🔬 condensed matter

Non-equilibrium dissipative stabilization of s- and d-wave superconductivity

This paper reports that weak coupling to thermal baths, even at temperatures well above the equilibrium critical temperature, can stabilize a nonthermal pairing gap in both s- and d-wave BCS superconductors by maintaining a highly non-equilibrium steady state described by a generalized Gibbs ensemble.

Original authors: Aleksey Lunkin, Yury Holubeu, Denis Golež, Zala Lenarčič

Published 2026-09-10

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CC BY 4.0

Original authors: Aleksey Lunkin, Yury Holubeu, Denis Golež, Zala Lenarčič

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of quantum materials, certain substances can conduct electricity with absolutely no resistance, a phenomenon known as superconductivity. This state usually appears only when a material is cooled to temperatures far below freezing, where the chaotic jiggling of atoms settles down enough for electrons to pair up and move in perfect unison. For decades, scientists believed that once a material warmed up past a specific critical temperature, this delicate pairing would break apart, and the superconducting state would vanish forever. The prevailing view was that you could not have a superconductor if the environment was too hot; the thermal energy would simply overwhelm the electrons, scattering them apart.

However, the rules of the quantum world are not always dictated by temperature alone. Recent advances have shown that by carefully manipulating how a system interacts with its surroundings, researchers can sometimes create ordered states that should not exist under normal conditions. This field, known as non-equilibrium physics, explores what happens when a system is constantly driven or connected to different environments, preventing it from ever settling into a standard, calm state. The question has been whether such active manipulation could not just tweak a material, but actually stabilize a superconducting state even when one of the surrounding environments is scorching hot compared to the material's usual limits.

A team of researchers in Slovenia has now demonstrated that this is indeed possible. By modeling a superconductor connected to two separate heat baths—one cool and one significantly hotter than the material's normal melting point for superconductivity—they found that the system could settle into a stable, superconducting state that defies equilibrium expectations. In their simulations, the superconductor was placed between these two distinct thermal environments. While a single hot bath would normally destroy the superconducting state, the interplay between the two baths created a unique, steady condition. The researchers found that the superconductor maintained a robust pairing of electrons, keeping the energy gap open and the state stable, even though one of the baths was at a temperature well above the critical threshold where superconductivity usually dies.

The mechanism behind this stability relies on a subtle balance of energy exchange. The system is coupled to the baths so weakly that the electrons do not instantly thermalize to the temperature of either bath. Instead, the electrons settle into a specific distribution of energy states that is neither hot nor cold, but a distinct, non-thermal arrangement. The researchers describe this as a "greenhouse effect" for quantum particles. Just as a greenhouse traps heat to create a specific climate inside, the weak coupling to the two baths traps the electrons in a configuration where they remain paired. This happens because the baths act in concert to shuffle the electrons into a state where they are less likely to break apart, effectively stabilizing the superconducting order against the heat of the hotter bath.

This effect was shown to work for two different types of superconductors. The first is the conventional type, where the electron pairs form a simple, spherical symmetry. The second is the unconventional type, which has a more complex, four-lobed shape and is found in high-temperature superconducting materials. In both cases, the simulations revealed that the superconducting state could survive and even thrive in regions of temperature where it would normally be impossible. The researchers mapped out the conditions under which this happens, showing that as long as one bath remains below a certain critical point, the system can sustain the superconducting gap even if the other bath is much hotter. The stability is not a fleeting moment but a permanent steady state, maintained as long as the connection to the two baths persists.

The study suggests that this stabilization is not a minor fluctuation but a significant, robust effect. The size of the superconducting gap in this non-equilibrium state is comparable to what is seen in standard, cold superconductors, indicating that the material is fully functional. The researchers also noted that this stability holds true even when the connection to the baths is very weak, which is crucial because it means the effect does not require massive amounts of energy to drive the system. This distinguishes their finding from other methods that try to induce superconductivity using strong external pulses, which often heat the material so much that the effect is lost. Here, the delicate balance of two different temperatures does the work, allowing the system to find a stable path that avoids the thermal destruction usually associated with high heat.

The implications of this work extend beyond just understanding how electrons behave. The researchers propose that this principle could be tested in real-world experiments using cavities, which are structures that trap light and electromagnetic waves. By placing a superconducting material inside such a cavity, scientists could engineer the environment to mimic the two thermal baths described in the model. This would allow for the creation of superconducting states in conditions that were previously thought to be too hostile. The work opens a new avenue for controlling quantum materials, suggesting that by carefully designing how a system exchanges energy with its surroundings, we can stabilize exotic states of matter that are otherwise hidden from view.

While the results come from a theoretical model and computer simulations rather than a physical experiment, the framework used is based on well-established principles of quantum mechanics and statistical physics. The researchers used a method that tracks how the energy levels of the electrons change over time, ensuring that the results are consistent with the laws governing open quantum systems. They also checked that their findings were not dependent on a specific choice of parameters, confirming that the effect is a general feature of this type of system. The study provides a clear roadmap for how dissipative engineering—using the flow of energy and entropy to create order—can be used to manipulate the fundamental properties of matter.

In the end, this work challenges the long-held assumption that heat is always the enemy of superconductivity. It shows that under the right conditions, a hot environment can be part of the solution rather than the problem. By connecting a material to two different thermal worlds, it is possible to create a third, stable world where superconductivity persists. This discovery adds a new chapter to the story of how we control quantum matter, moving from simply cooling things down to actively engineering the flow of energy to keep the quantum world alive.

Technical Summary: Non-equilibrium Dissipative Stabilization of s- and d-wave Superconductivity

Problem Statement
Stabilizing superconductivity beyond its equilibrium regime is a central challenge in non-equilibrium state engineering. Traditional active driving approaches, such as light-induced transitions, often inject large amounts of energy, leading to strong heating that restricts their applicability. While cavity engineering offers a route to modify electromagnetic modes and alter quantum states, it is currently constrained by weak coupling regimes and long-wavelength limitations. Furthermore, existing theoretical frameworks for driven superconductors, such as generalized Eliashberg theory, often rely on non-Markovian treatments that yield only weak deviations from equilibrium or require linearization, limiting the description of large non-thermal effects. The authors address the question of whether a superconductor can be stabilized in a non-thermal steady state with a finite order parameter even when coupled to a thermal bath at a temperature well above the equilibrium critical temperature ( $T_c$ ), without the strong heating associated with active driving.

Methodology
The authors model a BCS superconductor (both s-wave and d-wave symmetries) coupled to two thermal baths at distinct temperatures, $T_1$ and $T_2$ . The system is described using a Lindblad master equation, where the superconducting material is treated within a mean-field approximation (quadratic self-consistent problem) and the baths induce transitions via Lindblad dissipators that obey detailed balance at their respective temperatures.

Key methodological features include:

Weak Coupling Regime: The coupling strength to the baths ( $\gamma_0$ ) is assumed to be much smaller than the zero-temperature gap ( $\Delta_0$ ), implying slow state evolution.
Generalized Gibbs Ensemble (GGE) Ansatz: Instead of assuming the system relaxes to a standard thermal Gibbs ensemble, the authors approximate the time-dependent density matrix as a GGE. This ensemble is parametrized by time-dependent chemical potentials ( $\mu_{p,\sigma}$ ) associated with Bogoliubov quasiparticles, treating these quasiparticle occupations as approximately conserved quantities on intermediate timescales.
Self-Consistent Evolution: The evolution of quasiparticle occupation numbers is governed by rate equations derived from the Lindblad terms, which include processes for quasiparticle creation, annihilation, and scattering. The superconducting gap $\Delta_p(t)$ is updated self-consistently based on the instantaneous quasiparticle distribution.
Scattering Limits: The study primarily focuses on the limit of weak intrinsic scattering ( $\gamma_1 \ll \gamma_0$ ), where the GGE description holds. They also examine the limit of strong intrinsic scattering ( $\gamma_1 \approx \gamma_0$ ), where the system is described by a single effective temperature determined by energy exchange rates.

Key Contributions and Results
The paper reports the existence of a non-thermal pairing gap in BCS superconductors coupled to two thermal baths, even when one bath is at a temperature significantly higher than the equilibrium $T_c$ .

Non-Equilibrium Stabilization (s-wave): For s-wave superconductors, the authors find that a steady state with a finite gap exists in broad regions where $T_1 \neq T_2$ , specifically when one bath is below $T_c$ and the other is above it (e.g., $T_2 < T_c < T_1$ ). This is in contrast to the thermal case, where the gap vanishes if the effective temperature exceeds $T_c$ . The mechanism relies on the "greenhouse effect," where weak driving and dissipation lead to an $O(1)$ effect due to the non-thermal reshuffling of quasiparticle populations.
Non-Equilibrium Stabilization (d-wave): Similar stabilization is observed for d-wave superconductors. Notably, the region of finite superconducting order extends over a much broader range of temperatures compared to the s-wave case. The steady state retains a finite order parameter despite broad quasiparticle distributions with peaks in the nodal regions.
Nature of the Transition: The phase transition from the normal to the superconducting state in this dissipative setup is found to be of second order. Analytical expansion near the transition line confirms a square-root dependence of the gap on the temperature difference ( $\Delta \sim \sqrt{\tilde{T}_c - T_1}$ ), contrasting with first-order transitions reported in some other driven-dissipative studies.
Mechanism Distinction: The authors distinguish their mechanism from the traditional Eliashberg effect. While Eliashberg effects rely on weak driving and scale with the inelastic relaxation rate, the effect reported here is parametrically strong ( $O(1)$ ) and independent of the weak coupling strength $\gamma_0$ in the steady state. The deviation from thermal occupation is determined entirely by the bath temperatures.
Robustness: The results are shown to be robust against variations in the relative strength of scattering versus creation/annihilation processes in the bath coupling (parameter $\alpha$ ).

Significance and Claims
The paper claims to establish a generic framework for the dissipative engineering of symmetry-broken states, using superconductivity as a paradigmatic example. The core significance lies in demonstrating that weak, non-thermal dissipation can induce a strong ( $O(1)$ ) redistribution of quasiparticle occupations, thereby stabilizing superconducting order in regimes where it is absent in equilibrium.

The authors propose that this "greenhouse effect" applies to both conventional and unconventional superconductors. They suggest that this mechanism could be realized in cavity-based experiments, such as those involving Fabry-Pérot cavities, split-ring resonators, or surface plasmon polaritons, where the electromagnetic environment can be engineered to induce the requisite bath couplings. The work aims to open avenues for manipulating quantum phases without the strong heating associated with active driving, potentially extending to other symmetry-broken phases like charge-density waves or magnetic orders. The authors remain modest, noting that while the GGE approximation is well-tested for integrable systems, future work involving advanced diagrammatic tools could be necessary to include fluctuations beyond the mean-field treatment.

Technical Summary: Non-equilibrium Dissipative Stabilization of s- and d-wave Superconductivity

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