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
Tc. 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.
The coupling between a quantum particle's intrinsic angular momentum and its center-of-mass motion gives rise to the so-called helicity states that are characterized by the projection of the spin onto the direction of momentum. In this paper, by unfolding the superfluid-density tensor into its intrahelicity and interhelicity components, we reveal that the latter contribution is directly linked with the total quantum metric of the helicity bands. We consider both Rashba and Weyl spin-orbit couplings across the BCS-BEC crossover and show that the geometrical interhelicity contribution is responsible for up to a quarter of the total superfluid density. We believe this is one of those elusive effects that may be measured within the highly tunable realm of cold Fermi gases.
We investigate the topological phase transition of Kitaev spin liquid in an external magnetic field by calculating the Berry curvature and the Fubini-Study metric. Employing Jordan-Wigner transformation and effective perturbative theory to transform the Hamiltonian into fermionic quadratic form, the Berry curvature is calculated by choosing the effective magnetic field as the parameter, and we find that the
xy component of the Berry curvature has the same behavior around the critical lines with the phase diagram. Especially, we relate the Berry curvature with the derivative of effective magnetic susceptibility which can be regarded as the signature of topological phase transition. Moreover, we related the second nonlinear susceptibility with the interband Berry connection. Then we analytically calculate the generalized Berry curvature in the mixed state called the mean Uhlmann curvature, which can be related with the spectral function. The curves where the Uhlmann curvature changes temperature with different coupling constants reveal that it will have a peak at
T=0.2K when adjusting
Jx from 0.2 to 0.3 with
Jy=0.25, in which the first-order derivative is discontinuous, revealing the characteristic of thermodynamic phase transition from the
B phase quantum spin liquid state to the magnetic ordered state. Last, we analytically calculate the quantum geometric tensor in the effective magnetic-field space whose imaginary part is the Berry curvature and the real part is the Fubini-Study metric, and we find that the
zz component of the Fubini-Study metric and the phase diagram are highly correlated, which will peak at the cross point of three phases, and then the Fubini-Study metric is extended to the finite temperature, called the Bures metric, which has the same behavior of the Uhlmann curvature at
T=0.2K in
B phase.
We consider a weakly interacting Bose-Einstein condensate that is loaded into an optical lattice with a two-point basis and described by a two-band Bose-Hubbard model with generic one-body and two-body terms. By first projecting the system onto the lower Bloch band and then applying the Bogoliubov approximation to the resultant Hamiltonian, we show that the inverse effective-mass tensor of the superfluid carriers in the Bogoliubov spectrum has two physically distinct contributions. In addition to the usual inverse band-mass tensor that is originating from the intraband processes within the lower Bloch band, there is also a quantum-geometric contribution that is induced by the two-body interactions through the interband processes. We also discuss the conditions under which the latter contribution is expressed in terms of the quantum-metric tensor of the Bloch states, i.e., the natural Fubini-Study metric on the Bloch sphere.
Geometric Quantum Mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. At the heart of that is a quantity called Quantum Geometric Tensor (or Fubini-Study metric), which is a complex tensor with the real part serving as the Riemannian metric that measures the `quantum distance', and the imaginary part being the Berry curvature. Following a physical introduction of the basic formalism, we illustrate its physical significance in both the adiabatic and non-adiabatic systems.
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.