Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions

arXiv:2604.18777 · cond-mat.supr-con · Submitted 2026-04-20 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions".

Mira: This work investigates the interplay between superconductivity and nematic order in a two-dimensional electron gas model incorporating competing pairing and quadrupolar forward-scattering interactions.

Kai: First, who's behind it and why it matters.

Title and authors: Mira: So we're looking at "Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions," which sounds like it's tackling how different types of ordering—superconductivity versus nematicity—can fight or even team up.

Kai: Yeah, I was checking the title, and it seems to focus on that specific 2D electron gas setup where you have competing pairing and quadrupolar forward-scattering interactions. It sounds like they're trying to find a minimal way to describe these intertwined phases in correlated electron systems.

Lev: From my side, I’m thinking this is interesting because the model uses a mean-field approach to determine the phase diagrams as functions of interaction strengths and temperature, which is exactly what we need before we even think about running anything on real hardware.

Mira: Exactly, and what's compelling about the setup is that they include both s-wave and d-wave superconducting channels alongside the nematic order parameter. This immediately tells us that the physics isn't just looking at a single type of pairing; it’s exploring how different pairing symmetries interact with this new nematic tendency.

Kai: It seems to be setting up a framework where symmetry dictates the phase structure, which is key for understanding things in materials like cuprates. I wonder what kind of material they are simulating when they talk about this 2D electron gas model?

Lev: If they can map out these phase diagrams accurately, it gives us a very clear roadmap for what conditions might lead to specific states, which is helpful when trying to design experiments or even quantum simulations.

The paper's summary: Kai: The core idea of the paper, "Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions," is to investigate the interplay between superconductivity, specifically s-wave and d-wave channels, and nematic order driven by quadrupolar forward-scattering interactions.

Mira: They're essentially using a mean-field formulation to solve coupled self-consistent equations to compute the free energy density of this system, which allows them to determine the phase diagrams based on interaction strengths and temperature.

Kai: The main finding they highlighted was that at zero temperature, the nematic order competes strongly with d-wave superconductivity, leading to a direct first-order phase transition between those two phases.

Lev: A first-order transition is important because it means there’s a clear jump in the order parameter when you cross that line, which gives us a very distinct feature to look for in any experimental setup or simulation.

Mira: But they also found that when you consider the s-wave pairing interaction alongside this nematic order, something different happens; it allows for a coexistence phase where an anisotropic Fermi surface develops alongside a uniform superconducting gap.

Kai: So, to put it simply, the paper shows how the presence of these competing interactions—pairing and quadrupolar scattering—creates complex phase diagrams with both direct competition and simultaneous existence of orders.

Lev: That coexistence aspect is really telling because it suggests that in some materials, you don't just have one ordered state dominating; you might see a mixture of them happening at the same time under specific conditions.

The paper's improvements: Mira: The authors suggest several ways to look at the system more deeply, starting with how they define the nematic order parameter based on whether you're in a continuous rotational case or an Ising-Nematic scenario.

Kai: They specifically mention that the distinction between these two scenarios matters because it affects what we call Goldstone modes, which is important for understanding the low-energy physics of these ordered states.

Lev: That’s a practical point; knowing whether you're dealing with continuous symmetry or Ising-Nematic behavior changes how you model the excitations, which is crucial when thinking about running this on actual hardware where we have to deal with finite degrees of freedom.

Mira: They also point out that they consider the effect of the underlying lattice perturbatively when studying the continuous rotation symmetric case, which lets them comment on how real lattice effects might modify their mean-field results.

Kai: And they do discuss how these interactions are defined through specific form factors, which guides the symmetry considerations for both pairing and quadrupolar scattering.

Lev: From an error correction perspective, if you were to try to simulate this, understanding the precise symmetries of the nematic order parameter would be vital for designing stabilizers that correctly capture those degrees of freedom.

Conclusion: Mira: To wrap up the paper "Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions," it really emphasizes how quadrupolar interactions naturally couple superconducting and nematic degrees of freedom, leading to a wide variety of phase diagrams.

Kai: The main points are that they see first-order transitions when nematicity competes directly with d-wave superconductivity, and coexistence phases emerge when s-wave pairing is involved alongside the nematic order.

Lev: For me, the most important implication is that this study gives us a concrete map of the parameter space—interaction strengths and temperature—that dictates whether we see a pure state or a complex mixture of orders in these systems.

Mira: It provides a clear theoretical basis for materials discovery by showing how tuning interaction parameters can lead to specific intertwined phases, which is really helpful for guiding experimentalists.

Kai: It's about seeing how symmetry shapes the outcome, and understanding those symmetry-driven transitions is essential groundwork for any future work on these correlated systems.

Lev: Overall, this paper gives us a very useful theoretical tool to guide where we should focus our experimental efforts when probing the complex phase behavior of 2D systems with competing orders.

Nei Lopes, * Guilherme da Silva do Vale, Daniel G. Barci

Centro Brasileiro de Pesquisas Físicas · Departamento de Física Teórica, Universidade do Estado do Rio de Janeiro

cond-mat.supr-con

Submitted: 2026-04-20

Updated: 2026-04-20

Comments: 8 pages, 5 figures

Journal ref: Phys. Rev. B114, 165106 (2026)

DOI: 10.1103/93xz-rklx

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 83/100

The gist: This work investigates the interplay between superconductivity and nematic order in a two-dimensional electron gas model incorporating competing pairing and quadrupolar forward-scattering

Key concepts

Quadrupolar Forward-Scattering Interactions
These are specific types of interactions in the electron system that are invariant under a certain symmetry transformation (k -> -k). They represent the quadrupolar nature of the interaction and are crucial because they couple or mediate the relationship between nematic order and superconducting pairing, which is key to understanding their interplay.
Nematic Order
Nematic order refers to an anisotropic state in a material where the system possesses directional dependence but no net spontaneous rotational symmetry breaking like a standard magnetic ordering. In this model, it is represented by quadrupolar components that describe how the electron density or pairing symmetry is distorted anisotropically.
Coexistence Phase
This phase occurs when two different orders—such as s-wave superconductivity and nematic order—are simultaneously present in the system at low temperatures. The paper finds this happens when nematic order competes with s-wave superconductivity, resulting in a state where a uniform superconducting gap exists across an underlying anisotropic (nematic) Fermi surface.
Mean-Field Formulation
This is a mathematical technique used to simplify the complex many-body problem by replacing the interactions between individual electrons with average effective fields. It allows researchers to derive coupled self-consistent equations for all relevant order parameters, such as the superconducting gap and nematic components, making it possible to study the resulting phase structure.

Terminology

Summary

This work investigates the interplay between superconductivity and nematic order in a two-dimensional electron gas model incorporating competing pairing and quadrupolar forward-scattering interactions. It matters because it provides a minimal framework to describe intertwined nematic and superconducting phases in correlated electron systems, highlighting how symmetry and interaction strength shape the phase structure, which is relevant for understanding phenomena in cuprates and other low-dimensional materials.

Model Setup

The study considers a two-dimensional electron gas with interactions in both the particle-particle (superconducting) and particle-hole (nematic) channels. The Hamiltonian includes kinetic energy terms, superconducting pairing interactions, and quadrupolar forward-scattering interactions. The explicit forms of these interactions are guided by symmetry considerations:

  1. Quadrupolar forward-scattering interactions are invariant under the transformation k → −k, which characterizes nematic symmetry and coincides with that of a d-wave superconducting order parameter.

  2. The interaction form factors are defined as:

((

Mean-Field Formulation

The mean-field approach leads to five coupled self-consistent equations corresponding to the different components of the order parameters: one s-wave superconducting component ∆s, two d-wave components ∆s,c d and ∆s d, and two nematic components ∆s,c n. The effective chemical potential and the anisotropic superconducting gap are momentum-dependent:

  1. The effective chemical potential is given by µ(θ) = µ − 2 (∆s n f s k + ∆c n f c k).

  2. The anisotropic superconducting gap reads ∆k(θ) = ∆s + ∆c d f c k + ∆s d f s k.

  3. The free energy density is given by F = −2T X k ln(1 + e −βEk(θ)) + X k (ϵk − µ(θ) − Ek(θ))− (∆s)2 Vs− (∆s d) squared + (∆c d) squared Vd− (∆s n) squared + (∆c n) squared f squared.

Zero-Temperature Phase Diagrams

The zero-temperature phase diagrams reveal a rich interplay between the orders:

  1. Competition between s- and d-wave superconductivity at T = 0, neglecting quadrupolar interactions (f2 = 0), shows that negative values with increasing Vs (Vd) stabilize the s-wave (d-wave) superconducting phase, separated by a first-order phase transition line.

  2. The interplay between d-wave superconductivity and nematic order at T = 0, setting Vs = 0, shows that increasing Vd stabilizes a d-wave superconducting phase for f2 > −0.4, while the nematic phase remains stable for f2 < −0.4.

  3. The interplay between s-wave superconductivity and nematic order at T = 0, setting Vd = 0, shows that increasing f2 stabilizes a nematic phase for small Vs, and sufficiently large Vs leads to a coexistence phase where a uniform superconducting gap develops on an anisotropic (nematic) Fermi surface.

Finite-Temperature Effects

At finite temperatures, quadrupolar interactions promote the emergence of additional superconducting components, giving rise to regimes where s-wave, d-wave, and nematic orders coexist.

  1. For fixed Vs = Vd = 1.0 and varying f2, the system undergoes a first-order phase transition into a coexistence phase where s-wave, d-wave, and nematic orders are simultaneously present at low temperatures.

  2. Upon increasing temperature, superconducting order is progressively suppressed, yielding a purely nematic phase, which eventually undergoes a continuous transition to the disordered state at higher temperature.

Symmetry and Coexistence

The results highlight the crucial role of symmetry in determining phase structure:

  1. When nematicity competes with d-wave superconductivity (sharing the same rotational symmetry), the system exhibits a direct first-order transition between these two phases.

  2. When nematic order competes with s-wave superconductivity (belonging to a different symmetry class), a coexistence phase emerges, characterized by a uniform superconducting gap on an anisotropic (nematic) Fermi surface.

Conclusion

The paper demonstrates that quadrupolar interactions provide a natural mechanism for coupling superconducting and nematic degrees of freedom, leading to rich phase diagrams that include first-order and continuous transitions, coexistence regions, and multicritical behavior. The resulting phase diagrams exhibit a rich interplay of competing and coexisting orders, particularly at low temperatures. The analysis is based on a mean-field treatment of a rotationally invariant system in two-dimensions. While long-range nematic order at finite temperature is prohibited by the Mermin–Wagner theorem, the mean-field phase diagrams provide a qualitatively reliable description when considering weak coupling to an underlying lattice.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper, Competition and coexistence of superconductivity and nematic order in a two-dimensional electron gas with quadrupolar interactions, which provides a minimal theoretical framework for understanding the interplay between superconductivity (s-wave and d-wave) and nematic order in correlated electron systems.

Here are the specific improvements that can be made to AI systems based on this research, along with what these improved systems can achieve:

The core improvement lies in integrating the physics of competing symmetries (nematic vs. superconducting) into machine learning models for materials discovery and condensed matter simulation. This paper provides a blueprint for constructing physically motivated, symmetry-aware generative models and predictive tools.

Here are the specific improvements:

  1. [Improvement] Development of Symmetry-Constrained Generative Models for Novel Correlated States (Inspired by Section II & III).

  2. [Improvement] Creation of Multi-Scale Phase Diagram Predictors with Topological Transition Mapping (Inspired by Section III, Figures 1–5).

  3. [Improvement] Implementation of Dynamic Critical Exponent Prediction in Quantum Critical Regimes (Inspired by Section III, Figure 2 and Text).

  4. [Improvement] Training of Machine Learning Potentials for Anisotropic Fermi Surfaces (Inspired by the anisotropic dispersion in Eq. 2.16).

The resulting improved AI systems can perform the following specific tasks:

  1. [Improved AI System Capability] The system can act as a discovery engine for novel electronic materials by generating candidate crystal structures or doping configurations that are predicted to exhibit specific intertwined phases (e.g., coexistence of s-wave SC and nematic order) based on calculated interaction strengths derived from the model parameters in Section II.

  2. [Improved AI System Capability] The system can predict the ground state phase diagram (Fig. 1, 2, 3) for a given set of interaction constants and temperature regimes, allowing researchers to quickly identify whether a material under certain conditions will favor an isotropic liquid crystal phase, a pure d-wave SC state, or a coexistence regime.

  3. [Improved AI System Capability] The system can precisely map the transition lines (first-order vs. second-order) and identify critical points (QCPs) in the parameter space defined by interaction strengths and temperature, significantly reducing the experimental search space for tuning critical parameters like Landau parameters or coupling constants.

  4. [Improved AI System Capability] The system can predict whether a material will exhibit a nematic transition of the Kosterlitz-Thouless type (finite T, weak lattice coupling) versus a mean-field transition, guiding experimentalists on the expected nature of fluctuations in 2D systems.

  5. [Improved AI System Capability] The system can generate high-fidelity, anisotropic electronic structure models (using potentials trained on Eq. 2.16) that accurately reflect the momentum-dependent chemical potential and gap anisotropy, enabling better prediction of transport properties (like magnetoresistance) in materials exhibiting nematicity, such as Sr3Ru2O7.

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