Flat-band Fulde-Ferrell-Larkin-Ovchinnikov State from Quantum Geometric Discrepancy

TL;DR

This work identifies quantum geometric discrepancy (QGD) as a generic mechanism to stabilize FFLO states in flat-band systems, tying finite-momentum pairing to the anomalous quantum distance (AQD) between paired Bloch states. By projecting interactions onto a narrow flat band and analyzing pairing susceptibility, the authors derive a simple criterion Tc(q)/T0 = 1 - \bar{d}^2_{\,\mathcal{A}}(q) that selects the optimal pairing momentum $\boldsymbol{Q}$ from the AQD landscape. Using a tunable two-orbital $\zeta$-lattice model, they demonstrate a QGD-driven, first-order transition from BCS to FFLO with a Lifshitz point and show that finite bandwidth and inversion-breaking terms modify but do not erase the geometric mechanism. Self-consistent mean-field calculations corroborate the analytic predictions, showing phase diagrams and free-energy landscapes consistent with the AQD-based criterion. The work further explores robust extensions to triangular lattices, helical-like states with diode effects, and inter-flavor FFLO in pseudo-Landau levels, highlighting the broad applicability of geometry-driven flat-band superconductivity.

Abstract

We propose a new scheme for realizing Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) Cooper pairing states within flat bands, in contrast to the conventional paradigm such as the Zeeman effect. Central to our scheme is the concept of ``quantum geometric discrepancy'' (QGD) that measures differences in the quantum geometry of paired electrons and drives the flat-band FFLO instability. Remarkably, we find that this instability is directly related to a quantum geometric quantity known as ``anomalous quantum distance'', which formally captures QGD. To model both QGD and the anomalous quantum distance, we examine a flat-band electronic Hamiltonian with tunable spin-dependent quantum metrics. Utilizing the band-projection method, we analyze the QGD-induced FFLO instability from pairing susceptibility. Furthermore, we perform mean-field numerical simulations to obtain the phase diagram of the BCS-FFLO transition, which aligns well with our analytical results. Our work demonstrates that QGD offers a general and distinctive mechanism for stabilizing the flat-band FFLO phase.

Figures (11)

• Figure 1: Schematic illustrations of (a) conventional FFLO from the Zeeman effect, and (b) flat-band FFLO from QGD. In (a), an external $B$ field induces spin-split Fermi surfaces, leading to the formation of Cooper pairs with finite momentum $\bm{Q}$, where $|\bm{Q}| \propto B/v_F$. In (b), the finite momentum $\bm{Q}$ is stabilized when the quantum metrics of the paired electrons are different ($\mathcal{G}_\uparrow \neq \mathcal{G}_\downarrow$) within flat bands.
• Figure 2: (a) The $\zeta$-lattice with spin-dependent long-range hoppings. $\zeta_{\sigma}$ denotes the quantum metric of spin-$\sigma$ sector. (b) Energy dispersions of the $\zeta$-lattice model in flat-band limit ($W=0$) and with finite bandwidth ($W\neq 0$). Band gap $\Delta_{\mathrm{gap}}$ is much lager than bandwidth $W$.
• Figure 3: (a) $T_{c\bm{Q}}(\eta)$ (dashed blue), $T_{c0}(\eta )$ (solid red), and the corresponding $Q(\eta )$ (solid green) calculated from Eq. \ref{['critem']} as a funtion of $\eta$. The orange dot labels the Lifshitz point. For comparison, the dashed green line shows $Q(\eta)$ calculated from the self-consistent mean-field (MF) theory. (b) $T_{c\bm{q}}(\eta )/T_0$ from Eq. \ref{['critem']} for $\boldsymbol{q} = q(\hat{x}+\hat{y}),\, q\in \left [ 0,\pi \right ]$, at $\eta=0$ and $\eta=0.25$, respectively. In our calculations, we set $\zeta_{\downarrow}=3$ and $\zeta_{\uparrow}=\zeta_{\downarrow}(1+\eta)/(1-\eta)$.
• Figure 4: The BCS-FFLO phase diagram, without (a) and with (b) band dispersion. The dashed black line labels $T_{c0}(\eta)$, while the solid red line labels $T_{c\boldsymbol{Q}}(\eta)$. The BCS and FFLO phases are colored in light blue and yellow, respectively. Parameters used here: $U = 0.4t$, $\nu = 0.5$, $\zeta_{\downarrow} = 3$, $W/U=0$ in (a) and $W/U=0.12$ in (b). (c) (d) (e): The free energy density landscapes of three highlighted dots in (a) at $T = 0.1T_0$ and different $\eta$ ($\eta = 0.10$, $0.23$, and $0.29$, respectively).
• Figure 5: (a) 1D SSH-like model with nearest-neighbor hopping $u$ and $v$, second nearest-neighbor hopping $t_a$ and $t_b$, and third nearest-neighbor hopping $w$. (b) The band structure of the model when $\phi = 0$ (black solid line) and $\phi = 0.1\pi$ (red dashed line). (c) The distributions of quantum metric $g(k)$ at $\phi = 0$, $0.01\pi$, and $0.05\pi$ (labeled in black, red, and blue lines, respectively). (d) The critical temperatures of the most favored FF states and the BCS state are plotted as functions of $\phi$ in red and blue lines, respectively. The results from self-consistent mean-field (MF) calculations are denoted by the solid lines, and the analytical results from Eq. \ref{['critem']} (PS) are denoted by the dashed lines. (e) The favored finite-momentum $Q$ with varying $\phi$. Parameters used here: $v = |w| = t_a = t_b = 0.4t, u = -t, U = 0.1t, \nu=0.5.$