Critical point of the transition between $s_\pm$ and $s_{++}$ states of a two-band superconductor with nonmagnetic impurities

This paper demonstrates that the transition between $s_\pm$ and $s_{++}$ superconducting states in a two-band model with nonmagnetic impurities evolves from a smooth crossover at high temperatures to a first-order phase transition at low temperatures, thereby establishing a critical end point on the temperature-impurity scattering rate phase diagram that suggests the possibility of a quantum phase transition.

The Gist

Imagine a superconductor as a bustling dance floor where electrons pair up and move in perfect unison. In certain "two-band" superconductors (like some iron-based materials), there are two different groups of dancers, and they have a specific rule for how they move together: sometimes they move in perfect sync, and other times, they move in opposite directions (one group steps forward while the other steps back).

This paper explores what happens when you throw a "party crasher" into the mix: impurities (tiny bits of dirt or disorder in the material). The researchers wanted to know: How does the dance change when the floor gets messy, and does the temperature of the room matter?

Here is the breakdown of their discovery using simple analogies:

1. The Two Dance Styles: $s_{\pm}$ vs. $s_{++}$

• The $s_{\pm}$ State (The Opposites): Think of this as a dance where Group A and Group B are doing a "tug-of-war" rhythm. When one group is happy (positive), the other is sad (negative). This is the natural state for these materials when they are clean.
• The $s_{++}$ State (The Sync): This is when both groups decide to dance in the same direction, feeling the same "vibe" (positive).
• The Switch: The paper shows that if you add enough "dirt" (impurities) to the dance floor, the dancers might suddenly switch from the "tug-of-war" style to the "sync" style.

2. The Temperature Factor: Hot vs. Cold

The researchers found that the way this switch happens depends entirely on how cold the room is.

• At High Temperatures (The Smooth Slide): Imagine the dancers are energetic and a bit sloppy. If you start adding dirt, they don't suddenly snap into a new formation. Instead, they slowly, smoothly transition from the old dance to the new one. It's like a crossover—a gradual shift where you can't point to an exact moment where the change happened.
• At Low Temperatures (The Snap): Now, imagine the dancers are frozen in place, moving with precision. If you add dirt here, they don't slide into the new dance. They snap. One moment they are doing the old dance, and the very next moment, they are doing the new one. This is a First-Order Phase Transition. It's like a light switch: it's either OFF or ON, with no dimming in between.

3. The "Critical End Point" (The Tipping Point)

This is the most exciting part of the discovery. The researchers found a specific spot on their map (a graph of Temperature vs. Amount of Dirt) where the behavior changes.

• The Metaphor: Think of a mountain peak.
  • On one side of the peak (high heat), the transition is a gentle slope (crossover).
  • On the other side (low heat), the transition is a sheer cliff (sudden snap).
  • The very top of the mountain is the Critical End Point. It is the exact temperature where the "gentle slope" stops and the "cliff" begins.

4. The Quantum Mystery (The Zero-Temperature Limit)

The researchers pushed their calculations to the absolute limit: Absolute Zero (the coldest possible temperature).

• The Finding: As they made the "dirt" stronger and stronger, the "mountain peak" (the Critical End Point) kept getting lower and lower.
• The Extrapolation: Even though they couldn't calculate exactly at Absolute Zero (due to mathematical limits), their data suggested the peak would eventually hit the ground.
• The Implication: If the peak hits the ground, it means there is a Quantum Critical Point. This is a state where the material changes its fundamental nature not because of heat, but purely because of the amount of disorder, even at absolute zero. It's like the material is "frozen" in a state of constant change, driven by quantum mechanics rather than thermal energy.

Summary Analogy

Imagine you are trying to change a group of people from wearing Red Shirts to Blue Shirts.

• In a warm room: You tell them to change, and they slowly swap shirts one by one. It's a smooth process.
• In a freezing cold room: You tell them to change, and they all freeze. Then, suddenly, everyone swaps shirts at the exact same second. It's a chaotic, sudden event.
• The Critical Point: There is a specific "Goldilocks" temperature where the behavior shifts from "smooth swap" to "sudden snap."
• The Quantum Twist: The authors suspect that if you make the room extremely cold and the "dirt" on the floor very specific, this sudden snap could happen even at absolute zero, creating a new, mysterious state of matter that exists only in the quantum realm.

Why does this matter?
Understanding these transitions helps scientists design better superconductors (materials that conduct electricity with zero resistance). If we can control these "snaps" and "smooth slides," we might be able to build more efficient power grids or faster computers that work at higher temperatures.

Technical Summary

Here is a detailed technical summary of the paper "Critical point of the transition between s± and s++ states of a two-band superconductor with nonmagnetic impurities" by Vadim A. Shestakov and Maxim M. Korshunov.

1. Problem Statement

The paper investigates the thermodynamic nature of the transition between two distinct superconducting order parameter states in multiband superconductors (specifically iron-based pnictides and chalcogenides):

• $s_{\pm}$ state: A sign-changing order parameter arising from spin fluctuations.
• $s_{++}$ state: A sign-preserving order parameter.

The study focuses on how nonmagnetic impurities induce a transition between these states. While previous work established that this transition depends on impurity scattering rates and temperature, the precise thermodynamic character of the transition (whether it is a smooth crossover or a first-order phase transition) and the existence of a critical endpoint where these behaviors merge remained to be fully characterized. The authors aim to map the phase diagram in the "Temperature" vs. "Impurity Scattering Rate" plane and determine if a Quantum Phase Transition (QPT) occurs at zero temperature.

2. Methodology

The authors employ a theoretical framework based on a two-band model with isotropic superconducting order parameters within each band.

• Thermodynamic Potential: The core of the analysis is the calculation of the change in the Grand thermodynamic potential ($\Delta\Omega = \Omega_S - \Omega_N$), also known as the Landau free energy, between the superconducting and normal states.
• Formalism:
  • The calculation utilizes the Luttinger-Ward expression generalized for multiband systems.
  • The system is treated using Matsubara Green's functions in combined band and Nambu space.
  • Impurity scattering is modeled using the $T$-matrix approximation (equivalent to the non-crossing diagram approximation).
  • The impurity potential is split into intraband ($v$) and interband ($u$) components, characterized by the ratio $\eta = v/u$. The study assumes interband scattering dominance ($\eta = 0$).
• Key Parameters:
  • $\sigma$ (Generalized Scattering Cross-section): Ranges from 0 (Born limit, weak scattering) to 1 (Unitary limit, strong scattering).
  • $\Gamma_a$ (Impurity Scattering Rate): Proportional to impurity concentration.
  • Eliashberg Equations: Solved self-consistently to determine the renormalized frequency ($\tilde{\omega}_{\alpha n}$) and order parameter ($\tilde{\phi}_{\alpha n}$).
• Thermodynamic Derivatives:
  • Entropy ($\Delta S$): Derived from the first derivative of $\Delta\Omega$ with respect to temperature.
  • Specific Heat ($\Delta C$): Derived from the second derivative of $\Delta\Omega$ with respect to temperature.
  • Phase Transition Criteria: A smooth $\Delta\Omega$ implies a crossover; a "kink" (non-analyticity) in $\Delta\Omega$ implies a first-order transition, manifested as a jump in entropy and a divergence in specific heat.

3. Key Contributions

• Thermodynamic Characterization: The authors definitively characterize the $s_{\pm} \to s_{++}$ transition not just as a change in gap symmetry, but as a thermodynamic phase transition with distinct orders depending on temperature.
• Identification of a Critical End Point (CEP): They identify a specific point in the $(\Gamma_a, T)$ phase diagram where the nature of the transition changes from a first-order phase transition (at low $T$) to a smooth crossover (at high $T$).
• Quantum Critical Point (QCP) Hypothesis: Through extrapolation of the CEP to zero temperature, the paper proposes the existence of a Quantum Critical Point driven by disorder, suggesting a quantum phase transition between the two states.
• Role of Scattering Strength: The study clarifies how the scattering strength parameter ($\sigma$) shifts the location of the CEP, showing that the transition temperature is maximal in the Born limit and decreases as scattering strength increases.

4. Results

• Low Temperature Regime ($T < T_{CEP}$):
  • In the Born limit ($\sigma = 0$) and near it, the Grand potential $\Delta\Omega$ exhibits a kink along a specific line of impurity scattering rates.
  • This kink corresponds to an abrupt sign change in the gap function of one band ($\Delta_b$).
  • Thermodynamically, this manifests as a discontinuous jump in entropy and a divergence in electronic specific heat, confirming a first-order phase transition.
  • In this regime, multiple solutions to the Eliashberg equations coexist.
• High Temperature Regime ($T > T_{CEP}$):
  • The kink in $\Delta\Omega$ disappears. The transition becomes a smooth crossover.
  • The Eliashberg equations yield a unique solution.
  • The specific heat peak associated with the transition splits into two peaks that move apart as temperature increases, and the gap changes sign smoothly.
• Critical End Point (CEP):
  • The CEP occurs at $T_{CEP} \approx 0.07 T_{c0}$ in the Born limit.
  • As the scattering strength $\sigma$ increases, $T_{CEP}$ decreases monotonically. For $\sigma = 0.18$, $T_{CEP}$ drops to $\approx 0.01 T_{c0}$.
  • The CEP location in the $(\Gamma_a, T)$ plane shifts to lower temperatures and slightly lower scattering rates as $\sigma$ increases.
• Extrapolation to Zero Temperature:
  • Due to the limitations of the Matsubara technique (discrete frequencies), the authors extrapolated the CEP trajectory to $T=0$.
  • The extrapolation suggests the CEP converges to a Quantum Critical Point (QCP) at $T=0$ with a scattering rate $\Gamma^Q_{a} \approx 1.15 T_{c0}$ and $\sigma_{QCP} \approx 0.21$.
  • This implies that at absolute zero, a quantum phase transition between $s_{\pm}$ and $s_{++}$ states can be driven solely by increasing impurity concentration.

5. Significance

• Fundamental Physics: The work provides a rigorous thermodynamic proof that disorder-induced transitions in multiband superconductors can be first-order, challenging the view that they are always continuous crossovers.
• Experimental Guidance: The specific heat divergence and entropy jumps predicted at the first-order transition line offer clear experimental signatures for detecting this transition in iron-based superconductors (e.g., via proton irradiation to tune disorder).
• Quantum Criticality: The proposal of a disorder-driven Quantum Critical Point connects the physics of superconducting order parameter transitions to the broader field of quantum criticality, drawing parallels with metamagnetic transitions in other materials like Sr$_3$Ru$_2$O$_7$.
• Model Refinement: The study refines previous findings regarding the "smearing" of the transition at higher scattering strengths, showing that the abrupt transition persists to lower temperatures than previously thought, provided the scattering strength is within a specific range.