Maldacena–Qi Model: Coupled SYK Wormhole

• The Maldacena–Qi model is a coupled two-site SYK system deformed by a bilinear interaction, realizing an eternal traversable wormhole in nearly-AdS2 gravity.
• Its low-temperature regime exhibits a wormhole phase with global AdS2 geometry, while the high-temperature phase transitions to disconnected black holes.
• Tunneling spectroscopy reveals discrete, equally spaced resonances linked to the emergent SL(2) symmetry and low-energy reparametrization modes.

Searching arXiv for the cited Maldacena–Qi papers and related tunneling work. arXiv_search query: "(Maldacena et al., 2018) Eternal traversable wormhole Maldacena Qi" arXiv_search query: "(Zhou et al., 2020) Tunneling through an Eternal Traversable Wormhole" The Maldacena–Qi model is a coupled two-site Sachdev–Ye–Kitaev system introduced by Maldacena and Qi as a solvable realization of an eternal traversable wormhole in nearly-$AdS_2$ gravity. In its standard form, it consists of two identical SYK Hamiltonians, labeled left and right, deformed by a relevant bilinear interaction. In the low-temperature regime, the coupling gaps one combination of the two reparametrization modes and stabilizes a ground state dual to global $AdS_2$; in the high-temperature regime, the dominant saddle is a pair of disconnected black holes. The model therefore provides a setting in which the same large-$N$ quantum mechanics exhibits both wormhole and black-hole phases, together with an explicit boundary diagnostic in tunneling spectroscopy (Maldacena et al., 2018, Zhou et al., 2020).

1. Hamiltonian formulation

In the complex-fermion version, the Hamiltonian is the sum of two SYK$_4$ systems and a left–right bilinear coupling,

$H = H_L + H_R + H_{\rm int},$

with

$$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$

and

$$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$

The quartic couplings $J_{ij;kl}$ are independent Gaussian variables with

$$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$

and the large-$N$ limit is taken with $AdS_2$0 and $AdS_2$1 fixed (Zhou et al., 2020).

In the original Majorana formulation, the model is

$AdS_2$2

where each SYK Hamiltonian contains $AdS_2$3-body random interactions,

$AdS_2$4

In this form, the left–right bilinear is a relevant deformation, and at leading order in the melonic large-$AdS_2$5 expansion it enters the LR self-energy as

$AdS_2$6

(Maldacena et al., 2018).

A common shorthand is to regard the model as “two coupled SYKs,” but the defining feature is specifically the bilinear inter-copy coupling. That coupling is not a peripheral perturbation: it is the ingredient that stabilizes the low-temperature wormhole ground state.

2. Large-$AdS_2$7 dynamics and low-energy effective theory

After disorder averaging, the large-$AdS_2$8 solution is formulated in terms of bilocal fields

$AdS_2$9

and conjugate self-energies $N$0. At leading order, the two-point functions satisfy self-consistent Dyson equations in the melonic limit, and physical observables become self-averaging at $N$1 (Maldacena et al., 2018, Zhou et al., 2020).

At energies well below the SYK interaction scale, the relevant degrees of freedom are reparametrization modes. For a single copy, the Schwarzian action takes the form

$N$2

where

$N$3

is the Schwarzian derivative and $N$4 (Zhou et al., 2020).

For the coupled system, the low-energy action is a Schwarzian-plus-coupling theory,

$N$5

with $N$6 and $N$7 (Maldacena et al., 2018). In the two-site model, the coupling $N$8 gaps one linear combination of the two reparametrizations. The remaining collective sector is the one identified with the wormhole throat in the nearly-$N$9 dual.

This effective theory isolates the “gravitational” subsector shared by the coupled-SYK and nearly-$_4$0 descriptions. In that sense, the gravity dual is not added phenomenologically; it emerges from the same soft sector that controls the low-energy large-$_4$1 dynamics.

3. Nearly-$_4$2 dual and traversable wormhole geometry

In the gravity description, the model is dual to Jackiw–Teitelboim gravity with matter fields and a boundary interaction between the two asymptotic regions. Maldacena and Qi constructed a nearly-$_4$3 solution describing an eternal traversable wormhole. The wormhole is supported by negative null energy generated by quantum fields under the influence of the external coupling between the two boundaries (Maldacena et al., 2018).

In global $_4$4 coordinates,

$_4$5

the backreaction of the matter stress tensor modifies the dilaton profile without producing a horizon. The bulk remains causal between the two boundaries, and the two sides are in direct contact through a static throat (Maldacena et al., 2018).

In the low-temperature limit $_4$6, the dominant saddle of the two-site model is the global $_4$7 geometry, with boundary times related by a constant redshift and

$_4$8

in the complex-fermion version (Zhou et al., 2020). This regime is the wormhole phase. By contrast, the high-temperature phase is dual to two disconnected $_4$9 Rindler patches, corresponding to two black holes rather than a single connected geometry (Zhou et al., 2020).

A frequent misconception is to identify the low-temperature state with a thermofield double of two decoupled systems without qualification. The more precise statement is that the exact ground state of the coupled SYK model at small $H = H_L + H_R + H_{\rm int},$0 is very close to a thermofield double of the two decoupled SYKs at an effective temperature $H = H_L + H_R + H_{\rm int},$1, while the coupled system itself possesses a distinct interacting Hamiltonian and a stabilized traversable wormhole ground state (Maldacena et al., 2018).

4. Symmetry, spectrum, and correlators

The low-temperature bulk geometry has an exact $H = H_L + H_R + H_{\rm int},$2 isometry. In the complex-fermion analysis, this symmetry fixes the small-fluctuation spectrum to a tower of equally spaced poles,

$H = H_L + H_R + H_{\rm int},$3

which is the signature later seen in the tunneling observables (Zhou et al., 2020).

In the low-energy effective theory, expanding around the symmetric solution $H = H_L + H_R + H_{\rm int},$4 gives an $H = H_L + H_R + H_{\rm int},$5-singlet “boundary graviton” sector with potential

$H = H_L + H_R + H_{\rm int},$6

which has a stable minimum. The same expansion yields $H = H_L + H_R + H_{\rm int},$7-multiplets of matter modes of weight $H = H_L + H_R + H_{\rm int},$8, with exact global-time spectrum

$H = H_L + H_R + H_{\rm int},$9

rescaling on the boundary to

$$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$0

(Maldacena et al., 2018).

Boundary correlators reflect this discrete structure. For a matter operator of weight $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$1,

$$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$2

while

$$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$3

The same correlators imply exponential decay at late real times and are reproduced by bulk propagation through a global $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$4 wormhole (Maldacena et al., 2018).

The significance of the equally spaced poles is specific. They are not merely finite-size resonances; in the bulk interpretation they are normal-mode resonances of global $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$5, and their spacing is fixed by $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$6 symmetry (Zhou et al., 2020).

5. Phase structure, thermodynamics, and ensemble dependence

The model has two canonical phases. At low temperature, the preferred saddle is the eternal traversable wormhole, in which the two SYK boundaries are connected in the bulk. At high temperature, the preferred saddle is a pair of disconnected black holes, with each SYK dual to an $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$7 Rindler patch (Zhou et al., 2020).

At fixed $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$8, the temperature-driven transition between these phases is first order. The transition temperature satisfies $$H_{L} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{L,i}^\dagger c_{L,j}^\dagger\,c_{L,k}\,c_{L,l}, \qquad H_{R} \;=\;\sum_{i<j<k<l}J_{ij;kl}\,c_{R,i}^\dagger c_{R,j}^\dagger\,c_{R,k}\,c_{R,l},$$9, and numerically one finds

$$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$0

for small $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$1 by locating the crossing point of the free energies $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$2 and $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$3 (Zhou et al., 2020). Maldacena and Qi describe the same phenomenon as a Hawking–Page-like transition between a low-temperature thermal $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$4 phase and a high-temperature black-hole phase (Maldacena et al., 2018).

The canonical low-temperature partition function is dominated by the ground-state energy,

$$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$5

with entropy small and zero at $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$6 up to boundary-graviton degeneracy. The high-temperature phase has large entropy, $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$7, where $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$8 is the extremal entropy of each black hole (Maldacena et al., 2018).

An apparent tension between “first-order transition” and “continuous connection” is resolved by ensemble dependence. In the canonical ensemble there is a first-order transition, but in the microcanonical ensemble $$H_{\rm int} \;=\;\mu\,\sum_{i=1}^N\Bigl(c^\dagger_{L,i}c^{}_{R,i}+c^\dagger_{R,i}c^{}_{L,i}\Bigr).$$9 is smooth and monotonic, so the low- and high-temperature phases are continuously connected (Maldacena et al., 2018). This distinction is central to the thermodynamics of the model.

6. Tunneling spectroscopy and boundary diagnostics

A direct probe of the Maldacena–Qi wormhole phase is obtained by coupling each SYK site to a non-interacting lead of density of states $J_{ij;kl}$0, with tunnel rate

$J_{ij;kl}$1

The relevant boundary objects are the retarded Green’s functions $J_{ij;kl}$2, $J_{ij;kl}$3, and the Schwinger–Dyson equation for the leads gives an exact tunneling probability $J_{ij;kl}$4 together with the current

$J_{ij;kl}$5

(Zhou et al., 2020).

In the wormhole phase at $J_{ij;kl}$6, the conformal retarded functions can be written in closed form in terms of

$J_{ij;kl}$7

and the resulting tunneling probability exhibits narrow resonances at

$J_{ij;kl}$8

The differential conductance $J_{ij;kl}$9 correspondingly shows peaks at $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$0 (Zhou et al., 2020).

In the black-hole phase $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$1, to leading order in $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$2,

$$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$3

and the tunneling probability becomes a single peak near $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$4 that scales as $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$5 (Zhou et al., 2020). No equally spaced structure survives, because the disconnected black-hole geometry lacks the global $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$6 box that produces the tower of normal modes.

The bulk and boundary calculations agree in both phases. In the wormhole geometry, solving the Dirac equation in global $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$7, matching ingoing waves at the two asymptotic boundaries to the flat-space leads, and imposing smoothness reproduces the boundary tunneling formula once the infrared cutoff is identified as

$$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$8

In the black-hole geometry, a perturbative matching on two disconnected Rindler patches reproduces the same high-temperature expression for $$\overline{J_{ij;kl}}=0, \qquad \overline{|J_{ij;kl}|^2}=\frac{2J^2}{N^3},$$9 (Zhou et al., 2020).

The experimental implication advanced in the tunneling analysis is precise: equally spaced peaks in $N$0 at $N$1 would directly probe the $N$2 spectrum of nearly-$N$3, whereas the disconnected phase shows only a broad zero-bias peak suppressed by $N$4. Potential realizations mentioned include cold-atom or solid-state quantum dots engineered to mimic complex SYK interactions, each dot coupled to metallic leads (Zhou et al., 2020).