Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.08336

Quantum geometry of collective pairing fluctuations in the superfluid weight of multiband superconductors

R. N. Kalkan, M. Iskin

Published Oct 6, 2026• 0 citations
arXivHTMLPDF

Abstract

Beyond the frozen-pairing contribution, the superfluid weight of a multiband superconductor contains a correction from the self-consistent relaxation of the pairing amplitudes under a phase twist. We show that this correction has a quantum-geometric origin, not in the space of Bloch states but in the space of collective pairing fields. For an attractive multiband Hubbard model at the mean-field level, the thermodynamic Hessian governing the pairing response coincides with the static Gaussian pair-fluctuation kernel at zero momentum at every temperature below $T_c$. Since gauge invariance relates a phase twist to a finite center-of-mass momentum of the pair field, the superfluid weight is set by the small-momentum curvature of the eigenvalue branch that evolves from the Goldstone mode, and the relaxation correction is a stiffness-weighted quantum metric of the corresponding soft eigenvector on the manifold of pairing configurations. The same construction yields the Ginzburg-Landau gradient coefficient with the identical geometric correction, hence the coherence-length tensor, and, with the low-frequency dynamics of the kernel, the pair-mass tensor. Lieb-lattice calculations verify the equivalence of the thermodynamic, fluctuation-kernel, and soft-mode formulations and the independence of the superfluid weight from the orbital embedding.

DOI
10.48550/arxiv.2610.08336

Related papers

Structure factors and quantum geometry in multiband BCS superconductors

M. Iskin

We consider multiband BCS superconductors that exhibit time-reversal symmetry and uniform pairing, and analyze their dynamic density and spin structure factors using linear-response theory within the mean-field BCS-BEC crossover framework at zero temperature. Our results for the multi-orbital Hubbard model satisfy the associated f-sum rules in several limits. In particular, in the strong-coupling limit, they coincide with those of a weakly-interacting Bose gas of Cooper pairs, where the low-energy collective Goldstone modes serve as Bogoliubov phonons. We further reveal that the quantum-geometric origin of the low-energy structure factors, along with related observables such as the superfluid-weight tensor and the effective-mass tensor of Cooper pairs, can be traced all the way back to the effective-mass theorem for Bloch bands in this limit. As an illustration, we investigate the pyrochlore-Hubbard model numerically and demonstrate that the Goldstone modes are the only relevant collective degrees of freedom in the flat-band regime.

arXiv:2503.10085•Physical review. B./Physical review. B•Jul 14, 2025•4 citations

Extracting quantum-geometric effects from Ginzburg-Landau theory in a multiband Hubbard model

M. Iskin

We first apply the functional-integral approach to a multiband Hubbard model near the critical pairing temperature and derive a generic effective action that is quartic in the fluctuations of the pairing order parameter. Then we consider time-reversal-symmetric systems with uniform (i.e., at both low momentum and low frequency) pairing fluctuations in a unit cell and derive the corresponding time-dependent Ginzburg-Landau (TDGL) equation. In addition to the conventional intraband contribution that depends on the derivatives of the Bloch bands, we show that the kinetic coefficients of the TDGL equation have a geometric contribution that is controlled by both the quantum-metric tensor of the underlying Bloch states and their band-resolved quantum-metric tensors. Furthermore, we show that thermodynamic properties such as the London penetration depth, Ginzburg-Landau (GL) coherence length, GL parameter, and upper critical magnetic field have an explicit dependence on quantum geometry.

arXiv:2304.03613•Physical review. B./Physical review. B•Jun 12, 2023•23 citations

Quantum-metric contribution to the pair mass in spin-orbit-coupled Fermi superfluids

M. Iskin

As a measure of the quantum distance between Bloch states in the Hilbert space, the quantum metric was introduced to solid-state physics through the real part of the so-called geometric Fubini-Study tensor, the imaginary part of which corresponds to the Berry curvature measuring the emergent gauge field in momentum space. Here, we first derive the Ginzburg-Landau theory near the critical superfluid transition temperature and then identify and analyze the geometric effects on the effective mass tensor of the Cooper pairs. By showing that the quantum-metric contribution accounts for a sizable fraction of the pair mass in a surprisingly large parameter regime throughout the BCS--Bose-Einstein condensate crossover, we not only reveal the physical origin of its governing role in the superfluid density tensor but also hint at its plausible roles in many other observables.

arXiv:1801.09388•Physical Review A•Mar 28, 2018•38 citations

Pair size and quantum geometry in a multiband Hubbard model

M. Iskin

We study the size of two-body bound states and Cooper pairs within a multiband Hubbard model that features time-reversal symmetry and uniform pairing on a generic lattice. Our analysis involves (i) an exact calculation of the localization tensor to determine the size of lowest-lying two-body bound state in vacuum, and (ii) an evaluation of the analogous tensor to estimate the average size of Cooper pairs within the mean-field Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover theory at zero temperature. Beyond the conventional intraband contribution that depends on Bloch bands, we show that pair size also has a geometric contribution governed by the quantum-metric tensor of the Bloch states and their band-resolved quantum-metric tensors. As a concrete example, we investigate the pyrochlore-Hubbard model numerically and demonstrate that, while the pair size diverges in the weakly interacting BCS regime of dispersive bands, it remains finite and relatively small in the flat-band regime, even for infinitesimal interaction, perfectly matching the exact two-body result in the dilute limit.

arXiv:2409.14921•Physical review. B./Physical review. B•Jan 3, 2025•6 citations

Coherence length and quantum geometry in a dilute flat-band superconductor

M. Iskin

To explore the influence of quantum-geometric effects on the Ginzburg-Landau coherence length in a dilute flat-band superconductor, we adopt a BCS-BEC crossover approach to the multiband pyrochlore-Hubbard model near the critical temperature for superconductivity. Our self-consistent formulation for this three-dimensional lattice benchmarks very well against the so-called zero-temperature coherence length, demonstrating the monotonic decay of the coherence length to zero as the interaction strength increases. Additionally, we show that the effective mass of the many-body bound states (i.e., Cooper pairs) is nearly identical to that of the lowest-lying two-body bound states in the dilute flat-band limit.

arXiv:2407.08449•Physical review. B./Physical review. B•Oct 8, 2024•8 citations

Quantum Information Geometry of Multicomponent Superconducting Fluctuation Transport

Zi-Ting Sun, Ying-Ming Xie, Naoto Nagaosa

Quantum geometry underlies many electronic responses, but its transport signatures have so far been established mainly for pure single-particle Bloch states. Whether collective many-body fluctuations possess a measurable quantum geometry remains largely unexplored. Here we show that superconducting fluctuation transport provides a direct probe of quantum information geometry in collective many-body matter. Starting from a multicomponent time-dependent Ginzburg-Landau theory in the Gaussian fluctuation regime, we identify the equilibrium density matrix of fluctuating Cooper pairs as the static pair propagator, which defines a positive mixed-state manifold in momentum space. The geometry of this manifold is directly measurable through paraconductivity: the longitudinal paraconductivity is governed by the quantum Fisher information of superconducting fluctuation modes, while the fluctuational anomalous Hall effect is governed by the mean Uhlmann curvature, the mixed-state counterpart of Berry curvature. This correspondence further yields geometric bounds between these two transport components, with no direct analogue in normal electronic transport. Applied to chiral superconducting fluctuations in quarter-metal systems motivated by rhombohedral multilayer graphene, a symmetry-allowed Lifshitz invariant generates finite mean Uhlmann curvature and logarithmically enhances the anomalous Hall conductivity above the critical temperature. Our results establish collective superconducting fluctuations as an experimentally accessible transport probe of mixed-state quantum information geometry.

cond-mat.mes-hallCross-listedarXiv:2606.15928•Jun 16, 2026
Beyond the frozen-pairing contribution, the superfluid weight of a multiband superconductor contains a correction from the self-consistent relaxation of the pairing amplitudes under a phase twist. We show that this correction has a quantum-geometric origin, not in the space of Bloch states but in the space of collective pairing fields. For an attractive multiband Hubbard model at the mean-field level, the thermodynamic Hessian governing the pairing response coincides with the static Gaussian pair-fluctuation kernel at zero momentum at every temperature below TcT_cTc​. Since gauge invariance relates a phase twist to a finite center-of-mass momentum of the pair field, the superfluid weight is set by the small-momentum curvature of the eigenvalue branch that evolves from the Goldstone mode, and the relaxation correction is a stiffness-weighted quantum metric of the corresponding soft eigenvector on the manifold of pairing configurations. The same construction yields the Ginzburg-Landau gradient coefficient with the identical geometric correction, hence the coherence-length tensor, and, with the low-frequency dynamics of the kernel, the pair-mass tensor. Lieb-lattice calculations verify the equivalence of the thermodynamic, fluctuation-kernel, and soft-mode formulations and the independence of the superfluid weight from the orbital embedding.
We consider multiband BCS superconductors that exhibit time-reversal symmetry and uniform pairing, and analyze their dynamic density and spin structure factors using linear-response theory within the mean-field BCS-BEC crossover framework at zero temperature. Our results for the multi-orbital Hubbard model satisfy the associated f-sum rules in several limits. In particular, in the strong-coupling limit, they coincide with those of a weakly-interacting Bose gas of Cooper pairs, where the low-energy collective Goldstone modes serve as Bogoliubov phonons. We further reveal that the quantum-geometric origin of the low-energy structure factors, along with related observables such as the superfluid-weight tensor and the effective-mass tensor of Cooper pairs, can be traced all the way back to the effective-mass theorem for Bloch bands in this limit. As an illustration, we investigate the pyrochlore-Hubbard model numerically and demonstrate that the Goldstone modes are the only relevant collective degrees of freedom in the flat-band regime.
We first apply the functional-integral approach to a multiband Hubbard model near the critical pairing temperature and derive a generic effective action that is quartic in the fluctuations of the pairing order parameter. Then we consider time-reversal-symmetric systems with uniform (i.e., at both low momentum and low frequency) pairing fluctuations in a unit cell and derive the corresponding time-dependent Ginzburg-Landau (TDGL) equation. In addition to the conventional intraband contribution that depends on the derivatives of the Bloch bands, we show that the kinetic coefficients of the TDGL equation have a geometric contribution that is controlled by both the quantum-metric tensor of the underlying Bloch states and their band-resolved quantum-metric tensors. Furthermore, we show that thermodynamic properties such as the London penetration depth, Ginzburg-Landau (GL) coherence length, GL parameter, and upper critical magnetic field have an explicit dependence on quantum geometry.
As a measure of the quantum distance between Bloch states in the Hilbert space, the quantum metric was introduced to solid-state physics through the real part of the so-called geometric Fubini-Study tensor, the imaginary part of which corresponds to the Berry curvature measuring the emergent gauge field in momentum space. Here, we first derive the Ginzburg-Landau theory near the critical superfluid transition temperature and then identify and analyze the geometric effects on the effective mass tensor of the Cooper pairs. By showing that the quantum-metric contribution accounts for a sizable fraction of the pair mass in a surprisingly large parameter regime throughout the BCS--Bose-Einstein condensate crossover, we not only reveal the physical origin of its governing role in the superfluid density tensor but also hint at its plausible roles in many other observables.
We study the size of two-body bound states and Cooper pairs within a multiband Hubbard model that features time-reversal symmetry and uniform pairing on a generic lattice. Our analysis involves (i) an exact calculation of the localization tensor to determine the size of lowest-lying two-body bound state in vacuum, and (ii) an evaluation of the analogous tensor to estimate the average size of Cooper pairs within the mean-field Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover theory at zero temperature. Beyond the conventional intraband contribution that depends on Bloch bands, we show that pair size also has a geometric contribution governed by the quantum-metric tensor of the Bloch states and their band-resolved quantum-metric tensors. As a concrete example, we investigate the pyrochlore-Hubbard model numerically and demonstrate that, while the pair size diverges in the weakly interacting BCS regime of dispersive bands, it remains finite and relatively small in the flat-band regime, even for infinitesimal interaction, perfectly matching the exact two-body result in the dilute limit.
To explore the influence of quantum-geometric effects on the Ginzburg-Landau coherence length in a dilute flat-band superconductor, we adopt a BCS-BEC crossover approach to the multiband pyrochlore-Hubbard model near the critical temperature for superconductivity. Our self-consistent formulation for this three-dimensional lattice benchmarks very well against the so-called zero-temperature coherence length, demonstrating the monotonic decay of the coherence length to zero as the interaction strength increases. Additionally, we show that the effective mass of the many-body bound states (i.e., Cooper pairs) is nearly identical to that of the lowest-lying two-body bound states in the dilute flat-band limit.
Quantum geometry underlies many electronic responses, but its transport signatures have so far been established mainly for pure single-particle Bloch states. Whether collective many-body fluctuations possess a measurable quantum geometry remains largely unexplored. Here we show that superconducting fluctuation transport provides a direct probe of quantum information geometry in collective many-body matter. Starting from a multicomponent time-dependent Ginzburg-Landau theory in the Gaussian fluctuation regime, we identify the equilibrium density matrix of fluctuating Cooper pairs as the static pair propagator, which defines a positive mixed-state manifold in momentum space. The geometry of this manifold is directly measurable through paraconductivity: the longitudinal paraconductivity is governed by the quantum Fisher information of superconducting fluctuation modes, while the fluctuational anomalous Hall effect is governed by the mean Uhlmann curvature, the mixed-state counterpart of Berry curvature. This correspondence further yields geometric bounds between these two transport components, with no direct analogue in normal electronic transport. Applied to chiral superconducting fluctuations in quarter-metal systems motivated by rhombohedral multilayer graphene, a symmetry-allowed Lifshitz invariant generates finite mean Uhlmann curvature and logarithmically enhances the anomalous Hall conductivity above the critical temperature. Our results establish collective superconducting fluctuations as an experimentally accessible transport probe of mixed-state quantum information geometry.