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Superfluidity in topologically nontrivial flat bands

Sebastiano Peotta, Päivi Törmä

Nature Communications · 2015 · CC BY

doi:10.1038/ncomms9944 (opens in a new tab)

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Abstract

Topological invariants built from the periodic Bloch functions characterize new phases of matter, such as topological insulators and topological superconductors. The most important topological invariant is the Chern number that explains the quantized conductance of the quantum Hall effect. Here we provide a general result for the superfluid weight Ds of a multiband superconductor that is applicable to topologically nontrivial bands with nonzero Chern number C. We find that the integral over the Brillouin-zone of the quantum metric, an invariant calculated from the Bloch functions, gives the superfluid weight in a flat band, with the bound Ds⩾|C|. Thus, even a flat band can carry finite superfluid current, provided the Chern number is nonzero. As an example, we provide Ds for the time-reversal invariant attractive Harper–Hubbard model that can be experimentally tested in ultracold gases. In general, our results establish that a topologically nontrivial flat band is a promising concept for increasing the critical temperature of the superconducting transition.

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Figures2

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Figure 1 Superfluid transport and Wannier functions.
Full legend
(a) Localized Wannier functions are obtained from the Bloch functions of a set of bands, called a composite band2425. To have superfluidity in a flat band, the pairing takes place only in a subset of the bands within the composite band, for example, in a single flat band. While the Wannier functions built from the Bloch functions of the band where pairing takes place are delocalized due to the nonzero Chern number25 C≠0 (b) the Wannier functions of the composite band are exponentially localized (c). We show that the superfluid weight Ds in a flat band is given by the Brillouin-zone average of the quantum metric2627, which is the real part of an invariant. (d) The imaginary part of gives the Chern number C. The positivite semidefiniteness of leads to the bound Ds⩾|C| on the superfluid weight.
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Figure 2 Composite bands and Wannier functions.
Full legend
(a) We consider lattices in three and two dimensions (2D in the figure). Periodic boundary conditions (PBC) are used. The lattice contains Nc unit cells and each unit cell contains Norb sites/orbitals. The vectors ai (i=1, 2, 3) are the fundamental vectors of the Bravais lattice14 while the vectors bα (α=1,…,Norb) are the positions of the centres of the orbitals (Wannier functions) within a unit cell. A single orbital is specified by a triplet (or pair) of integers i=(ix, iy, iz) and by the sublattice index α and is centred at the position vector riα=ixa1+iya2+iza3+bα. (b) The band structure is obtained by solving the Schrödinger equation with periodic potential V(r)=V(r+ai). It consists of the band dispersions ɛnk, with n the band index and ħk the lattice quasimomentum, and the periodic Bloch functions gnk(r)=gnk(r+ai) (Bloch functions for brevity) obtained from the Bloch plane waves. We consider a composite band, that is, a subset of contiguous bands well separated in energy from other bands. The Chern numbers Cn for individual bands calculated from the Bloch functions may be nonzero (such as the flat band n=2 in the figure), but their sum equals zero. The Chern number refers to spin-resolved bands since the spin along a quantization axis (conventionally the z axis) is conserved. (c) The Wannier functions, defined as the Fourier transform of the Bloch functions, allow us to derive a tight-binding Hamiltonian that reproduces exactly a single band or a composite band of the original continuum Hamiltonian (see Supplementary Note 2). Since individual bands may be topologically nontrivial with nonzero Chern numbers, their Wannier functions wn(r) are not exponentially localized25, and the Peierls substitution in the effective Hamiltonian is therefore not justified1516. (d) By constructing Wannier functions as linear superpositions of Bloch waves of all bands in the composite band, exponentially localized Wannier functions wα(r) can be created. The mixing of the different bands is provided by the unitary matrix Uα,n(k). This justifies the Peierls substitution for a composite band.
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