Chiral SYK Model: Tunable Chaos & Scaling

• Chiral SYK model is a quantum system that uses low-pass filtering to restrict interactions to a single fermionic chirality, enabling a focused study of strong interactions.
• It features emergent hyperscaling with tunable dynamical critical exponents and anomalous dimensions derived from filtered Schwinger–Dyson equations.
• Coupling external probe fermions reveals modified scaling dimensions and conserved higher-spin currents, linking controlled chaos to holographic dualities.

The Chiral SYK Model is a class of strongly interacting quantum mechanical (and, in higher dimensions, quantum field theoretic) systems constructed to isolate and paper the interplay between chirality, randomness, strong interactions, emergent scaling, hyperscaling, and universal features such as many-body chaos and higher-spin symmetry. These models generalize the original Sachdev-Ye-Kitaev (SYK) model by enforcing chiral structure—typically by ensuring that only fermions of a definite handedness participate in the SYK interaction—via nonlocal filtering or symmetry projections. Their analysis reveals novel phases characterized by distinctive infrared scaling behavior (hyperscaling), a tunable chaotic regime, emergent higher-spin conservation in certain limits, and a bridge to higher-dimensional holographic dualities.

1. Incorporation of Chirality: Projection, Filtering, and Model Construction

The essential innovation in the chiral SYK construction is the restriction of the SYK-type random all-to-all interaction to a single chiral sector (for example, left-movers) of fermions—while the complementary sector remains free or weakly interacting. Consider a lattice of $N$ Majorana (or Dirac) fermions $\chi^{i,a}$ per site $i$ ($a=1,..,N$). The kinetic term yields both left- ($L$) and right- ($R$) moving modes; to isolate a chiral sector, the model defines “filtered” fermion operators via a nonlocal convolution in position space:

$\eta^i_a = \sum_j \widetilde{F}(i-j) \chi^{j, a}$

with the filter kernel $\widetilde{F}$ chosen so that, in momentum space, only modes with $k\approx 0$ and a prescribed sign (i.e., low-momentum left-movers) have support. Typical filters include scaling forms $F(k) \sim |k|^{-\gamma}$ or a Gaussian cutoff $F(k) = \exp(-\pi^2 k^2/\widehat{D}^2)$. This filtering procedure defines two disjoint low-pass sectors:

$$\eta_{\text{L}}^i = A\sum_j \widetilde{F}(i-j)\chi^{j,a} \quad , \quad \eta_{\text{R}}^i = \hat{A}\sum_j \exp(-\dotsc)(-1)^{i-j}\chi^{j,a}$$

The SYK-type interaction (with random couplings $J_{i,abcd}$) is only turned on for the filtered $\eta_{\text{L}}$ variables:

$$\mathcal{L}_{\text{int}} \sim \sum_i J_{i,abcd}\,\eta^{(L)}_{i,a}\eta^{(L)}_{i,b}\eta^{(L)}_{i,c}\eta^{(L)}_{i,d}$$

The result is that only a single chiral sector is strongly coupled, while the remaining sector is essentially free—a design yielding a model with protected chirality in its low-energy states (Berkooz et al., 2016).

2. Low-Pass Filters, Chiral Protection, and Emergent Hyperscaling

The low-pass filter serves two key purposes:

• Chiral Protection: Suppression of high-momentum modes (where lattice fermions generally mix left and right movers) ensures that only long-wavelength, single-chirality modes are subject to strong interactions. This maintains an effective chiral sector even in the presence of a lattice regularization that would otherwise break chirality.
• IR Hyperscaling: At low energies, the filtered Schwinger–Dyson equations for the collective propagators acquire nontrivial scaling structure. Define filtered propagators as

$$\widetilde{G}_k(\tau) = F(k)^2 G_k(\tau)\quad;\quad \widetilde{G}(\tau) = \sum_k \widetilde{G}_k(\tau)$$

The filtered SD equations become \begin{align*} \widetilde{\Sigma}(\tau) &= \frac{A^8 J^2}{L^3} [\widetilde{G}(\tau)]^3\ \widetilde{G}(\omega) &= \sum_k \frac{1}{ F^{-2}(k)(-i\omega + E_k) - \widetilde{\Sigma}(\omega) } \end{align*} For scaling filters $F(k)\sim |k|^{-\gamma}$ and linearized dispersion, the IR solution develops nontrivial anomalous dimension $\Delta$ and a dynamical critical exponent $z$ (Berkooz et al., 2016): $$\Delta = \frac{1+4\gamma}{2(1+8\gamma)}\ ;\quad z = \frac{2\gamma+1}{6\Delta-1} = \frac{1}{2} + 4\gamma$$ This "hyperscaling" reflects anisotropic scaling between time and space: $$\omega \rightarrow \lambda^{-z}\omega \quad ,\quad p\rightarrow\lambda^{-1}p$$ The exponent $z$ is tunable by the filter parameter $\gamma$.

3. Scaling of Probes: Probing the Chiral SYK Core

A further generalization involves coupling an external (probe) fermion $\rho$ to the chiral SYK core. The probe can be filtered (to match the chiral regime of the core) and interacts via mixed vertices: $$\mathcal{L}_{\rho,\eta} \sim \sum \widehat{J} [F(k)\rho]\ldots[F(k)\eta^a]$$ The corresponding SD equations for the probe's self-energy yield IR anomalous dimensions dependent on $\gamma$: $$\Delta_1 - 1 = -\frac{2\gamma}{1 + 4\gamma} \ ;\quad \Delta_2 - 1 = \frac{1 + 2\gamma}{1 + 4\gamma}$$ This indicates that the scaling dimension of the probe operator is directly controlled by the degree of chiral filtering in the system, a concrete manifestation of how chiral and hyperscaling structures are imprinted on external probes (Berkooz et al., 2016).

4. Symmetry, Conservation Laws, and Tuning of Chaos

By imposing interactions only on one chiral sector, the model acquires a conserved chiral charge and a symmetry protecting the left-mover subspace. The presence of this symmetry constrains the dynamics and, importantly, modifies the system’s chaotic behavior. In the presence of conserved chiral charge (or, in the complex SYK, U(1) charge), the Lyapunov exponent for the growth of out-of-time-ordered correlators (OTOCs) is suppressed compared to the maximally chaotic value, and can be tuned by the strength of symmetry breaking (Bhattacharya et al., 2017). This ability to interpolate between maximal scrambling (generic SYK) and nonchaotic phases (by enforcing exact chiral symmetry or an effective chiral chemical potential) makes the chiral SYK and related models a platform for exploring the boundaries of quantum chaos and ergodicity.

5. Infrared Universality: Connection to Higher-Spin Symmetry

Contours of the IR fixed point in chiral SYK models are sensitive to UV regularization (filter choice) and interaction pattern, but strong filtering—especially in the regime where the parameter controlling the ratio of Fermi to chiral multiplets approaches a limiting value—leads to emergent higher-spin symmetry (Peng, 2018, Ahn et al., 2018). In these limits the IR four-point function reveals an infinite tower of approximately (or, at the endpoints, exactly) conserved higher-spin currents. Correspondingly, the Lyapunov exponent governing many-body chaos can vanish as the model approaches these "higher-spin symmetric" points. This demonstrates a mechanism by which certain abstract symmetry structures—known from integrable or tensionless string theories—materialize in strongly disordered, chiral quantum matter.

6. Realization, UV Completions, and Applications

The lattice regularized chiral SYK model and its filtering constructions are flexible and can be adapted to various physical contexts. Non-random (deterministic) tensor representations exist that flow to the same long-wavelength physics, and chiral filtering can be engineered in quasicrystals, coupled edge modes of higher-dimensional topological phases, or arrays of mesoscopic quantum dots (Peng, 2018, Ahn et al., 2018). The ability to probe the chiral SYK core with external fermions opens up applications to quantum simulation of black hole phenomenology, operator spreading, and designer non-Fermi liquids with tunable scaling and chaos properties.

---

Summary Table: Key Parameters and Their Roles

Parameter / Feature	Role in Chiral SYK Model	Consequence
$\gamma$ (filter exponent)	Controls low-pass filtering	Tunes hyperscaling exponent $z$ and anomalous dimension $\Delta$
$\eta^{(L)}$, $\eta^{(R)}$	Projected chiral fermions	Only one sector interacts strongly
$J$ (interaction strength)	SYK-like random 4-fermion coupling	Sets interaction scale (core SYK physics)
External probe fermion $\rho$	Test chiral sector	Probe inherits chiral scaling dimensions
Conserved chiral charge	Imposed symmetry	Modifies/suppresses chaos, tunes OTOC Lyapunov exponent
Higher-spin symmetry endpoint	$\mu \to 1/q^{+}$ or $\mu \to +\infty$	Emergence of infinite conserved higher-spin tower, vanishing Lyapunov exponent

7. Implications and Outlook

Chiral SYK models, by engineering chirality using low-pass filtering and projected interactions, demonstrate that the interplay of symmetry restriction, disorder, and strong interaction enables access to novel universality classes not found in either conventional field theory or integrable models. Hyperscaling, tunable chaos, emergent higher-spin conservation, and nontrivial probe responses illustrate a phase space rich in physics akin to both holographic gravity (through black hole–like scrambling and non-Fermi liquid scaling) and to highly symmetric (e.g., tensionless string) CFTs. The ability to interpolate between chaotic and nonchaotic phases, modify operator scaling at will, and realize higher-spin symmetry endpoints connects these chiral SYK models to deep issues in quantum many-body theory, quantum chaos, and quantum gravity (Berkooz et al., 2016, Peng, 2018, Ahn et al., 2018).