Ephemeral Superconductivity Atop the False Vacuum
Gal Shavit Stevan Nadj-Perge Gil Refael




Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
Walter Burke Institute of Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA

Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
T. J. Watson Laboratory of Applied Physics, California Institute of
  Technology, 1200 East California Boulevard, Pasadena, California 91125, USA

Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA

§ ABSTRACT 
A many body system in the vicinity of a first-order phase transition may get trapped in a local minimum of the free energy landscape.
These so-called false-vacuum states may survive for exceedingly long times if the barrier for their decay is high enough.
The rich phase diagram obtained in graphene multilayer devices presents a unique opportunity to explore transient superconductivity on top of a correlated false vacuum.
Specifically, we consider superconductors which are terminated by an apparent first-order phase transition to a correlated phase with different symmetry.
We propose that quenching across this transition leads to a non-equilibrium ephemeral superconductor, readily detectable using straightforward transport measurements.
Besides enabling a simple detection scheme, the transient superconductor also generically enhances the false vacuum lifetime, potentially by orders of magnitude.
In several scenarios, the complimentary effect takes place as well: superconductivity is temporarily emboldened in the false vacuum, albeit ultimately decaying.
We demonstrate the applicability of these claims for two different instances of superconductivity terminated by a first order transition in rhombohedral graphene.
The obtained decay timescales position this class of materials as a promising playground to unambiguously realize and measure non-equilibrium superconductivity.


Ephemeral Superconductivity Atop the False Vacuum
    Gil Refael
    September 9, 2024
=================================================





§ INTRODUCTION




Phase transitions in many-body correlated systems may often be succinctly described by an appropriate classical or quantum field theory <cit.>.
The equilibrium many-body ground state is identified by the global minimum of the free-energy associated with this description.
However, other locally-stable minima may exist, albeit with a higher energy density.

These minima act as the “false-vacuum” (FV) of the system, and may be long-lived due to their metastable nature.
A system can generically get trapped in the FV state when it is quenched through a first-order phase transition.
Supercooling and superheating of water are well-known classical examples of this phenomenon <cit.>, yet quantum systems such as spin chains <cit.>, superconducting wires <cit.>, and atomic superfluids <cit.> 
have been show to exhibit metastable phases and FV decay.

Further, the FV concept itself has originated in the context of cosmology, where it may have truly dire implications <cit.>.

In recent years, graphene multilayers have emerged as an exciting platform with a high degree of tunability to study correlated electron phenomena, topological phases, unusual superconductivity, and their interplay <cit.>.
A recurring theme in these systems is the peculiar vicinity of the superconducting phases to symmetry-breaking phase transitions.
In several cases, the superconducting dome itself is terminated by an abrupt transition where the Fermi surface undergoes significant reconstruction <cit.>, which in some instances is strongly indicative of a first-order phase transition <cit.>.
Recently, similar phenomenology was observed in twisted bilayer WSe_2, where hysteretic behavior was observed at the boundary between superconductivity an a correlated phase <cit.>.


In this work, we propose these materials as a platform for realizing and exploring out-of-equilibrium superconductivity, which exists as a metastable phase on top of the FV manifold of the symmetry-broken phase.


This extraordinary non-equilibrium metastable state arises in the vicinity of the true vacuum symmetry-broken phase.
A useful heuristic of the sort of scenarios we discuss is presented in Fig. <ref>.
In this generic phase diagram, superconductivity and a correlated phase are in close proximity, separated by a first-order transition line.
Specifically, we are interested in cases where the correlated phase preempts superconductivity and 
overtakes it.

Thus, after a sudden quench from the superconducting phase across the transition (black arrow in Fig. <ref>), there exists a possibility of a long-lived transient superconductor, realized on top of the FV.
Alternatively, we also consider quenching from a parent normal phase directly into the suppressed superconductor through a first-order transition (green arrow).
Such a protocol may allow one to 
reveal buried underlying

superconductivity in such systems, masked by competing phases.

We underline the regimes where FV superconductivity are most relevant and experimentally accessible.
This is accomplished by combining microscopic calculations for two candidate materials, rhombohedral trilayer graphene (RTG) and Bernal-stacked bilayer graphene (BBG), and a field theoretical description of the FV decay phenomena.
We estimate the expected lifetimes of the ephemeral superconductors to be of the order of ∼ 100 nanoseconds, enabling straightforward detection methods, relying on time-resolved transport measurements.
Remarkably, the unusual presence of superconductivity in the FV state is what enables such simplified detection schemes in a solid-state setting.
Superconductivity provides an unambiguous transport signal – a delay between a current driven through the system and the appearance of a voltage drop.

Furthermore, we show that the incompatibility between the correlated symmetry-broken phase and the superconductor non-trivially enhances the stability of the FV and its lifetime.
As we show, this is a generic feature in scenarios where a subordinate phase develops on top of a “primary” false vacuum.
This may be understood as a result of magnification of the surface tension between the true and false vacuum states of the system.
The strength of surface tension plays a major role in determining the energetics of the FV decay.

Interestingly, the FV superconductivity may actually survive at higher temperatures compared to its equilibrium counterpart on the other side of the transition.
This transient enhancement of superconductivity comes at a cost of incurring a finite lifetime.
Generically, what drives the symmetry-breaking transition which terminates the superconductor is the density of states (DOS) near the Fermi level ν.
Clearly, it is also an important factor in determination of the superconducting properties.
For example, conventionally the superconducting transition temperature T_c∝exp(-1/uν) (u is the pairing strength).
For weak-coupling superconductors, uν≪ 1, T_c is especially sensitive to ν.
In the FV, the superconductor temporarily experiences a higher DOS while the correlated phase is suppressed.
This facilitates favorable superconducting properties, potentially beyond those available at equilibrium under similar conditions. 

The rest of the paper is organized as follows.
In Sec. <ref> we present the general theoretical scheme which we use to estimate the decay of the FV, and how its lifetime depends on properties of the relevant incompatible phases.
We then apply these phenomenological tools to microscopic calculations performed for the graphene devices in Sec. <ref>, considering different symmetry-breaking possibilities, and different transition scenarios.
We propose a simple experimental transport-based scheme to characterize the transient superconductivity in Sec. <ref>.
Finally, we summarize our results and conclude the discussion in Sec. <ref>.



§ FALSE VACUUM DECAY


We consider scenarios where as a function of some tuning parameter, r (which can be a magnetic field, an electric displacement field, pressure, etc.), a system undergoes a first-order phase transition with an order parameter ϕ̂.
Within this ϕ̂-ordered phase, it is further assumed that the disordered (e.g., paramagnetic) phase

remains a metastable local minimum of the effective free energy.

Let us also examine the consequences of an additional phase, with order parameter Ψ̂, which (i) also condenses in the vicinity of the ϕ̂ phase transition, and (ii) is incompatible with the ϕ̂ phase.
Generally, we will be interested in the case where the energy scales associated with ϕ̂ dominate over those of Ψ̂, and the ϕ̂ order is the equilibrium ground state, and the so-called “true vacuum”.


The scenario we describe above may be captured by the following Ginzburg-Landau free energy functional, 

    F	   =∫ d^2x[σ/2|∇ϕ|^2+16gϕ^2(ϕ-1)^2-Bϕ^2]
       +∫ d^2x[κ/2|∇Ψ|^2-a(r)/2|Ψ|^2+b/4|Ψ|^4] 
       +λ/2∫ d^2x|Ψ|^2ϕ^2.

Clearly, the first-order transition is governed by the parameter denoted as B.
When B<0, ϕ=0 (the paramagnetic phase where the spontaneous ordering of ϕ̂ does not yet occur)

minimizes the free energy in the absence of Ψ̂.
At B=0, the minima at ϕ=0 , 1 are degenerate, and the ordered phase takes over at B>0, where ϕ≈ 1 becomes the global minimum.
In the metastable regime B<B_c=16g, there exists a finite energy barrier between the two minima, whose strength is g right at the transition.
The stiffness σ gives rise to a surface tension between domains of ϕ=0 and ϕ≈ 1,

    J_ϕ = ∫_0^ϕ_>0 dϕ√(2σ V(ϕ)),

where V(ϕ)=16gϕ^2(ϕ-1)^2-Bϕ^2, and ϕ_>0 is the maximal ϕ for which V(ϕ)>0.
The correlation length is calculated by optimizing the energy of a domain-wall for B=0 (Appendix <ref>), ξ_ϕ=√(σ/8g).

In the second part of F, a(r) is associated with the second-order transition.
When a>0, the decoupled uniform solution is |Ψ|=Ψ_0√(a/b).
Here the appropriate correlation (coherence) length is ξ_Ψ=√(κ/(2a)), and the surface tension is 
J_Ψ = 8/3ξ_Ψa^2/4b.
Finally, λ>0 couples the two sectors, and precludes ϕ̂-Ψ̂ order coexistence when it is sufficiently large as compared to the energy densities g and a.






 §.§ Critical droplet analysis

We consider the system described by Eq. (<ref>), initialized to the ordered (superconducting) Ψ_0 phase, yet tuned to metastability, a>0 and 0<B<B_c.
The true vacuum is ϕ≈1, yet the initial state of the system is a locally stable minimum of the free energy.
We schematically evaluate the characteristic time necessary for the system to decay to the true vacuum, adapting the methods developed by Langer <cit.> and Coleman <cit.>.

Neglecting quantum fluctuations (whose role we later discuss), 
relaxation is facilitated by thermal fluctuations of ϕ̂ droplets (or “bounces” <cit.>) within the Ψ̂ bulk, see illustration in Fig. <ref>a.
The energy of a droplet with radius R may be evaluated in the so-called thin-wall limit (R≫ξ_ϕ) by <cit.> 

    E_ droplet   =π R^2 f_ true
        -π(R+δ R)^2  f_ meta
       +2π R J_ϕ + 2π(R+δ R) J_Ψ,

where f_ true=-B, f_ meta=-a^2/4b, and δ R ≈(ξ_ϕ+ξ_Ψ)/2, due to the necessary suppression of the Ψ̂ preceding the ϕ̂ droplet formation.
The droplet experiences an effective force ∝∂ E_ droplet/∂ R, pushing it towards either expansion or collapse, thus determining its fate.
Thus, the droplet energy threshold, i.e., the droplet energy at which it would tend to overtake the system, will be given by 

    E_ thresh.=E_ droplet(R_c),
 
and

    R_c=J_ϕ+J_Ψ-δ Rf_ meta/f_ meta-f_ true,

determined by ∂ E_ droplet/∂ R |_R_c=0 (Fig. <ref>b).
The expression for R_c accounts for two effects, which are solely due to the ordered Ψ̂ phase, on top of the FV.

First, the denominator in Eq. (<ref>) is made smaller  by the presence of the finite condensation energy of Ψ̂.
In our regime of interest, as we discuss below, one usually finds |f_ true|≫|f_ meta|, and this effect is of vanishing importance.

Second, the additional J_Ψ in the numerator, associated with the surface tension of the secondary (superconducting) phase, is reasonably expected to be rather small compared to J_ϕ, the surface tension contribution of the dominant correlated order, yet not negligible.
This is because the surface tension is roughly proportional to the product of the correlation length and the energy scale associated with the ordered phase.
While the latter might be much bigger for ϕ̂ as compared to Ψ̂, this difference is usually somewhat compensated by the ration of correlation lengths, ξ_Ψ≫ξ_ϕ.
In the regime where the denominator effect is negligible, one may approximate the threshold droplet radius
R_c≈ J_ϕ/B (1+c J_Ψ/J_ϕ), where c is an order-1 numerical factor which depends on microscopic details.
As we demonstrate for the particular systems we consider below, this mechanism indeed facilitates a more stable false-vacuum in the presence of the secondary Ψ̂ phase, which in our case is a superconductor.



 §.§ Effects of quantum fluctuations


Our analysis utilizes the classical Ginzburg-Landau functional, Eq. (<ref>), producing a free-energy barrier towards nucleation and eventual FV decay.
Let us discuss the role of quantum fluctuations on the phenomenon described here.
In the zero temperature limit, β=1/T→∞, any finite energy barrier completely annihilates decay through thermal fluctuations.
However, fluctuations due to the quantum nature of the ϕ̂ field may overcome this limitation.

Setting aside Ψ̂ for a moment, one interprets the ϕ̂ part of Eq. (<ref>) as the classical limit of the imaginary-time action
S_ϕ=∫_0^β dτ∫ d^dx 
[ρ/2(∂_τϕ)^2+σ/2|∇ϕ|^2+V(ϕ)].
This corresponds to the imaginary-time path integral partition function
Z=∫ Dϕ e^-S_ϕ.
In the ρ→∞ limit, variations along the τ dimension are suppressed, and one recovers the classical limit (with a prefactor of β replacing ∫ dτ integration).

Performing similar nucleation calculus in Euclidean space-time (see Appendix <ref>), one finds a temperature scale, T_Q, below which the quantum decay pathway is dominant,

    T_Q=3/32Bξ_ϕ/J_ϕτ_ϕ^-1,

with τ_ϕ=√(ρ/8g).
We stress that below T_Q the FV phenomenon persists, yet the lifespan of the metastable phase saturates, and remains roughly the same as the temperature is lowered further.

Notably, at any finite temperature, a small enough bias B exists such that the quantum decay is much less efficient.
The reason for this dependence on the bias B stems from the effective higher dimension of the quantum problem, where the surface of the droplet is d-dimensional, whereas classically it is (d-1)-dimensional.
As a consequence (Appendix <ref>), the threshold energy scales as B^-d in the quantum case (B^1-d classically).
Thus, when the bias between the false and true vacuum is small enough, the quantum decay pathway is highly disfavored.
Without loss of generality, we henceforth assume this is the case for our detailed microscopic analysis below.

Another interesting possibility occurs in an intermediate temperature regime, where quantum behavior dominates the ϕ̂ sector, yet Ψ̂  (now introduced back in our discussion), remains effectively classical due to its much longer correlation time τ_Ψ.
The relevant temperature regime is
τ_ϕ/τ_Ψ<T/T_Q<1.
Heuristically, it takes considerably lower temperatures (as compared to T_Q) to saturate the effects of the Ψ̂ field redistribution energetics.

This regime thus enables further enhancement of the FV stability by the competing sub-leading order, as discussed in Appendix <ref>.
Once more, this enhancement takes place even if the discrepancy between 
f_ true
and
f_ meta
is of orders of magnitude, due to combination of the correlation length effect ξ_Ψ≫ξ_ϕ (already discussed above), and a ∝ T_Q/T prefactor to the relative Ψ̂ contribution.
Thus, the lower the temperature in this regime, the larger the FV life-time enhancement becomes due to the ordering competition.



§ REALIZATION IN GRAPHENE MULTILAYERS


We consider first-order phase boundaries observed in both BBG and RTG separating correlated symmetry-broken phases and superconductivity, either incipient or fully formed.
In all cases, we follow the same methodology allowing us to extract the relevant energy densities and surface tensions.

We begin with an Hamiltonian of the form
 
    H= H_0 + H_ int,

where H_0 describes the appropriate band structure, and 
H_ int=U_c/Ω∑_ qρ_ qρ_ -q 
is the electron-electron interaction, which we take as short-range for simplicity (ρ_ q is the momentum-q component of the density in the relevant band, U_c is the interaction strength, and Ω is the system area).
At a given density n_ tot, we compute the free energy as a function of the relevant order parameter ϕ̂, 

    F(ϕ) = ⟨ H⟩_ HF,(ϕ,n_ tot)
        -
        ⟨ H⟩_ HF,(0,n_ tot),

where ⟨⟩_ HF,(ϕ,n) is the Hartree-Fock expectation value for ϕ̂=ϕ at fixed density n.
We use the zero-temperature expression, as the temperature is assumed to be far below the ϕ̂ phase transition.
Contributions to Eq. (<ref>) due to fluctuations around the mean-field solution, as well as due to other competing instabilities are beyond the current scope of this work.

As the energy scales associated with the order parameter jumps are comparable to the Fermi energy (see below) we will approximate the correlation length as the inter-particle separation, i.e., ξ_ϕ≈ n_ tot^-1/2.
The stiffness is thus approximated by σ≈ 8 ξ_ϕ^2 F^*, with the barrier height F^*=max F( ϕ∈[0,ϕ_0 ]).
ϕ_0 is the global minima of F, and we approximate B ≈ -F(ϕ_0).

In the normal state, we evaluate two crucial quantities regarding superconductivity.
Namely, the critical temperature T_c, and the superconducting condensation energy, playing the role of a^2/(4b) in our discussion above, Eq. (<ref>).
The superconducting coherence length is taken as a phenomenological parameters from experiments, and assumed to scale with T_c in the conventional manner.
The details of our superconducting calculations are presented in Appendix <ref>, where we use general considerations and avoid making assumptions on the origin of the pairing glue.
Moreover, we refrain from pinpointing the exact mechanisms by which the transition into the correlated phase extinguishes superconductivity, and keep our discussion as general as possible.
(In Appendix <ref>. we discuss a curious scenario where the transition suppresses the superconducting phase through a combination of substantial DOS reconstruction and retardation effects.)

Intervalley coherence in rhombohedral trilayer grpahene (RTG).—
A promising candidate to observe the phenomenon introduced here is the superconductor region denoted as SC1 in Ref. <cit.>.
Its boundary in the n_ tot–D plane (where D is the perpendicular displacement field) coincides with a transition to a symmetry-broken correlated phase, which appears to be first-order <cit.>.
Though the nature of the correlated phase terminating SC1 has not been confirmed, it is somewhat constrained.
The lack of spin and orbital ferromagnetism suggests either spin-valley locking, an intervalley-coherent phase (IVC), or combination thereof.
Theoretical studies have shown the IVC to be robust and ubiquitous throughout the phase diagram <cit.>, steering our focus to the IVC case for simplicity.



In Fig. <ref>a we plot the Hartree-Fock energy landscape as a function of the IVC order parameter and displacement field.
Moving from low to high fields, one clearly observes a region of metastability: the global minimum is at ∼ 3 meV, whilst the normal state remains locally stable.
At high enough values of D, the normal state finally becomes unstable.

Next, we compute the false-vacuum decay threshold, shown in Fig. <ref>b.
As expected, close to the transition point it diverges, due to a vanishing energy difference between the FV and the true vacuum, driving the critical droplet size increasingly larger [Eq. (<ref>)].
Notably, the threshold for decay becomes much smaller when the FV state is not superconducting, due to the surface tension contribution of the superconductor.
As shown in the inset, the relative strength of J_Ψ increases, signaling two effects.
The first is a decrease in the ordered phase surface tension J_ϕ [Eq. (<ref>)] as the bias between the true and false vacuum increases.
The second is a result of enhanced superconducting T_c, owing to an enhancement of DOS near the Fermi level as one moves deeper into the ordered state.

We now turn to estimate the life-time of this false vacuum,

    τ_ decay∼τ_0 e^β E_ thresh.,

where τ_0 is the much debated <cit.> fluctuation time-scale prefactor.
For simplicity, we take the worst-case-scenario, and estimate it as the time two electrons separated by a correlation length can “know about eachother”,
τ_0≈ξ_ϕ / v_F ≈ 10^-13 sec.
Taking a reasonable β E_ thresh.∼ 15-20 leads to time scales
τ_ decay∼ O(100   nsec - 10 μ sec).

Stoner blockade in Bernal bilayer graphene (BBG).—
The phase diagram of BBG has been shown to hosts a multitude of sharp phase transitions, as well as superconductivity in the presence of either an in-plane magnetic field or proximity to WSe_2 <cit.>.
The phenomenology is suggestive of a vicinity of a superconductive phase in the absence of these two perturbations, where the formation of a competing correlated phase suppresses it <cit.>.
We consider an alternative route to circumvent the presumed correlated phase, by quenching across the transition.

We focus on the vicinity of the magnetic-field-induced superconducting regime, where phenomenology is consistent with a fourfold to twofold degenerate Stoner transition.
The Hartree-Fock free energy as a function of the polarization order parameter is shown in Fig. <ref>a.
A metastable regime is clearly developed near the transition.
The normal-state energetic are quite similar to those obtained for the RTG case, and lead to similar energy threshold, even when accounting for a lower T_c of the superconductor observed in experiments (see Appendix <ref>).

There is a notable difference in this case compared to the RTG scenario above.
Here, the system is in the normal state after the quench.
It decays to the correlated phase with the characteristic τ_ decay, yet superconductivity may form more rapidly, as it does not need to overcome an energy barrier.
We estimate the superconductor formation rate as γ_Ψ̂∼Δ_0, i.e., roughly proportional to the equilibrium superconducting gap <cit.>.
Combining the decay rate γ=τ_ decay^-1, γ_Ψ̂, and the decay rate in the absence of superconductivity γ_0, we may approximate the probability of the system being in the superconducting false vacuum (Appendix <ref>),

    p_ false(t)=γ_Ψ̂/γ_Ψ̂+γ_0-γ[1-e^-(γ_Ψ̂+γ_0-γ)t]e^-γ t.

As demonstrated in Fig. <ref>b, at early times superconductivity builds up at a rate ∼γ_Ψ̂ to some finite fraction, and decays at long times at the rate γ.
Our calculations indicate superconductivity should remain visible up to O(100) nsec timescales.



In principle, for the specific scenario described in this section, where the system is quenched rapidly through a superconducting phase transition, one should take into account the formation of topological defects (vortices) due to the Kibble-Zurek mechanism <cit.>.
Physically, one expect that following the quench, independently coherent domains of size ∼ξ_Ψ form,
and coalesce at a time-scale
τ_GL≈π/(8|T_c-T|)^-1 <cit.>, the Ginzburg-Landau relaxation time <cit.>.
We note that τ_GL is of the same order of γ_Ψ̂^-1.
The relaxation dynamics of these defects, as well as oscillations in the order parameter magnitude <cit.> are presumed to play a secondary role, and are thus beyond the current scope of this work.




§ DETECTING EPHEMERAL SUPERCONDUCTIVITY

 

Thus far we have demonstrated the stabilization of the false vacuum by superconductivity, as well as the potential enhancement of the superconducting state in the metastable regime.
We now explore another intriguing consequence of the physics studied in this work.
Namely, the superconducting nature of the false vacuum make the phenomenon readily accessible to transport measurements.

Consider an experimental setup similar to the one pioneered in Ref. <cit.>, depicted in the inset of Fig. <ref>.
The tuning parameter in the system, e.g., displacement field, is quenched from time -t_ quench to t=0.
At t=0 a current pulse is fed to the device, and the transient voltage response is recorded.
At a time t=t_ delay voltage drop is eventually observed.
originally, the experiment was tailored to measure the relaxation time of equilibrium superconductors, and thus no signal would appear if the amplitude of the current pulse is less than the superconducting critical current I_c.
Here, the situation is made significantly richer by the decay of superconductivity itself.

Under some simple yet reasonable assumptions (Appendix <ref>), we find the delay as the time at which f(t_ delay)=0, with the initial conditions f(0)=1, where f obeys the time-evolution,  

    τ_GL∂_tf=-e^2t/τ_ decay(I/I_c)^24/27f^3+f(1-f^2).




In Fig. <ref> we demonstrate the dependence of the voltage signal delay time on the amplitude of the injected current.
If the false vacuum state is stable enough, a noticeable uptick in the delay is made clearly visible for currents slightly below the equilibrium critical values.
At relatively small values of the probe current, the expected delay is of the same order as the false-vacuum decay time τ_ decay.



§ CONCLUSIONS


Ephemeral superconductivity is a fascinating non-equilibrium state of matter, which is yet to be fully clarified <cit.>.
Here, we explore an unusual striking phenomenon: Superconductivity developing on top of a correlated false-vacuum manifold.
In contrast with previous scenarios where superconductivity is terminated by a first-order phase transition <cit.>, the transient superconductor is temporarily protected by the correlated FV.
Furthermore, the FV can facilitate the formation of a superconductor followed bu its decay (Fig. <ref>b), in otherwise superconductivity-excluded areas of the equilibrium phase diagram.
The proposed non-equilibrium superconductor is incompatible with the true many-body ground state of the system, and is thus ephemeral albeit metastable.

We elucidated how this FV superconductor can remarkably and significantly enhance the false-vacuum lifespan, possibly by a few orders of magnitude.
This enhancement occurs even if the superconducting condensation energy is negligible as compared to that of the correlated true vacuum, as we demonstrated by utilizing a general phenomenological framework [Eq. (<ref>)].
This unique effect is rooted in a notable surface tension contribution of the superconductor to the nucleation of critical threshold droplets, which facilitate the decay <cit.>.
The prospect of FV stabilization by a generic secondary order, which is adversarial to the corresponding true vacuum, is left for future work, as well as a broader view on quantum effects and the hybrid quantum-classical regime.

We demonstrated how the ephemeral superconducting state can be realized in multilayer graphene devices.
These systems ubiquitously show the coexistence of superconducting regions in the phase diagram, and apparent first-order phase transitions to correlated symmetry-broken phases <cit.>.
On occasions, these transitions actually terminate the superconducting dome, making such devices an ideal platform to explore the intertwined false-vacuum physics.
Combining Hartree-Fock calculations for the symmetry-broken transition energetics, theory of the superconductivity transition, and phenomenology obtained from experiments, we estimate the relevant FV  lifetime for observing the FV decay to be ≳ O(100  nsec),
rendering the decay process as much slower as compared to recently observed transient superconducting phenomena <cit.>. 

Another intriguing prospect in these graphene systems, is that competition with a correlated phase transition arguably undermines the potential equilibrium superconducting orders.
(This is reminiscent of the competing phases in high-T_C superconductors, e.g., <cit.>.)
The multilayer graphene superconductors were observed to be stabilized by some perturbations, e.g., magnetic fields or proximity to a non-trivial substrate.
However, we have pointed out a different route for bringing these low-lying superconductors to light: transiently suppressing the symmetry-broken state by quenching the system through the first-order transition.
The motivation for such a process to be feasible is simple.
The same factor that participates in driving the transitions, the DOS near the Fermi energy, is the one that is expected to enhance superconductivity when given the chance (see related discussion in Refs. <cit.>).

An important consequence of ephemeral superconductivity “dressing” the metastable false-vacuum is that direct experimental detection of its decay becomes simple and experimentally accessible.
The long time-scales for the superconducting life-times mentioned above are in fact conducive to straightforward transport measurements.
Resolution of ∼ 1 nsec in the voltage-delay experimental scheme we detail in Sec. <ref> can readily be obtained.
Tracking the temporal signal due to the decay of the superconductor, and as a consequence the establishment of the equilibrium ground-state, may be achieved in these graphene systems (as well as in another candidate system, twisted WSe_2 <cit.>), without the need of sophisticated light-based apparatus.
Moreover, conclusive transport signatures of the kind we discussed obviate many of the ambiguities associated with photo-measurements of non-equilibrium superconductors <cit.>. 
These facts, along with the ability to electrically tune them across the phase diagram, make highly-correlated multilayer graphene devices ideal playgrounds for manipulating and probing the false-vacuum decay in the context of solid-state systems.


We thank John Curtis, Thomas Weitz, and Erez Berg for enlightening discussions.
S.N.-P. acknowledges support from the National Science Foundation (grant number DMR-1753306). G.R. and S.N.-P. also acknowledge the support of the Institute for 
Quantum Information and Matter, an NSF Physics Frontiers 
Center (PHY-2317110).
GS acknowledges support from the Walter Burke Institute for Theoretical Physics at Caltech, and from the Yad Hanadiv Foundation through the Rothschild fellowship.
Part of this work was done at the Aspen Center for Physics, which is supported by the NSF grant PHY-1607611.


aapmrev4-2






§ PHENOMENOLOGICAL MODEL



 §.§ Surface tension calculations

Given the free energy

    F_ϕ=∫ d^2x[σ/2|∇ϕ|^2+16gϕ^2(ϕ-1)^2],

we calculate the energy associated with a domain-wall solution of extent ξ, ϕ(x)=1/2( tanhx/ξ+1) per unit length along the domain wall direction L_y,

    F_ϕ/L_y=[σ/8ξ+gξ]∫_-∞^∞dz1/cosh^4z.

Minimizing with respect to ξ, we obtain the correlation length
ξ_ϕ=√(σ/8g), and the surface tension
J_ϕ^0=8/3gξ_ϕ=√(8σ g/9).

When an additional bias is added to F_ϕ favoring the ordered phase this surface tension is only approximately correct.
The bias always tends to lower the surface tension, and can generally be calculated by integrating over the effective equations of motion, i.e.,

    J_ϕ = ∫_0^ϕ_>0 dϕ√(2σ V(ϕ))
        = 4gξ_ϕ∫_0^ϕ_>0 dϕ√(V(ϕ)/g),

where V(ϕ)=16gϕ^2(ϕ-1)^2-Bϕ^2, and ϕ_>0 is the maximal ϕ for which V(ϕ)>0.
In the right hand side of Eq. (<ref>) we have expressed the surface tension in terms of the parameters g and ξ_ϕ, which we can later input from either phenomenology or mean-field calculations.
One easily verifies that for B=0 one recovers exactly J_ϕ^0.


Repeating the first part of the calculation above, now for the Ψ̂ part, with the free energy

    F_Ψ=∫ d^2x[κ/2|∇Ψ|^2-a/2|Ψ|^2+b/4|Ψ|^4],

we obtain the correlation length
ξ_Ψ = √(2κ/a), and the surface tension
J_Ψ = 8/3ξ_Ψa^2/4b.



 §.§ Explicit results from the phenomenological model


In Fig. <ref> we demonstrate how the critical bubble radius and the value of the energy threshold depend on the metastability bias parameter B.
Clearly, as the ordered ϕ≈ 1 phase becomes increasingly favorable, true vacuum bubbles become easier to nucleate.
The impact of the competing Ψ̂ phase, and the enhancement of metastability due to its presence are quite clear – both R_c and E_ thresh. increase substantially as compared to the case of its absence. 






 §.§ Quantum regime

It is instructive to explore the false vacuum decay of the ϕ̂ phase in arbitrary dimensionality d, and with possibly important quantum fluctuations. 
For that end, we introduce the imaginary time action

    S_ϕ=∫ dτ∫ d^dx[4gτ_ϕ^2(∂_τϕ)^2+4gξ_ϕ^2|∇ϕ|^2+16gϕ^2(ϕ-1)^2-Bϕ^2],

where we have re-written the σ term from Eq. (<ref>) in terms of the correlation length, and τ_ϕ can be thought of as a correlation time characterizing quantum fluctuations. 
For example, in the limit τ_ϕ→∞, fluctuations with respect to τ freeze out, and we may ignore the (∂_τϕ)^2 contribution entirely.
Thus one replaces the integration over the Lagrangian ∫ dτ L_ϕ, by a simple factor of the “length” of the system in imaginary-time, β L_ϕ, i.e., we recover the classical expression.

To simplify the analysis, we rescale the action S_ϕ as the following, x→ xξ_ϕ, τ→ττ_ϕ,

    S_ϕ/τ_ϕξ_ϕ^d=∫ d^d+1x[4g|∇_d+1ϕ|^2+16gϕ^2(ϕ-1)^2-Bϕ^2].

With this isotropic form, one may readily employ the bubble formalism we use in the main text. 
The action barrier of forming a true vacuum bubble is approximately given by

    S_ bubble/τ_ϕξ_ϕ^d=π^d+1/2/Γ(d+1/2+1)[-R^d+1B+(d+1)R^dJ̃_̃ϕ̃],

where Γ is Euler's gamma function, and the rescaled surface tension is J̃_̃ϕ̃=16g∫_0^ϕ_>0dϕ√(ϕ^2(ϕ-1)^2-B/16gϕ^2).
The critical (dimensionless) R̃_̃c̃ is then found to simply be 

    R̃_̃c̃=dJ̃_̃ϕ̃/B.

We thus find the action barrier for the formation of the critical bubble,

    S_ Q=π^d+1/2d^d/Γ(d+1/2+1)τ_ϕ(J̃_̃ϕ̃/B)^dξ_ϕ^dJ̃_̃ϕ̃.

Going through the same steps for the classical case (τ_ϕ→∞), we find the classical barrier under the same conditions,

    S_ C=π^d/2(d-1)^d-1/Γ(d/2+1)β(J̃_̃ϕ̃/B)^d-1ξ_ϕ^dJ̃_̃ϕ̃.

One clearly sees that due to the different dependence on J̃_̃ϕ̃/B, the classical energy threshold is much lower close to the transition (B≪J̃_̃ϕ̃), and thus thermal fluctuations dominate. 
At low enough temperatures however, the β prefactor can become significant, and quantum decay of the false vacuum “short-circuits” the classical path. 
We may estimate the temperature below which this becomes important,

    T_Q≈Γ(d+1/2+1)(d-1)^d-1/√(π)Γ(d/2+1)d^dB/J̃_̃ϕ̃τ_ϕ^-1→_d=23/32B/J̃_̃ϕ̃τ_ϕ^-1,

and on the right hand side we explicitly plugged in d=2, which is appropriate for our considerations in this work. 
The upshot here is that close enough to the transition, one may find a regime where the classical decay paths are most important.

Let us make another comment on scenarios where τ_ϕ may be rather pathological. 
In cases where the order parameter commutes with the Hamiltonian, on a mean-field level one expects to find τ_ϕ^-1=0, and the false vacuum can only decay through thermal fluctuations.
This is the case for example for conventional Stoner transitions, and for the transverse field Ising model with vanishing transverse field. 
Realistically, due to corrections and phenomena beyond our simplified descriptions, τ_ϕ^-1 may be finite, yet one expects it to be rather suppressed as compared to other generic symmetry breaking transitions, e.g., an IVC order.



 §.§ Hybrid regime

In the presence of both the ϕ̂ and Ψ̂ phases in the false vacuum decay problem, there exists another regime of interest.
Assume that the temperature is below T_Q, which is set by the properties of the ϕ̂ action, yet still significantly higher than the temperature where variations of Ψ in imaginary time τ are enabled.
Namely,

    T_Qτ_ϕ/τ_Ψ<T<T_Q.


Let us simplify the analysis in this regime by limiting ourselves to the d=2 case.
The critical bubble will assume an anomalous shape here.
Its core would be comprised of an ellipsoid with radius ∼R̃_̃c̃ξ_ϕ and height ∼2R̃_̃c̃τ_ϕ (we have simplified even more, by assuming the Ψ̂ action has little effect on the value of the critical R̃_̃c̃).
However, this ellipsoid would be embedded inside a cylinder of height β and radius ∼R̃_̃c̃ξ_ϕ+δ R of the intermediate ϕ=0, Ψ=0 state, akin to the corona in Fig. <ref>a.
Since τ_Ψ is too large, the Ψ̂ sector still attempts to avoid variations in the imaginary time direction, leading to the cylindrical shape of the domain wall.
Under these assumptions, the action barrier S_Q^Ψ now reads

    S_Q^Ψ=S_Q[1+8T_Q/T(1+ξ_Ψ/ξ_ϕB/4J̃_̃ϕ̃)^2a^2/4b/B].

Notice that even if the ratio a^2/4b/B is rather small (as in the scenarios we considered in the main text), a significant stabilization of the false vacuum decay may occur due to the combination of the large prefactors 8T_Q/T and ξ_Ψ/ξ_ϕ.
For example, taking the reasonable parameters B=0.4J̃_̃ϕ̃, ξ_Ψ=20ξ_ϕ, and T_Q=5T one finds 
S_Q^Ψ=S_Q[1+360a^2/4b/B].
It thus becomes plausible in this regime that the Ψ̂ contribution to the metastability becomes comparable and possibly dominates over the ϕ̂ contribution, even if the discrepancy in energy scales is very large.





§ MICROSCOPIC CALCULATIONS




 §.§ Crystalline graphene band structure


We begin by calculating the the non-interacting band structure of biased Bernal-stacked bilayer graphene. 
Expanded around the valley K/K' points in momentum space, the Hamiltonian is <cit.> 

    H_BLG=∑_𝐤,τ,sc_τ s𝐤^†h_τ(𝐤)c_τ s𝐤,

with c_τ s𝐤=(A_1,τ s𝐤,B_1,τ s𝐤,A_2,τ s𝐤,B_2,τ s𝐤)^T,
where X_i,τ s𝐤 annihilates an electron on sub-lattice
X in layer i, with spin s, and momentum 𝐤 near
the valley τ. The matrix h_τ is given by

    h_τ(𝐤)=[       U/2  v_0π_τ^* -v_4π_τ^*   -v_3π_τ;    v_0π_τ    U/2+Δ'       γ_1 -v_4π_τ^*;   -v_4π_τ       γ_1   -U/2+Δ'  v_0π_τ^*; -v_3π_τ^*   -v_4π_τ    v_0π_τ      -U/2 ],

with π_τ=τ k_x+ik_y, and the parameters v_i=√(3)/2aγ_i,
a=0.246 nm, γ_0=2.61 eV, γ_1=361 meV, γ_3=283
meV, γ_4=138 meV, and Δ'=15 meV <cit.>.
The interlayer potential difference U is approximately U≈-dD/ϵ,
where the interlayer distance is d≈0.33nm, ϵ≈4.3,
and D is the displacement field. 
We diagonalize H_BLG at each momentum, and extract the dispersion relation of the lowest-lying valence band, which we denote by ϵ_τ,𝐤.

For the rhombohedral trilayer graphene calculations, we follow a similar method.
We diagonalize the Hamiltonian adapted from Refs. <cit.>,

    h_τ( k)=[  Δ+δ_1+δ_2      γ_2/2     v_0π^†     v_4π^†       v_3π          0;      γ_2/2 -Δ+δ_1+δ_2          0     v_3π^†       v_4π       v_0π;       v_0π          0      Δ+δ_2        γ_1     v_4π^†          0;       v_4π       v_3π        γ_1      -2δ_2     v_0π^†     v_4π^†;     v_3π^†     v_4π^†       v_4π       v_0π      -2δ_2        γ_1;          0     v_0π^†          0       v_4π        γ_1     -Δ+δ_2 ],

where Δ is the interlayer potential difference (proportional to the displacement field D.
Notice we have written the Hamiltonian in the basis (A_1,B_3,B_1,A_2,B_2,A_3), where A_j/B_j correspond to different graphene sublattices in layer j.
Here, we use the adopt the parameters γ_0=3.1 eV, γ_1=0.38 eV, γ_2=-15 meV, γ_3=-0.29 eV, γ_4=-141 meV, δ_1=-10.5 meV, δ_2=-2.3 meV.
Frot the trilayer calculations in this work, we once again are only interested in the dispersion relation of the lowest lying valence band.





 §.§ IVC phase transition

We consider the phase transition at a given total density n from a normal symmetric phase, where each of the four spin-valley flavors has occupation n/4, and between an intervalley coherent (IVC) phase. 
The non-interacting spectrum of the relevant valence band at the τ=± valley is ϵ_τ( k), and depends on the value of the displacement field. 

For simplicity, we assume the IVC is doubly degenerate, and no further symmetry breaking. Denoting the momentum-independent IVC order parameter Δ_ IVC, we can calculate the mean field free-energy with respect to the normal state,


    F_ IVC(Δ_ IVC)=2[F_ kin(Δ_ IVC)-F_ kin(Δ_ IVC=0)+ΩΔ_ IVC^2/g],

with Ω the system volume, g the relevant coupling constant
in the IVC channel, the factor of 2 accounts for the two-fold degeneracy,
and

    F_ kin(Δ_ IVC)=∑_𝐤,iE_𝐤,i Θ[μ(Δ_ IVC)-E_𝐤,i].

Here, the mean field energies are 

    E_𝐤,1/2=ϵ_++ϵ_-/2±√((ϵ_+-ϵ_-/2)^2+Δ_ IVC^2),

and the chemical potential μ(Δ_ IVC) is
determined self-consistently by the relation 
2/Ω∑_𝐤,iΘ[μ(Δ_ IVC)-E_𝐤,i]=n.



 §.§ Stoner transition

Here, we outline the calculation of the free energy in the vicinity of the first-order phase transitions we consider in this work.
We simplify, by assuming the relevant energetics do not change much in the sub-Kelvin temperature ranges (where superconductivity is measured in experiments).
This is justified if the transition temperature associated with these transitions is on the order of several Kelvin or higher, which appears to be experimentally consistent <cit.>.

Assuming a short-range density-density interaction of strength U, the system can lower its energy on a mean-field level by re-distributing the electronic densities between different flavors with density n_i

    E_ int = U/2[(∑_i n_i)^2 - ∑_i n_i^2].

We will mostly be interested in transitions from a fully flavor-symmetric 2d-fold degenerate state, to a d-fold degenerate state.
Concretely, in graphene systems with valley degeneracy, d=2, and the transition we focus on corresponds to, e.g., spin polarization or spin-valley locking.
Thus, fixing the total density n_ tot, one has d flavors at density n_+=n_ tot/2d+δ, and d flavors with n_-=n_ tot/2d-δ.
Thus, the contribution from the spontaneous polarization to the free energy density is

    F_ int/d = - U δ^2.


Whereas the interaction U favors the Stoner transition, the kinetic energy associated with the band disfavors it.
This can be quantified by the energy cost associated with populating (depopulating) states with higher (lower) energies as compared to the non-interacting Fermi level,






    F_ kin/d=[∫_E_F^E_+ - ∫_E_-^E_F]
         dϵ N(ϵ)
         (ϵ-E_F),

where N is the non-interacting DOS, and the Fermi energies are obtained by the relations
∫_-∞^E_F dϵ N(ϵ) =n_ tot/2d, 
∫_-∞^E_± dϵ N(ϵ) =n_ tot/2d±δ. 



 §.§ Superconductivity calculations

The nature of the superconducting order parameter, as well as the origin of the pairing glue, is not yet resolved in the graphene materials we discuss in the main text.
Further more, different materials and different superconducting regions in the phase diagram of the same device may have different properties and origins.
We thus refrain from an attempt to clarify such important questions regarding these superconductors, and keep our discussion as general as possible.

We assume some retardation energy scale, ω^*, at which an attractive interaction in the Cooper channel g_ att. is introduced, and restrict ourselves to a superconducting gap with trivial symmetry.
Accounting for the suppression of the initial repulsive interaction in the Cooper channel U_0 by the Anderson-Morel mechanism <cit.>, we solve the following equation for T_c,

    (g_ att.-U_0/1+U_0ℓ) D=1,

where

    ℓ = ∫_ω^*^∞ dξ N(ξ)/|ξ|,
              D(T_c)= ∫_0^ω^*dξ N(ξ)tanhξ/2T_c/ξ,

and N(ξ) is the DOS of at a distance ξ away from the Fermi level.
We note that in the case of a four-fold degenerate normal state (spin and valley degenerate phase), the initial U_0 is actually U_0≈ U_ intra-U_ inter, with U_ intra (U_ inter) being the intra- (inter-) valley interaction strength.
This is a well-known consequence of multiband superconductivity enhanced by the interband interactions.

At a given inverse temperature β=1/T the gain in free energy due to condensation of the superconducting phase is given by

    F_Δ   =|Δ|^2/g_ eff.-4∫_0^ω^*dξ N(ξ)(√(ξ^2+|Δ|^2)-|ξ|)
       -8/β∫_0^ω^*dξ N(ξ)log(1+e^-β√(ξ^2+|Δ|^2)/1+e^-β|ξ|),

where the effective coupling constant is precisely the one in Eq. (<ref>), i.e.,
g_ eff.=g_ att.-U_0/1+U_0ℓ.
Minimizing F_Δ with respect to Δ would produce the analog of a^2/(4b) from Eq. (<ref>).
We note that one may also expand Eq. (<ref>) to find suitable expressions for a and b, yet this expansion becomes less reliable for extracting the condensation energy in the regime T≪ T_c.

In order to estimate the behavior of the coherence length, we make use of a phenomenological parameter, ξ_0(T_c,0).
It is the zero-temperature coherence length expected for a superconductor with critical temperature T_c.
We assume here that the relation between the coherence length and the critical temperature remains the same within our regime of interest (although T_c may change considerably).
Employing conventional Ginzburg-Landau relations, we may approximate,

    ξ_Δ(T_c,T)≈ξ_0(T_c,0)√(T_c,0^2/T_c(T_c-T)).

Therefore, calculation of T_c, in combination with the aforementioned phenomenological parameter, will give us ξ_Δ at any temperature.





 §.§ Extended data


Let us put the first-order transitions we study in the main text in the context of a broader region of the experimental phase diagram, i.e., in a broad regime of total density and displacement field.
In Fig. <ref> we highlight the phase boundaries for the IVC transition in RTG (<ref>a), and for the Stoner transition in BBG (<ref>b), for the same parameters we explore the stability of the false vacuum in.
Additionally, we explore a different region of the RTG phase diagram where a Stoner-like transition may occur (lower density and displacement fields, Fig. <ref>c).
It is clear that on the level of our mean field analysis, the first-order nature of the transition is quite robust over a broad range of variables.



As stated in the main text, we do not specify how superconductivity is suppressed by the spontaneously formed order.
Here, however, we elaborate on such a possible mechanism near an IVC transition in RTG.
The mechanism can be most readily understood by examination of the DOS near the Fermi level, before or right after the IVC order condenses.
As shown in Fig. <ref>a, the DOS right at the Fermi energy remains mostly unchanged, so a dramatic effect would seem peculiar.
Moreover, it appears that a large portion of the DOS has actually moved closer to the Fermi level, which should naively only help superconductivity.

The answer to this puzzle is encoded in the calculation of ℓ, which determines how the initial Coulomb repulsion is screened and renormalized in the Cooper channel.
The renormalization ℓ is solely determined by the DOS outside the ω^* window around the Fermi level, where the retarded attraction becomes effective.
Right outside this retardation window, the prominent DOS feature in the IVC phase is a substantial “pseudogap” present between the IVC-split bands.
Thus, following the IVC transition the effective interaction
g_ eff.=g_ att.-U_0/1+U_0ℓ
plummets, making the superconductor phase virtually undetectable, see Fig.<ref>b.



Finally, for the transitions explored in the main text, as well as the RTG Stoner transition, we provide some extended results regarding the metastability diagrams (left column of Fig. <ref>), the superconducting parameters needed for extracting the energetics relevant to the superconducting phase (Fig. <ref>, center column), and the size of the free-energy barriers and the role of the secondary superconducting order in determining them (right column of Fig. <ref>).
Qualitatively, the results share great similarities.
This suggests that the sort of physical phenomena we discuss in this work is quite robust, and should appear quite ubiquitously in theses strongly correlated graphene systems.






§ HEURISTIC MASTER EQUATION

Let us consider a heuristic approach to the state of the system, where three points of interest exist in phase space.
The first, is the true ground state of the system, to which we assign the occupation probability p_ true.
The second, is the false vacuum state, with p_ false.
Lastly we have also the normal phase, where both of the phases, i.e., ϕ̂ and Ψ̂, are disordered, to which we assign p_0.
We may estimate the state of the system as a function of time by considering the time evolution

    d/dt[  p_ true;      p_0; p_ false ]=[         0       γ_0         γ;         0 -γ_Ψ̂-γ_0         0;         0      γ_Ψ̂        -γ ][  p_ true;      p_0; p_ false ].

Here, γ corresponds to τ_ decay^-1 which we compute in the presence of the secondary competing order (so one always expects γ_0>γ), γ_0 is the decay rate in the absence of this competition, and γ_Ψ̂ is the time it takes the Ψ̂ phase to form in the system starting from the normal state (assuming the absence of the true vacuum order).

Essentially, there are two scenarios of interest, depending on the initial conditions at the time of the quench. 
In the first case, the system starts in the false vacuum state,
(p_ true(0),p_0(0),p_ false(0))=(0,0,1).
This is the scenario relevant for example in RTG next to the supposed IVC transition, when the system is quenched from the superconducting state into the IVC regime.
In that case, p_0 becomes irrelevant and we recover the expected decay,

    p_ false(t) = e^-γ t = e^-t/τ_ decay.


In the second more complicated case, the initial conditions are
(p_ true(0),p_0(0),p_ false(0))=(0,1,0).
The system is quenched from a normal phase into a regime where formation of the Ψ̂ order competes with the false vacuum.
The solution for p_ false is less simple yet quite straightforward,

    p_ false(t)=γ_Ψ̂/γ_Ψ̂+γ_0-γ[1-e^-(γ_Ψ̂+γ_0-γ)t]e^-γ t.


In the interesting scenario, where the superconducting-like phase is formed first and γ_Ψ̂ is the largest rate, the false vacuum first forms in a time-scale ∼γ_Ψ̂^-1, occupies a fraction determined by the ratio of the different rates, and then goes on to decay with the characteristic time-scale τ_ decay as before.




§ ORDER PARAMETER RELAXATION

Let us estimate the manner by which the superconducting order parameter relaxation would manifest in measurements of t_d, the delay time between a current driven through the system, and the appearance of a voltage signal.
Our starting point is the time-dependent Ginzburg-Landau equation <cit.> (neglecting the role of the sub-leading thermodynamic order parameter fluctuations),

    -a/2τ_GL(∂_t+2ieφ)Ψ=δ F/δΨ^*.

Here, F is the functional introduced in Eq. (<ref>) (disregarding the coupling to the ϕ̂-sector, to be considered shortly), and φ is the scalar potential.
We rewrite the equation in terms of the normalized order parameter Ψ≡√(b/a)ψ (a>0 in the ordered phase), and find

    τ_GL(∂_t+2ieφ)ψ=2ξ_Ψ^2∂_x^2ψ+ψ(1-|ψ|^2).

Notice we assumed the possibility of current flowing solely in the x̂ direction with density

    j_s=ie/m^*a/b(ψ^*∂_xψ-ψ∂_xψ^*),

and m^* is the effective mass.

We employ the ansatz ψ(x)=f e^iθ(x), assuming that f remains fairly constants over length scales ≫ξ_Ψ.
The current may then be expressed as a function of ∂_x θ,
j_s = -2e/ma/b∂_xθ[1-2ξ_Ψ^2(∂_xθ)^2], and maximizing over this variable allows one to obtain the critical current

    j_c=-e/m^*a/b4/3√(6)ξ_Ψ.


We now combine our ansatz, the definition of the current operator, Eq. (<ref>), and the real part of Eq. (<ref>), and find the following connection between the current density and the relaxation of the order parameter,

    τ_GL∂_tf=-(j_s/j_c)^24/27f^3+f(1-f^2).

In the standard experiment demonstrated in Ref. <cit.>, one obtains the delay time by integrating Eq. (<ref>) over the change of f from unity to zero,

    t_d^0/τ_GL = ∫^1_0 df/(I/I_c)^24/27f^3-f+f^3,

where I and I_c are the applied current and critical current.
The delay time t_d^0 diverges when approaching I_c from the I>I_c side, and is infinite (by definition) for I<I_c.

Next, we modify Eq. (<ref>) as to be appropriate for our considered scenario.
As a first order approximation, we introduce this modification via the critical current alone.
We assume an independent temporal decay of the superfluid density |Ψ_0|^2=a/b with a time scale τ_ decay≫τ_GL, facilitating a similar decay of the critical current.
As a result, the modified evolution we consider is

    τ_GL∂_tf=-e^2t/τ_ decay(j_s/j_c)^24/27f^3+f(1-f^2).

The validity of Eq. (<ref>) becomes questionable around j_s→ 0, as one finds relaxation times much longer than τ_ decay, violating our working assumption.
This is due to the fact that Eq. (<ref>) implicitly assumes the dominance of the GL-induced relaxation over the decay.
Thus, it is valid strictly in the regime

    j_s≫ j_c√(τ_GL/τ_ decay).























































