗ model of Marcus and Vandoren. In the latter, the effective action is derived and branches of conformal solutions are discussed. The parameter space of SYK∗ is investigated for large and small M . The former reveals a phase separation and a N critical point in the T−M plane is claimed to exist.
We show that a variant of the Gross-Neveu Yukawa model with disorder provides a real, nonsupersymmetric generalization of the Sachdev--Ye Kitaev (SYK) model to three dimensions. The model contains MMM real scalar fields and NNN Dirac (or Majorana) fermions, interacting via a Yukawa interaction with a local Gaussian random coupling in three dimensions. In the limit where MMM and NNN are both large, and the ratio M/NM/NM/N is held fixed, the model defines a line of infrared fixed points parametrized by M/NM/NM/N, reducing to the Gross-Neveu vector model when M/N=0M/N=0M/N=0. When M/NM/NM/N is nonzero, the model is dominated by melonic diagrams and gives rise to SYK-like physics. We compute the spectrum of single-trace operators in the theory, and find that it is real for all values of M/NM/NM/N.
We discuss a 1+1 dimensional generalization of the Sachdev-Ye-Kitaev model. The model contains N Majorana fermions at each lattice site with a nearest-neighbour hopping term. The SYK random interaction is restricted to low momentum fermions of definite chirality within each lattice site. This gives rise to an ordinary 1+1 field theory above some energy scale and a low energy SYK-like behavior. We exhibit a class of low-pass filters which give rise to a rich variety of hyperscaling behaviour in the IR. We also discuss another set of generalizations which describes probing an SYK system with an external fermion, together with the new scaling behavior they exhibit in the IR.