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arXiv:2109.06000 · Physical Review A

Effective-mass tensor of the two-body bound states and the quantum-metric tensor of the underlying Bloch states in multiband lattices

M. Iskin

Published Feb 9, 2022• 18 citations
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Abstract

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.

DOI
10.1103/physreva.105.023312

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By considering an on-site attraction between a spin-\ensuremath↑\ensuremath{\uparrow}\ensuremath↑ and a spin-\ensuremath↓\ensuremath{\downarrow}\ensuremath↓ 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.
Assuming an onsite attraction between a spin-↑\uparrow↑ and a spin-↓\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. As a nontrivial illustration we analyze the two-body problem in a Kagome lattice with nearest-neighbor hoppings, and show that the exact relation provides a perfect benchmark.
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.
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.
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.
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.
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.