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arXiv:1810.09870 · Physical review. B./Physical review. B

Quantum metric and effective mass of a two-body bound state in a flat band

Päivi Törmä, Long Liang, Sebastiano Peotta

Published Dec 26, 2018• 146 citations
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Abstract

We consider two-body bound states in a flat band of a multiband system. The existence of pair dispersion predicts the possibility of breaking the degeneracy of the band and creating order, such as superconductivity. Within a separable interaction potential approximation, we find that the finiteness of the effective mass of a bound pair is determined by a band-structure invariant, which in the uniform case becomes the quantum metric. The results offer a simple foundation to understand and predict flat-band superconductivity. We propose an experiment to test the interaction-induced pair motion.

DOI
10.1103/physrevb.98.220511

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We consider two-body bound states in a flat band of a multiband system. The existence of pair dispersion predicts the possibility of breaking the degeneracy of the band and creating order, such as superconductivity. Within a separable interaction potential approximation, we find that the finiteness of the effective mass of a bound pair is determined by a band-structure invariant, which in the uniform case becomes the quantum metric. The results offer a simple foundation to understand and predict flat-band superconductivity. We propose an experiment to test the interaction-induced pair motion.
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.
In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.
We consider the two-body problem in a periodic potential, and study the bound-state dispersion of a spin-$\ensuremath{\uparrow}$ fermion that is interacting with a spin-$\ensuremath{\downarrow}$ fermion through a short-range attractive interaction. Based on a variational approach, we obtain the exact solution of the dispersion in the form of a set of self-consistency equations, and apply it to tight-binding Hamiltonians with on-site interactions. We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversal symmetry, and show that the lowest-energy bound states disperse quadratically with momentum, whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlying Bloch states. In particular, we apply our theory to the Mielke checkerboard lattice, and study the special role played by the interband processes in producing a finite effective mass for the bound states in a nonisolated flat band.
By considering an on-site attraction between a spin-$\ensuremath{\uparrow}$ and a spin-$\ensuremath{\downarrow}$ fermion in a multiband tight-binding lattice, here we study the two-body spectrum and derive an exact relation between the inverse of the effective-mass tensor of the lowest bound states and the quantum-metric tensor of the underlying Bloch states. In addition to the intraband (or the so-called conventional) contribution that depends only on the single-particle spectrum and the interband (or the so-called geometric) contribution that is controlled by the quantum metric, our generalized relation has an additional interband contribution that depends on the so-called band-resolved quantum metric. All of our analytical expressions are applicable to those multiband lattices that simultaneously exhibit time-reversal symmetry and fulfill the condition on spatially uniform pairing. As a nontrivial illustration we analyze the two-body problem in a kagome lattice with nearest-neighbor hoppings, and we show that the exact relation provides a perfect benchmark.
Superconductivity in flat-band systems, governed by quantum metric of Bloch states rather than the BCS framework, exhibits unique phenomena due to the vanishing electron group velocity. Here, we propose the vortex states and vortex size as direct probes to explore the quantum geometry effects in flat-band superconductors. We show that flat-band vortex bound states are sharply localized near the vortex core, and the energy gap between the lowest two bound states is on the order of the bulk superconducting gap. Both the spatial spread and energy scales of bound states are controlled by the flat-band's quantum metric length. Moreover, the vortex size at zero temperature, set by the quantum metric length, is atomic in scale and independent of interaction strength. Near TcT_cTc​, the vortex size corresponds to the Ginzburg-Landau coherence length, diverges as ξ∼Tc/(Tc−T)ξ0ξ\sim \sqrt{T_c/(T_c-T)}ξ_0ξ∼Tc​/(Tc​−T)​ξ0​, where ξ0ξ_0ξ0​ depends linearly on the quantum metric length. Thus, the quantum metric serves as the lower bound for vortex state spread and vortex size. We also introduce perturbations to make the flat band dispersive, and distinguish flat-band vortices from BCS-like vortices. Our results establish vortices as universal probes of quantum geometry in flat-band superconductors.
The formation of bound bosonic pairs of fermions, followed by their Bose-Einstein (quasi) condensation (BEC), is a foundational mechanism of superconductivity. At low filling, flat-band superconductivity is well captured by this mechanism provided the flat band is separated from the occupied lower band by an energy gap. However, particularly high TcT_cTc​ values are anticipated when the non-interacting flat and lower bands touch---as in the prototypical attractive Lieb-lattice model studied here---invalidating the conventional picture: while interactions might protect the bound state by opening a gap, no small parameter guarantees the separation of the bound-state energy from this gap or occupied-band excitations, leaving the pair's fate uncertain. Based on a controlled-precision numerical protocol---which we demonstrate to be essential in this fundamentally non-perturbative problem---we show that the BEC mechanism, underpinned by an interaction-induced gap, is generically robust and remarkably efficient: fermions doped into the flat band form bound pairs within this gap with an anomalously light effective mass, enabling an exceptionally high TcT_cTc​.