---
title: 'Multicopy SYK Model: Multi-Flavor Quantum Systems'
url: https://www.emergentmind.com/topics/multicopy-syk-model
type: topic
---

# Multicopy SYK Model: Multi-Flavor Quantum Systems

The multicopy SYK model encompasses a broad class of quantum many-body systems generalizing the Sachdev–Ye–Kitaev (SYK) paradigm to multiple flavors, indices, spatial sites, or bands. These models are central to ongoing research in quantum chaos, non-Fermi-liquid physics, and holography, and provide a highly controlled large-$N$ limit that reveals emergent conformal symmetry and connections to nearly-AdS$_2$ gravity. Here, "multicopy" subsumes tensor SYK models with multiple flavors, multi-replica or multi-site SYK arrays, and models with matrix or band structure. Key variants include the two-flavor $O(N)^3$ tensor SYK model [1811.04330], the two-indices SYK model [1809.06667], spatial arrays of SYK dots (“SYK chains” and beyond) [1705.03956], and generalizations to N-replica or multi-band systems [2106.02384, 1805.00870]. 

## 1. Model Structures and Hamiltonians

**Two-Flavor (Tensor) SYK Model.** The archetypal multicopy $O(N)^3$ tensor SYK model features two Majorana triplet fields, $\psi_1^{abc}$ and $\psi_2^{abc}$, and a quartic Hamiltonian 
\[
H = \frac{\beta g}{2} \sum_{α_1...α_4} I_1(α_i) \psi_1^{α_1}\psi_1^{α_2}\psi_2^{α_3}\psi_2^{α_4}
+ \frac{g}{4} \sum_{f=1}^2 \sum_{α_1...α_4} I_2(α_i) (\psi_f^{α_1}\psi_f^{α_2}\psi_f^{α_3}\psi_f^{α_4}),
\]
where $\beta$ tunes the strength of mixed-flavor interactions. In component form, the indices $a,b,c=1...N$ and the contraction tensors $I_1$, $I_2$ encode the $O(N)^3$ invariance. For $\beta=0$ the flavors decouple, while $\beta=1$ yields a rotated basis with two identical single-flavor Hamiltonians at halved coupling. The limit $\beta\to\infty$ reproduces the (non-disordered) Gross–Rosenhaus two-flavor SYK-type model [1811.04330].

**Two-Indices SYK Model.** This model considers $N$ sites, each with $M$ Majorana fermions $\chi_{i\alpha}$, and the Hamiltonian
\[
H = \frac{1}{\sqrt{2M}} \sum_{i<j} J_{ij} \left( \chi_{i\alpha}\chi_{j\alpha} \right)\left( \chi_{i\beta}\chi_{j\beta} \right)
\]
with independently Gaussian $J_{ij}$ ($\langle J_{ij}^2\rangle=J^2/N$). This two-indices construction enables a $1/M$ expansion at large $N$ and is closely related to quantum spin liquid and spin glass physics [1809.06667].

**Spatially Extended and Multi-Band Models.** Generalizations to $d+1$ dimensions and multi-band (flavor) structure are realized by placing $N$-flavor fermions at each of $L$ sites or adding additional bands. The generic action reads
\[
S = \int d\tau \left[ \sum_{ia\alpha} \chi^a_{i\alpha} \partial_\tau \chi^a_{i\alpha}
- \frac{i^{q/2}}{q!} \sum_{i_1...i_q,a_1...\alpha_q} J^{a_1...\alpha_q}_{i_1...i_q} \chi_{i_1}^{a_1 ...} ... \chi_{i_q}^{a_q ...}
\right]
\]
with $J$ coupling tensors obeying Gaussian statistics, possibly with long-range spatial or temporal correlations [1705.03956, 1805.00870]. In multi-band/bi-flavor systems, inter-flavor couplings of arbitrary form can be introduced, and the kernel $D_{ab}(x,\tau)$ defines interaction structure.

## 2. Large-$N$ Dynamics, Schwinger–Dyson Equations, and Marginality

In the large-$N$ limit, multicopy SYK models exhibit dominance by "melonic" diagrams, preserving solvability owing to the recursive nature of self-energy diagrams. For each flavor or site $a$, one defines bilocal Green's functions $G_a(\tau,x)$ and self-energies $\Sigma_a(\tau,x)$, leading to coupled Schwinger–Dyson equations, e.g.,
\[
G(i\omega)^{-1} = -i\omega - \Sigma(i\omega), \qquad \Sigma(\tau) = \lambda\, G(\tau)^3
\]
for suitable coupling $\lambda$ encasing the model-specific parameter dependence, such as $(3\beta^2+1)g^2N^3$ for the two-flavor tensor model [1811.04330]. At strong coupling, the conformal ansatz $G(\tau)\sim \operatorname{sgn}(\tau)/|\tau|^{2\Delta}$ applies, with $\Delta$ fixed by consistency. In the two-flavor tensor model, $\Delta=1/4$ and the normalization is set by $b^4(3\beta^2+1)g^2N^3=1/(4\pi)$. 

The $1/M$ expansion in the two-indices SYK model is exactly marginal: corrections renormalize the amplitude but preserve the infrared scaling [1809.06667].

## 3. Four-Point Kernel, Spectral Structure, and Operator Dimensions

The $O(1/N)$ connected four-point functions in multicopy models are determined by ladder kernels with flavor matrix structure. In the two-flavor tensor model, the kernel
\[
K_{(αβ),(γδ)}(t,t';t_3,t_4) = -g^2N^3 M_{(αβ),(γδ)} G(t_3-t)G(t_4-t')G(t-t')^2
\]
features $M_{(αβ),(γδ)}$ that decomposes into two $2\times 2$ flavor blocks: "same-to-same" and "same-to-mix" channels, whose eigenvalues and eigenoperators (symmetric/antisymmetric diagonal, even/odd mixed) correspond to distinct conformal primaries [1811.04330]. 

The scaling dimensions $h$ are determined as the real roots of transcendental equations $g_i(h)=1$, where $g_i(h)$ are kernel eigenvalues that depend on the flavor structure and model parameters (specifically $\beta$ in the two-flavor tensor case). For every channel,
\[
g_1(h) = -\tfrac32 \frac{\tan\bigl( \tfrac\pi2(h-1/2) \bigr)}{h - 1/2}
\]
and analogous expressions for $g_2$, $g_3$, $g_4$ with $\beta$-dependent coefficients. Each $g_i(h)$ determines an infinite tower of operator dimensions (for $h\to \infty$) whose precise offsets encode the effects of inter-flavor or inter-copy couplings.

Complex scaling dimensions emerge for certain parameter regimes, destabilizing the conformal phase. Notably, in the two-flavor tensor model, the $O_3$ mixed operator acquires complex $h$ for $\beta > 1$ or $\beta < 0$, signaling the breakdown of stability [1811.04330].

## 4. Emergent Symmetries, Conformal Dynamics, and the Schwarzian Limit

Multicopy SYK models at low energies exhibit emergent reparametrization ($\mathrm{SL}(2,\mathbb{R})$) invariance, weakly broken by UV terms. In the strict infrared, the soft mode is parametrized by reparametrization of time, $f(\tau)$, resulting in the universal Schwarzian effective action
\[
S[f] = -N \frac{\alpha_S}{J} \int d\tau\, \{ f(\tau), \tau \},
\]
with $\{f,\tau\}$ the Schwarzian derivative [1811.04330, 1809.06667, 2106.02384]. In the multicopy context, e.g., N-flavor models or N-replica chains, the low-energy effective theory consists of a sum of Schwarzian actions, one for each copy,
\[
I_{\rm bdy} = -\frac{1}{8\pi G} \sum_{i=1}^N \int du_i\, \phi_{r,i}(u_i)\, \operatorname{Sch}(t_i(u_i), u_i)
\]
[2106.02384]. Only when the couplings preserve ultra-locality in time and space does the full conformal/diffeomorphic invariance survive. Algebraically decaying or nonlocal disorder ($\alpha \neq 0$, $\beta \neq 0$) explicitly breaks this to a non-universal action for Goldstone modes [1705.03956].

## 5. Quantum Chaos, Spin-Glass Physics, and Instabilities

The out-of-time-ordered correlator (OTOC) in these models is governed by the spectrum of the four-point kernel. In the strict large-$N$ (and large-flavor or large-band) limit, the OTOC can be power-law rather than exponentially growing, with zero Lyapunov exponent in certain regimes (notably at $N=\infty$ and/or $M=\infty$ in the two-indices model) [1809.06667]. Quantum chaos and a positive Lyapunov exponent emerge only at subleading orders in $1/N$ and $1/M$. 

In replica-extended models, replica off-diagonal modes can condense at low temperature, producing a quantum spin glass (QSG) instability. The critical temperature for QSG in the two-indices SYK model is nonperturbatively suppressed,
\[
T_{QSG} \simeq J \exp\left(-\sqrt{\frac{\pi}{2}M}\right)
\]
and is avoided for $N < \exp(\sqrt{\pi M/2})$ [1809.06667]. Models with inter-flavor couplings out of the range $0 < \beta < 1$ in the two-flavor tensor model develop complex operator dimensions, indicating instability toward non-unitary or non-conformal phases [1811.04330].

## 6. Holographic Duality, Multi-Schwarzian Boundary Action, and Quantum Gravity Correspondence

The large-$N$/melonic dominance and emergent low-energy Schwarzian structure of multicopy SYK models motivate a duality with nearly-AdS$_2$ gravity, in particular, multi-dilaton (or multi-Jackiw–Teitelboim) gravity with one dilaton field per copy or flavor. This correspondence is made explicit in multi-replica extensions:
- Each SYK replica is dual to a boundary Schwarzian quantum mechanics, and inter-replica couplings correspond to bulk potentials coupling the dilaton fields [2106.02384].
- The partition function can be realized in several dual ways: as the disorder-averaged SYK partition function, as a Hartle–Hawking wavefunction in multi-JT gravity, or via spin-foam (BF theory) quantization.
- The BF quantization formalism demonstrates that the number of dynamical bulk degrees of freedom matches the number of SYK copies, and the spin-foam sum embodies the full multi-replica dynamics.

This correspondence enables analog gravity models for condensed matter systems, such as multilayer graphene with edge-state dynamics mapped to multi-Schwarzian boundary theories [2106.02384].

## 7. Transport, Thermodynamics, and Limiting Regimes

In spatially extended and multi-band multicopy SYK models, the thermodynamic and dynamical exponents are continuously tunable via the correlation parameters ($\alpha$, $\beta$) in the interaction kernel. For a (flavor-diagonal) action 
\[
D(x,\tau) = \frac{F}{|\tau|^{2\alpha}|x|^{2\beta}}
\]
the infrared scaling
\[
G(\omega,k) \sim (i\omega - B k^z)^{-\eta},\qquad \eta=1-\frac{1-2\alpha}{q-2},\quad z = \frac{d(q-2)+2\beta}{2-2\alpha}
\]
obtains, with the special case $z\to\infty$ recapturing the ultra-local, 0+1D SYK limit with finite entropy as $T\to 0$ [1805.00870]. Away from this limit, maximally chaotic behavior, finite ground state entropy, and universality of transport break down. Optical conductivity and specific heat display power-law scaling with exponents determined by model parameters. Notably, in higher dimensions or with long-range disorder, transport becomes non-diffusive or even gapped, in contrast to the diffusive regime of the local SYK chain [1705.03956].

## Summary Table of Key Features in Representative Multicopy SYK Variants

| Model Variant                | Defining Hamiltonian / Coupling                | Emergent Symmetry          | Spectral Stability           |
|------------------------------|------------------------------------------------|----------------------------|------------------------------|
| Two-Flavor Tensor SYK        | $O(N)^3$, 2 flavors, $\beta$-tuned quartic     | SL(2,$\mathbb R$), Schwarzian | Stable for $0<\beta<1$       |
| Two-Indices SYK              | 2-site indices, $M$-flavors per site           | SL(2,$\mathbb R$), Schwarzian | QSG for $T<T_{QSG}$, see text|
| N-Replica/Multi-JT Gravity   | $N$ independent SYK, intercopy potential $V$   | Sum of $N$ Schwartzians    | Stability set by $V$-structure|
| Multicopy (Lattice) SYK      | $L$ sites, long-range correlated $J$           | Explicitly broken conf.    | No maximal chaos for $d>0$   |

## References

- Large $N$ Tensor and SYK Models [1811.04330]
- Two indices Sachdev-Ye-Kitaev model [1809.06667]
- New Massive JT Multi-Gravity and N-Replica of SYK Models [2106.02384]
- Thickening and sickening the SYK model [1705.03956]
- Seeking to develop global SYK-ness [1805.00870]

Source: https://www.emergentmind.com/topics/multicopy-syk-model