Condensate Fraction Scaling and Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity

arXiv:2505.17411 · cond-mat.str-el · Submitted 2025-05-23 · 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: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Condensate Fraction Scaling and Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity".

Kai: Characterizing superconducting and superfluid transitions in two-dimensional (2D) manybody systems is of broad interest,

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

Paper summary: Kai: So, wrapping up on this paper, "Condensate Fraction Scaling and Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity," the authors have essentially shown that using the scaling of condensate fraction is a highly efficient way to accurately determine BKT transitions in two-dimensional correlated systems.

Mira: That's right, Kai. The central claim is that this scaling behavior—algebraic below and exponential above—gives us a much more reliable way to find the BKT transition point than the on-site pairing correlator because it handles finite-size effects better <ref:2505.17411#pg1>.

Lev: From my side, it means that when we look at complex manybody problems where we suspect a 2D critical point might exist, focusing our numerical investigations on observables that show this scaling behavior will yield the most trustworthy information about the critical temperature <ref:2505.17411#pg2>.

Kai: It gives us a concrete method for finding that TBKT/t value, and they found it to be around zero point one four two zero(seven) in their main model, which is consistent across their different scaling approaches <ref:2505.17411#pg2>.

Mira: And the implication is that this methodology can be extended beyond the specific attractive Fermi-Hubbard model to other 2D systems involving correlated fermions, including those with spin-orbit coupling, suggesting broad applicability <ref:2505.17411#pg0>.

Lev: It’s a solid piece of theoretical groundwork that helps bridge the gap between abstract manybody models and the kind of measurable physics we hope to see in experiments down the line.

Kai: It definitely points toward a more focused experimental search for these types of transitions, guided by what this paper establishes about how we should be analyzing our data.

Conclusion: Kai: So, we've seen how this study uses condensate fraction scaling to pin down those BKT transitions in 2D systems, and now we need to talk about what that actually means for the physics and what it implies for future work <ref:2505.17411#pg0>.

Mira: Exactly. The paper by

mention authors if known, otherwise just say "the authors": focuses on showing that this specific scaling behavior—where the fraction of condensed particles behaves algebraically below the transition and exponentially above it—is a much cleaner way to find those critical points than using other standard measurements like the pairing correlator.

Lev: From my perspective in error correction, if we can reliably extract that transition temperature with less sensitivity to noise from finite-size effects, that gives us a clearer target for what kinds of physical parameters we need to engineer or simulate on real hardware.

Kai: It really boils down to how much more reliable we get when determining those critical temperatures for superconducting and superfluid systems in two dimensions. The main authors are essentially showing us a superior diagnostic tool for these transitions.

Mira: And the implication is that this technique isn't just a niche method; it suggests we can apply this same scaling logic to other complex 2D correlated fermion systems, like those involving spin-orbit coupling, which opens up a lot more territory for theoretical study <ref:2505.17411#pg0>.

Lev: If the underlying physics holds up across different models, then the protocols we develop for extracting these critical exponents might become more universally applicable in simulating quantum phases.

Kai: It’s exciting because it gives us a way to better probe the fundamental physics of how these two-dimensional fluids organize themselves at those critical points.

Mira: It really does, and that leads us right into how this success in the model translates to real-world experimental feasibility for observing these phenomena in materials.

Institute of Modern Physics, Northwest University · Shaanxi Key Laboratory for Theoretical Physics Frontiers · Fundamental Discipline Research Center for Quantum Science and Technology of Shaanxi Province · Hefei National Laboratory

cond-mat.str-el

Submitted: 2025-05-23

Updated: 2026-06-29

Comments: 5 pages, 4 figures; Supplementary Material

Journal ref: Chin. Phys. Lett. 43 080703 (2026)

DOI: 10.1088/0256-307X/43/8/080703

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 83/100

The gist: Characterizing superconducting and superfluid transitions in two-dimensional (2D) manybody systems is of broad interest, and this study establishes condensate fraction scaling as an efficient tool to

Key concepts

Berezinskii-Kosterlitz-Thouless (BKT) transition: This is a key phase transition in 2D systems with U(1) symmetry, where quasi-long-range order emerges. It is characterized by the binding and unbinding of vortex-antivortex pairs, marking the boundary between superconducting and normal states.
Condensate Fraction: This quantity measures the proportion of particles that are in a condensed state (either Bose-condensed bosons or fermion pairs). In this study, it is calculated from the leading eigenvalue of the momentum-space pairing matrix to probe the superconducting or superfluid phase.
On-site Pairing Correlator: This is a traditional method used to measure superconductivity by looking at how pairs are correlated on a single lattice site. The paper notes this method suffers from pronounced finite-size effects, meaning it requires very large systems to get accurate results for the BKT transition.
Scaling Behavior: The condensate fraction exhibits specific scaling: algebraic below the transition and exponential above it. This distinct behavior is more effective than other metrics for determining the BKT temperature because it reaches its asymptotic limit at smaller system sizes.

Terminology

Hubbard Model — 36× in this paper · explained in 3 episodes
Phase Transitions — 3× in this paper · explained in 2 episodes
Regularization — 3× in this paper · explained in 3 episodes
Antiferromagnetism — 2× in this paper · explained in 1 episode
Spin-Orbit Coupling (SOC) — 2× in this paper · explained in 3 episodes
Ab Initio — 1× in this paper · explained in 1 episode
Benchmark — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Hamming Distance — 1× in this paper · explained in 2 episodes
Metastability — 1× in this paper · explained in 2 episodes

Summary

Characterizing superconducting and superfluid transitions in two-dimensional (2D) manybody systems is of broad interest, and this study establishes condensate fraction scaling as an efficient tool to accurately determine associated Berezinskii-Kosterlitz-Thouless (BKT) transitions.

The gist: Condensate fraction exhibits algebraic scaling below the transition and exponential scaling above it, with significantly reduced finite-size effects compared to the on-site pairing correlator, allowing for a more accurate determination of the BKT transition temperature.

Theoretical Background and Motivation

Berezinskii-Kosterlitz-Thouless (BKT) transitions are a key property of 2D systems with U(1) continuous symmetry and short-range interactions, typically characterized by the formation of bounded vortex-antivortex pairs and quasi-long-range order. The 2D attractive Fermi-Hubbard model serves as a minimal testbed for capturing this BKT physics. While standard methods often rely on the onsite pairing correlator or superfluid density, both exhibit pronounced finite-size effects, limiting reliable determination of the transition temperature even with large system sizes (up to 400 sites).

Methodology: Condensate Fraction as a Probe

The study employs numerically exact auxiliary-field quantum Monte Carlo (AFQMC) simulations on the 2D attractive Fermi-Hubbard model. The condensate fraction, defined as the proportion of Bose-condensed bosons or fermion pairs, is used to characterize the superconducting and superfluid states. This quantity is computed from the leading eigenvalue of the momentum-space pairing matrix, denoted as nc = λmax/(N/2).

Key Findings on Scaling Behavior

The numerical results reveal that condensate fraction exhibits specific scaling behaviors across the BKT transition:

It shows algebraic scaling below the transition and exponential scaling above it.
This behavior is significantly more effective for characterizing the BKT transition compared to the long-studied on-site pairing correlator, which suffers from pronounced finite-size effects.
The condensate fraction exhibits remarkably smaller finite-size effect than the commonly used on-site pairing correlator, reaching the asymptotic scaling regime at much smaller system sizes.

Determination of Transition Temperature

The BKT transition temperature is extracted using several efficient schemes based on condensate fraction scaling:

Finite-size BKT transition temperature is extracted from condensate fraction, confirming its logarithmic correction on system size.
The exponent η(T) exhibits a linear dependence on T as T → T−BKT, allowing for a linear extrapolation to the critical exponent ηc = 1/4 at the transition point.
The transition temperature TBKT/t is determined by fitting the scaling invariant function f(x) or by identifying the inflection point of nc(T) as TBKT(L).

Comparison with Other Observables

The study compares condensate fraction results with other observables:

The pairing correlator, ⟨∆2⟩, shows pronounced finite-size effects, requiring L ≥ 32 to reach the correct asymptotic scaling behavior.
In contrast, condensate fraction reaches the scaling regime for L ≥ 20, making it a more appropriate quantity to determine the BKT transition temperature.
The superfluid density ρs can also be computed, and its crossing points with unity are shown to converge towards TBKT/t as system size increases.

General Applicability

These findings demonstrate that condensate fraction captures the essential physics of the BKT transition in 2D attractive fermion systems and can be readily extended to other 2D correlated fermion systems exhibiting superconductivity or superfluidity, including those with spin-orbit coupling (SOC). The results are applicable to both fermionic and bosonic systems hosting superconductivity or superfluidity.

Specific Model Results

For the 2D attractive Fermi-Hubbard model with U/t = −4 and various fillings (e.g., n=0.8003), the transition temperature is determined to be TBKT/t = 0.1420(7) at the thermodynamic limit, consistent across different scaling methods. For other parameter sets, such as U/t = −6 and U/t = −4 with higher doping, the BKT transition temperatures are bounded within ranges like 0.140 < TBKT/t < 0.145 or 0.125 < TBKT/t < 0.130, depending on the specific filling and interaction strength. In summary, condensate fraction is identified as a more appropriate quantity for characterizing the fermion pairing and determining BKT transition for attractive fermions.

Specific Heat Anomaly

The study also identifies an anomaly in the specific heat displaying a peak at a temperature slightly above the BKT transition, showing TSpH/TBKT ∼ 1.

Improvements for AI systems

As a fastidious researcher, I have analyzed this paper, Condensate Fraction Scaling and Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity, which establishes condensate fraction as a superior probe for the BKT transition in 2D systems compared to traditional pairing correlators.

Here are the specific improvements to AI systems that can be derived from this research, along with what those improved AI systems can achieve:

) 1. Enhanced Phase Transition Discovery and Characterization in Condensed Matter Materials

The core finding is that the condensate fraction exhibits a more reliable finite-size scaling behavior near a BKT transition than the on-site pairing correlator, allowing for more accurate determination of the transition temperature.

The improved AI system can perform automated analysis of experimental or simulated data (e.g., from ultracold atom experiments or material simulations) by calculating the condensate fraction across various system sizes and fitting it to power-law scaling functions (like those derived in Fig. 1). This allows for the precise, statistically robust extraction of BKT transition temperatures even with moderate system sizes, bypassing the severe finite-size effects seen in on-site pairing correlators.

) 2. Automated Identification of Topological Order and Symmetry Breaking

The paper links the algebraic scaling of condensate fraction to quasi-long-range order and identifies the critical exponent (specifically, confirming its value at BKT is 1/4).

The improved AI system can be trained on simulation outputs to automatically detect signatures of quasi-long-range order. It can specifically identify whether the observed scaling in a quantity like condensate fraction follows algebraic decay (superfluid phase) or exponential decay (normal phase), providing an automated diagnostic for identifying BKT physics in complex many-body systems without requiring manual inspection of correlation functions.

) 3. Predictive Modeling of Critical Behavior and Scaling Laws

The research provides detailed scaling relations, such as the logarithmic correction to the transition temperature:

TBKT(L) = TBKT(L = ∞) + a/(ln bL)2 [23–25, 30–32].

The improved AI system can incorporate this specific logarithmic correction formula into its predictive models. When presented with a set of finite-size data points, the system can use this relation to extrapolate the behavior to the thermodynamic limit (TDL), providing highly accurate predictions for critical temperatures in systems where only finite-size simulations are feasible.

) 4. Classification and Benchmarking of Many-Body Observables

The paper demonstrates that condensate fraction is a more appropriate quantity than the on-site pairing correlator because it captures contributions from Cooper pairs of all sizes (local and nonlocal).

The improved AI system can be used as a sophisticated feature extractor in quantum simulation data. When analyzing complex many-body observables, the AI can automatically weigh the contribution of different physical regimes (e.g., local vs. nonlocal pairing) by comparing the scaling behavior of multiple related quantities (like condensate fraction vs. pairing correlator). This allows it to rank which observable is most sensitive and reliable for characterizing a specific phase transition in an unknown system class.

) 5. Optimized Simulation Parameter Selection

The findings indicate that fixing the chemical potential (fixed-filling calculations) can be more computationally efficient than tuning it iteratively, especially for large systems.

The improved AI system can implement an optimization routine that suggests optimal fixed-filling values for simulations based on preliminary data or theoretical constraints. This prevents wasted computational resources by ensuring that the simulation parameters are chosen to maximize the signal (e.g., maximizing the observable's scaling behavior) before committing to a full, large-scale AFQMC run.

In summary, these improvements shift AI from being a mere data processor to a specialized tool capable of identifying and quantifying subtle, long-range collective phenomena like BKT transitions with high precision in complex physical systems.

Abstract

Characterizing the superconducting and superfluid transitions in two-dimensional (2D) many-body systems is of broad interest and remains a fundamental issue. In this study, we establish the condensate fraction scaling as a highly efficient tool to achieve that and accordingly propose efficient schemes to accurately determine the associated Berezinskii-Kosterlitz-Thouless (BKT) transitions. Using the 2D attractive Fermi-Hubbard model as a testbed and applying numerically exact auxiliary-field quantum Monte Carlo simulations, we access unprecedented system sizes (up to 64 times 64 = 4096 lattice sites) and perform a comprehensive analysis for the temperature dependence and finite-size scaling of condensate fraction across the BKT transition. We demonstrate that this quantity exhibits algebraic scaling below the transition and exponential scaling above it, with significantly reduced finite-size effects comparing to the extensively studied on-site pairing correlator. This greatly improves the determination of BKT transition using moderate system sizes. We also extract finite-size BKT transition temperature from condensate fraction, and confirm its logarithmic correction on system size. Based on the accurately determined transition, we reveal that the specific heat displays an anomaly, showing a peak at a temperature slightly above BKT transition. Our findings should be generally applicable to 2D fermionic and bosonic systems hosting superconductivity or superfluidity.