Josephson wormhole in coupled superconducting Yukawa-SYK metals

We show that two Yukawa-SYK models with a weak tunneling contact can have an exotic hybrid superconducting thermofield-double-like state that is holographically dual to a traversable wormhole connecting two black holes with charged scalar hair. The hybrid superconducting thermo-field-double/wormhole state is distinguishable by anomalous scaling of revival oscillations in the fermionic Green’s function, but also in a unique Andreev-revival in the anomalous Green’s function. The existence of this TFD/wormhole state surprisingly shows that the some quantum critical effects can survive the phase transition to superconductivity. This Andreev-revival is in principle an accessible signature of the transition to the TFD/wormhole phase detectable in the ac-Josephson current.

💡 Research Summary

The paper investigates whether a traversable wormhole phase, previously identified in coupled SYK models, can survive when the individual systems become superconducting. The authors focus on the Yukawa‑SYK (YSYK) model, a zero‑dimensional quantum dot containing N flavors of spin‑½ fermions coupled to M flavors of bosons through random four‑body interactions. By choosing the interaction matrix from the Gaussian Orthogonal Ensemble (GOE) they preserve time‑reversal symmetry, which allows the uncoupled YSYK to develop a superconducting ground state (α = 0). Two identical YSYK dots are then coupled via a weak single‑electron tunneling term λ c†₁c₂ + h.c., while sharing the same disorder realization.

In the large‑N, M limit the dynamics are captured by coupled Schwinger‑Dyson equations for a 4 × 4 fermionic Green’s function matrix ˆG and a bosonic propagator ˆD. The self‑energies Σ and anomalous self‑energies Φ depend on the disorder‑averaged coupling g², the tunneling λ, and the parameter α that distinguishes the metallic (α = 1) from the superconducting (α = 0) regime. The equations are solved numerically both on the Matsubara axis and after analytic continuation to real frequencies using Anderson iteration.

For the metallic case (α = 1) the phase diagram reproduces the known SYK‑wormhole physics: at temperatures above the SYK coherence scale T_SYK ≈ g²/ω₀² the system behaves as two essentially free fermion dots (labelled 2BH for “two black holes”). When T < T_SYK and the tunneling λ is below a critical value λ_WH, the system undergoes a first‑order transition to a thermofield‑double (TFD) state that is holographically dual to a traversable wormhole (WH). In this phase the diagonal Green’s function G_d(τ) decays exponentially, defining a gap E_gap that scales as λ^{1/(2‑2Δ_f)} (with Δ_f≈0.42, giving an exponent ≈0.86). The spectral function shows a ladder of equally spaced low‑energy peaks E_n = E_gap(1 + n/Δ), leading to revival oscillations in the time‑domain transmission amplitude T_{αβ}(t). These revivals are interpreted as a particle repeatedly traversing the wormhole.

When the GOE disorder is used (α = 0) each isolated YSYK becomes a superconductor below a critical temperature T_c ≈ T_SYK. Introducing the same tunneling λ produces a richer phase diagram. For very large λ the system remains a conventional gapped metal with no superconductivity because the density of states at zero energy vanishes. For small λ the superconducting gap Δ dominates and the system is in a standard superconducting phase (labelled SC). As λ is increased, pair‑breaking tunneling suppresses Δ; near a critical tunneling strength λ_c ≈ λ_WH the superconducting order collapses. Remarkably, in this regime the spectral function exhibits additional resonances that match those of the metallic TFD/WH state. The anomalous (Andreev) Green’s function F_{ab}(τ) also displays a pronounced revival peak at the same frequency, which the authors term an “Andreev‑revival”. Holographically this corresponds to an electron entering the wormhole and emerging as a coherent hole.

The Andreev‑revival has a direct experimental signature: it enhances the AC Josephson current I(ω) at the revival frequency. The authors derive that the current contains a term proportional to λ · F(ω), leading to a sharp peak in the frequency‑dependent Josephson response. Thus, measuring the AC Josephson effect in a weakly coupled YSYK‑based Josephson junction provides a feasible route to detect the wormhole phase.

Thermodynamically, the free energy F(T) shows a discontinuous derivative at the 2BH ↔ WH transition, confirming its first‑order nature, and a small hysteresis is observed when the system is annealed from high to low temperature and back. The full phase diagram in the (λ, T) plane contains four distinct regions: free‑fermion (FF), metallic SYK (2BH), wormhole (WH), superconducting (SC), and a hybrid Josephson‑wormhole phase (SC‑WH) that appears just before superconductivity is destroyed by strong tunneling.

Finally, the paper discusses realistic platforms where such physics could be realized. Disordered graphene flakes in strong magnetic fields, engineered quantum dot arrays, or high‑T_c cuprate‑like strange metals that exhibit SYK‑type quantum criticality are suggested as candidates. By fabricating a weak tunneling contact between two such systems and performing high‑resolution AC Josephson spectroscopy, one could observe the predicted revival peaks and thereby witness a table‑top realization of a traversable wormhole dual to a strongly correlated quantum critical state. This work thus bridges condensed‑matter superconductivity, quantum chaos, and holographic gravity, opening a concrete experimental pathway to probe exotic gravitational phenomena in solid‑state laboratories.