Glass-induced enhancement of superconducting $T_c$: Pairing via dissipative mediators

TL;DR

This paper investigates how dissipative mediators influence superconductivity, revealing that non-local dissipation can enhance the critical temperature ($T_c$) up to a maximum before destroying superconductivity.

Contribution

It introduces a novel mechanism where dissipation in non-local mediators can increase $T_c$, contrasting with the traditional view that dissipation always suppresses superconductivity.

Findings

Non-local dissipation can enhance $T_c$ up to a critical point.

Local dissipation reduces the superconducting transition temperature.

The maximum $T_c$ occurs when dissipation matches the mediator's spatial stiffness energy scale.

Abstract

With substantial evidence of glassy behavior in the phase diagram of high $T_c$ superconductors and its co-existence with superconductivity, we attempt to answer the question: what are the properties of a superconducting state where the force driving cooper pairing becomes dissipative? We find that when the bosonic mediator is local, dissipation acts to reduce the superconducting critical temperature ($T_c$). On the other hand, contrary to na\"{i}ve expectations, $T_c$ behaves non-monotonically with dissipation for a non-local mediator -- weakly dissipative bosons at different energy scales act coherently to give rise to an increase in $T_c$ and eventually destroy superconductivity when the dissipation exceeds a critical value. The critical value occurs when dissipative effects become comparable to the energy scale associated with the spatial stiffness of the mediator, at which point, $T_c$ acquires a maximum. We outline consequences of our results to recent proton irradiation experiments (M. Leroux et al.,~\cite{Welp2018}) on the cuprate superconductor La$_{2-x}$Ba$_x$CuO$_4$ (LBCO) which observe a disorder induced increase in $T_c$ even when the transition temperature of the proximate charge density wave (CDW) is unaffected by the presence of irradiation. Our mechanism is a novel way to raise $T_c$ that does not require a `tug-of-war' -type scenario between two competing phases.

Equations

$$\displaystyle D(\tau)\equiv\bigg{[}\langle S_{i\mu}(\tau)S_{i\mu}(0)\rangle\bigg{]}\sim\frac{1}{\tau^{2}},$$

$$S[\Psi ,{\Psi }^{\ast }]$$

$$S_0[\Psi ,{\Psi }^{\ast }]$$

$$D(\text{q},i{\omega }_n)=\frac{\alpha }{\kappa q^2+{\omega }_n^2+2\eta |{\omega }_n|+M^2}.$$

$$\Delta (i{\omega }_n,\text{k})=\frac{|g{|}^2}{\beta V}\sum _{\text{q},{\omega }_m}\frac{D(\text{q},i{\omega }_n-i{\omega }_m)\Delta (i{\omega }_m,\text{k}+\text{q})}{{\omega }_m^2+{\xi }_{\text{k}+\text{q}}^2+\Delta (i{\omega }_m,\text{k}+\text{q}{)}^2},$$

$$1$$

$$1=\lambda {\int }_{-\infty }^{\infty }d\xi \left(\frac{1}{4T({\xi }^2+{\Delta }^2)}-\frac{tanh\frac{\sqrt{{\xi }^2+{\Delta }^2}}{2T}}{2({\xi }^2+{\Delta }^2{)}^{3/2}}\right).$$

$$\sum _m\frac{1}{(|{\omega }_m{|}^2+x^2)|{\omega }_m{|}^2}=\frac{x-2Ttanh\frac{x}{2T}}{4x^3{T}^2}$$

$$\frac{1}{\lambda }$$

$$a(T)=\frac{1}{2}\left[\psi \left(2,\frac{3}{2}+\frac{\beta \Lambda }{2\pi }\right)-\frac{1}{2}\psi (2,\frac{1}{2})\right]$$

$$u(T)$$

$$v$$

$$F(x)$$

$$G(x)$$

$$\displaystyle\frac{1}{\bar{\lambda}}=\frac{1}{2\pi T}\Bigg{[}\frac{\bar{\eta}^{\prime}\pi^{2}}{1+\bar{\eta}^{\prime 2}}-\frac{\bar{\eta}^{\prime}\pi^{4}(3+\bar{\eta}^{\prime 2})}{12(1+\bar{\eta}^{\prime 2})}\left(\frac{\Delta}{2\pi T_{c}}\right)^{2}+...\Bigg{]}$$

Full text

I Introduction

In $s$-wave superconductors (SCs) where the quasiparticle excitation spectrum is fully gapped and has a constant sign of the pairing form factor across the Fermi surface, Anderson’s magic theorem keeps the critical temperature ($T_{c}$) robust to weak, non-magnetic impurities. In higher angular momentum SCs ($p$-,$d$-wave etc) or SCs where the sign of the gap changes across parts of the Fermi surface (such as $s_{\pm}$ pnictide SCs), $T_{c}$ is drastically suppressed with the addition of impurities$-$magnetic or otherwise Balatsky et al. (2006). These effects hold in the independent disorder limit and in the absence of electron correlations.

At a collective level when electron correlations are taken into account, randomness can yield several interesting phases of matter Lee and Ramakrishnan (1985). Amongst these is the spin glass (SG) phase widely observed in the phase diagram of many strongly correlated systems like high $T_{c}$ SCs Chou et al. (1995); Wakimoto et al. (2000); Cordero et al. (2001); Hunt et al. (2001); Julien (2003); Sternlieb et al. (1990); Mitrović et al. (2008); Imai and Hirota (2018); Julien et al. (1999); Baek et al. (2012); Wu et al. (2013); Dioguardi et al. (2013); Rømer et al. (2013); Ryu et al. (2015a, b); Grinenko et al. (2012); Yadav and Paulose (2010). The SG phase exhibits a remarkable phenomenology Mydosh (2015) – a transition into the SG defined by a broad cusp in the specific heat, a split in the DC magnetization at the SG transition depending on whether the SG phase is field cooled (FC) or zero field cooled (ZFC), linear temperature dependence of the AC susceptibility peak, and aging. Theoretically, SGs are described by an order parameter where the spin average on each site is non-vanishing but goes to zero when averaged over the lattice Read et al. (1995). Important to the discussions that follow, spin correlators at the SG critical point follow a power law of the form Bray and Moore (1985); Miller and Huse (1993); Read et al. (1995); Dalidovich and Phillips (1999)

[TABLE]

which in frequency space reads $D(\omega)\sim|\omega|$. Here, $S_{i\mu}$ is the $\mu-$th component of the spin at site $i$, and the angular and square brackets denote thermal and site averages respectively. The linear frequency dependence of the spin correlators indicates that dissipative dynamics is a necessary – albeit not sufficient – ingredient of SGs.

In this work, we explore the robustness of $T_{c}$ and properties of a superconducting state where the dynamics of the pairing mediator is rendered dissipative due to collective disorder (in the aforementioned sense). To this end, we add to the Lagrangian describing the mediator a dissipative term Caldeira and Leggett (1983); Chakravarty et al. (1987, 1988); Wagenblast et al. (1997) $\Delta L=\sum_{\textbf{k},\omega_{n}}\eta|\omega_{n}||\Psi(\textbf{k},\omega_{n})|^{2}$. Here k and $\omega_{n}$ are the momenta and Matsubara frequencies, $\eta$ is a measure of dissipation, and $\Psi(\textbf{k},\omega_{n})$ is the bosonic field. We find that dissipative effects generally act to suppress $T_{c}$ when the mediator is local. This occurs because dissipation has the effect of reducing the attractive interaction mediating cooper pairs. However, contrary to naïve expectations, $T_{c}$ behaves non-monotonically with dissipation for a non-local mediator. In this scenario, weakly dissipative bosons at different energy scales act coherently to give rise to an increase in $T_{c}$ and eventually destroy superconductivity when the dissipation exceeds a critical value. The critical value occurs when the dissipation parameter, $\eta$, becomes comparable to the energy scale associated with the velocity of the mediating bosons (or the spatial stiffness); at this crossover, $T_{c}$ acquires a maximum value. We also study the effects of dissipative mediator on the ratio $\frac{2\Delta(0)}{T_{c}}$ and the heat capacity jump at the superconducting transition and find departures from values predicted by BCS theory.

II Experimental basis

We now make our case for a dissipative or ‘glassy’ mediator from experiments on a variety of high $T_{c}$ SCs. The SG phase has been observed extensively in the under-doped and regions proximate to superconductivity in the phase diagrams of both the cuprate Chou et al. (1995); Wakimoto et al. (2000); Cordero et al. (2001); Hunt et al. (2001); Julien (2003); Sternlieb et al. (1990); Mitrović et al. (2008); Imai and Hirota (2018); Julien et al. (1999); Baek et al. (2012); Wu et al. (2013) and iron based superconductors Dioguardi et al. (2013); Rømer et al. (2013); Ryu et al. (2015a, b); Grinenko et al. (2012); Yadav and Paulose (2010). Existing evidence is also spread over several techniques such as DC magnetization Chou et al. (1995); Grinenko et al. (2012); Yadav and Paulose (2010); Ryu et al. (2015a, b); Wakimoto et al. (2000), NMR/NQR Dioguardi et al. (2013); Imai and Hirota (2018); Hunt et al. (2001); Baek et al. (2012); Wu et al. (2013); Julien et al. (1999), $\mu$SR Sternlieb et al. (1990); Mitrović et al. (2008) and neutron scattering Sternlieb et al. (1990); Rømer et al. (2013). Given the strong evidence of a SG phase and its proximity to the superconducting dome in high $T_{c}$ SCs, it is already reasonable to consider its effect on the pairing problem. Additionally, there is ample experimental evidence lending credence to a dissipative character of fluctuations that mediate Cooper pairing. First, disorder causes the $d$-electron spins (Cu spins in the cuprates and Fe spins in the iron superconductors) to exhibit glassy behavior and not the dopant spins Julien (2003); Chou et al. (1995); Dioguardi et al. (2013); Grinenko et al. (2012) (although in certain iron based systems, it is the dopant spins become glassy Nadeem et al. (2015)). Second, SG and SC phases actually co-exist in a variety of high $T_{c}$ SCs Julien et al. (1999); Dioguardi et al. (2013); Grinenko et al. (2012). This indicates a strong inter-mixing of properties of the two phases, similar to what is expected in the context of other mean-field orders (such as density waves) acquiring a glassy behavior Mross and Senthil (2015). Third, neutron scattering and NMR/NQR measurements in the cuprate SCs La2-xSrxCuO2 (LSCO) and La2-xBaxCuO4 (LBCO) have found a direct ‘slowing’ of spin fluctuations in the vicinity of glassy orders Hunt et al. (2001); Rømer et al. (2013); Sternlieb et al. (1990); Wu et al. (2013). Finally, early theoretical predictions on doping La2CuO4 clearly point to a frustration induced glassy behavior of the Cu $d$-orbital spins in the phase diagram Aharony et al. (1988).

Hence, the notion of a dissipative pairing mediator in high temperature superconductors has firm foundations in both experiment and theory. As will be argued later in this paper, non-local dissipative mediators can help throw light on recent proton irradiation experiments Leroux et al. (2018) on LBCO which observe a disorder induced increase in $T_{c}$ even when the transition temperature of the proximate charge density wave (CDW) is unaffected by the presence of radiation disorder. The mechanism we propose in this paper forms an alternative way to raise $T_{c}$ of a superconductor that does not require a ‘tug-of-war’ -like scenario between two competing phases.

III Model and gap equation

We begin by writing the conjectured model for the bosonic propagator. The total action consists of a free part $S_{0}[\Psi,\Psi^{*}]$ and a dissipative part $S_{dis}[\Psi,\Psi^{*}]$ defined by

[TABLE]

where $\kappa$ is the spatial stiffness or energy scale associated with the boson velocity, and the squared mass, $M^{2}$, is proportional to the inverse correlation length. As outlined in the introduction, we take the dissipative term to be form $S_{dis}[\Psi,\Psi^{*}]=\sum_{\textbf{k},\omega_{n}}(2\eta~{}|\omega_{n}|)|\Psi(\textbf{k},\omega_{n})|^{2}$ in Fourier space with the various quantities defined previously. With this total action, the bosonic propagator, $D(\textbf{q},i\omega_{n}-i\omega_{m})$, takes the form

[TABLE]

Here $q=|\textbf{q}|$ and $\alpha$ is a constant with dimensions of energy that can be absorbed into an effective coupling constant (similar to spin fluctuations; see for example Bennemann and Ketterson (2008)).

$T_{c}$, local case ($\kappa=0$): We choose a quadratic electron dispersion $\xi_{\textbf{k}}$ and gap function denoted by $\Delta(i\omega_{n},\textbf{k})$. Substituting $D(\textbf{q},i\omega_{n}-i\omega_{m})$ into the gap equation,

[TABLE]

and assuming an isotropic, frequency independent $s$-wave gap (defined by the $\textbf{k}=0$ value and denoted by $\Delta$ henceforth), the equation determining $T_{c}$ (setting $\Delta=0$) reduces to $1=\pi\lambda T\sum_{\omega_{m}<\Lambda}\frac{1}{|\omega_{m}|\left(\omega_{m}^{2}+2\eta|\omega_{m}|+M^{2}\right)}$ where $T$ is set to $T_{c}$. Here $\beta$ is the inverse temperature, $g$ is the interaction strength, $\lambda\equiv N(0)|g|^{2}\alpha$ is the coupling constant, $\Lambda$ is the high energy cut-off and $N(0)$ is the density of states at the Fermi energy. The sum over $\omega_{m}$ can be performed exactly to yield the equation for $T_{c}$ as ($c.c$ is complex conjugate)

[TABLE]

where $\bar{M}\equiv\sqrt{M^{2}-\eta^{2}}$ and $\psi(x)$ denotes the digamma function. The solutions for $T_{c}$ as a function of the parameters $M$ and $\eta$ are shown in the left panels of Figs 1 and 2. The reduction in $T_{c}$ as a function of the mass and dissipation can be intuitively understood by taking the limit of $M\gg T,\eta$ and $\eta\gg T,M$ respectively. In these limits, $\eta$ and $M^{2}$ can be factored out of the Matsubara sum which, in effect, reduces the coupling constant $\lambda$ and hence suppresses $T_{c}$.

$T_{c}$, non-local case ($\kappa\neq 0$): We can make similar assumptions on the superconducting gap for the $\kappa\neq 0$ case. To maintain analytical tractability and focus on the effect of dissipation parameter $\eta$, we will later set the mass (now renormalized by the chemical potential; we use the same symbol for ease of notation) to zero. We can now substitute the bosonic propagator with $\kappa\neq 0$ back into the gap equation Eq 2. The resulting energy integral can be solved exactly by the method of residues and takes the form $\int_{-\infty}^{\infty}\frac{d\xi}{(\xi^{2}+r^{2})(\kappa\xi+s)}=\frac{\pi s}{(\kappa^{2}r^{2}+s^{2})r}$, where $r^{2}=\omega_{m}^{2}+\Delta^{2}$ and $s=\omega_{m}^{2}+\eta^{\prime}|\omega_{m}|+M^{2}$. Performing the remaining Matsubara sum we obtain the equation for $T_{c}$ as

[TABLE]

where $\eta^{\prime}\equiv 2\eta$. The solution for $\bar{T}_{c}=T_{c}/\kappa$ is plotted in the right panel of Fig 2 as a function of of $\bar{\eta}^{\prime}=\eta^{\prime}/\kappa$. As is evident, for the case of a non-local mediator, $T_{c}$ behaves non-monotonically with dissipation and rises up to $40\%$ of the initial $\eta=0$ value. This happens because weakly dissipative bosons at different energy scales act coherently to give rise to an increase in $T_{c}$ but eventually destroy superconductivity for large dissipation. The critical value occurs when the dissipation parameter is of the order of the stiffness constant ($2\eta\sim\kappa$); at this point, $T_{c}$ acquires a maximum with respect to $\eta$. This physics follows from the energy integral leading to Eq. 4 above. To see this, notice that the role of the stiffness parameter $\kappa$ is to induce non-monotonicity in an ‘effective’ coupling constant as a function of $\eta$ – while $\eta$ acts only to reduce the effective coupling constant for the local case, the energy integral (leading to Eq. 4) for the non-local mediator forces the gap equation to acquire dissipative contributions that both increase and decrease the effective coupling constant. Consequently, this translates into a non-monotonic behavior in $T_{c}$.

IV Gap and specific heat jump

We now study the variation of the gap with temperature and the specific heat jump at $T_{c}$. In Fig 3 we plot the temperature dependence of the superconducting gap as a function of the dissipation and mass parameters for $\kappa=0$. Both $\eta$ and $M$ reduce the zero temperature gap $\Delta(0)$ and $T_{c}$; however, dissipation (mass) has a greater (smaller) effect on $T_{c}$ compared to $\Delta(0)$. Hence, the BCS ratio $\frac{\Delta(0)}{T_{c}}$ increases (decreases) with the dissipation (mass) parameter. To get an analytical handle for the gap near $T_{c}$, we begin with the case of $\eta=M=0$ where the gap equation becomes

[TABLE]

We have made use of the summation identity

[TABLE]

above. We next expand for small gaps near $T_{c}$ to obtain

[TABLE]

where

[TABLE]

is weakly temperature dependent in the limit of $\beta\Lambda\rightarrow\infty$, $b\simeq\frac{31}{32}\frac{\beta_{c}^{4}\xi(5)}{\pi^{4}}$ and $\psi(n,x)$ is the $n$-th order digamma function. Setting the gap to zero in Eq 7, we can read off the dependence of $T_{c}$ on the coupling as $T_{c}\sim\sqrt{\lambda}$, which grows faster than the conventional BCS relation. The temperature dependence of the gap can be derived as $\Delta^{2}(T)=\frac{2a(0)T_{c}\pi^{4}}{4\pi^{2}(31/32)\xi(5)}\left(T_{c}-T\right)$, and therefore, the normalized specific heat jump at $T_{c}$ is ($\gamma=2\pi^{2}N(0)/3$ is the normal state specific heat) $\frac{\Delta C}{\gamma T_{c}}=\frac{3a(0)}{4\xi(5)}\left(\frac{32}{31}\right)\simeq 6$, which is greater than the BCS value. Similarly, in the limit where the dissipation is much larger than the temperature and mass ($\eta|\omega_{m}|\gg|\omega_{m}|^{2},M^{2}$), we have $\frac{1}{\lambda}\simeq\frac{u(T)}{4\pi T\eta}-\frac{v}{8\eta\pi^{4}T_{c}^{3}}~{}\Delta^{2}$. Here

[TABLE]

and $H(z)$ is the Harmonic number. $T_{c}$ can be evaluated again by setting $\Delta=0$ and we see that, in this limit, $T_{c}\sim\frac{\lambda}{\eta}$. The cross-over from $T_{c}\sim\sqrt{\lambda}$ to $T_{c}\sim\lambda$ as a function of $\eta$ is shown in Fig 1(right). The temperature dependence of the gap can be evaluated from above as $\Delta(T)^{2}=\frac{2\pi^{5}T_{c}}{v}(T_{c}-T)$; hence, the specific heat jump at $T_{c}$ takes the value $\frac{\Delta C}{\gamma T_{c}}=\frac{3\pi^{3}}{v}\sim 3.64$ which is again greater than the BCS value.

On the other hand, expanding the gap equation for a non-local mediator ($\kappa\neq 0$) in the limit of $\eta=M\rightarrow 0$, we obtain $\frac{\kappa^{2}}{\lambda}\simeq F\left(\frac{\kappa}{2\pi T}\right)-G\left(\frac{\kappa}{2\pi T}\right)\tilde{\Delta}^{2}$. The dimensionless functions $F(x)$, $G(x)$ and $\tilde{\Delta}$ are defined as

[TABLE]

$\tilde{\Delta}=\frac{\Delta}{2\pi T}$, and $\gamma_{E}$ is the Euler gamma constant. In the limit $x\ll 1$, the functions $F(x)$ and $G(x)$ satisfy the property $F(x)=C_{1}x^{2}$ and $G(x)=C_{2}x^{2}$, where $C_{1}$ and $C_{2}$ are numerical constants. The dependence of $T_{c}$ on $\lambda$ goes as $T_{c}\sim\sqrt{\lambda}$ and the temperature dependence of the gap takes the form $\Delta(T)^{2}=\frac{8\pi^{2}C_{1}T_{c}}{C_{2}}(T_{c}-T)$. This implies that the specific heat jump is $\frac{\Delta C}{\gamma T_{c}}\sim 5.6$, again larger than the BCS value. However, in the limit $\eta|\omega_{m}|\gg|\omega_{m}|^{2},M^{2}$ the expansion of the gap equation gives

[TABLE]

where $\bar{\lambda}=\lambda/\kappa$. Setting $\Delta=0$, we see that $T_{c}\sim\bar{\lambda}$ and the temperature dependence of the gap is given by $\Delta(T)^{2}=\frac{24T_{c}(T_{c}-T)(1+\bar{\eta}^{\prime 2})}{(3+\bar{\eta}^{\prime 2})}$. Hence, the specific heat jump is (weakly) dependent on the dissipation parameter and is given by $\frac{\Delta C}{\gamma T_{c}}=\frac{36(1+\bar{\eta}^{\prime 2})}{\pi^{2}(3+\bar{\eta}^{\prime 2})}$. For small $\bar{\eta}^{\prime}$, the normalized specific heat jump is $\simeq 1.2$ and is smaller than the BCS value consistent with specific heat experiments in under-doped cuprates Tallon and Loram (2001) and the pnictides Abdel-Hafiez et al. (2012).

V Discussions and experiments

Several theoretical works have explored mechanisms that yield an enhancement of $T_{c}$ with disorder strength. These phenomena range from competition of superconductivity with a proximate density wave phase Grest et al. (1982); Psaltakis (1984); Fernandes et al. (2012); Mishra and Hirschfeld (2016), multiorbital effects Gastiasoro and Andersen (2017), local inhomogeneities in the pairing interactions and mediators  Rømer et al. (2018); Martin et al. (2005); Arrigoni and Kivelson (2003); Loh and Carlson (2007); Aryanpour et al. (2006, 2007) to localization Feigel?man et al. (2007); Burmistrov et al. (2012, 2015); Mayoh and García-García (2015). A few works have also explored the interplay between glassy phases and superconducting $T_{c}$ Seki (1995); Galitski and Larkin (2002). In Galitski and Larkin (2002), the authors study a spin-glass formed by RKKY interactions between paramagnetic spins in a superconductor, and find an interaction driven enhancement of $T_{c}$ for a fixed impurity density. As a function of impurity concentration, however, the authors find that the $T_{c}$ decreases monotonically. Ref. Seki (1995) also finds a similar decrease in $T_{c}$ due to a reduction of the effective interaction induced by a SG phase that does not take into account the role of dissipation explicitly. Our results can alternatively viewed from the perspective of the well studied spin-fermion model Monthoux et al. (1991); Monthoux and Pines (1992); Schmalian et al. (1999); Abanov and Chubukov (2000); Abanov et al. (2003) where the dissipation parameter is proportional to the inverse spin-fluctuation frequency $\omega_{SF}$. In all of these works, $\omega_{SF}$ and the correlation length $\xi$ parameters are held fixed for different materials (see Table I of Ref. Monthoux et al. (1991)) at $T=T_{c}$. Although these works do not study the effect of $\omega_{SF}$ and $\xi$ on $T_{c}$, legitimate questions can be raised with regards to whether these quantities can be varied by an experimentally controlled tuning knob. In the case relevant to the present context, the effect of disorder on $\omega_{SF}$ and $\xi$ needs further examination, and perhaps the current works brings forth the need for a microscopic understanding of the relationship between disorder strength and the dissipation parameters $\eta$ and $\omega_{SF}$ . In the following paragraphs we argue for the applicability of the mechanism presented in this paper to the cuprates.

We begin by emphasizing that the change in $T_{c}$ in our work is due to modification of the ‘effective’ coupling by dissipation, and is unrelated to pair-breaking effects originating from lowering translational symmetry (say due to magnetic/non-magnetic inhomogeneities, like those summarized in Ref. Balatsky et al. (2006)). Hence, it can be intuited that qualitative aspects of our conclusions must hold for higher angular momentum pairing as well (albeit with tedious calculations). This can be more readily seen by noting that when the summand in the gap equation (in Eq 2) is decomposed into its partial fractions, there is always at least one (non-zero) term present where the dissipation parameter contributes to enhance the effective coupling strength (similar to Eq  4). This term(s) generally competes with other terms which suppress $T_{c}$, but gives rise to a $T_{c}$ increase when the dissipation is weak enough. Furthermore, according to our proposal, disorder acts as an external tuning knob of the parameter $\eta$; hence, increased irradiation leads to larger dissipation. Recent magnetization and tunnel diode (penetration depth) experiments Leroux et al. (2018) on proton irradiated LBCO at $\frac{1}{8}$ doping found up to a 50% increase in $T_{c}$ as a function of radiation dosage. An increased dosage above a critical value gradually suppressed $T_{c}$ until the eventual destruction of superconductivity. LBCO also hosts a rich phase diagram with evidence of density wave orders (CDW, SDW Kim et al. (2008); Hücker et al. (2011); Miao (2017) as well as spin- glass behavior Cordero et al. (2001) in conjunction with superconductivity in the under-doped regime. Hence, it is natural to anticipate an influence of these phases on superconductivity and examine their implications to $T_{c}$ variation as a function of disorder. Of the aforementioned existing mechanisms of $T_{c}$ enhancement proposed in literature, a competition-based scenario between superconductivity and a density wave order seems the most promising at first sight – especially given the close proximity of the CDW phase to the superconducting dome. Indeed, this was the point of view first suggested by Leroux and co-workers in Leroux et al. (2018). However, a closer examination of the data points to detaills that render this mechanism debatable. First, assuming that x-ray scattering is primarily sensitive to long-range CDW order Foo , the CDW transition temperature seems unaffected by irradiation Leroux et al. (2018). But a mechanism involving the competition between two mean field phases necessary involves a tug-of-war scenario where one phase gains stability at the expense of its competitor Fernandes et al. (2012). Second, it is unclear how non-magnetic disorder affects two different mean field phases (CDW, SDW, SC etc) asymmetrically in a parameter independent manner, except under very specific circumstances Fernandes et al. (2012); Mishra and Hirschfeld (2016) which do not necessarily hold in the case of LBCO and cuprates. Third, other non-magnetic impurities are well known to kill $d$-wave superconductivity monotonically Balatsky et al. (2006). Thus a consistent picture which distinguishes proton and electron irradiation with other point impurities like $Zn$ at a microscopic level is absent. Finally, from Anderson’s theorem, one can expect that a $T_{c}$ enhancement that occurs through a competition based scenario must be more prevalent in $s$-wave SCs rather than higher angular momentum SCs which are far less robust to non-magnetic impurities. Experiments on the $s$-wave superconductor 2H-NbSe2, however, draw conclusions that are mixed at best Mutka (1983); Tsang et al. (1975); Cho et al. (2018). Hence, a reasonable explanation for non-monotonic $T_{c}$ dependence as a function of disorder in LBCO must necessarily involve a mechanism that does not depend on the competition of two mean-field like phases. The proposed mechanism in this paper, along with the experimental evidence provided in the introduction, forms a feasible alternative that fits experiments.

In conclusion, motivated by the close proximity of glassy phases to the superconducting dome in high $T_{c}$ SCs, we explored the role of dissipation on superconducting properties such as $T_{c}$, the temperature dependence of the gap, BCS ratio and the specific heat jump at $T_{c}$. We found that when the mediator is local, dissipation acts to reduce the effective coupling constant and $T_{c}$ monotonically. On the other hand, when the mediator is non-local, two competing effects of dissipation determine the $T_{c}$ variation – first, the dissipative contributions of individual bosons at a given energy that act to suppress $T_{c}$, and second, collective contributions where dissipation acts to connect bosons at different energy scales that combine coherently to increase the effective coupling and $T_{c}$. The former (latter) contribution dominates when the dissipation parameter is greater (lesser) than the bosonic spatial stiffness, i.e., $\eta>\kappa$ ($\eta<\kappa$); $T_{c}$ peaks when these two scales are comparable to each other. We also studied the effects of a dissipative mediator on the ratio $\frac{2\Delta(0)}{T_{c}}$ and the heat capacity jump at $T_{c}$, and found departures from values predicted by BCS theory. In particular, the specific heat jump at $T_{c}$ acquires a value smaller than that predicted by BCS theory when the mediator is both dissipative and non-local, consistent with experiment. We pointed out consequences of our results to recent proton irradiation experiments in LBCO Leroux et al. (2018) where superconducting $T_{c}$ is enhanced with increased radiation disorder despite a robust CDW transition temperature, and concluded that one does not require a ‘tug-of-war’ like scenario between two competing phases to enhance superconductivity.

Acknowledgements: We thank P. J. Hirschfeld and P. W. Phillips for discussions. This work is supported by the DOE grant number DE-FG02-05ER46236.