You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: |
    There are many systems in which two order parameters compete with each other. Of particular interest are systems in which these order parameters are both unconventional. In this contribution, we examine the representative example of a d-wave superconductor in the presence of a d-wave density wave, which has been suggested as a model for the pseudogap phase in the high-$\rm
    T_c$ superconductors. The physical properties of unconventional superconductivity in the presence of an anisotropic charge density wave are investigated within mean field theory. This model describes many features that were anticipated by an earlier phenomenological treatment of Tallon and Loram. In addition, the quasiparticle density of states in the presence of these two order parameters is calculated, which should be accessible by scanning tunneling microscopy.
author:
- Stephan Haas
- Kazumi Maki
- Thomas Dahm
- Peter Thalmeier
title: Anatomy of Gossamer Superconductivity
---

Introduction
============

The discovery of unconventional high-temperature superconductivity by Bednorz and Müller[@bednorz] in 1986 took many in the community by surprise. Subsequent developments and some of the initial confusions are documented in the textbook by Enz.[@enz] Among many others, Anderson[@anderson] pointed out in 1987 that high-$\rm T_c$ superconductivity should be understood in terms of a competition between a Mott insulating state in the limit of zero hole doping and a superconducting state at optimal doping. Meanwhile, experimental evidence for the d-wave nature of high-$\rm T_c$ superconductivity was found,[@damascelli; @harlingen; @tsuei] and general features of the phase diagram were established.[@tallon] However, the nature of the pseudogap phase is still being debated. However, now there is increasing consensus that this regime can be described by in terms of a d-density-wave.[@benfatto; @chakravarty; @dora]

Recently, Laughlin proposed a very intuitive formulation regarding how to deal with superconductors close to a Mott insulating state. [@laughlin]. This “gossamer superconductivity" is characterized by a very small superfluid density and spectral weight of the superconducting quasiparticles. [@zhang] In this contribution, we will apply Laughlin’s idea to a more general situation: unconventional superconductivity in the presence of unconventional density wave (UDW) order. More precisely, for high-$\rm T_c$ cuprates we study d-wave superconductivity in the presence of a d-density wave. [@dora] This could either be a charge density wave or a spin density wave order parameter, although the distinction between these two possibilities is not of importance for the following discussion. Here, we wish to examine whether some features of the pseudogap phase can follow from the proposed model, as it was suggested by Tallon and Loram[@tallon] based on a phenomenological approach. Furthermore, the recently observed weakening of superconductivity in the $\rm \kappa - (ET)_2$ salts when the sample is rapidly cooled may be interpretetd within the model we are presenting in this paper. [@pinteric]

When both the superconductivity and the density wave are conventional, this competition is not of great interest as long as the system can be reduced to a single band or a single Fermi surface. In this case, there is typically little room for coexistence, and one of the order parameters, usually the superconductivity, dominates over the the other one.[@yamaji] A similar statement holds if only one of the two competing order parameters is conventional. However, the situation changes completely when two unconventional order parameters are considered [@thalmeier]. If we look at heavy fermion systems and organic superconductors, we see signs for gossamer superconductivity almost everywhere, for example in CeCu$_2$Si$_2$ [@thalmeier], URu$_2$Si$_2$ [@ikeda; @virosztek], UPd$_2$Al$_3$, CeRhIn$_5$, CeCoIn$_5$ [@izawa; @bianchi], and the $\kappa$-(ET)$_2$ salts [@pinteric; @miyagawa]. Therefore gossamer superconductivity as defined above may be a pervading feature in high-$\rm T_c$ cuprates, heavy fermion superconductors and organic superconductors.

Here, we report calculations for nodal superconductivity in the presence of nodal density wave order. For simplicity, we limit ourselves to quasi-two-dimensional systems with cylindrical Fermi surfaces. We denote the superconducting and density wave order parameters by $\Delta_1(\vec{k})$ and $\Delta_2(\vec{k})$, respectively, with angular dependence $\Delta_1(\vec{k}) = \Delta_1 f(\vec{k})$ and $\Delta_2(\vec{k}) = \Delta_2 g(\vec{k})$, where $\Delta_1$ and $\Delta_2$ are the maximum values of the corresponding order parameters. Further, we limit ourselves to $f=\cos (2 \phi)$ (i.e. $d_{x^2-y^2}$-wave) while for $g$ we consider the two cases (a) $g=\cos (2 \phi)$ and (b) $g=\sin (2 \phi)$. The first case describes many features of the superconductivity in the pseudogap phase in high-$T_c$ cuprates, while we conjecture that the second case applies for the glassy phase of $\kappa$-(ET)$_2$X with X=Cu\[N(CN)$_2$\]Br.

Quasiparticle Green Function
============================

Let us consider a one-band quasi-two-dimensional Fermi surface where a nodal superconductor and a nodal density wave coexist. As a model Hamiltonian we can think of a generalized Hubbard model [@dora2]. The Green function is given in $4 \times 4$ spinor space as $$G^{-1} ( \vec{k}, E ) = E - \xi \rho_3 \sigma_3 - \eta \rho_3
- \Delta_1 f \sigma_1 \rho_3 - \Delta_2 g \rho_1 \sigma_3$$ where the Pauli matrices $\rho_i$ and $\sigma_i$ are operating on the particle-hole space and the $\vec{k}$ and $\vec{k} \pm \vec{Q}$ space, respectively, where $\vec{Q}$ is the nesting vector for the nodal density wave. We have assumed here that the UDW is a charge density wave. The Green function possesses poles at the quasiparticle energies $$E = \pm \sqrt{\left( \sqrt{\xi^2 + \Delta_2^2 g^2} 
\pm \eta \right)^2 + \Delta_1^2 f^2}$$ which is well known in the literature. [@chakravarty; @thalmeier] Here, $\eta$ is the imperfect nesting term. Less well-known, perhaps, is the quasiparticle density of states (DOS) given by

$$\frac{N \left( E \right)}{N_0} \equiv  G \left( E \right) = 
\frac{1}{2} \sum_\pm \mbox{Re} 
\left\{
\left\langle \frac{|E|}{\sqrt{\left( \sqrt{E^2-\Delta_1^2 f^2} \pm \eta
\right)^2
- \Delta_2^2 g^2}} 
 \right\rangle
\right\},$$

where the sum has to be taken over the two terms with $\pm \eta$. Here, $\langle .... \rangle$ denotes the angular average over $\phi $. In the limit $\eta \rightarrow 0$ this expression reduces to a very familiar form $$G \left( E \right) = \mbox{Re } \left\langle 
\frac{\left| E \right|}{\sqrt{E^2-\Delta_1^2 f^2-\Delta_2^2 g^2}} \right\rangle$$ In particular, for case (a) when $g=f=\cos(2\phi)$ we have $$G \left( E \right) = \left\{ \begin{array}{cl} 
\frac{2}{\pi} x K \left( x \right) & \mbox{for } x < 1 \\ 
\frac{2}{\pi} K \left( x^{-1} \right) & \mbox{for } x > 1 
\end{array} \right.$$ where $x=E/\sqrt{\Delta_1^2 + \Delta_2^2}$ and $K$ is the complete elliptic integral of the first kind. This has the same form as in $d$-wave superconductors,[@won] with the only exception that the $d$-wave gap has to be replaced by the total gap $\sqrt{\Delta_1^2 + \Delta_2^2}$. On the other hand, for case (b) when $f=\cos(2\phi)$ and $g=\sin (2 \phi)$ the DOS develops a complete gap below $E=\min(\Delta_1,\Delta_2)$ and we find $$G \left( E \right) = \left\{ \begin{array}{cl} 
0 & \mbox{for } E < \Delta_1 \\
\frac{2}{\pi} \frac{\left| E \right|}{\sqrt{E^2-\Delta_1^2}} y K \left( y
\right) & \mbox{for } y < 1 \\ 
\frac{2}{\pi}  \frac{\left| E \right|}{\sqrt{E^2-\Delta_1^2}}
K \left( y^{-1} \right) & \mbox{for } y > 1 
\end{array} \right.$$ where $y=\sqrt{(E^2-\Delta_1^2)/(\Delta_2^2-\Delta_1^2)}$. Here we have assumed $\Delta_1<\Delta_2$. The corresponding DOS with $\Delta_1/\Delta_2=$ 0.1, 0.3, and 0.5 is shown in Fig. 1(b). Except for the opening of the energy gap with increasing $\Delta_1$, the DOS looks very similar to the case of $\Delta_1$ =0, with a characteristic logarithmic singularity at $ |E| = \Delta_2$.

![ Density of states of a gossamer superconductor, (a) with $f = g = \cos( 2 \phi )$, and (b) with $f = \cos (2 \phi )$ and $g = \sin (2 \phi )$. ](fig2 "fig:"){width="6.5cm"} ![ Density of states of a gossamer superconductor, (a) with $f = g = \cos( 2 \phi )$, and (b) with $f = \cos (2 \phi )$ and $g = \sin (2 \phi )$. ](fig1 "fig:"){width="6.5cm"}

As we shall see in the following, for case (a) a non-vanishing $\eta$ is crucial for the presence of two order parameters. In Fig. 1(a) we show the corresponding DOS for the choices $\Delta_1$ = 0 ($\eta =0$), $\Delta_1$ = 0.1$\Delta_2$ ($\eta = 0.15 \Delta_2$), and $\Delta_1$ = 0.3$\Delta_2$ ($\eta = 0.4 \Delta_2$). Due to the presence of two order parameters, the logarithmic singularity at $ |E| = \Delta_2$ splits into two singularities, which are now located at $ |E| = \sqrt{(\Delta_2 \pm \eta )^2 + \Delta_1^2 }$. We expect this double-peak structure to be accessible to scanning tunneling microscope measurements in the pseudogap phase of the high-$\rm T_c$ cuprates.

Very recently, new measurements of the DOS in underdoped Bi2212 were reported.[@yurgens] Indeed, they show a clear double-peak structure. Unfortunately, it was pointed out later that the secondary peak is a likely artifact of the heating process.[@zavaritsky; @yurgens2] However, based on the present model calculation we conclude that a second peak should indeed be visible in a careful experiment.

Gap Equations
=============

For case (a) with $ f = g$, the gap equations are given by $$\begin{aligned}
\lambda_1^{-1} &=& 2 \pi T \sum_{n}  \mbox{Re} \Bigg\langle 
\frac{2 f^2}{\sqrt{\left( \sqrt{\omega_n^2 +
\Delta_1^2 f^2} + i \eta \right)^2 + \Delta_2^2 f^2}} 
\left( 1 + \frac{i \eta}{\sqrt{\omega_n^2 +\Delta_1^2 f^2}}
\right) \Bigg\rangle \\
\lambda_2^{-1} &=& 2 \pi T \sum_{n}  \mbox{Re} \Bigg\langle
\frac{2 f^2}{\sqrt{\left( \sqrt{\omega_n^2 +
\Delta_1^2 f^2} + i \eta \right)^2 + \Delta_2^2 f^2}}
\Bigg\rangle 
\label{gapeq}\end{aligned}$$ for $\Delta_1$ and $\Delta_2$ respectively. Here, $\lambda_1$ and $\lambda_2$ are the dimensionless coupling constants for superconductivity and density wave. Also, it is assumed here that $\lambda_1 < \lambda_2$, because otherwise there is no chance for the UDW to occur. Furthermore, for the survival of a finite superconducting component we also need $\eta \neq 0$. Then, for small $\eta$ one finds $\Delta_1(0)/\Delta_2(0) \sim (\eta / \Delta_2(0))^{3/2}$, where $\Delta_1(0)$ and $\Delta_2(0)$ are the corresponding order parameters in the zero-temperature limit. Moreover, for the energy gap that can be observed by scanning tunneling microscopy we find at low temperatures $0.5 (\sqrt{(\Delta_2 + \eta)^2 + \Delta_1^2} +
\sqrt{(\Delta_2 - \eta)^2 + \Delta_1^2}) \simeq 
\Delta_2 ( 1 + \Delta_1^2/(2(\Delta_2^2 - \eta^2 )))$. Thus, in the limit $\Delta_1 / \Delta_2 \ll 1$ and $|\eta |/\Delta_2 \ll 1$, scanning tunneling microscopy will still identify $\Delta_2(0)$. Therefore, the puzzling result that the detected gap $\Delta (0) \simeq
2.14 \rm T_c$, where $\rm T_c$ is the transition temperature of the pseudogap phase, can be naturally interpreted in terms of a d-density wave.[@dora2; @oda; @kugler] Finally, since the corresponding superfluid density at zero-temperature is given by $$\rho_S (0,\eta ) = \frac{\Delta_1^2 (0)}{\Delta_2^2 (0) + \Delta_1^2 (0)},$$ there should be a substantial reduction of this quantity in the coexistence regime.

For the case (b) where $f \neq g$ the gap equations are now given by $$\begin{aligned}
\lambda_1^{-1} &=& 2 \pi T \sum_{n}  \mbox{Re} \Bigg\langle 
\frac{2 f^2}{\sqrt{\left( \sqrt{\omega_n^2 +
\Delta_1^2 f^2} + i \eta \right)^2 + \Delta_2^2 g^2}} 
\left( 1 + \frac{i \eta}{\sqrt{\omega_n^2 +\Delta_1^2 f^2}}
\right) \Bigg\rangle \\
\lambda_2^{-1} &=& 2 \pi T \sum_{n}  \mbox{Re} \Bigg\langle
\frac{2 f^2}{\sqrt{\left( \sqrt{\omega_n^2 +
\Delta_1^2 f^2} + i \eta \right)^2 + \Delta_2^2 g^2}}
\Bigg\rangle 
\label{gapeq}\end{aligned}$$ In contrast to case (a), here we always encounter coexistence of two order parameters, even when $\eta = 0$. In particular, for the case $\eta = 0$, Eqs. 10 and 11 can be solved at zero temperature. $$\begin{aligned}
\Delta_2 (0) + \Delta_1 (0) & = & 2 \sqrt{E_1 E_2} \exp \left[ 
-(\lambda_1^{-1} + \lambda_2^{-1} )/2 \right] \\
\frac{\Delta_2 (0) - \Delta_1 (0)}{\Delta_2 (0) + \Delta_1 (0)} & = &
\lambda_1^{-1} - \lambda_2^{-1} ,\end{aligned}$$ where $E_1$ and $E_2$ are cut-off energies. Furthermore, the zero-temperature superfluid density is given by $$\begin{aligned}
\rho_s (0,0) = \frac{\Delta_1 (0)}{\Delta_2 (0) + \Delta_1 (0) }.\end{aligned}$$ Although the superfluid density is reduced in the present case, the reduction is not as severe as it is for case (a).

Concluding Remarks
==================

In this paper, we have analyzed the properties of a nodal superconductor in the presence of a nodal density wave. Our preliminary results look very consistent with experimental observations that have been reported for the pseudogap region of the high-$\rm T_c$ cuprates. For example, in contrast to the large quasiparticle energy gap observed by scanning tunneling microscopy, the corresponding superfluid density and the specific heat associated with superconductivity are much reduced.[@tallon] The present model calculation appears to describe these features consistently, although more measurements on this issue are clearly necessary. The application of this model to other potential “gossamer superconductors" is promising.

Acknowledgements
================

We have greatly benefited from discussions with Balazs Dora, Attila Virosztek, Yuji Matsuda, Marko Pinterić, Silvia Tomić, and Hyekyung Won. TD and KM are grateful for the hospitality of the Max-Planck Institute for Chemical Physics of Solids and the Max-Planck Institute for the Physics of Complex Systems at Dresden, where parts of this work were performed. SH acknowledges financial support by the National Science Foundation DMR-0089882.

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