Geometric Superfluid Weight in Quasicrystals

Abstract

We study the geometric contribution to the superfluidity in quasicrystals in which the conventional momentum-space quantum geometric tensor cannot be defined due to the lack of translational invariance. Based on the correspondence between the momentum and magnetic flux, we introduce the flux-space quantum metric in finite-size closed systems and reveal its contribution to the superfluid weight in quasicrystalline superconductors. As a toy model, we study the attractive Hubbard model on the Fibonacci quasiperiodic stub lattices that host flat energy spectra even in the presence of quasiperiodic hoppings. In the weak-coupling limit, we establish the relation between superfluid weight and the flux-space quantum metric in quasicrystal superconductors with flat energy spectra. Moreover, by analyzing the spread of Wannier functions, we propose a general fluctuation mechanism that explains how quasiperiodicity modulates the integrated flux-space quantum metric. Our theory provides a general way to examine the effect of the quantum geometry in systems lacking translational symmetry.

Figures (3)

• Figure 1: (a) Illustration of magnetic fluxes threading a 2D torus (reducing to a ring in the 1D case), used to formulate the flux-space quantum metric. (b) Structure of the 2D Amman-Beenker (AB)-QC composed of two sublattices, featuring two hopping amplitudes, $(1+\delta)t$ and $(1-\delta)t$. (c) Schematic of the 1D Stub-QC, which consists of three sites in each unit cell and becomes quasiperiodic by modulating the vertical hopping amplitudes according to the Fibonacci sequence.
• Figure 2: (a) $D_{s,\alpha\alpha}$ of the AB-QC as a function of $U/t$ for different $\delta$. Solid (dashed) lines represent the $xx$ ($yy$) component. (b) Comparison of $D_{ps,\alpha\alpha}$ and $D_{s,\alpha\alpha}$ in the small-$U$ regime. Dotted lines corresponds to $D_{ps,xx}$ and $D_{ps,yy}$, while the thick solid and dashed lines denote the results in (a). $\kappa_{\alpha\alpha}$ and $\kappa_{\alpha\alpha}^{qm}$ for the 2D AB-QC (c), 2D Lieb-QC (d), and 1D Stub-QC (e), plotted as functions of $\delta$ at $U=0.01$.
• Figure 3: (a) Top panel: Spread function of maximally localized Wannier functions for the 1D Stub-QC at $\delta=0.4$. Bottom panel: Spatial profiles of the Wannier functions for the 1D Stub-QC. Dashed vertical lines indicate the positions of the Wannier centers. The color represents the correspondence between the two panels. (b) $Q^{\rm FS}_{{\rm f},xx}$, $\langle\Omega^{xx}_{\rm f}\rangle_V$, and $\mathcal{Q}^{\rm RS}_{{\rm f},xx}$ of the flat-energy sector as functions of $\delta$ for the 1D Stub-QC. (c) $g^{\rm FS}_{{\rm f},xx,i}$ of the 1D Stub-QC at several different values of $\delta$, with the inset showing the energy gap between the flat-energy sector and the other sectors as a function of $\delta$.