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

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Pith's one-line read Condensate fraction finds the BKT transition from small lattices

desk verdict Solid AFQMC study showing condensate fraction is a far better finite-size probe of the BKT transition than the on-site pairing correlator; central claim holds up, though the L≥20 shortcut needs an explicit L_min robustness test. read the letter →

arxiv 2505.17411 v2 pith:B7I5YFN2 submitted 2025-05-23 cond-mat.str-el

Yuan-Yao He This is my paper · ORCID
classification cond-mat.str-el
keywords condensatefractionBerezinskii-Kosterlitz-ThoulesstransitionattractiveHubbardmodelauxiliary-fieldquantumMonteCarlofinite-sizescalingtwo-dimensionalsuperconductivityspecificheatanomaly
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes the condensate fraction—the fraction of fermions that form condensed pairs—as a sharp probe of the Berezinskii–Kosterlitz–Thouless (BKT) transition in two-dimensional superconductors and superfluids. Using the 2D attractive Fermi–Hubbard model and numerically exact auxiliary-field quantum Monte Carlo simulations on lattices up to 64×64, the author shows that the condensate fraction decays algebraically with system size below the transition and exponentially above it. Because this quantity has far weaker finite-size effects than the commonly used on-site pairing correlator, the BKT transition temperature can be determined accurately from lattices as small as L=20. The paper reports $T_\mathrm{BKT}/t = 0.1420(7)$ for $U/t=-4$, $\mu/t=0.25$, and finds a specific-heat peak at about $1.10\,T_\mathrm{BKT}$.

What carries the argument

The load-bearing object is the condensate fraction $n_c = \lambda_\mathrm{max}/(N/2)$, the largest eigenvalue of the momentum-space pairing matrix $M_{kk'} = \langle \hat{\Delta}_k^\dagger \hat{\Delta}_{k'} \rangle - \langle c^\dagger_{k\uparrow}c_{k\uparrow}\rangle\langle c^\dagger_{-k\downarrow}c_{-k\downarrow}\rangle$ divided by the number of pairs. Its eigenvector encodes the full pair wave function, including nonlocal Cooper pairs. The argument runs on the algebraic-to-exponential crossover of $n_c$ with linear system size $L$: $n_c \propto L^{-\eta}$ below the BKT temperature with $\eta$ reaching $1/4$ at the transition, and $n_c \propto e^{-\alpha L}$ above it. That crossover, rather than the noisy pairing correlator, is what locates the transition.

What would settle it

Compute the condensate fraction for a model with an exactly known BKT transition, such as the 2D XY model or the 2D Bose–Hubbard model at a known filling, and test whether the inflection-point scheme with $b=1$ reproduces the accepted $T_\mathrm{BKT}$ from lattices of size $L=20$ to $64$; a systematic offset would indicate the heuristic mapping is biased.

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Extended reading notes

Core claim

The central discovery is that the condensate fraction $n_c$, defined from the largest eigenvalue of the momentum-space pairing matrix, obeys the same algebraic size scaling as the pairing correlator in the quasi-ordered phase, $n_c \propto L^{-\eta}$ with $\eta \to 1/4$ at the transition, but with much smaller subleading corrections. Formally, the on-site pairing correlator carries a subleading $L^{-2}$ correction that dominates at the moderate sizes accessible to simulation, whereas $n_c$ includes Cooper pairs of all sizes and therefore scales cleanly from $L \approx 20$ upward. The crossover between algebraic and exponential scaling brackets the transition, and extrapolating the extracted exponent $\eta(T)$ to $\eta_c=1/4$ yields the transition temperature; the same $T_\mathrm{BKT}$ is obtained from data collapse and from the logarithmic finite-size scaling of $T_\mathrm{BKT}(L)$. The specific heat shows a peak slightly above the transition, at about $1.10\,T_\mathrm{BKT}$.

Load-bearing premise

The paper identifies the inflection point of the condensate-fraction temperature curve as the finite-size BKT transition temperature and extrapolates it using $T_\mathrm{BKT}(L) = T_\mathrm{BKT}(\infty) + a/(\ln bL)^2$ with $b$ fixed to 1; this mapping and fitting choice are heuristic and, if inappropriate, would shift the reported $T_\mathrm{BKT}$.

Editorial extensions

If this is right

  • Accurate BKT temperatures can be extracted from lattices as small as $L=20$, roughly an order of magnitude cheaper than the sizes needed for the on-site pairing correlator.
  • The scheme transfers directly to other 2D fermionic systems, including spin-orbit-coupled models and dilute Fermi gases, where the pairing correlator and superfluid density become vanishingly small.
  • For 2D bosonic systems, the condensate fraction from the single-particle density matrix provides a competitive alternative to superfluid-density-based methods.
  • The specific-heat anomaly at about $1.1\,T_\mathrm{BKT}$ gives experimentalists a precursor signature for locating the BKT transition in optical lattices.
  • The logarithmic correction $T_\mathrm{BKT}(L) = T_\mathrm{BKT}(\infty) + a/(\ln bL)^2$ is confirmed for a correlated fermion system, supporting the BKT finite-size scenario.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because $n_c$ collects pairing weight at all momenta, the small finite-size corrections observed here may be generic for order parameters defined from the leading eigenvalue of a reduced density matrix, not special to the Hubbard model; testing this on the 2D XY model would separate the two possibilities.
  • In dilute 2D Fermi gases, where both the pairing correlator and superfluid density vanish in finite systems, $n_c$ may be the only practical observable; a two-step extrapolation in $L$ and particle number, as outlined in the paper, could give the first unbiased $T_\mathrm{BKT}$ in the continuum limit.
  • The inflection-point method, if validated against exactly solvable BKT models, could be automated as a black-box estimator for transition temperatures in future tensor-network and Monte Carlo studies of 2D superconducting and superfluid models.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies the 2D attractive Fermi-Hubbard model with finite-temperature auxiliary-field quantum Monte Carlo, reaching system sizes up to 64x64 (4096 sites). It defines the condensate fraction n_c from the largest eigenvalue of the momentum-space pairing matrix and shows numerically that n_c scales algebraically with L below the BKT transition and exponentially above it, with markedly smaller finite-size effects than the on-site pairing correlator. From the n_c scaling the paper extracts the BKT transition temperature for U/t=-4, mu/t=0.25 by three methods: linear extrapolation of the exponent eta(T) to eta_c=1/4, data collapse of n_c L^{1/4}, and extrapolation of finite-size transition temperatures T_BKT(L), yielding consistent values around T_BKT/t=0.142. A specific-heat anomaly with a peak near 1.10 T_BKT is also reported, and the methods are argued to be applicable to other 2D fermionic and bosonic superfluids.

Significance. If the central scaling claim holds, the condensate fraction provides a practical and accurate observable for determining BKT transitions in 2D correlated systems, with substantially smaller finite-size corrections than the widely used on-site pairing correlator. The paper's strengths include numerically exact simulations at unprecedented lattice sizes for this model, small statistical errors, extensive supplementary results for multiple fillings and interaction strengths, and a consistent cross-check of T_BKT across three independent fitting schemes. The claimed general applicability to other fermionic and bosonic systems is plausible but remains an extrapolation from the Hubbard-model evidence presented.

major comments (4)
  1. [Fig. 1(a); SM Sec. IIIE] The central claim that L>=20 is sufficient to determine T_BKT accurately is not quantitatively tested. The supplemental material explicitly states that for U/t=-4, mu/t=1.25 the condensate fraction shows finite-size effects for L<=28 (SM Sec. IIIE), yet for the main parameter set no stability analysis with respect to the minimum system size L_min is shown. All three T_BKT estimates use the scaling assumption with a chosen L_min (L>=20 for n_c in Fig. 1(a), L=20-64 for the data collapse in Fig. 2(a), and the n_c(T) inflection points in Fig. 3), so a systematic bias from residual finite-size corrections would shift all three estimates together. The paper should report how eta and T_BKT vary as L_min is changed (for example L_min=16, 20, 24, 28) and include this spread in the final uncertainty of T_BKT=0.1420(7).
  2. [Fig. 3(a) and (b)] The identification of the inflection point of n_c(T) as the finite-size BKT transition temperature T_BKT(L) is heuristic and is not derived or independently justified. The agreement with the eta(T) scaling and data-collapse results is encouraging, but the logarithmic-correction extrapolation T_BKT(L)=T_BKT(infinity)+a/(ln bL)^2 and the associated claim that the logarithmic correction is 'confirmed' rest entirely on this mapping. The paper should either provide a derivation or a systematic comparison with alternative finite-size definitions of T_BKT(L), such as the crossing point of n_c L^{1/4} curves at eta=1/4. In addition, the reported error bars for T_BKT(infinity) from Fig. 3(b) exclude the fitting choice of fixing b=1; the inset with b free is mentioned as consistent, but the sensitivity should be quantified.
  3. [SM Eq. (13) and main text around Fig. 1] The paper derives the subleading L^{-2} correction for the on-site pairing correlator (SM Eq. 13), but no analogous derivation is given for the condensate fraction n_c. The main-text argument that n_c includes Cooper pairs of all sizes explains physically why local-pair fluctuations are reduced, but it does not rule out subleading corrections from momentum-space discretization or from the normalization lambda_max/(N/2). Because the central advantage of n_c over the pairing correlator is precisely its small finite-size corrections, this assumption is load-bearing for all three T_BKT estimates. The authors should either supply a derivation of the leading and subleading finite-size behavior of n_c or perform a direct multi-term fit (for example n_c = A L^{-eta} (1 + c L^{-2} + ...) with varying L_min) to demonstrate the absence of significant subleading terms for L>=20.
  4. [Inset of Fig. 1(a)] The linear extrapolation of eta(T) to eta_c=1/4 is described as a limit approached 'from below', but the text reports eta=0.304(2) at T/t=0.145, which is above the stated transition interval 0.140<T_BKT/t<0.145. If T/t=0.145 is included in the linear fit, the result would be biased by data that the paper itself says should be in the crossover regime to exponential decay. The manuscript should state exactly which temperatures enter the linear fit and justify the exclusion or inclusion of T/t=0.145.
minor comments (4)
  1. [Throughout] There are numerous typographical errors and misspellings, including 'attracive', 'diffrent', 'the the', 'inlcuding', 'questiones', 'capabilty', 'seperated', and 'tunning'. A careful proofreading pass is needed.
  2. [Fig. 3(b) and inset] The main text fixes b=1 in the formula T_BKT(L)=T_BKT(infinity)+a/(ln bL)^2, while the inset shows a fit using c+a/(ln bL)^2. The relationship between these parametrizations and why b=1 is the preferred choice should be stated more clearly.
  3. [Sec. IV (SM)] In the fixed-filling results, some statements such as 'the result of T_BKT/t ~ 0.10 should be obtained' for L=20 are not tied to a specific figure or table; giving the actual fitted value and its uncertainty would make the comparison quantitative.
  4. [Abstract and introduction] The phrase 'condensate fractionscaling' is broken across a line in the abstract and appears without a space in several places; this is a formatting issue, not a substantive one.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: BKT scaling forms are external templates, TBKT values are fits to directly measured condensate-fraction data, and the central scaling/finite-size claims are empirical observations with independent cross-checks.

full rationale

The paper does not claim to derive BKT theory from the condensate fraction; it imports standard BKT scaling forms as external inputs — ηc = 1/4, the correlation-length scaling variable x = L exp[−A(T/TBKT − 1)^−1/2], and the logarithmic finite-size correction TBKT(L) = TBKT(∞) + a/(ln bL)^2 — and applies them to a directly measured observable, nc = λmax/(N/2), defined via the leading eigenvalue of the pairing matrix. The algebraic-versus-exponential scaling of nc is read directly from log-log plots, and the exponent η(T) is extracted without using TBKT as an input. The three TBKT estimates are outputs of fits within BKT theory, not renamed fit parameters: the η(T) linear extrapolation uses measured η values and the external critical value ηc = 1/4; the data collapse uses fixed ηc = 1/4 and a fitted TBKT that must make the data collapse onto a single curve; and the TBKT(L) extrapolation uses the inflection point of the measured nc(T) and the standard logarithmic form. These procedures are consistency checks against BKT theory, not self-referential reductions. The claim of reduced finite-size effects is also empirical: the paper compares power-law fits for L ≥ 20 (nc) versus L ≥ 32 (⟨Δ2⟩), with relative deviations of about 3% versus 10% at L = 20, and this is not a fitted-input-called-prediction. Self-citations (e.g., Refs. [42, 43, 60]) supply methodology and prior applications, not the load-bearing justification for the current scaling result; the subleading L^−2 correction for ⟨Δ2⟩ is derived within the Supplemental Material (SM Eq. 13), not merely cited. Heuristic choices such as identifying the inflection point of nc(T) as TBKT(L) and fixing b = 1 are systematic-uncertainty/correctness risks rather than circularity, because the paper provides independent cross-checks with ⟨Δ2⟩ scaling, superfluid density, and the bounded range 0.140 < TBKT/t < 0.145. Overall, the central derivation chain is self-contained and externally benchmarked; no step reduces to its own input by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central result rests on standard BKT scaling relations and on the numerical observation that the condensate fraction follows the same finite-size scaling as the pairing correlator. The only ad hoc assumption is that nc inherits the same leading exponent with reduced subleading corrections, which is verified numerically. All transition temperatures are extracted through fits using these assumed scaling forms.

free parameters (4)
  • Amplitude A in BKT scaling variable = 1.267(20) for U/t=-4, μ/t=0.25 (Fig. 2a); other values in supplemental
    Fitted from data collapse in Fig. 2(a); enters the scaling variable x = L exp[-A(T/TBKT-1)^{-1/2}].
  • Coefficient a in TBKT(L) log-correction = Not quoted explicitly (fitted in Fig. 3b)
    Fitted in TBKT(L) = TBKT(∞) + a/(ln bL)^2.
  • Constant b in TBKT(L) log-correction = 1 (fixed by hand)
    b is fixed to 1 to stabilize the fit, following Refs. 24, 31, 32; leaving it free gives noisier results (inset of Fig. 3b).
  • Polynomial coefficients for f(x) in data collapse = Not reported
    f(x) is a polynomial in x fitted by least squares; the order and coefficients are not given, so the fit is not fully reproducible.
assumptions (6)
  • domain assumption BKT theory: algebraic decay with exponent η ≤ 1/4 below the transition, η = 1/4 at TBKT
    Used to identify the transition from the exponent scaling (Fig. 1 inset) and to fix ηc = 1/4 in the data collapse (Fig. 2).
  • domain assumption Nelson-Kosterlitz universal jump: ρs(T→TBKT-) = 2TBKT/π
    Used in SM Sec. III D to extract TBKT from the crossing of ρs×(π/2T) with unity.
  • domain assumption BKT finite-size scaling form: ⟨Δ²⟩L^{1/4} = f(L exp[-A(T/TBKT-1)^{-1/2}])
    Adopted for data collapse in Fig. 2 and SM; also applied to nc.
  • domain assumption Logarithmic finite-size correction: TBKT(L) = TBKT(∞) + a/(ln bL)^2
    Used in Fig. 3(b) for the thermodynamic-limit extrapolation.
  • ad hoc to paper Condensate fraction has the same leading scaling exponent as the on-site pairing correlator
    Assumed in SM Sec. II D and supported numerically; the analytic subleading correction is derived only for ⟨Δ²⟩, not for nc.
  • standard math Trotter error is eliminated by Δτ→0 extrapolation
    Stated in SM Sec. II B, but the extrapolation data are not shown.

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Cite this review

Pith. "Pith review of Condensate Fraction Scaling and Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity." pith.science (2026). https://pith.science/paper/B7I5YFN2

@misc{pith2026250517411,
  author       = {Pith},
  title        = {Pith review of: Condensate Fraction Scaling and Berezinskii-Kosterlitz-Thouless Transition of Superconductivity and Superfluidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B7I5YFN2}},
  note         = {Machine review of arXiv:2505.17411}
}
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 {\it 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 {\it 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.

Figures

Figures reproduced from arXiv: 2505.17411 by the authors.

Figure 1
Figure 1. FIG. 1. The finite-size scaling and log-log plots of (a) conden [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Data collapse for (a) condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Finite-size BKT transition temperature [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Specific heat [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The acceptance ratio of the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Demonstration of condensate fraction and pair wave function of 2D attractive Hubbard model for a 64 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. AFQMC results of condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Rescaled results of [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Rescaled results of [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Data collapse for (a) condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The results of (a) total energy per site [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: plots all the results of condensate fraction nc and pairing correlator ⟨∆2 ⟩ for U/t = −4, µ/t = 0.60. As shown in panel (b), the algebraic scaling in superfluid phase and exponential decaying in normal state of nc is clear, which results in the bounded range of the t…
Figure 14
Figure 14. Figure 14: FIG. 14. Data collapse for (a) condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. superfluid density [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Specific heat [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: plots all the results of condensate fraction nc and pairing correlator ⟨∆2 ⟩ for U/t = −4, µ/t = 1.25. As shown in panel (b), the algebraic scaling in superfluid phase and exponential decaying in normal state of nc is clear, which results in the bounded range of the t…
Figure 18
Figure 18. Figure 18: FIG. 18. Data collapse for (a) condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Specific heat [PITH_FULL_IMAGE:figures/full_fig_p021_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. The fermion filling (the main plots) and double occupancy (the insets) versus temperature and system size for all [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]
Figure 21
Figure 21. Figure 21: plots all the results of condensate fraction nc and pairing correlator ⟨∆2 ⟩ for U/t = −4,⟨nˆ⟩ = 0.50. As shown in panel (b), the algebraic scaling in superfluid phase and exponential decaying in normal state of nc is clear, which results in the bounded range of the t…
Figure 22
Figure 22. Figure 22: FIG. 22. Condensate fraction [PITH_FULL_IMAGE:figures/full_fig_p024_22.png]

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