Controlled pairing symmetries in a Fermi-Hubbard ladder with band flattening

Abstract

Band flattening has been identified as key ingredient to correlation phenomena in Moiré materials and beyond. Here, we examine strongly repulsive fermions on a ladder -- a minimal platform for unconventional $d$-wave pairing -- and show that flattening of the lower band through an additional diagonal hopping term produces non-Fermi liquid behavior, evidenced by the violation of Luttinger's theorem, as well as axial $d$-wave pairing correlations. Alternatively, plaquette ring exchange can also generate pairing, albeit with a distinct diagonal $d$-wave pairing symmetry. Hence, our finding showcases a competition of different unconventional pairing channels, and demonstrates via a simple model how band geometry can induce fermionic pairing. This offers broadly relevant insights for correlated flat-band systems, ranging from ultracold atoms to strongly interacting electrons in solids.

Figures (5)

• Figure 1: (a) Schematic representation of the extended $t-J-K$ model [cf. Eq. \ref{['eq:model']}]. Electrons are allowed to hop between all neighboring sites, including horizontal, vertical, and diagonals. The exchange interaction is tied to the hoppings via $J=4t^2/U$. The ring-exchange interaction acting on a single plaquette is shown. (b-d): Single-particle bands for (b) $t_d=0$, (c) $t_d=-0.4$, and (d) $t_d=-1$ (in units of $t=t_h=t_v$).
• Figure 2: (a,b,d,e) Momentum distribution $\langle \hat{n}_{\uparrow}(\bm{q}) \rangle$ for $L_x=30$, and fixed (a-c) $t_d=-0.1$ and (d-f) $t_d=-0.3$. (a) Shows FL behavior at $K=0.5$, whereas panel (b) NFL behavior at $K=1.4$. Both (d) at $K=0.65$ and (e) at $K=1.2$ showcase distinct NFL behavior. In panels (c,f) the absolute value of the derivative $\partial \langle \hat{n}_{\uparrow}(q_x,0) \rangle/\partial q_x$ is plotted as a function of $q_x$ and $K$ for $q_y=0$. Dotted lines in (c,f) mark the $K$ values which are shown in panels (a,b,d,e). The Fermi wave vector $k_F$ is represented by horizontal red dashed lines.
• Figure 3: Spectral function analysis for $t_d=-0.25$, $q_y=0$, and $L_x=36$. Two representative values are shown in (a) $K=0.8$ and (b) $K=1.2$. (c) Gap $\delta$ and (d) corresponding momentum $k_L$, schematically shown in the (a)-(b), as a function of $K$. Vertical lines correspond to FL-NFL transition points.
• Figure 4: Pair correlations for the two $d$-wave channels for $t_d=-0.25$, $L_x=30$, and three distinct values of $K$, representative of the three phases. (a) Diagonal (alias $d_{xy}$) and (b) axial (alias $d_{x^2-y^2}$) $d$-wave pair correlations, as schematically illustrated in the top panels. Dashed curves represent the best numerical fit between power-law and exponential. Correlations are renormalized to the first point to better illustrate the decay laws.
• Figure 5: Finite-size phase diagram depicting the three main phases observed: a Fermi-liquid phase (FL) and two non-Fermi-liquid (NFL) phases with distinct $d$-wave pairing symmetries. The color background shows the eigenvalue gap in the dominant pairing channel, i.e. $d_{x^2-y^2}$ (green) or $d_{xy}$ (red). As indicated by the symbols, the phase boundaries can also be obtained from inspection of the momentum distribution (MD) and the dynamical spin structure factor (DSSF), where critical values $K_{c1}$ and $K_{c2}$ mark the transitions between the different phases. The grid of $K$ values in the calculation yields horizontal error bars.