Solvable Strong-coupling Quantum Dot Model with a Non-Fermi-liquid Pairing Transition

TL;DR

This paper introduces the Yukawa-SYK model, a solvable random interacting quantum dot system exhibiting non-Fermi liquid behavior and a Kosterlitz-Thouless quantum-critical pairing transition, controlled by large-$M$, large-$N$ limits.

Contribution

It presents a solvable model with exotic pairing and non-Fermi liquid phases, revealing conditions for pairing and critical behavior in strongly interacting quantum dots.

Findings

Pairing occurs only in specific regions of the (M,N) parameter space.

Fermion incoherence can prevent Cooper pair formation despite attractive interactions.

The phase transition exhibits Kosterlitz-Thouless quantum-critical behavior.

Abstract

We show that a random interacting model exhibits solvable non-Fermi liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between $M$ quantum dots each hosting $N$ flavors of fermions and $N^2$ bosons that self-tunes to criticality at low energies. The diagrammatic expansion is controlled by $1/MN$, and the results become exact in a large-$M$, large-$N$ limit. We find that pairing only develops within a region of the $(M,N)$ plane --- even though the pairing interaction is strongly attractive, the incoherence of the fermions can spoil the forming of Cooper pairs, rendering the system a non-Fermi liquid down to zero temperature. By solving the Eliashberg equation and the renormalization group equation, we show that the transition into the pairing phase exhibits Kosterlitz-Thouless quantum-critical behavior.

Figures

Equations

$$M\le \sqrt{2N},M,N\to \infty .$$

$$H_0=\frac{i}{(MN{)}^{1/2}}\sum _{ij,\alpha \beta }^{N,M}{t}_{\alpha \beta }{\varphi }_{ij}{c}_{i\alpha }^{†}{c}_{j\beta }+\frac{1}{2}\sum _{ij}^N\left({\pi }_{ij}^2+m_0^2{\varphi }_{ij}^2\right).$$

$$⟨t_{\alpha \beta }{t}_{{\alpha }^{\prime }{\beta }^{\prime }}⟩={\omega }_0^3({\delta }_{\alpha {\alpha }^{\prime }}{\delta }_{\beta {\beta }^{\prime }}+{\delta }_{\alpha {\beta }^{\prime }}{\delta }_{\beta {\alpha }^{\prime }}).$$

$$\Sigma (\omega )=$$

$$\Pi (\Omega )=$$

$$\Pi (0)=2{\omega }_0^3\frac{M}{N}\int \frac{d\Omega }{2\pi }\frac{1}{(i\Omega {)}^2}$$

$$\Sigma (\omega )=$$

$$\tilde{\Pi }(\Omega )\equiv$$

$$\int \frac{d\Omega \mathrm{sgn}(\omega -\Omega )}{|\Omega {|}^{1-2x}|\omega -\Omega {|}^x}=-\frac{{\Gamma }^2(-x)}{2\Gamma (-2x)}|\omega {|}^x\mathrm{sgn}(\omega )$$

$$\int \frac{d\omega \mathrm{sgn}(\omega +\Omega /2)\mathrm{sgn}(\omega -\Omega /2)}{|\omega +\Omega /2{|}^x|\omega -\Omega /2{|}^x}$$

$$=|\Omega {|}^{1-2x}\frac{{\Gamma }^2(-x)}{2\Gamma (-2x)}\frac{1+\sec \pi x}{1/x-2}+\mathrm{div}$$

$$A^{2}B=\alpha (x),$$

$$\Sigma (\omega )=$$

$$\tilde{\Pi }(\Omega )=$$

$$\Sigma (\omega )$$

$$\Sigma (\omega )=$$

$$\tilde{\Pi }(\Omega )=$$

$$\Phi (\omega )=$$

$$\Phi (\omega )=\frac{2}{\alpha (x)M}{\int }_{\Delta }^{\bar{\omega }}\frac{d{\omega }^{\prime }}{2\pi }\frac{\Phi ({\omega }^{\prime })}{|\omega -{\omega }^{\prime }{|}^{1-2x}|{\omega }^{\prime }{|}^{2x}}.$$

$$\Phi (\omega )=$$

$$\alpha (x_{cr})M_{cr}=\frac{2^{2x_{cr}}\Gamma (\frac{1}{2}-x_{cr})\Gamma (x_{cr})}{{\pi }^{3/2}}.$$

$$\Phi (\omega )={\omega }^{-(1-2x)/2}\cos [\beta \log \omega +\varphi ],$$

$$\tan (\beta \log \Delta +\varphi )=\frac{2\beta }{1-2x},\tan (\beta \log \bar{\omega }+\varphi )=-\frac{2\beta }{1-2x}.$$

$$\Delta \sim \bar{\omega }\exp \left(-\frac{\pi }{\beta }\right)=\bar{\omega }\exp \left(-\frac{\gamma M^{3/4}{N}^{-1/4}}{\sqrt{2-M^2/N}}\right),$$

Full text

Introduction.— The pairing problem for non-Fermi liquids (nFL) is a fascinating open issue in condensed matter physics Monthoux et al. (2007); *scal2; Abanov et al. (2001); *acs; boo ; Fratino et al. (2016); rev ; Fradkin et al. (2010); Bok et al. (2016); Shibauchi et al. (2014); Combescot (1995); *Bergmann; *Bergmann2; *ad; *Marsiglio_88; *Marsiglio_91; *Karakozov_91; Bonesteel et al. (1996); Abanov et al. (2001); *acs; Son (1999a); Lee (2009a); Moon and Sachdev (2009); *moon_2; Metlitski and Sachdev (2010a); *max_last; Raghu et al. (2015a); Fradkin et al. (2010); Monthoux et al. (2007); *scal2; Efetov et al. (2013); Lederer et al. (2015); Wang et al. (2017a); Tsvelik (2017); Vojta and Sachdev (1999); Khveshchenko and Shively (2006); Lee et al. (2018); *we_last_D; Vilardi et al. (2019); Gerlach et al. (2017); *berg_2; *berg_3; Haule and Kotliar (2007); *kotliar2; Fratino et al. (2016); *tremblay_2; Georges et al. (2013); *georges2; *Bergmann; *Bergmann2; *ad; *Marsiglio_88; *Marsiglio_91; *Karakozov_91; Abanov et al. (2001); *acs; Metlitski et al. (2015a); Raghu et al. (2015a); Wang et al. (2017a); Mandal (2016); Wang et al. (2016); Lee et al. (2018); *we_last_D; Abanov et al. (2018); Wu et al. (2018); Phillips et al. (2019). In a general context, nFL behavior often occur via electron interactions mediated by gapless bosonic modes Metlitski and Sachdev (2010b, a); Metlitski et al. (2015a); Raghu et al. (2015a); Lederer et al. (2015); Lawler et al. (2006); Lawler and Fradkin (2007), rendering the electrons incoherent. Such gapless bosons typically arise in the vicinity of a quantum-critical point (QCP) or in gauge theories. The same interaction is usually also strongly attractive in some pairing-symmetry channel. The fermionic incoherence and the strong attractive interaction compete in determining whether the ground state is superconducting. However, the analytical solution of this problem is challenging since there is no natural small parameter in the problem to allow for a controlled calculation for the nFL behavior as well as for the pairing problem. Moreover, the two effects are of comparable strength, lacking a theoretical tuning parameter for the interplay between the nFL and superconductivity. One workaround is to extend the problem to a large-$N$ limit. Within this limit the vertex corrections to the interaction is suppressed by $1/N$, and one can solve for the nFL behavior analytically via self-consistent Schwinger-Dyson equations. Conveniently, the $1/N$ factor also serves as an effective dimensionless coupling constant in the pairing problem. The pairing problem in various large $N$ models for a Fermi surface (FS) coupled to critical bosons have been intensively studied. Interestingly, in a class of these models Abanov et al. (2001); Raghu et al. (2015a); Wang et al. (2018), as a function of $N$, the system at $T=0$ can either be in a pairing phase, or remain at the normal state, separated by a quantum-critical point. The latter situation is particularly striking — contrary to BCS theory where even an infinitesimal attractive interaction drives a Fermi liquid to a superconducting state, the incoherence of the nFL state destroys superconductivity, even if the attractive pairing interaction is strong. However, certain large-$N$ extensions become uncontrolled in two spatial dimensions Lee (2009b); Metlitski and Sachdev (2010b, a), and the quantum critical point for pairing can only be accessed for FS’s in fractional spatial dimensions $d=3-\epsilon$ ($0<\epsilon<1$) Raghu et al. (2015a); Mandal (2016), the physical meaning and effective realization of which are unclear. It naturally raises the question whether the pairing QCP may be an artifact of the fractional spatial dimensions.

Recent years have witnessed a remarkable revival of interest in the Sachdev-Ye-Kitaev (SYK) models Sachdev and Ye (2015); Kitaev ; Sachdev (2015); Kitaev and Suh (2018), due to its property of maximal quantum chaos Maldacena et al. (2016) and its connection with quantum black holes. These models describe random four-fermion interactions within a quantum dot of $N$ fermionic particles. Despite being strongly interacting, these models can be solved in the large-$N$ limit and exhibit nFL behavior Sachdev and Ye (2015); Song et al. (2017); Chowdhury et al. (2018); Wu et al. (2019); Can and Franz (2019). In the pairing problem for the SYK model as well as its lattice variants, the nFL was found to be generally unstable to pairing Bi et al. (2017); Wu et al. (2018); Patel et al. (2018); Kim et al. (2019); Chowdhury and Berg (2019); Cheipesh et al. (2019) in the presence of a small attractive interaction.

In this Letter we study a new solvable random interacting model with more exotic nFL pairing behaviors. This model describes $M$ quantum dots each with $N$ flavors of fermions. The fermions are coupled by a random Yukawa term to a matrix boson with a generic bare mass, and we thus refer to this model as the Yukawa-SYK model. We show that the coupling with the fermions makes the boson critical, independent of its bare mass. The critical behavior is similar to that recently obtained for a model proposed in Ref. Patel and Sachdev, 2018, where instead of Yukawa coupling a minimal coupling to a compact dynamical $U(1)$ gauge boson was introduced. However, we show that there the compactness of the gauge field actually confines the fermions and spoils the nFL behavior in that model. sup ; Patel Like the SYK model, we show that this model is solvable in the large-$N,M$ limit and exhibit nFL behavior. As one varies the ratio $N/M$, the exponent of the conformal fermionic self-energy interploates between 0 (same as in a noninteracting disordered electron system) and 1/2 (same as in the SYK model). We also determine the nonuniversal overall scale of the nFL self-energy, by matching the UV and IR properties. Remarkably, the large-$N,M$ limit also allows for an analytical solution of the pairing gap equation in the nFL regime. By solving the gap equation as an integral equation we show that the pairing phase only develops for

[TABLE]

Outside this range, the system remains a nFL down to zero temperature, even if the attractive interaction is singularly strong. Moreover, the pairing gap near the pairing QCP $M_{cr}=\sqrt{2N_{cr}}$ follows a Kosterlitz-Thouless (KT) scaling form and represents an infinite-order phase transition [see Eq. (18)], similar to that found in Ref. Raghu et al., 2015a in fractional dimensions. This KT scaling near the pairing QCP can be understood as coming from the annihilation of two renormalization group (RG) fixed points Kaplan et al. (2009); Hartnoll et al. (2018), which we show in Ref. sup, .

**The Yukawa-SYK model.—**We consider a random interacting model of $M$ quantum dots each with $N$ flavors of fermions $(c)$ coupled with a critical boson $(\phi)$ through a random Yukawa term. The Hamiltonian is given by

[TABLE]

where $\alpha,\beta\in(1,M)$ are indices for the quantum dots and $i,j\in(1,N)$ are $SO(N)$ flavor indices within a cluster. When taking the large $N,M$ limit one can vary the ratio $M/N$. Here $\pi_{ij}=\dot{\phi}_{ij}/g$ is the canonical momentum of the boson field $\phi_{ij}$. As we will see, the infrared (IR) dynamics of the boson field completely comes from its coupling with the fermions. However its bare dynamics is important for fixing the energy scale of low-energy dynamics. Note that the $i$ factor in the first term and Hermiticity of the Hamiltonian indicates that $\phi_{ij}\equiv-\phi_{ji}$ (assuming $t_{\alpha\beta}=t_{\beta\alpha}$). This will be important for the pairing problem. The random coupling amplitudes satisfy a Gaussian distribution where

[TABLE]

This model in Eq. (2) is similar to that recently studied in Ref. Patel and Sachdev, 2018, which has a similar $N,M$ assignment. There the fermions are randomly coupled to a compact gauge field and the authors obtained a nFL behavior at low-energies, much similar to our results below. However, we show in Ref. sup, that in the compact gauge theory fluctuations of the gauge field actually drives the system into a confined phase Patel .

Despite being a toy model far from realistically describing known condensed-matter systems, we shall see it exhibits interesting low-energy behaviors that are universal and independent of details, just like the SYK models Sachdev and Ye (2015); Kitaev ; Sachdev (2015); Kitaev and Suh (2018). (For example, one can construct an alternative model free of randomness that behaves essentially the same sup ). Further, we show below the normal state and the pairing state [Eqs. (4, 12)] are described by the Eliashberg equations widely adopted to describe electron-phonon superconductors and quantum-critical superconductors. Wang et al. (2017a) From this perspective the intricate structure of the Hamiltonian is merely a tool to make the various usual theoretical simplifications quantitatively controlled, much the same way as the usual large-$N$ approximation.

**Normal state analysis.—**To leading order in $1/(NM)$, the self-consistent Schwinger-Dyson equations for the bosonic and fermionic self-energies (in Euclidean time at $T=0$) are

[TABLE]

where $D(\Omega)=1/\left[\Omega^{2}+m_{0}^{2}+\Pi(\Omega)\right]$, and $G(\omega)=-1/\left[i\omega+\Sigma(\omega)\right]$. We show the corresponding diagrams in Fig. 1. The fact that these diagrams are to leading order in $1/(NM)$ can be seen using a double-line formalism, keeping track of both $i,j$ (solid line in Fig. 1 (b)) and $\alpha,\beta$ (dashed line) indices; all other diagrams are suppressed by $1/(NM)$.

As a first attempt, we evaluate the diagrams with bare propagators, and perturbation theory immediately fails: the integral

[TABLE]

is strongly infrared divergent, which would drive the bosonic mass $m^{2}=m_{0}^{2}+\Pi(0)$ towards negative. In this situation, the boson usually condenses described by a mean-field theory. Nevertheless, we are in 0d and fluctuation effects typically destroy any order. Large-$M$ limits, $M$ being the number of fermions, have often been invoked to suppress flucutation by $1/M$ and can stabilize order even in 0d. However, as for each boson there are $M/N$ fermions, one can check that in our case fluctuation effects of the boson are $O(N/M)$, in general not small. The fact that the bosons and the fermions have comparable amount of fluctuations spoils a simple mean-field solution. Instead, such a balance of fluctuation effects gives rise to a new, critical solution for the ground state. In this state, the divergence in the bare bubble (5) is cut off at the low energy, the bosonic mass $m^{2}$ gets renormalized to zero, and makes the fermions a nFL at low-energies, which is in turn consistent with the cutoff in the bare fermion bubble.

We analyze the solution for which at low energies, $\Pi(0)=m_{0}^{2}$ (criticality), $\Omega^{2}/g\ll\Pi(\Omega)-\Pi(0)$ and $\omega\ll\Sigma(\omega)$. We can take the conformal nFL ansatz for the self energies (at $T=0$):

[TABLE]

and the power $x$ can be determined by using the following integrals

[TABLE]

where “div” stands for a divergent constant to be regularized by high-energy physics. Defining $\alpha(x)=-{\Gamma^{2}(-x)}/[{4\pi\Gamma(-2x)}]$, self-consistency of Eq. (4) requires Patel and Sachdev (2018)

[TABLE]

We take $0<x<1/2$ to guarantee $B>0$, a requirement of causality 111Note that the equation for $x$ is similar to that of the supersymmetric SYK model at $\mathcal{N}=1$ Fu et al. (2017). However there the restriction $B>0$ leads to $1/2<x<1$, and the solution for $x$ yields $x=2/3$, consistent with supersymmetry. There the $0<x<1/2$ solution breaks the supersymmetry and is unstable, opposite to the present model.. For $N\ll M$, $x\to 0$ and for $N\gg M$, $x\to 1/2$.

Low-energy physics alone does not determine the prefactors $A$ and $B$, which is quite different from finite-dimensional models where the feedback effect from the nFL to the critical boson is negligible. Abanov et al. (2003) To fix them, one needs to examine the system behavior at high energies. For Eq. (6) to hold, self-energies should be dominant over the bare terms in the propagators. This requires $\omega,\Omega\ll\bar{\omega}\equiv\min(\omega_{F},\omega_{B})$, where from Eq. (6) $\omega_{F}\equiv\omega_{0}A^{1/(1-x)}$ and $\omega_{B}\equiv\omega_{0}B^{1/(1-2x)}$.

If $\omega_{F}\ll\omega_{B}$, $\omega_{F}$ serves as an effective infrared cutoff for the otherwise-divergent integral in Eq. (5). From $m_{0}^{2}-\int_{\omega_{F}}{\omega_{0}^{3}d\omega}/{\omega^{2}}=0$ we get $\omega_{F}\sim{\omega_{0}^{3}}/{m_{0}^{2}}.$ The requirement $A^{2}B=\alpha(x)$ yields

[TABLE]

where $c$ is a nonuniversal $O(1)$ constant. From this $\omega_{B}^{1+2x}=m_{0}^{3}/\omega^{1-2x}$, and the $\omega_{B}\gg\omega_{F}$ translates to $\omega_{0}\ll m_{0}$, which we denote as the weak-coupling region.

If $\omega_{B}\ll\omega_{F}$, $\Sigma(\omega)$ takes a different form for $\omega\gg\omega_{B}$. In this regime, the fermions have much higher energies than the typical boson energy $\omega_{B}$, and the latter behaves like disorder:

[TABLE]

The new scale $\sqrt{\omega_{0}^{3}/\omega_{B}}$ serves to cut off the divergence in Eq. (5), and thus $\omega_{B}\sim{m_{0}^{4}}/{\omega_{0}^{3}}$. Below the $\omega_{B}$ scale, $\Sigma,\Pi$ restore the forms in Eq. (6). Using self-consistency and $\omega_{B}=\omega_{0}B^{1/(1-2x)}$

[TABLE]

where $d=O(1)$, and we see that at $\omega\sim\omega_{B}$, $\tilde{\Pi}(\Omega)=\Omega^{2}$ by definition and that $\Sigma(\omega)$ smoothly crosses over between Eqs. (10, 11). The condition $\omega_{B}\ll\omega_{F}$ implies $\omega_{0}\gg m_{0}$, which we denote as the strong-coupling regime.

Remarkably, we have shown that for arbitrary values of $m_{0}$, the system behavior self-tunes to criticality (see also Ref. Esterlis and Schmalian, 2019), in sharp contrast of how similar systems behave at finite dimensions. We discuss the self-energies at higher frequencies in both regimes in Ref. sup, .

The Pairing problem.—

The diagram for the gap equation of the pairing problem are shown in Fig. 2. We consider inter-dot pairing with $\alpha\neq\beta$. Defining the pairing vertex $\sim\Phi_{\alpha\beta}c^{\dagger}_{i\alpha}c^{\dagger}_{i\beta}$, fermion statistics requires $\Phi_{\alpha\beta}=-\Phi_{\beta\alpha}\equiv\Phi$. 222We have assumed phase coherence between Cooper pairs with different $\{\alpha,\beta\}$. This can be ensured by turning on an arbitrarily small attractive pair-hopping interaction. Indeed the interaction mediated by exchanging $\phi_{ij}$ is attractive in this channel [note the $i$ factor in Eq. (2)]. The Eliashberg equation for pairing is given by

[TABLE]

Compared with the normal state analysis, the right hand side of (12) is suppressed by $1/M$, as only the internal flavor index $i$ is summed over. Restricting to even-frequency pairing, one can verify that in our model the interactions for all other pairing channels are repulsive.

In this work we focus on the pairing problem at $T=0$. To find the pairing gap up to an $O(1)$ coefficient, it suffices to solve the linearized gap equation in the presence of an infrared cutoff $\Delta$, roughly the magnitude of the pairing gap $\sim\omega\Phi(\omega)/\left[\omega+\Sigma(\omega)\right]$. We place an effective UV cutoff at the nFL energy scale $\bar{\omega}=\min(\omega_{F},\omega_{B})$, and pairing comes from physics below this scale. The gap equation becomes

[TABLE]

Note that the nonuniversal $c,d$ dependence in Eqs. (9, 11) has disappeared.

In sharp contrast with the BCS pairing, the integral in (13) does not explicitly contain a logarithmic IR divergence Abanov et al. (2001), at least with a constant $\Phi$. In fact Eq. (13) is very similar to the gap equation in quantum-critical pairing problems Abanov et al. (2001); Wang et al. (2016, 2018), and to see the pairing instability we need to solve the full integral equation. We extend the domain of $\omega^{\prime}$ to $(0,\infty)$ and split the integral on the right hand side:

[TABLE]

Note that the first integral alone can be matched with the left hand side by using a power law ansatz

[TABLE]

where $0<\operatorname{Re}y<1-2x$. We then need to make sure the other two integrals in (14) vanish for external frequencies $\Delta\ll\omega\ll\omega_{0}$. However, they could only vanish if $\Phi(\omega^{\prime})$ is oscillatory in those intervals. This requires $y$ to be complex. Abanov et al. (2001); Wang et al. (2016)

One can show from Eq. (15) that for large $\alpha(x)M$, $y$ takes real values, and a complex $y$ is only possible for small enough $\alpha(x)M$. The critical value for $\alpha(x)M$ is given by the minimum value of the right hand side of (15) for a real $y$, which is reached at $y=(1-2x)/2$. We then find the critical value to be

[TABLE]

Together with Eq. (8), this gives a line of critical values $(M_{cr},N_{cr})$. We will see that this region denotes the pairing phase at $T=0$. In the limit $N,M\to\infty$ where our results become exact, one can verify that the critical values satisfy $M_{cr}=\sqrt{2N_{cr}}\ll N_{cr},$ and for $M>\sqrt{2N}$ the system remains a nFL down to zero temperature.

For $M\leq\sqrt{2N}$, the solution for $y$ is complex, $y=(1-2x)/2\pm i\beta$. Near the critical pairs $(N,M)_{cr}$, $\beta$ scales as $\beta\propto\sqrt{\lambda(N,M)-\lambda_{cr}(N,M)},$ where $\lambda(N,M)\equiv 1/(\alpha M)$. The power-law ansatz can be rewritten as

[TABLE]

where $\phi$ is a free parameter. With this form for $\Phi(\omega^{\prime})$, requiring the second and third integrals in (14) to vanish determines the value of $\phi$ and $\Delta$. Using the fact that the external frequency $\Delta\ll\omega\ll\omega_{0}$, we obtain

[TABLE]

The solution of $\Phi(\omega)$ that does not change sign within $(\Delta,\omega_{0})$ maximizes the condensation energy. Requiring this we get at small $\beta$,

[TABLE]

where $\gamma$ is an $O(1)$ number. Indeed, this region with $M\leq\sqrt{2N}$ corresponds to a pairing phase. Furthermore, we see that near the pairing QCP $\Delta$ onsets via an infinite-order phase transition similar to a KT transition. Altland and Simons (2010) The connection to the KT transition can be made clear in an RG framework: this exotic KT scaling of the pairing QCP comes from the merger of two fixed points Raghu et al. (2015a), which we explain in Ref. sup, . This scaling was also found for quantum-critical pairing models Abanov et al. (2001); Raghu et al. (2015a), as well as in some holographic models Kaplan et al. (2009); Hartnoll et al. (2018). This is the main result of this work.

It is again important to address whether the above mean-field result for pairing is destroyed by fluctuations of the order parameter $\Phi_{\alpha\beta}$. In the present case there are $M^{2}$ boson species coupled to $NM$ fermion species. Therefore the flucutation effects of $\Phi_{\alpha\beta}$ beyond its mean-field theory is $O(M/N)$ (compare with the $O(N/M)$ obtained previously for the fluctuation effects of $\phi_{ij}$). In the region $M\leq\sqrt{2N}$, such effects are suppressed.

Conclusion.— The interplay between nFL and pairing has been a long standing open issue due to the lack of a natural control parameter. We have shown in an exactly solvable large-$N$ random interacting model that the opposite tendencies of fermionic incoherence and strong attraction from the same interaction lead to remarkable consequences — for a large range of $(N,M)$, the nFL behavior completely spoils the Cooper pairing, despite the pairing interaction mediated by critical bosons is singularly strong. Only for some values that asymptote to $M_{cr}\leq\sqrt{2N_{cr}}$, the system enters a pairing phase. By solving the Eliashberg equation, we have shown that the $T=0$ critical point between the pairing phase and the nFL phase exhibit a KT scaling behavior. Unlike previous models exhibiting this behavior that requires a fractional spatial dimension, the present model has a well-defined base manifold. It will be interesting to explore its experimental and numerical realizations.

An interesting question is the quantum chaotic behavior across the pairing QCP. The low-energy conformal invariance indicates that the nFL state should saturate the upper bound of Lyapunov exponent $\lambda_{L}$ and is dual to a quantum black hole Maldacena et al. (2016); Kim et al. (2019), just like the SYK model. If so, it will be interesting to see how the coefficient of $\lambda_{L}$ behave across the pairing QCP. Qualitatively, we expect $\lambda_{L}$ to drop due to the formation of the condensate. We postpone a full analysis of $\lambda_{L}$ to future studies. Another interesting question for future work is the interpretation of the KT scaling of the superconducting transition from the perspective of the 2d classical bulk theory.