Maldacena–Qi Coupled SYK Model

• The Maldacena–Qi coupled SYK model is a two-site extension of the SYK model featuring a bilinear left–right interaction that manifests traversable wormhole physics in nearly AdS₂.
• It leverages large‑q techniques and exact diagonalization to uncover distinct thermodynamic phases, including a gapped wormhole phase and an ergodic black-hole phase with precise spectral and entanglement characteristics.
• Time-dependent couplings and Floquet dynamics in the model enable controlled studies of quantum chaos, quenches, and protocols for high-fidelity quantum state preparation.

The Maldacena–Qi coupled SYK model is a two-site extension of the Sachdev–Ye–Kitaev model in which two identical SYK systems are coupled by a bilinear left–right interaction. It was proposed as a microscopic, UV-complete model whose low-temperature sector is dual to an eternal traversable wormhole in nearly ${\rm AdS}_2$, and it has since become a standard setting for studying thermofield-double structure, Hawking–Page-like transitions, spectral statistics, entanglement, and real-time holographic dynamics in a controllable many-body system (Maldacena et al., 2018).

1. Hamiltonian, conventions, and discrete structure

In its standard form, the model consists of two identical $q$-body SYK Hamiltonians, one for the left system and one for the right system, together with a bilinear coupling,

$$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$

Equivalent notation often writes $H=H_L+H_R+\mu S$, with

$S=i\sum_j \psi_{L,j}\psi_{R,j}.$

The disorder couplings are Gaussian with zero mean, and their variance scales as $J^2/N^{q-1}$. The literature uses two closely related counting conventions: some works assign $N$ Majorana fermions to each side, while others assign $N/2$ Majoranas to each side and reserve $N$ for the total number of Majoranas in the doubled system (Alet et al., 2020, García-García et al., 2019).

The inter-site coupling $\mu$ is the defining deformation. In the SYK language it is a relevant bilinear deformation; in the holographic interpretation it couples the two boundaries of the dual nearly-$q$0 description. Because exchanging left and right is equivalent to $q$1, one may restrict to $q$2 without loss of generality in the symmetric model (Alet et al., 2020).

A nontrivial discrete structure appears in the identical-coupling case. Besides the overall chirality $q$3, the full Hamiltonian commutes with

$q$4

so $q$5 is conserved modulo $q$6. Accordingly, the Hamiltonian splits into four decoupled $q$7 sectors. This additional symmetry is operationally important because level statistics must be computed after restricting to a single symmetry sector; otherwise spectral diagnostics are contaminated by trivial degeneracy structure rather than intrinsic many-body dynamics (García-García et al., 2019).

2. Ground state, thermofield-double structure, and large-$q$8 solution

A central question is whether the ground state of the coupled model is a thermofield double of the two uncoupled SYK copies. The relevant family of trial states is

$q$9

or, in an equivalent convention,

$$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$0

One then defines either the optimal inverse temperature $$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$1 or the maximal overlap $$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$2 (Alet et al., 2020, García-García et al., 2019).

Large-$$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$3 analysis gives an explicit relation between coupling, effective inverse temperature, and gap. Introducing parameters $$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$4 and $$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$5 by

$$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$6

the large-$$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$7 solution yields $$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$8 and a spectral gap

$$H=H_{\rm SYK}^L+H_{\rm SYK}^R+i\,\mu\sum_{j}\chi_j^L\chi_j^R.$$9

For small $H=H_L+H_R+\mu S$0, the leading large-$H=H_L+H_R+\mu S$1 behavior gives $H=H_L+H_R+\mu S$2, while refined expansions give $H=H_L+H_R+\mu S$3. Exact diagonalization pushed to $H=H_L+H_R+\mu S$4 for $H=H_L+H_R+\mu S$5 and $H=H_L+H_R+\mu S$6 for $H=H_L+H_R+\mu S$7 found that the finite-$H=H_L+H_R+\mu S$8 gap extrapolated with $H=H_L+H_R+\mu S$9 agrees with the large-$S=i\sum_j \psi_{L,j}\psi_{R,j}.$0 prediction at the $S=i\sum_j \psi_{L,j}\psi_{R,j}.$1–$S=i\sum_j \psi_{L,j}\psi_{R,j}.$2 level for moderate $S=i\sum_j \psi_{L,j}\psi_{R,j}.$3. The same study found that the ground-state overlap with the optimal TFD is typically $S=i\sum_j \psi_{L,j}\psi_{R,j}.$4–$S=i\sum_j \psi_{L,j}\psi_{R,j}.$5, and that the entanglement entropy of the ground state approaches the thermal entropy of the TFD with $S=i\sum_j \psi_{L,j}\psi_{R,j}.$6 corrections (Alet et al., 2020).

Finite-$S=i\sum_j \psi_{L,j}\psi_{R,j}.$7 exact diagonalization emphasizes a more nuanced picture. In the accessible size range $S=i\sum_j \psi_{L,j}\psi_{R,j}.$8, the overlap $S=i\sum_j \psi_{L,j}\psi_{R,j}.$9 remains $J^2/N^{q-1}$0 but develops a pronounced minimum near $J^2/N^{q-1}$1. For weak coupling, the coefficients of the exact ground state in the $J^2/N^{q-1}$2 basis decay as a power law,

$J^2/N^{q-1}$3

rather than with the exponential $J^2/N^{q-1}$4 characteristic of an exact TFD. By contrast, in the strong-coupling regime the ground state approaches the lowest-eigenvalue eigenstate of the spin operator $J^2/N^{q-1}$5, which is an exact infinite-temperature TFD with $J^2/N^{q-1}$6. These results imply that “ground state $J^2/N^{q-1}$7 TFD” is quantitatively accurate, but its quality depends on coupling, system size, and the diagnostic used to define closeness (García-García et al., 2019).

3. Thermodynamic phases and finite-temperature response

The coupled model exhibits two principal equilibrium regimes. For weak coupling and low temperature, $J^2/N^{q-1}$8, the system is in a gapped “wormhole” or “graviton-gas” phase. In this regime the free energy $J^2/N^{q-1}$9 is approximately flat up to temperatures of order the gap. For sufficiently large temperature or sufficiently strong coupling, the system crosses into a “two black-hole” phase in which the gap closes smoothly and the spectrum becomes fully ergodic (García-García et al., 2019).

Large-$N$0 Schwinger–Dyson analysis finds a first-order Hawking–Page transition line $N$1 for $N$2, terminating at a critical coupling

$N$3

Above $N$4, the first-order transition becomes a sharp but smooth crossover. Exact diagonalization indicates that the change in low-energy spectral statistics occurs near the same coupling scale, and that the thermodynamic transition sharpens with increasing system size for small $N$5, while evolving into crossover behavior as $N$6 increases (García-García et al., 2019).

Real-time Schwinger–Dyson studies of the finite-temperature spectral function resolve the microscopic content of these phases. In the low-temperature phase, the single-particle spectral function is described by weakly interacting fermions with a renormalized single-particle gap and a discrete set of sharp quasiparticle peaks. The matter excitations follow

$N$7

with $N$8 fixed by the Schwarzian-plus-coupling saddle. The imaginary-time Green’s function shows a gap $N$9 that is nearly temperature-independent until the transition is reached. Above the transition temperature $N/2$0, the sharp peaks merge into a broad continuum, and the spectral response resembles that of two weakly coupled black holes. For small $N/2$1, the critical line obeys

$N/2$2

consistent with Schwarzian and large-$N/2$3 expectations (Qi et al., 2020).

In holographic terms, the low-temperature phase is associated with a global $N/2$4 eternal wormhole, whereas the high-temperature phase is associated with two disconnected $N/2$5 black holes coupled only through matter. The coupled SYK model therefore provides a microscopic realization of a Hawking–Page-like competition between connected and disconnected bulk geometries (Qi et al., 2020).

4. Quantum chaos, level statistics, and spectral organization

The most direct spectral probe of quantum chaos in this model is the short-range level statistics after unfolding. Two standard observables are the nearest-neighbor spacing distribution $N/2$6 and the adjacent-gap ratio

$N/2$7

For Poisson statistics, $N/2$8 and $N/2$9; for GOE statistics, $N$0 and $N$1 (García-García et al., 2019).

In the wormhole regime $N$2, the low-lying tail of the spectrum deviates strongly from random-matrix behavior. The level spacing distribution becomes close to Poisson, and $N$3 approaches the Poisson value. In the black-hole regime, by contrast, the full spectrum is well described by GOE statistics. Near $N$4, the lowest levels become GOE-correlated rather abruptly, indicating a chaos transition that tracks the endpoint of the Hawking–Page line. One implication is that traversable-wormhole physics and quantum-chaotic ergodicity are not synonymous: the wormhole phase is not fully ergodic in the random-matrix sense, even though each uncoupled SYK copy is maximally chaotic in other diagnostics (García-García et al., 2019).

The spectral density also changes qualitatively across coupling regimes. At strong coupling, $N$5, the averaged spectral density

$N$6

breaks into $N$7 blobs centered on the eigenvalues $N$8 of the spin operator $N$9. Each blob has weight of order $\mu$0 and width $\mu$1. At weaker coupling these blobs merge into a continuous density without clear substructure. This reorganization reflects the increasing dominance of the one-body bilinear term over the SYK interaction as $\mu$2 grows (García-García et al., 2019).

5. Time-dependent couplings, quenches, and holographic real-time dynamics

The model admits a nontrivial extension to time-dependent left–right coupling,

$\mu$3

which can be analyzed systematically at large $\mu$4 on a closed Schwinger–Keldysh contour. After disorder averaging, the leading large-$\mu$5 saddle is governed by bilocal fields $\mu$6 and $\mu$7, and can be reformulated in terms of a complex reparameterization $\mu$8. Introducing

$\mu$9

one obtains a classical Hamiltonian

$q$00

with equations $q$01 and $q$02. This description interpolates between eternal-black-hole and traversable-wormhole trajectories within a single dynamical framework (Lensky et al., 2020).

A distinguished protocol is the “rescued black hole.” One starts from a thermofield-double state of the decoupled system, evolves for some time with $q$03, then turns on a carefully tuned time-dependent coupling so that the trajectory lands exactly at the fixed point corresponding to the coupled ground state. In the low-energy dual, this interpolates from a two-sided Rindler black hole to the global $q$04 vacuum. Comparison between the large-$q$05 solution and the low-energy Schwarzian description shows that if the rescue begins sufficiently late, even low-energy observables exhibit an $q$06 discrepancy from the naive JT-plus-free-matter effective theory. This was interpreted as evidence that the low-energy description fails near the inner horizon and may encode a two-dimensional analog of singularity formation (Lensky et al., 2020).

Periodic driving introduces a complementary nonequilibrium regime. One may modulate either the bilinear coupling,

$q$07

or the SYK coupling $q$08, and define a Floquet Hamiltonian by

$q$09

The driven phase diagram in the $q$10 plane contains a stable traversable-wormhole region with a finite gap, an unstable black-hole-like region in which the gap closes and the spectrum becomes continuous, and resonance structures that enhance or suppress traversability. The same setup allows computation of a drive-dependent Lyapunov exponent $q$11, with peaks near Floquet resonances and suppressed chaos in other regions. These results suggest that the competition between connected and disconnected phases can be controlled dynamically rather than only thermodynamically (Mimó, 20 Oct 2025).

6. Variants, extensions, and related research programs

Several deformations preserve core Maldacena–Qi phenomenology while modifying the microscopic symmetry content. In the imbalanced model,

$q$12

adiabaticity arguments and exact diagonalization indicate that the ground-state character remains close to the TFD over a broad range of imbalance. The many-body gap is essentially constant for small $q$13, and the low-energy wormhole phase retains a gap and revival dynamics. In the maximally imbalanced limit $q$14, one interacting SYK system is coupled to free Majorana zero modes. This limit preserves some signatures of wormhole physics, such as a nonzero gap and revivals, but loses others, including the Hawking–Page first-order transition, and was therefore described as marginal rather than a full-fledged wormhole dual (Haenel et al., 2021). A distinct $q$15-symmetric generalization replaces Majoranas by complex fermions with separate chemical potentials on the two sides. In the zero-temperature, small-coupling limit and for small average chemical potential, the ground state is a $q$16 eternal wormhole equivalent to a TFD; at higher chemical potential the system transitions to a black-hole phase, and the IR dynamics acquires both Schwarzian and $q$17 phase modes (Zhang, 2020).

Supersymmetric and non-Hermitian generalizations probe which features of the coupled model are structurally robust. Off-diagonal couplings between supersymmetric SYK copies can be chosen to preserve both supersymmetry and solvability in $q$18, $q$19, and proposed $q$20 versions. Their low-energy limits are governed by super-Schwarzian actions, and the model admits supersymmetric TFD states, transmission amplitudes, and multi-side OTOC diagnostics (Zhang et al., 2024). A non-Hermitian deformation obtained by a pseudo-Hermitian similarity transformation changes the left and right hopping amplitudes asymmetrically but leaves the energy spectrum, the gap scaling $q$21, the ground-state entanglement, the Schwarzian effective potential, and the free energy invariant with respect to the non-Hermiticity parameter $q$22 (Cai et al., 2022).

Recent work has also connected the model to questions about hidden order parameters and baby universes. A large-$q$23 formalism for the stabilizer Rényi entropy in the coupled SYK model found a series of first-order transitions as temperature is varied, including an intrinsic jump of the stabilizer Rényi entropy that does not appear in conventional thermodynamic observables, suggesting that “magic” can diagnose phase structure invisible to the free energy (Zhang et al., 22 Sep 2025). In another direction, a version of the Maldacena–Qi phase transition has been used in constructions of baby-universe states in JT gravity, where the coupled model supplies the wormhole branch and its first-order competition with a black-hole saddle (Sasieta et al., 28 Nov 2025). In the double-scaling limit, the partition function of the coupled model develops three saddle points—two thermal disks, a thermal $q$24 cylinder, and a cylinder with a handle—and the corresponding chord-diagram construction yields a Hartle–Hawking state with genuine tripartite entanglement between the two exterior regions and a baby-universe Hilbert space (Sontag et al., 6 May 2026).

The model has also become a target for quantum-state preparation protocols. For the $q$25 coupled SYK Hamiltonian, feedback-based algorithms such as FALQON and TR-FALQON were found to stall when initialized in trivial product states, whereas a hybrid ITE–TR-FALQON protocol incorporating an imaginary-time step reached fidelities $q$26 by $q$27–$q$28 layers for $q$29, and reproduced the von Neumann and Rényi entropies of the exact TFD state with high precision in simulations with $q$30 Majoranas per side averaged over $q$31 disorder realizations (Pexe et al., 2 Jul 2026). This computational line of work treats the Maldacena–Qi model not only as a holographic toy model but also as a benchmark for preparing strongly entangled thermal-like states on quantum hardware.