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Superfluid Weight of Strongly Inhomogeneous Superconductors

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Pith's one-line read The paper derives a linear-response correction to the superfluid weight from the pairing potential's response to a vector potential and shows it is essential for inhomogeneous superconductors, including setting the unpinned vortex-lattice…

desk verdict A clear and useful derivation of the pairing-potential response correction to superfluid weight, with a solid numerical benchmark but a genuine gap on the vortex-lattice null space. read the letter →

arxiv 2505.00069 v2 pith:YJ4I3D6J submitted 2025-04-30 cond-mat.supr-con

Jonathan Schirmer , Enrico Rossi This is my paper · ORCID
classification cond-mat.supr-con
keywords superfluidweightstiffnessinhomogeneoussuperconductivitypairingpotentialresponselineartheoryvortexlatticeBogoliubov-deGennesBerezinskii-Kosterlitz-Thoulesstransition
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 a linear-response formula for the superfluid weight $D^{(s)}_{\mu\nu}$ of a superconductor whose pairing potential $\Delta$ varies strongly in space. The central claim is that the standard expression, which treats $\Delta$ as fixed when a vector potential $A$ is applied, misses a correction $\delta D^{(s)}_{\mu\nu}$ coming from the way $\Delta$ itself responds to $A$. For a periodic modulation of the pairing, the correction is quantitatively necessary; for an unpinned vortex lattice it is qualitative, turning the uncorrected finite stiffness into the expected value $D^{(s)}_{\mu\nu}=0$. The paper verifies the formula by comparing it with direct numerical second derivatives of the free energy with respect to $A$. Because the superfluid weight sets the Berezinskii-Kosterlitz-Thouless temperature in two dimensions, the correction matters for predictions of phase coherence in inhomogeneous thin films.

What carries the argument

The central object is the pairing-potential response $\delta\Delta_m/\delta A_\mu|_0$, the change each site's complex gap undergoes when a slow static vector potential is applied. It is computed from the singular linear system $K\,\delta\Delta=C$ (Eqs. 12-15), where the kernel $K$ is the pairing susceptibility including the $-1/U$ term and $C_\mu$ couples the kinetic-current vertex to pair creation; inverting via a pseudoinverse from a singular-value decomposition projects out the global phase mode of $\Delta$. This response enters the current operator as off-diagonal particle-hole blocks and produces the correction $\delta D^{(s)}_{\mu\nu}=2\,\mathrm{Re}\sum_m (C_\mu)_m\,\delta\Delta_m^*/\delta A_\nu$. The paper also shows the same result follows from vertex corrections and Ward identities, which is what guarantees gauge invariance of the final tensor.

What would settle it

Compute the null space of $K$ (Eq. 12) for a vortex-lattice configuration with no pinning: if the dimension exceeds one, the pseudoinverse projects out a physical mode and the predicted $D^{(s)}_{\mu\nu}=0$ would not survive under a different gauge choice or finite-size extrapolation. Alternatively, evaluate $D^{(s)}$ by the finite-difference free-energy method at larger system sizes and check whether the cancellation persists as the vortex lattice is translated by a small amount.

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Extended reading notes

Core claim

Within Bogoliubov-de Gennes mean-field theory for an $s$-wave superconductor with arbitrary spatial inhomogeneity, the paper derives $D^{(s)}_{\mu\nu}=D^{(s,0)}_{\mu\nu}+\delta D^{(s)}_{\mu\nu}$, where $D^{(s,0)}$ is the usual current-correlation expression and $\delta D^{(s)}_{\mu\nu}=2\,\mathrm{Re}\sum_m (C_\mu)_m\, \delta\Delta_m^*/\delta A_\nu$. The vector $\delta\Delta/\delta A$ is not an independent input: it solves the singular linear system $K\,\delta\Delta=C$ (real and imaginary parts coupled), with $K$ having one zero mode corresponding to the global $U(1)$ phase of $\Delta$, so the physical solution is obtained by pseudoinverse. The paper argues that this correction is the collective-mode, or anomalous vertex, contribution required by gauge invariance when inhomogeneities mix longitudinal and transverse responses, and demonstrates numerically for two two-dimensional examples that the full formula coincides with the free-energy second derivative while the uncorrected formula does not. In the vortex-lattice example the full formula gives $D^{(s)}_{\mu\nu}=0$ for an unpinned array, matching known results, whereas the uncorrected expression is finite.

Load-bearing premise

The load-bearing premise is that the linear-response kernel $K$ has exactly one zero mode, the global phase of the pairing potential, so that the pseudoinverse yields the physical response $\delta\Delta/\delta A$; for a vortex lattice, the vortices' ability to slide without energy cost could create additional zero or near-zero modes, and the paper checks this case by numerical agreement with the free-energy derivative instead of by analyzing the null space.

Editorial extensions

If this is right

  • For any strongly inhomogeneous superconductor, the superfluid weight requires solving the gap's response to the probe field; the standard uncorrected formula is quantitatively wrong when the pairing modulation is strong.
  • In two dimensions, because the Berezinskii-Kosterlitz-Thouless temperature is set by the superfluid weight, the correction shifts predicted $T_{\rm BKT}$ values for patterned and moiré superconductors.
  • An unpinned vortex lattice has exactly zero superfluid weight, so it produces no Meissner response; the uncorrected expression would falsely predict a finite response.
  • A periodic pinning potential restores a finite superfluid weight, and the corrected formula interpolates between zero and the uncorrected value as pinning strengthens.
  • Gauge invariance demands the correction: multiplying the pairing potential by a complex constant produces no total current, and the formula projects out such gauge artifacts.

Reading between the lines

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

  • A testable extension the paper does not run: use the corrected stiffness to predict $T_{\rm BKT}$ in inhomogeneous films and compare with measured sheet inductance or resistance peaks, isolating the $\Delta$-response contribution from the usual kinetic term.
  • The null-space assumption could be probed by adding weak disorder that breaks translational invariance; if additional near-zero modes of $K$ appear beyond the global phase, the exact cancellation for vortex arrays may become finite-size dependent.
  • The same linear-response machinery applies to flat-band superconductors where quantum-geometric contributions are large; the $\Delta$-response term could partially cancel or enhance those contributions, and the paper's formula provides a way to test which.
  • The direction dependence of the corrected tensor under an anisotropic pinning potential could serve as a probe of vortex-lattice orientation, since the unpinned cancellation is exact but any pinning symmetry breaking leaves a directional stiffness.
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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

2 major / 5 minor

Summary. The paper derives a linear-response expression for the superfluid weight D_s of strongly inhomogeneous superconductors, including the response of the pairing potential Δ to the probe vector potential A. The central formula is D_s = D_s^(0) + δD_s, with δD_s = 2 Re Σ_m (C_μ)_m δΔ_m^*/δA_ν, where δΔ/δA is obtained from the singular linear system K δΔ = C (Eqs. 12–16). The authors argue that the correction is negligible only for homogeneous pairing, significant for periodically modulated pairing, and essential for vortex lattices: for an unpinned vortex lattice the uncorrected D_s^(0) is finite while the full expression gives D_s = 0. The paper validates the formula by comparing with the numerically computed second derivative d²F/dA² of the free energy for both a periodic-potential superconductor and a vortex lattice.

Significance. If the result is correct, it is a practically useful and conceptually important contribution: it provides a concrete BdG-level formula for the superfluid weight in the presence of strong spatial variations of the pairing potential, including the vortex case where collective-mode corrections are qualitatively decisive. The derivation in the main text and Supplementary Material is explicit and self-contained, and the comparison with the independent free-energy second-derivative benchmark is a genuine strength. The gauge-invariance discussion, in particular the projection of the global phase mode, is thoughtful. The claim that the standard uncorrected expression is qualitatively wrong for an unpinned vortex lattice is physically expected from earlier work [66–68], which lends additional credibility to the numerical cancellation. The remaining major weakness is that the null-space structure of the linear-response kernel K in the vortex case is asserted rather than analyzed; this is addressable and does not, in my view, invalidate the central derivation.

major comments (2)
  1. [SM 'Inversion of Equation (12)'; Fig. 1(b)] The proof that K has exactly one zero mode is incomplete for the vortex-lattice case. In the unpinned case V0=0 the vortex lattice spontaneously breaks continuous translational symmetry, and a uniform translation of the vortex array is a zero-energy Goldstone deformation; the corresponding order-parameter change δΔ_m ∝ v·∇Δ_m is a periodic function on the magnetic supercell and hence lies in the space on which Eq. (12) acts. The manuscript asserts (SM, 'Inversion of Equation (12)') that K has rank one less than its dimension, reflecting only the global U(1) phase, but it provides no singular-value spectrum, no nullity count, and no overlap of C_μ with near-null vectors for the vortex case. This matters because the central vortex result D_s=0 is obtained as the complete cancellation D_s^(0)+δD_s=0 in Fig. 1(b): if the pseudoinverse projects out a physical translational mode, Eq. (16) will not yield the correct δD_s. The agreement with d²F/dA² shown by the yellow crosses is encouraging, but it is a single numerical benchmark, not an analysis of the kernel. The authors should report the singular-value spectrum of K in the vortex case, the dimension of the near-null space, the overlap ⟨C_μ|null⟩, and a system-size convergence study of the cancellation.
  2. [SM 'Gauge Invariance'; Eq. (93)] The finite-difference evaluation of δΔ_m/δA_μ for a vortex lattice is delicate because the vortex phase is singular and the numerical solution for Δ(Q) is not guaranteed to be a smooth function of Q. The authors correctly note that δD_s must be invariant under adding a large complex multiple z_i Δ_m, and they state that they verified numerically that Eq. (84) projects out such terms. However, no quantitative description of this verification is given (for example, the range of |z_i| tested or the residual variation of δD_s). Since the vortex-lattice conclusion rests on this projection in the finite-difference route, the paper should either present this check explicitly or base the vortex result on the SVD route together with the null-space analysis requested above.
minor comments (5)
  1. [Eq. (2) and SM Eq. (31)] The pairing contribution to the current is written as [δΔ c†c† − (1/2U)δ|Δ|² + H.c.]; because the bracket includes H.c., the total constant term is −(1/U)δ|Δ|², consistent with SM Eq. (31). The present notation invites a spurious factor-of-two concern; please write the total term explicitly as −(1/U)δ|Δ|² to avoid confusion.
  2. [Fig. 1(a) discussion] The sentence 'only the value of D_s obtained by taking into account the corrections due to δΔ_m/δA_μ = 0 agrees...' appears to contain a typo; the intended meaning is clearly δΔ_m/δA_μ ≠ 0, since the whole point is that the corrections are nonzero for V0 ≠ 0.
  3. [SM Fig. 2 caption] SM Fig. 2 refers to 'Fig. 2 of the main text', but the main text contains only Fig. 1; the caption should reference Fig. 1(b).
  4. [Reference [59]] Reference [59] has a formatting error: 'Phys. Rev. B 89, 014507 (2014)89, 014507 (2013)' should be cleaned up.
  5. [Main text around Eq. (17)] After Eq. (17), the text says 'for V0=0 the superconductor is homogeneous'; this is only true for the periodic-potential panel. In the vortex-lattice panel V0=0 still leaves the vortex-induced inhomogeneity. Consider clarifying to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the correction formula is derived from linear response and validated against an independent free-energy derivative; self-citations are background/setup only.

full rationale

No circular step is present. The central object δΔ_m/δA_ν is obtained by solving the linear system K δΔ = C (Eq. 12), which is itself derived from the linear-response formula Eq. (11); the correction Eq. (16) then follows by inserting the A-dependent part of the current vertex I_ν into the current-correlation expression Eq. (10). No parameter is fitted to reproduce D_s: the only numerical benchmark, d²F/dA² (yellow crosses in Fig. 1), is computed independently from self-consistent free energies and agrees with the formula, including the unpinned-vortex cancellation D_s = 0 that is also checked against Refs. [66-68]. Self-citations ([22], [37], [38], [70]) appear only as background or as the numerical vortex-lattice setup and are not load-bearing for the central identity. The unanalyzed null-space of K in the vortex case is a correctness/robustness caveat, not circularity.

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

The central formula rests on standard BdG and linear response with Peierls substitution. The paper-specific structural assumption is the single-zero-mode pseudoinverse of K for delta Delta/delta A; it is supported by a Ward identity for the global phase and by numerical agreement with the free-energy derivative, but the null space for vortex translations is not analyzed. No free parameters are fitted to the superfluid weight itself.

assumptions (6)
  • domain assumption Bogoliubov-de Gennes mean-field theory with a local s-wave pairing potential is a sufficient description of the superconducting state for the response calculation.
    The Hamiltonian (1) is the starting point; the paper explicitly says it does not address identification of the many-body ground state but uses BdG pragmatically.
  • domain assumption Linear response theory with a static, uniform vector potential coupled by Peierls substitution gives the superfluid weight.
    Eqs. (2)-(9) define D_s through the current response to A at q=0; this is the standard definition used throughout the field.
  • domain assumption The pairing potential is spatially periodic over a unit cell that can be chosen large enough to contain inhomogeneities or one flux quantum.
    SM Eq. (22) assumes Delta_{j+R} = Delta_j, and the main text enlarges the primitive cell accordingly for vortex lattices.
  • ad hoc to paper The linear-response kernel K has exactly one zero mode, corresponding to the global U(1) phase, so the pseudoinverse gives the physically relevant delta Delta/delta A.
    SM 'Inversion of Equation (12)'; this is load-bearing for vortex lattices and is supported only by a Ward identity for the global phase plus numerical checks.
  • domain assumption An unpinned vortex lattice has zero superfluid weight, as established in Refs. [66-68].
    Used in the main text as the benchmark for the vortex-lattice example; if this external result were wrong, the claim of qualitative failure of the uncorrected formula would shift.
  • domain assumption The second derivative of the free energy with respect to A, evaluated at the self-consistent Delta(A), defines the same superfluid weight as the linear response to a vector potential.
    Main text below Eq. (8); the numerical benchmark relies on this equality.

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Pith. "Pith review of Superfluid Weight of Strongly Inhomogeneous Superconductors." pith.science (2026). https://pith.science/paper/YJ4I3D6J

@misc{pith2026250500069,
  author       = {Pith},
  title        = {Pith review of: Superfluid Weight of Strongly Inhomogeneous Superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJ4I3D6J}},
  note         = {Machine review of arXiv:2505.00069}
}
read the original abstract

In this work, we obtain the expression, within the linear response approximation, that allows the direct calculation of the superfluid weight for strongly inhomogeneous superconductors. Using this expression, we find that, in general, the correction to the superfluid weight due to the response of the superconductor's pairing potential to the perturbing vector potential is important in superconductors with a strongly inhomogeneous pairing potential. We consider two exemplary cases: the case when strong inhomogeneities in the pairing potential are induced by a periodic potential, and the case when superconducting vortices are induced by an external magnetic field. For both cases we show that the correction to the superfluid weight due to the response of the paring potential to the perturbing vector potential can be significant, it must be included to obtain quantitatively correct results, and that for the case when vortices are present the expression of the superfluid weight that does not include such correction returns qualitatively wrong results.

Figures

Figures reproduced from arXiv: 2505.00069 by the authors.

Figure 1
Figure 1. (a) shows the calculated values of (1/2)Tr(D (s) µν ) for a 2D superconductor on a square lat￾tice with lattice constant a = 1 in the presence of the periodic potential V (rj ) = − V0 2  cos 2π M xj  + cos 2π M yj   (17) with period M. In Eq.(17) rj = (xj , yj ). The potential has the effect of modulating the density of electrons, as well as the amplitude of the order parameter ∆j , thus rendering the supercon… view at source ↗
Figure 2
Figure 2. FIG. 2. The energy as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Top row: the spatially averaged value of [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

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Reviewed August 16, 2026 · model on record in the stance chip above.