Geometric superfluid weight

Scope

Superfluid weight is a current or phase-twist response defined with a particular thermodynamic and electromagnetic convention. In a multiband superconductor, its conventional part is associated with intraband dispersion, while geometric terms involve the wavefunctions and interband matrix elements. These are parts of a specified response calculation, not two experimentally separated streams of Cooper pairs. R1 R3

A response kernel also depends on how frequency and wavevector approach zero. Scalapino, White and Zhang discuss distinct limiting procedures for diagnosing insulating, metallic and superconducting lattice states. A clean metal's ideal transport response should not be identified with the equilibrium Meissner response merely because both are called a stiffness. This entry checks that paper's original abstract; its full correlation-function derivation remains open. R6

Mathematical assumptions

For an isotropic superconducting sheet, define the phase-energy density by $f=\rho_s|\nabla\theta-(2e/\hbar)\mathbf A|^2/2$, with $\rho_s$ in joules. The sheet electromagnetic kernel is $K_\square=4e^2\rho_s/\hbar^2=1/L_\square$, where $L_\square$ is sheet kinetic inductance. Peotta and Törmä instead define a mass-normalized weight through $f=D_s p_s^2/8$, with $p_s=\hbar\nabla\theta$ in zero field. Thus their convention gives $\rho_s=\hbar^2D_s/4$ and $K_\square=e^2D_s$. These quantities have different units even when each is called superfluid weight. R1 R4

A pair phase written as $\theta=2\mathbf q\cdot\mathbf r$ introduces another factor: the curvature with respect to $\mathbf q$ is four times the curvature with respect to $\nabla\theta$. For a hypothetical $L_\square=10$ nH, the sheet convention above gives $\rho_s/k_B\approx0.7845$ K. Three illustrative inductances were checked with SI charge, Planck and Boltzmann constants. These are conversion examples, not measured superconducting transition temperatures. R1 R4

In a fixed orbital basis, differentiating $H|m\rangle=\varepsilon_m|m\rangle$ gives $\langle n|\partial_iH|m\rangle=\delta_{nm}\partial_i\varepsilon_m+(\varepsilon_m-\varepsilon_n)\langle n|\partial_i m\rangle$. Flat eigenvalues set the diagonal velocity to zero without forcing the interband matrix elements to vanish. Fifteen checks in an illustrative two-band model reproduce this identity and $g_{kk}=1/4$ for a rotating eigenvector; multiplying either eigenvector by a smooth phase leaves the metric unchanged. This checks a band identity, not superconductivity or a self-consistent current response. R3 R5

The familiar isolated-band expression $D_{s,ij}^{\rm geom}\propto\sum_{\mathbf k}(|\Delta|^2/E_{\mathbf k})\tanh[E_{\mathbf k}/(2k_BT)]g_{ij}(\mathbf k)$ displays a geometric structure, not a convention-free numerical formula. It requires stated pairing, band-isolation and symmetry assumptions, a wavevector integration measure and charge factors. The metric used in the checked Liang manuscript is twice the Wiki convention $g_{ij}=\operatorname{Re}\langle\partial_i u|(1-|u\rangle\langle u|)|\partial_j u\rangle$. Translate that normalization before comparing prefactors. R1 R3

The internal pairing fields must also respond to the twist. Under the time-reversal and equilibrium-minimum assumptions studied by Huhtinen and colleagues, the total free-energy curvature contains a negative-semidefinite relaxation correction to the frozen-field result. In their isolated flat-band uniform-pairing setting, the relevant integrated metric has minimal trace with respect to orbital positions. Equal pairing at zero twist does not justify freezing all relative phases at nonzero twist. A nonzero unrelaxed result can even become zero after relaxation. The scalar illustration in the flat-band article shows this possibility without claiming to reproduce the complete lattice response. R5

Experimental criteria

Measure and report the total response with its units, dimensionality, frequency, geometry and fitting model. Tanaka and colleagues obtain stiffness through microwave kinetic inductance with a parallel electrode-proximitized edge in the circuit model. Their conversion uses an assumed active-region aspect ratio equal to the device aspect ratio; the author-v2 definition $D_s=\ell/(W L_{\rm TBG})$ uses the separated bulk MATBG branch. This inverse sheet inductance is distinct from the combined parallel termination and from the entire resonator. The twist-angle comparison makes the branch subtraction explicit. A comparison with a conventional estimate can support a geometric interpretation without constituting a model-independent measurement of its fraction. R4

The selected author manuscript distinguishes a 20 mK refrigerator base from an approximately 40 mK wired-device temperature. Its roughly 1 K resistive scale uses a fitted half-resistance criterion. In supplementary Figure 12, the falling fit crosses half the zero-temperature intercept of the normal-state fit; the separately named zero-resistance scale is a fitted zero crossing. Neither the measurement temperature nor that resistive criterion should automatically be labelled a BKT transition. Full published-version and raw-data comparisons remain open. R4

For an effective isotropic two-dimensional BKT description, the jump involves the renormalized total sheet phase stiffness: $k_BT_{\rm BKT}=\pi\rho_s(T_{\rm BKT}^-)/2$. It does not determine a separate geometric component. Substituting a zero-temperature stiffness gives only an upper estimate if the stiffness at the transition does not exceed that zero-temperature value. With the hypothetical 10 nH example this estimate is about 1.232 K; it is neither a measured transition nor evidence that the BKT assumptions hold in a particular device. R4 R7

Limits and disputes

A nonzero Chern number can provide a geometric lower bound in specified models, but it is not necessary for every geometric contribution. The Chern-trivial Lieb-lattice example has a finite flat-band response under its interaction assumptions. A bound on an integrated metric is not automatically a numerical bound on a dimensionful measured stiffness: filling, interaction, normalization and the complete response matter. R1 R2 R5

Selected original passages and equations, the band-current identity and unit conversions have been checked. Complete scientific acceptance has not been reached. Full current-correlation and BKT derivations, self-consistent multiband response, remote-band convergence and experimental data audits remain open. Disorder, finite frequency, sample inhomogeneity and phase fluctuations can affect the comparison between a theoretical weight and a measured response. R3 R4 R5 R6 R7

Discovery implications

Report conventional, geometric and total response using a common convention, including internal-field relaxation. Check a band model's current operator and geometry before applying a flat-band formula, and retain the uncertainty of the experimental baseline. Geometry can constrain transport by pairs; it does not automatically supply the attraction, determine gap symmetry or ensure macroscopic phase coherence. R1 R3 R5

Related topics

Continue with quantum metric, flat-band superconductivity, flat-band pairing, Meissner effect and dimensionality and phase slips.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.