Spin-SYK Variants & Infrared Dynamics

• The Spin-SYK model encompasses various SYK-type constructions that replace fermionic variables with spin operators or SU(d) generators to capture diverse many-body phenomena.
• It employs distinct microscopic constructions—such as Pauli-spin, qudit, and higher-spin approaches—with all-to-all random interactions to explore strongly interacting quantum systems.
• Its infrared dynamics combine Schwarzian reparametrization with additional soft spin sectors, leading to maximal chaos and random matrix universality in spectral statistics.

Searching arXiv for recent and foundational papers on Spin-SYK and related variants. The term Spin-SYK model is used in the literature for several closely related extensions of the Sachdev–Ye–Kitaev framework in which spin degrees of freedom, spin symmetry, or spin-sector observables are central. In one usage, Majorana fermions are replaced by Pauli spins or hard-core bosons, yielding all-to-all random spin Hamiltonians such as the SpinXY4 model (Hanada et al., 2023). In another, the local variables are SU($d$) generators rather than fermions, leading to “qudit SYK” and other two-local generalizations (Hanada et al., 15 May 2025). A different line of work studies spinful, time-reversal-invariant SYK models of complex fermions with SU(2) symmetry and pairing channels (Wang et al., 2020), while the low-energy theory of SYK models with continuous global symmetry identifies the spin case with $G=\mathrm{SU}(2)$ and an infrared factorization into a Schwarzian mode and a rigid rotor on the group manifold (Liu et al., 2019). Yet another usage concerns “higher-spin SYK,” where the singlet bilinear tower realizes $hs[\lambda]$ or $ehs[\lambda]$ and admits higher-spin BF or Poisson-sigma-model bulk duals (Bekaert et al., 24 Sep 2025). The label therefore does not denote a single microscopic Hamiltonian, but a family of SYK-type constructions sharing random, strongly interacting, and often maximally chaotic low-energy dynamics.

1. Terminological scope and principal variants

The literature does not use Spin-SYK in a single uniform way. The most direct meaning is a literal spin replacement of the fermionic SYK variables: the SpinXY4 model replaces Majoranas by local Pauli operators, while preserving all-to-all random 4-local interactions (Hanada et al., 2023). A second meaning is a spinful SYK model, where the microscopic fermions carry SU(2) spin and the low-energy theory includes both Schwarzian and spin-rotor sectors (Liu et al., 2019). A third meaning is higher-spin SYK, in which the relevant “spin” is not microscopic SU(2) but an infinite tower of singlet bilinears organized by a higher-spin algebra and described holographically by higher-spin extensions of Jackiw–Teitelboim gravity (Bekaert et al., 24 Sep 2025). A fourth meaning is phenomenological: SYK-like scaling and operator growth in the spin sector of correlated electron models, especially the spin-freezing crossover of multi-orbital Hubbard systems (Tsuji et al., 2018).

Usage of “Spin-SYK”	Degrees of freedom	Defining structure
Pauli-spin / hard-core-boson model	Local $\sigma^{x,y}$ operators	All-to-all random 4-local spin interactions
SU($d$) or qudit SYK	SU($d$) generators $T_{i,\alpha}$	Random $q$-local couplings, including two-local cases
Spinful fermionic SYK	Complex spin-$1/2$ fermions $G=\mathrm{SU}(2)$0	SU(2)-invariant random interactions, pairing channels
Higher-spin SYK	Bilinear singlet tower	$G=\mathrm{SU}(2)$1 symmetry, BF/PSM bulk dual
Spin-sector SYK phenomenology	Hubbard-Hund local spin observables	$G=\mathrm{SU}(2)$2 and SYK-like OTOCs

This plurality is not merely terminological. It reflects distinct research programs: quantum simulation with spins or qudits, infrared effective theory with global symmetry, holographic higher-spin generalizations, and non-random condensed-matter realizations of SYK-like spin dynamics.

2. Microscopic constructions

In the Pauli-spin realization SpinXY4, the operators are defined by

$G=\mathrm{SU}(2)$3

and the Hamiltonian is an all-to-all random 4-local sum of terms $G=\mathrm{SU}(2)$4, where $G=\mathrm{SU}(2)$5 counts how many sites contribute both $G=\mathrm{SU}(2)$6 and $G=\mathrm{SU}(2)$7 components and supplies the Hermiticity factor. The random couplings are i.i.d. standard Gaussian, and the normalization is chosen so that energy, entropy, and characteristic time scales obey the standard SYK large-$G=\mathrm{SU}(2)$8 scaling. The model admits a hard-core-boson interpretation because $G=\mathrm{SU}(2)$9 act as creation and annihilation operators with on-site exclusion (Hanada et al., 2023).

The SU($hs[\lambda]$0) generalization, or qudit SYK, replaces Pauli matrices by generators $hs[\lambda]$1 of SU($hs[\lambda]$2), normalized by

$hs[\lambda]$3

with site operators $hs[\lambda]$4. Its $hs[\lambda]$5-local Hamiltonian is

$hs[\lambda]$6

where the couplings are independent zero-mean Gaussians and the variance is fixed by $hs[\lambda]$7, yielding

$hs[\lambda]$8

The $hs[\lambda]$9 case reproduces the usual spin-SYK. The same work also introduced “overlapping clusters” Majorana models with essentially two-local structure, for example

$ehs[\lambda]$0

with variance $ehs[\lambda]$1 chosen to match the energy variance of the original $ehs[\lambda]$2 SYK model (Hanada et al., 15 May 2025).

A distinct microscopic construction is the spin-1/2, time-reversal-invariant SYK model built from complex fermions $ehs[\lambda]$3. Its random interaction is

$ehs[\lambda]$4

with real couplings obeying

$ehs[\lambda]$5

This model preserves SU(2) and time reversal and becomes the basis for pairing, pseudogap, and superconducting extensions when supplemented by attractive interactions such as a negative-$ehs[\lambda]$6 Hubbard term (Wang et al., 2020).

3. Infrared structure, collective fields, and holographic descriptions

Across these variants, the low-energy description repeatedly reduces to Schwarzian dynamics plus additional soft sectors. In standard large-$ehs[\lambda]$7 SYK-like treatments one introduces bilocal fields $ehs[\lambda]$8 and $ehs[\lambda]$9 obeying Schwinger–Dyson equations

$\sigma^{x,y}$0

with an emergent reparametrization mode governed by

$\sigma^{x,y}$1

where $\sigma^{x,y}$2. For a bosonic operator $\sigma^{x,y}$3 of scaling dimension $\sigma^{x,y}$4, the conformal finite-temperature correlator takes the form

$\sigma^{x,y}$5

In the “spin-SYK” setting of spin-like composite operators, Schwarzian loop corrections reshape the retarded correlator and density of states and generate explicit formulas for local susceptibilities and higher-point OTOCs (Qi et al., 2018).

For SYK models with continuous global symmetry $\sigma^{x,y}$6, the infrared effective action factorizes as

$\sigma^{x,y}$7

where the group sector is a free particle on the group manifold,

$\sigma^{x,y}$8

For the spin case $\sigma^{x,y}$9, the group partition function is

$d$0

This establishes the infrared spin sector as a rigid rotor on SU(2), with energies $d$1 and a modified low-temperature thermodynamics and spectral form factor. The reparametrization mode still controls maximal chaos, and the global-symmetry modes do not increase the Lyapunov exponent beyond $d$2 (Liu et al., 2019).

The Pauli-spin SpinXY4 model has a different but related collective-field structure. After disorder averaging, the real-time path integral retains only $d$3 permutation symmetry rather than finite-coupling $d$4. The large-$d$5 description closes on an infinite tower of multi-local invariants

$d$6

In the strong-coupling limit an emergent $d$7 symmetry appears, the theory reduces to a symmetric bilocal field $d$8, and the collective action becomes

$d$9

The saddle equation is

$d$0

which is the bosonic-spin analogue of the SYK bilocal formalism (Hanada et al., 2023).

In higher-spin SYK, the bilinear singlet tower

$d$1

has scaling dimensions $d$2, and on the generalized free line with $d$3 one obtains $d$4. The bulk dual replaces ordinary JT gravity by a higher-spin BF or Poisson sigma model based on $d$5 or $d$6. Two realizations are emphasized: Model A, built from a deformation of $d$7 and formulated as a BF theory with a Lax pair, and Model B, a perturbatively local Poisson sigma model derived from a cyclic $d$8 structure. In both cases the scalar tower satisfies

$d$9

and the boundary dynamics is a higher-spin generalization of the Schwarzian action (Bekaert et al., 24 Sep 2025). Closely related bulk-side analyses derive generalized Schwarzian actions for Yang–Mills and $T_{i,\alpha}$0 higher-spin extensions directly from BF-formulated dilaton gravity and the Lee–Wald–Zoupas symplectic structure (González et al., 2018).

4. Chaos, level statistics, and the spin-glass question

The Pauli-spin SpinXY4 model is strongly chaotic in the random-matrix sense. The operator

$T_{i,\alpha}$1

commutes with the Hamiltonian, so parity is conserved and the Hilbert space splits into two blocks of dimension $T_{i,\alpha}$2. When same-site $T_{i,\alpha}$3 pairs are allowed, the finite-$T_{i,\alpha}$4 random-matrix universality class is GUE for any $T_{i,\alpha}$5. Numerically, the density of states over the bulk is nearly indistinguishable from SYK4 up to $T_{i,\alpha}$6, the nearest-neighbor spacing ratios agree with GUE except for a few of the lowest levels, and the spectral form factor exhibits long ramps very close to those of SYK4. Two-point functions distinguish parity-flipping and parity-preserving operators: $T_{i,\alpha}$7 behaves like the SYK Majorana correlator and lacks a ramp/plateau, whereas $T_{i,\alpha}$8 shows a ramp/plateau because it probes a single parity sector. Edwards–Anderson diagnostics suggest that a small number of low-energy states may show spin-glass-like tendencies, but over most of the spectrum the model is ergodic and RMT-like (Hanada et al., 2023).

The SU($T_{i,\alpha}$9) and two-local modifications reach a similar conclusion. For qudit SYK with $q$0 at $q$1 and $q$2 at $q$3, the spacing distribution $q$4 converges to the GUE Wigner surmise as $q$5 increases, the adjacent-gap ratio concentrates near the GUE value $q$6, and the spectral form factor shows a clear dip–ramp–plateau. For the overlapping-clusters model

$q$7

the expected random-matrix ensemble depends on $q$8: GOE for $q$9 and GUE for $1/2$0 after sector decomposition. Its density of states has soft edges, but the bulk still displays spectral rigidity and Wigner–Dyson statistics over a wide energy window (Hanada et al., 15 May 2025).

A recurring misconception is that SYK’s strong disorder should generically induce a spin-glass phase. For the $1/2$1 Majorana SYK model, this was explicitly tested and rejected. Replica off-diagonal condensation is controlled by a replicon mass $1/2$2; a naive nearly-conformal estimate suggests an exponentially small critical temperature, but for $1/2$3 this lies outside the conformal regime, and the correct Schwarzian-regime computation yields

$1/2$4

with $1/2$5, so the replica-symmetric saddle remains stable at large $1/2$6. Numerically, for $1/2$7 up to $1/2$8 Majoranas the edge statistics agree with RMT rather than Poisson, the spectral form factor shows a ramp and plateau, and the ground-state energy distribution is Gaussian rather than Tracy–Widom. The conclusion is that the $1/2$9 Majorana SYK model remains paramagnetic and ergodic, with no spin-glass phase (Gur-Ari et al., 2018).

5. Spin observables, susceptibilities, and pairing instabilities

Spin-sensitive observables in SYK-like models can be formulated through bosonic composite operators $G=\mathrm{SU}(2)$00 of infrared scaling dimension $G=\mathrm{SU}(2)$01. In that framework, the retarded two-point function receives explicit Schwarzian loop corrections, and the density of states

$G=\mathrm{SU}(2)$02

undergoes a transfer of spectral weight from the low-frequency quasiparticle peak to finite-frequency sidebands identified as Hubbard bands. The same Schwarzian theory gives an exponentially growing crossed OTOC,

$G=\mathrm{SU}(2)$03

and explicit local spin susceptibilities. At finite temperature,

$G=\mathrm{SU}(2)$04

For the marginal non-Fermi-liquid case $G=\mathrm{SU}(2)$05,

$G=\mathrm{SU}(2)$06

while higher derivatives give compact $G=\mathrm{SU}(2)$07 and $G=\mathrm{SU}(2)$08 expressions relevant to dynamical susceptibilities (Qi et al., 2018).

In the spin-1/2, time-reversal-invariant fermionic SYK model with onsite attractive interaction, the basic pairing operators are

$G=\mathrm{SU}(2)$09

Off-diagonal long-range order is diagnosed by an extensive largest eigenvalue $G=\mathrm{SU}(2)$10 of $G=\mathrm{SU}(2)$11. Exact diagonalization at half-filling for $G=\mathrm{SU}(2)$12 finds a finite critical attraction $G=\mathrm{SU}(2)$13 below which there is neither ODLRO nor a many-body gap beyond the small finite-size SYK gap; instead the system enters a pseudogap regime dominated by phase fluctuations. Those fluctuations are described by a quantum Kuramoto action,

$G=\mathrm{SU}(2)$14

with $G=\mathrm{SU}(2)$15 and critical coupling $G=\mathrm{SU}(2)$16. In the opposite limit $G=\mathrm{SU}(2)$17, the low-energy theory becomes a generalized Richardson model with

$G=\mathrm{SU}(2)$18

ground-state ODLRO eigenvalue

$G=\mathrm{SU}(2)$19

and a first-order thermal transition at

$G=\mathrm{SU}(2)$20

This is a genuine spinful SYK superconductivity problem rather than a spin-glass problem (Wang et al., 2020).

6. Physical realizations and condensed-matter analogues

A concrete spin-system realization was proposed in a Kitaev–$G=\mathrm{SU}(2)$21 chain. Starting from the Hamiltonian

$G=\mathrm{SU}(2)$22

with the four-Majorana parton representation

$G=\mathrm{SU}(2)$23

the 1D limit contains an extensive set of exact $G=\mathrm{SU}(2)$24 zero modes. For $G=\mathrm{SU}(2)$25, the pure Kitaev limit has a ground-state degeneracy scaling as $G=\mathrm{SU}(2)$26, first-order matrix elements of the $G=\mathrm{SU}(2)$27 perturbation vanish inside the zero-mode manifold, and in the anisotropic regime $G=\mathrm{SU}(2)$28 one can project to an effective quartic Hamiltonian

$G=\mathrm{SU}(2)$29

This yields a clean $G=\mathrm{SU}(2)$30 SYK interaction among the emergent low-energy Majoranas. The same work introduced spin-string proxies for Majorana OTOCs and reported exact-diagonalization data at $G=\mathrm{SU}(2)$31 and $G=\mathrm{SU}(2)$32: without perturbations the OTOC is nearly time-independent, while with perturbations it shows early-time growth suggestive of scrambling, albeit with strong finite-size effects (Zuo et al., 2024).

A different realization is not microscopic disorder, but spin-SYK phenomenology in Hund metals. In two- and three-orbital Hubbard models with finite Hund coupling $G=\mathrm{SU}(2)$33, dynamical mean-field theory combined with a continuous-time Monte Carlo impurity solver shows that the spin-freezing crossover hosts both a non-Fermi-liquid self-energy

$G=\mathrm{SU}(2)$34

and SYK-like OTOC dynamics. For the spin-sensitive operator

$G=\mathrm{SU}(2)$35

the two-orbital model at $G=\mathrm{SU}(2)$36, $G=\mathrm{SU}(2)$37, $G=\mathrm{SU}(2)$38, and $G=\mathrm{SU}(2)$39 exhibits short-time damping $G=\mathrm{SU}(2)$40 with $G=\mathrm{SU}(2)$41 for $G=\mathrm{SU}(2)$42, followed by a power-law tail $G=\mathrm{SU}(2)$43 for $G=\mathrm{SU}(2)$44. The three-orbital analogue near $G=\mathrm{SU}(2)$45 gives an exponent $G=\mathrm{SU}(2)$46. The time dependence approximately collapses as a function of $G=\mathrm{SU}(2)$47, and finite-$G=\mathrm{SU}(2)$48 complex-fermion SYK exact diagonalization yields a comparable long-time exponent $G=\mathrm{SU}(2)$49 when one identifies $G=\mathrm{SU}(2)$50 with the Hund coupling $G=\mathrm{SU}(2)$51. In this sense, the spin-freezing regime realizes spin-sector SYK dynamics in a translationally invariant, non-random setting (Tsuji et al., 2018).

These developments suggest that the phrase Spin-SYK model names a research area rather than a single theory. Its common thread is the transplantation of SYK logic—random strong interactions, solvable infrared structure, spectral chaos, and anomalous spin response—into settings where spin operators, spin symmetry, or higher-spin singlet sectors are fundamental.