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Pairing Instabilities of the YukawaSYK Models with Controlled Fermion Incoherence
by Wonjune Choi, Omid Tavakol, Yong Baek Kim
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Submission summary
Authors (as registered SciPost users):  Wonjune Choi 
Submission information  

Preprint Link:  scipost_202110_00021v1 (pdf) 
Date submitted:  20211013 17:40 
Submitted by:  Choi, Wonjune 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The interplay of nonFermi liquid and superconductivity born out of strong dynamical interactions is at the heart of the physics of unconventional superconductivity. As a solvable platform of the strongly correlated superconductors, we study the pairing instabilities of the YukawaSachdevYeKitaev (YukawaSYK) model, which describes spin1/2 fermions coupled to bosons by the random, alltoall, spin independent and dependent Yukawa interactions. In contrast to the previously studied models, the random Yukawa couplings are sampled from a collection of Gaussian ensembles whose variances follow a continuous distribution rather than being fixed to a constant. By tuning the analytic behavior of the distribution, we could control the fermion incoherence to systematically examine various normal states ranging from the Fermi liquid to nonFermi liquids that are different from the conformal solution of the SYK model with a constant variance. Using the linearized Eliashberg theory, we show that the onset of the unconventional spin triplet pairing is preferred with the spin dependent interactions while all pairing channels show instabilities with the spin independent interactions. Although the interactions strongly damp the fermions in the nonFermi liquid, the same interactions also dress the bosons to strengthen the tendency to pair the incoherent fermions. As a consequence, the onset temperature $T_c$ of the pairing is enhanced in the nonFermi liquid compared to the case of the Fermi liquid.
Current status:
Reports on this Submission
Report #2 by Yuxuan Wang (Referee 2) on 2021125 (Invited Report)
 Cite as: Yuxuan Wang, Report on arXiv:scipost_202110_00021v1, delivered 20211205, doi: 10.21468/SciPost.Report.3998
Strengths
Exactly solvability of the model which addresses an important and challenging theoretical problem.
Weaknesses
The discussion on the normalstate properties is a bit too terse; reference to a previous work should be accompanied by a more detailed discussion to make the paper more selfcontained.
Report
In this interesting work, the authors generalized the YukawaSYK model (equivalent to the lowrank SYK model) to allow for more structure in the random coupling constant (g_{ijk} is a Gaussian random variable in ij, but obeys a more complicated distribution in k). As a result, the authors obtain a rich phase diagram displaying Fermiliquid, nonFermi liquid, and superconducting phases. This work is an important contribution to the understanding of the myriad of phases of a stronglycorrelated system which usually defies analytical approaches. For this reason I think it meets the criteria for publication in SciPost.
I do have several questions which I hope they can clarify:
1. It would be helpful to compare the nonFermi liquid obtained here with that of the YukawaSYK model with a fixedvariance coupling. Do the Green's functions have similar scaling behavior? Is there a limiting case of $\eta$ for which the results go back to that with a fixed variance? Naively, one would think if \eta \to \infinity, the only possible value of \lambda is \lambda_{max}. It would also be nice if the authors can compare the properties of superconducting transition/state with those obtained in Ref. 36 and in Phys. Rev. B 104, 125120 (2021).
2. The author mentioned that for some parameter range the boson field can condense, leading to a constant contribution to the boson propagator. However, it has been argued in Ref. 28 (among several other works cited by the authors) the boson cannot condense even if its renormalized mass vanishes, essentially due to strong quantum fluctuations in 0d, but rather remains critical. I understand the authors are considering a different model, but it would be helpful if the author can provide more explanations on this point.
Report #1 by Anonymous (Referee 1) on 2021114 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202110_00021v1, delivered 20211104, doi: 10.21468/SciPost.Report.3792
Report
This is an interesting paper. The authors generalize the YukawaSYK model to allow for a continuous distribution of variances to the bosonic fields. This allows them to tune the normal state from Fermi to nonFermi behavior, and to study how this influences the critical temperature of the lowT superconducting state, and its nature (singlet or triplet).
The main conclusion is that Tc may be enhanced for the nonFermi liquid because even though the quasiparticles become less coherent, the same "knob" strengthens the coupling to bosons, and the latter wins.
I think this is a nice demonstration that this phenomenology is possible, in a model where an essentially exact solution can be found after selfaveraging.
Nevertheless, I do not see how this work meets any of the acceptance criteria (i) Detail a groundbreaking theoretical/experimental/computational discovery; (ii) Present a breakthrough on a previouslyidentified and longstanding research stumbling block; (iii) Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work; (iv) Provide a novel and synergetic link between different research areas.
This is the reason I suggest the paper be moved/accepted to SciPost Physics Core, whose criteria it meets better.
One optional suggestion for the authors: perhaps if you allow the masses of the bosons to also follow a distribution (instead of all being set to m) you might get an independent "knob" on the behavior of the bosons, and be able to tune the nonFermi character of the fermions independently of the strength of the boson glue?
Author: Wonjune Choi on 20220318 [id 2300]
(in reply to Report 1 on 20211104)
We first thank the anonymous referee for reviewing our manuscript. Although the referee questioned the qualification of our paper to be published in SciPost Physics, we respectfully disagree.
Anonymous referee writes:
I think this is a nice demonstration that this phenomenology is possible, in a model where an essentially exact solution can be found after selfaveraging.
Nevertheless, I do not see how this work meets any of the acceptance criteria (i) Detail a groundbreaking theoretical/experimental/computational discovery; (ii) Present a breakthrough on a previouslyidentified and longstanding research stumbling block; (iii) Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work; (iv) Provide a novel and synergetic link between different research areas.
This is the reason I suggest the paper be moved/accepted to SciPost Physics Core, whose criteria it meets better.
Our response:
The recurring theme in the field of the correlated superconductivity is understanding the competition between the nonFermi liquid behaviour and superconducting instability due to strong electronic interactions. Despite many theoretical studies have been done since the discovery of high $T_c$ cuprates, the answer to the question is still inconclusive because the controlled theories of strongly correlated systems are rare.
Our work (ii) presents a breakthrough on a previouslyidentified and longstanding research stumbling block by investigating the solvable models which allow the systematic theoretical studies on the competition between the nonFermi liquid behaviour and the pairing instability. The YukawaSYK model with the distribution of variances is a rare exactly solvable model which supports both the Fermi liquid, nonFermi liquids, and paired states at low temperatures. This can be contrasted to the previously published variants of the SYK model. They support only the nonFermi liquid states and the paired states. We would also like to highlight that our model does not introduce any ad hoc parameter to independently control the amount of fermion incoherence and pairing instability. The distribution of variances, which has a concrete physical meaning, impacts on both fermion pairing and fermion incoherence. Thus, our work examined the competition between nonFermi liquidness and superconductivity by controlling the common physical origin.
By examining the pairing instability of both Fermi and nonFermi liquid normal states, we could draw nontrivial but reliable conclusion that the nonFermi liquid normal states with more incoherent fermions can have stronger tendency to form Cooper pairs (i.e., higher transition temperature $T_c$) compared to the Fermi liquid. Given that our model is exactly solvable, our theoretical conclusion can motivate followup researches for more realistic and complex models for the superconductivity born out of nonFermi liquid normal states.
Furthermore, our work expand the scope of the SYK model researches by examining the pairing instabilities of the nonfastscrambling normal states of the SYK model in the systematic fashion. While most previous researches focused on the singlet pairing of the fast scrambling NFL states of the SYK model, our work called for attention to both spin singlet and triplet pairings of nonfastscrambling normal states of the YukawaSYK models.
With the one parameter family of the Fermi and nonFermi liquid states controlled by the parameter $\eta$, we discovered the leading spin triplet pairing instability born out of various amount of fermion incoherence. As our research first explored the leading triplet pairing instabilities from a single YukawaSYK quantum dot, there are plethora of followup questions such as possibility of topological spin triplet SYK superconductivity from some higher dimensional generalisation of our theory. Therefore, our work (iii) opened a new pathway in the field of the SYK superconductivity.
Anonymous referee writes:
One optional suggestion for the authors: perhaps if you allow the masses of the bosons to also follow a distribution (instead of all being set to m) you might get an independent "knob" on the behavior of the bosons, and be able to tune the nonFermi character of the fermions independently of the strength of the boson glue?
Our response:
We thank the referee for the optional suggestion to expand our research. Introducing the distribution of mass must be an interesting research direction to investigate the role of boson mass on the pairing instability and nonFermi liquid behaviour of the normal states. However, in the current work, the introduction of the variance of the coupling constant was sufficient to demonstrate how the fermion incoherence competes with the Cooper pair formation. Therefore, we will leave the extensive survey of our model with the distribution of the boson mass to the future work.
Author: Wonjune Choi on 20220318 [id 2299]
(in reply to Report 2 by Yuxuan Wang on 20211205)We thank Professor Yuxuan Wang for his sincere review and for considering our work suitable for publication in SciPost Physics.
Prof. Wang writes:
Our response:
Following the suggestion, we expand Section 4, which discusses the normal state properties of the YukawaSYK model, to make our paper more selfcontained. Below, we answer the questions in the referee's report.
Prof. Wang writes:
Our response:
The YukawaSYK models with a fixed variance coupling [Esterlis & Schmalian (2019) and Classen & Chubukov (2021)] demonstrate two distinct nontrivial normal states at finite temperatures: the quantum critical SYK nonFermi liquid (SYKNFL) normal state at low temperatures, and the impuritylike nonFermi liquid state at intermediate temperature window.
The SYKNFL state is the fast scrambling, conformal solution of the YukawaSYK model. The nonFermi liquid nature originates from strong bosonfermion interactions which dynamically tune the mass of bosons to zero [Wang (2020)]. The scaling property of the fermion Green function $G(i\omega_n)$ is dominated by the fermion selfenergy $\Sigma(i\omega_n) \sim i\mathrm{sgn}(\omega_n) \omega_n^{12\Delta}$ at low frequencies. The scaling dimension lies between $\frac{1}{4} < \Delta < \frac{1}{2}$ and depends on the ratio between the number of bosons ($M$) and fermions ($N$), $\gamma = M / N$.
The normal states of our model with the distribution of the variances $\rho(\lambda)$ show different scaling behaviour compared to the SYKNFL states. Both "Fermi liquid" ($\eta > 0$) and "nonFermi liquid" ($1 < \eta < 0$) states show the "impuritylike" behaviour, which is also observed from the fixedvariance model at intermediate temperature window [Esterlis & Schmalian (2019) and Classen & Chubukov (2021)]. The strong interactions do not tune the mass of the boson to zero. Instead, the bosons act as impurity centres that scatter fermions. Especially, as we can see from Eq. (23), the impuritylike scattering due to the Bose condensate $\varphi$ and the static uncondensed bosons $D_N(i\nu_n = 0)$ (the zero frequency part of the uncondensed boson's Green function) lead to the leading scaling dimension of the fermions $\Delta = \frac{1}{2}$, i.e., $G(i\omega_n) \sim i\mathrm{sgn}(\omega_n)$, for all $\gamma$ and $\eta > 1$. The subleading selfenergy corrections $\Sigma_N(i\omega_n) \sim i\mathrm{sgn}(\omega_n) \omega_n^{1+\eta}$ due to the impuritylike scattering from the uncondensed bosons introduce anomalously large quasiparticle decay $1/\tau_{\mathrm{life}} \sim T^{1+\eta}$ [Kim, Cao & Altman (2020a)].
While the impuritylike nonFermi liquid fixed point is not stable in the fixedvariance models, our model supports the impuritylike Fermi liquid and nonFermi liquid states as stable infrared fixed points at low temperatures. When it comes to the scaling property of the stable IR fixed point, our model with the distribution of the variances shows qualitatively distinct normal states compared to the previously studied fixedvariance models which support the SYKNFL state. Furthermore, by controlling the distribution $\rho(\lambda)$, our model provides systematic control over the fermion incoherence.
Prof. Wang writes:
Our response:
Since the model distribution $\rho \sim (\lambda_{\max}  \lambda)^\eta$ in Eq. (16) is a welldefined probability distribution only if $\eta > 1$, i.e., $\int_0^{\lambda_{\max}} \rho(\lambda) ~ d\lambda$ diverges if $\eta \leq 1$, we cannot naively extend our result to the limit $\eta \to \infty$. If we tried to extend the calculations for $\eta \leq 1$, the integral in Eq. (24) for the boson Green function diverges.
Although no limiting case of $\eta$ can fully reproduce the results of the model with a fixed variance, $\eta \to 1^{+}$ limit partially recovers the impuritylike NFL behaviour and its pairing instability in Esterlis & Schmalian (2019). The model distribution $\rho(\lambda)$ in Eq. (16) is designed to explore diverse singular properties of the boson Green function $D_N(i\nu_n)$ near zero frequency, $\nu_0 = 0$. The singular property of $D_N(i\nu_n)$ is determined by the function $f_{\eta}(x)$, whose asymptotic property is derived in Eq. (28). For $\eta > 1$, we can examine the boson Green function such that $f_{\eta}(x)$ decays slower than $\frac{1}{1x}$. To be more specific, our model distribution $\rho_\eta(\lambda)$ parameterized by $\eta > 1$ allowed us to explore the boson Green function
for $\nu_n$ near 0. Note that if we consider a fixed variance model, i.e., $\rho(\lambda) \sim \delta(\lambda\lambda_{\max})$, then
We can see that the lowfrequency behaviour of the fixed variance model $D_{\mathrm{fixed}}$ is equal to that of $D_\eta$ with $\eta \to 1^{+}$. (For unambiguous notation, we denote $D_\eta$ for the boson Green function with the distribution of variances and $D_{\mathrm{fixed}}$ for the boson Green function with a fixed variance.)
In addition, the asymptotic behaviour of the fermion selfenergy of our model $\Sigma_{\eta}$ approaches the selfenergy of the fixed variance model $\Sigma_{\mathrm{impurity}}$ for the impuritylike NFL state in Esterlis & Schmalian (2019):
as $\eta \to 1^{+}$.
Therefore, the impuritylike NFL state of the fixed variance coupling model can be considered as the limiting case of "$\eta \to 1^{+}$".
However, we should note that the stability of the lowenergy fixed point is dramatically changed: while the impuritylike fixed point is stable for $\eta > 1$, it is an unstable fixed point in case of the fixed variance model (see Figure 2 of Esterlis & Schmalian (2019)). The SYKNFL state becomes the stable IR fixed point in the fixed variance model.
Prof. Wang writes:
Our response:
Both Ref. 36 and Phys. Rev. B 104, 125120 (2021) discussed the pairing instabilities of the impuritylike NFL state in the strongly coupled regime, i.e., $\rho(\lambda) \sim \delta(\lambda\lambda_{\max})$ with $\lambda_{\max} \gg m^3$. For the spinsinglet Yukawa coupling ($S_g \sim g_{ij,k} c_{i\alpha}^\dagger c_{j\alpha} \phi_k$ in Eq. (4)), the nature of the paired state and the pairing transition is qualitatively similar to the pairing instability of the impuritylike NFL state discussed in Ref. 36 and Phys. Rev. B 104, 125120 (2021). However, in case of the spintriplet Yukawa coupling ($S_g \sim g_{ij,k} c_{i\alpha}^\dagger \sigma^z_{\alpha\beta} c_{j\beta} \phi_k$), we found that the spintriplet pairing $F^{1,2}(\tau) \sim c_{i\uparrow}^\dagger(\tau) c_{i\uparrow}^\dagger(0) \pm c_{i\downarrow}^\dagger(\tau) c_{i\downarrow}^\dagger(0)$ at the distant time $\tau >0$ is the leading instability.
Most importantly, our work focused on how the transition temperature changes as we tune the fermion incoherence by controlling the distribution $\rho(\lambda)$. In the fixed variance model, changing the absolute magnitude of the variance $\lambda_{\max}$ or the mass of bosons $m$ does not influence the critical temperature $T_c$. This can be seen from the constant $T_c$ in Figure 5 of Esterlis & Schmalian (2019). Previous work demonstrated one particular example of how the impurityNFL state experiences pairing instability. However, in our theory, the fermion incoherence is strongly dependent on the distribution control parameter $\eta$. Hence, we could demonstrate how the fermion incoherence born out of the strong fermionboson interactions competes with the enhanced pairing tendency born out of the same physical origin. This could not be discussed in the previous studies due to the lack of controllability for the fermion incoherence.
Prof. Wang writes:
The BoseEinstein condensation (BEC) occurs when the number of excited states is bounded from above, i.e., $N_{\mathrm{excited}} < N_0$, at some temperatures below $T_{\mathrm{BEC}}$. If there were $N$ number of bosons, the remaining $N  N_0$ number of bosons are forced to be in the ground state since the Bose statistics limits the maximum number of the excited bosons. Then, the macroscopic number ($N  N_0 \gg 1$) of bosons are said to be condensed in the ground state.
For our model, it is difficult to calculate the maximum number of available excited states $N_0$ explicitly. Instead, we can demonstrate that the bosons inevitably become unstable (i.e., $m^2  \lambda_{\max}\Pi(0) < 0$) without the BoseEinstein condensation when $\eta > 0$. If there were no BEC ($\varphi = 0$), we will first show that the boson Green function $D(i\nu_n)$ is bounded from above when $\eta > 0$. Next, we will prove that the fermion selfenergy $\Sigma(i\omega_n)$ is bounded from above because $D(i\nu_n)$ is bounded from above. At last, when the fermion selfenergy is bounded from above, we can show that the boson selfenergy $\Pi(0)$ at zero frequency is bounded from below. The lower bound of $\Pi(0)$ is a function of temperature, and we will show that $m^2  \lambda_{\max} \Pi(0)$ inevitably becomes negative at some finite temperatures because of this lower bound. In short, we will prove that BEC must occur in $\eta>0$ model because "no BEC" implies the bounded boson Green function which results in the unstable bosons. Below, we mathematically demonstrate this idea.
When $\eta > 0$, we first demonstrate below that the contribution of uncondensed bosons to the boson Green function $D_N(i\nu_n)$ is bounded from above. From Eq. (25), we find
Using l'Hospital's rule, explicit calculations can show that $\lim_{x\to1} f_\eta(x) = 1/\eta.$ Since $f_\eta$ is a strictly increasing function of $x \in (\infty, 1]$, $f_\eta(x) \leq f_\eta(1)=1/\eta$. Thus, the above inequality holds. Note that if $\eta \leq 0$, $f_\eta(x)$ diverges at $x=1$.
Let us suppose that the bosons are not condensed, i.e., $\varphi = 0$. Then, from Eq. (18),
Note that the normal state fermion Green function without the BoseEinstein condensation has the canonical form:
Since $iG_0(i\omega_n)^2$ decays at least as fast as $1/n^2$, its sum over all Matsubara frequencies must be convergent. Hence, $\mathcal{S}$ must be a finite nonnegative real number. To be more specific, if $\Sigma_0(i\omega_n) \gg \omega_n$ as $\omega_n \to \infty$, $\sum_{n} i G_0(i\omega_n)^2$ converges because the square of the Green function decays faster than $1/n^2$. If $\Sigma_0(i\omega_n) \ll \omega_n$ as $\omega_n \to \infty$, then $\sum_{n} i G_0(i\omega_n)^2$ is also convergent because $iG_0(i\omega_n)^2$ decays as $1/n^2$.
When the fermion selfenergy is bounded from above, the boson selfenergy at zero frequency is bounded from below:
where $\psi^{(1)}$ is the polygamma function of order 1. Thus, $m^2  \lambda_{\max}\Pi(0) < 0$ if
Since $\mathcal{S}$ also depends on temperature $T$, one can question whether the above inequality can be satisfied for some finite temperatures $T>0$. To examine the existence of the solution of the inequality, let us first investigate the $\gamma = 0$ case for given $\eta$, $m$, and $\lambda_{\max}$. Since $\mathcal{S} = 0$ when $\gamma = 0$,
Thus, $m^2  \lambda_{\max} \Pi(0) < 0$ if $T < \lambda_{\max} / 2m^2$, i.e., the bosons become unstable at finite temperatures if there were no BoseEinstein condensation. Suppose $i\Sigma_0(i\omega_n)$ for finite $\gamma > 0$ is continuously connected to the $\gamma = 0$ limit, i.e., if we increase $\gamma$ from $0$, then $\mathcal{S}$ also continuously increases from zero. (By numerically solving SchwingerDyson equations, we confirmed that the selfconsistent Green functions and selfenergies for finite $\gamma > 0$ is continuously connected to the selfconsistent solution for $\gamma = 0$ (up to $\gamma = 1$). This is also previously confirmed by Kim, Cao & Altman (2020b).) Since $\psi^{(1)}(1/2+\mathcal{S}/2\pi)$ is an analytic function of $\mathcal{S} \geq 0$, the existence of the solution at $\gamma = 0$ implies the existence of the solution for some finite $\gamma > 0$.
In conclusion, the BoseEinstein condensation is necessary for the $\eta > 0$ model to avoid the instability of bosons. If the bosons are condensed ($\varphi \neq 0$), the bosons can be either critical or stable because the total boson Green function $D(i\nu_n) = (\beta \lambda_{\max}/\gamma) \varphi \delta_{n,0} + D_N(i\nu_n)$ is no longer bounded from above. Then the quantum fluctuations of the bosons can yield arbitrarily large fermion selfenergy. The sufficiently large fermion selfenergy would not result in too large boson selfenergy which can make the bosons unstable. Note that our conclusion is consistent with the previous work for the fixedvariance model. Since the boson Green function is not bounded from above for the fixedvariance model, the bosons can remain critical without the condensation [Wang (2020)]. We would also like to note that physics of our model is different from that of free Bose gas at $d=0$ because the bosons are strongly coupled to the massless fermions. The intertwined dynamics of the strongly correlated bosons and fermions can result in the Bose condensation even at $d=0$ under certain conditions.