Josephson wormhole in coupled superconducting Yukawa-SYK metals

TL;DR

This work shows that two coupled Yukawa-SYK metals with shared disorder can host a finite-temperature wormhole state that is holographically dual to a traversable wormhole, and that this TFD/WH phase persists even when superconductivity is present. By solving the bilocal Schwinger-Dyson equations with a tunneling term $\lambda$, the authors identify a metallic TFD/WH regime linked to a first-order 2BH→WH transition and reveal how the superconducting sector undergoes a cross-over to a Josephson wormhole, accompanied by distinctive Andreev-revival features in both the normal and anomalous Green's functions. The Andreev-revival peaks and the associated enhancements in the AC Josephson current provide experimentally accessible signatures of the TFD/WH state, offering a concrete route to observe holographic wormhole physics in tabletop devices. The results connect quantum criticality, holography, and superconductivity, and suggest experimental platforms such as disordered graphene and mesoscopic cuprate grains for testing wormhole physics in condensed matter systems.

Abstract

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.

Figures (10)

• Figure 1: Schematic phase diagram of two coupled YSYK systems without (metallic, left) and with superconductivity (right), and $g_{\text{SYK}}/\omega_0^{3/2}\simeq 0.5$ in units of the bare boson mass. The configuration of two decoupled, strongly correlated YSYK systems is holographically dual to two thermal black holes (labeled 2BH) . It is separated by a first order phase transition from a low temperature thermofield-double state holographically dual to a wormhole (WH). For coupled superconducting YSYK systems, the system first transitions to a superconducting state (2BH-SC) and crosses over to a novel Josephson wormhole state (SC-WH) close to the tunneling strength where superconductivity is suppressed due to pair-breaking tunneling. At high temperatures or large tunneling coupling the coupled system is in a free-fermion phase (FF) that smoothly connects.
• Figure 2: TFD/WH state signatures in two tunnel-coupled metallic YSYK models with $g/\omega_0^{3/2}=0.5$. A: The density of states exhibits the regular linear spacing characteristic of the TFD/WH. B: The gap in both the fermionic and bosonic Green's function shows the the expected $\lambda^{\frac{1}{2-2\Delta_f}}$ scaling with $1/(2-2\Delta_f)=0.86$. The inset shows the exponential behavior of the imaginary time Green's functions and the purple dashed lines indicate the region where the fit was performed. C: As $\lambda$ decreases, the position of the leading and subleading peaks approach the analytical prediction $E_n/E_{\text{gap}}=(1+\frac{1}{\Delta}n)$ indicated by the dotted lines. D: Revival oscillations of the transmission amplitude $T_{\alpha\beta}(t) = \frac{2}{\pi}|G_{\alpha\beta}^>(t)|$, with $iG_{\alpha\beta}^>(\omega) = -[1-n_F(\omega)]\text{Im}G^R_{\alpha\beta}(\omega)$. $T_d$ and $T_{od}$ are perfectly out of phase. The characteristic frequency is $\omega_{re}= \frac{p_1}{2\pi}$, where $p_1$ is the average spacing between the peaks in the spectral function. In the gravitational description, this represents a particle traversing/reemerging from the wormhole.
• Figure 3: The free energy of two coupled metallic YSYKs with $\lambda = 0.05$, $g/\omega_0^{3/2}=0.5$ shows a first order transition at low $T$ from a 2BH to a WH configuration. (Right) The first order nature also shows up as a hysteresis between the solutions obtained from annealing from high to low temperature ($F_{H-L}$), and low to high temperature ($F_{L-H}$).
• Figure 4: (Left) Diagonal fermionic spectral function in the super-conducting YSYK model without tunneling (dashed) and with weak tunneling $\lambda = 0.05$ (solid) at low temperature, well below $T_{TFD}$. Compared to the contactless superconductor, the tunneling contacted superconductor shows extra resonances. (Right) The additional resonances can be identified with the metallic TFD/WH state (dashed, green) by tuning towards $\lambda_c$. The superconducting resonance splits into two peaks that disappear in the metallic state and a peak that is a higher wormhole resonance.
• Figure 5: A: Free energy of the coupled superconducting YSYK with $\lambda = 0.05$, $g/\omega_0^{3/2}=0.5$. We observe a second-order transition to the strongly coupled superconducting state at $T_c$, and then at a lower $T<T_c$ a cross-over --- to our numerical accuracy --- from this 2BH-SC to a Josephson wormhole configuration. $T_{\rm WH}$ denotes the temperature where the metallic non-superconducting coupled YSYKs show the first order transition to the TFD/WH state. B: Andreev-revival peaks characteristic of the TFD/WH state show up in the anomalous Green's function in comparison to the contactless superconductor (blue, dashed). C: This is detectable as a large increase in amplitude Im$\Pi^R_F(\omega)$ in the cosine term of the AC Josephson current ( D) when the bias voltage is equal to a TFD/WH resonance.