Multicopy SYK Model: Multi-Flavor Quantum Systems

• Multicopy SYK models are quantum many-body systems that extend the original SYK paradigm by incorporating multiple flavors, indices, or bands.
• They employ diverse Hamiltonians—from tensor and two-index formulations to lattice constructions—to study quantum chaos and non-Fermi-liquid behavior.
• These models reveal large-N dynamics with emergent conformal symmetry, Schwarzian effective actions, and holographic dualities linking to nearly-AdS₂ gravity.

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 (Kim, 2018), the two-indices SYK model (Ye, 2018), spatial arrays of SYK dots (“SYK chains” and beyond) (Khveshchenko, 2017), and generalizations to N-replica or multi-band systems (Addazi et al., 2021, Khveshchenko, 2018).

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 (Kim, 2018).

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 (Ye, 2018).

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 (Khveshchenko, 2017, Khveshchenko, 2018). 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 (Kim, 2018). 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 (Ye, 2018).

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 (Kim, 2018).

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 (Kim, 2018).

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 (Kim, 2018, Ye, 2018, Addazi et al., 2021). 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,

$$I_{\rm bdy} = -\frac{1}{8\pi G} \sum_{i=1}^N \int du_i\, \phi_{r,i}(u_i)\, \operatorname{Sch}(t_i(u_i), u_i)$$

(Addazi et al., 2021). 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 (Khveshchenko, 2017).

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) (Ye, 2018). 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 $N < \exp(\sqrt{\pi M/2})$ (Ye, 2018). 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 (Kim, 2018).

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 (Addazi et al., 2021).
• 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 (Addazi et al., 2021).

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 $T\to 0$ (Khveshchenko, 2018). 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 (Khveshchenko, 2017).

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 $d>0$

• Large $N$ Tensor and SYK Models (Kim, 2018)
• Two indices Sachdev-Ye-Kitaev model (Ye, 2018)
• New Massive JT Multi-Gravity and N-Replica of SYK Models (Addazi et al., 2021)
• Thickening and sickening the SYK model (Khveshchenko, 2017)
• Seeking to develop global SYK-ness (Khveshchenko, 2018)