Superconductivity-induced spectral weight transfer due to quantum geometry

Letter

Superconductivity-induced spectral weight transfer due to quantum geometry

Junyeong Ahn and Naoto Nagaosa

Phys. Rev. B 104, L100501 – Published 7 September 2021

Abstract

Optical spectral weight transfer associated with the onset of superconductivity at high-energy scales compared with the superconducting gap has been observed in several systems such as high- $T_{c}$ cuprates. While there are still debates on the origin of this phenomenon, a consensus is that it is due to strong correlation effects beyond the BCS theory. Here, we show that there is another route to a nonzero spectral weight transfer based on the quantum geometry of the conduction band in multiband systems. We discuss applying this idea to the cuprates and twisted multilayer graphene.

Received 27 June 2021
Revised 18 August 2021
Accepted 23 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L100501

Physics Subject Headings (PhySH)

Superfluid density

Superconductors

BCS theory Bloch wave theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Junyeong Ahn ^1,* and Naoto Nagaosa^2,3,†

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
²RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
³Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan

^*junyeongahn@fas.harvard.edu
^†nagaosa@riken.jp

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Vol. 104, Iss. 10 — 1 September 2021

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Images

Figure 1
A simplified picture of the optical spectrum of metals in the normal and superconducting states. (a) Normal state. $E_{g}$ represents the energy of the excitation to the Fermi surface. There may be additional excitation channels to the Fermi surface from lower-energy levels, or from the Fermi surface to higher-energy levels, both of which we do not show here for simplicity. $p$ and $d$ represent two different electronic bands. (b) Superconducting state. $2 Δ$ indicates the superconducting gap. Both (1) intraband and (2) interband spectral weights are lost by the opening of the superconducting gap, while (3) new electron-hole mixing excitation channels open up.
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Figure 2
Optical conductivity of a Dirac fermion in normal and superconducting states in two dimensions. (a) Optical conductivity. Only regular (nonsuperfluid) parts are shown. The red dashed line and solid blue line correspond to the normal and superconducting states, respectively. The Drude conductivity is shown with a finite inverse scattering rate of $Γ = 0.01 E_{F}$ for its visual presentation, while we take the clean limit $Γ \to 0$ to calculate the interband conductivities. We consider an $s$ -wave gap with $Δ = 0.1 E_{F}$ in the superconducting state. The inset shows the spectrum of the Dirac fermion in the normal state. (b) Loss of the interband optical conductivity by the superconducting transition. All conductivities are calculated at zero temperature.
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Figure 3
Superconductivity-induced spectral weight transfer in a model of the $Cu O_{2}$ plane in cuprate superconductors. (a)–(d) Tight-binding model. (a) Location of $p_{x, y}$ and $d_{x^{2} - y^{2}}$ orbitals on the Lieb lattice. The unit cell is shown as a shaded rectangle. We consider the tight-binding model on this lattice with hopping up to the next-nearest neighbor Eq. (8). We take on-site energies $ε_{d} = 0$ eV and $ε_{p} = - 2.416$ eV and hopping amplitudes $t_{p p} = 751$ meV and $t_{p d} = 1.257$ eV. (b) Hole doping $x = 1 - n$ as a function of Fermi level $E_{F}$ . $n$ is the number of conduction electron per unit cell. The red and blue curve corresponds to the normal and superconducting states, where $Δ_{d} = 20$ meV in the latter. We take $E_{F} = 1.695$ eV in (d)–(g). This choice corresponds to hole doping $x = 0.1$ per unit cell from the half-filled conduction band. The inset shows $δ E_{F} = E_{F}^{SC} - E_{F}^{normal}$ needed to preserve the average charge, which changes sign at the van Hove singularity (VHS). (c) Band structure. (d) Fermi surface of a hole pocket centered at $(π, π)$ . (e) Interband optical conductivity in the normal state. Only the longitudinal part $σ^{x x} = σ^{y y}$ is shown. The contribution from the intraband Drude conductivity is not included here. (f) Superconductivity-induced reduction of the interband optical conductivity. We take $Δ_{d} = 20$ meV. (g) Spectral weight transfer corresponding to (e). For numerical calculations of the optical conductivity, we take finite broadening $ℏ Γ = 10$ meV for the Lorentzian.
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