The interplay of phase fluctuations and nodal quasiparticles: ubiquitous Fermi arcs in two-dimensional d-wave superconductors

TL;DR

The work shows that pseudogap formation and ubiquitous Fermi arcs in 2D nodal superconductors can arise from thermal (static) phase fluctuations, with two emergent length scales, $\xi(T)$ and $\xi_{\text{BCS}}(\mathbf{k})$, governing the arc evolution. A minimal phenomenological framework connects phase-fluctuation scattering to a self-energy $\Sigma(\mathbf{k},\omega)$ and, through the Green's function poles, to a temperature-driven transition from a gapped pseudogap to a metallic state, producing anisotropic Fermi arcs. The authors validate the mechanism with sign-problem-free DQMC on a Hubbard-like model with fluctuating $d$-wave pairing, observing a linear relation $\Gamma_{\text{pf}} \propto \xi^{-1}$ consistent with theory and demonstrating robustness against moderate interactions. The results suggest a universal mechanism for phase-fluctuation-driven Fermi arcs applicable to cuprates, FeSe thin films, twisted bilayer graphene, and cold-atom setups, beyond the cuprate family.

Abstract

We propose that the pseudogap and Fermi arcs can universally emerge due to thermal (static) phase fluctuations in the normal state of 2D nodal superconductors. By considering a minimal phenomenological model with spatially fluctuating superconducting pairings, we theoretically investigate the role of superconducting phase fluctuations in generic 2D superconductors with disorder-average technique. It is shown for nodal d-wave superconductors that phase fluctuations mediate the scattering of d-wave quasiparticles, smearing out the nodal quasiparticle gap and further leading to pseudogap and Fermi arcs. Moreover, the evolution of Fermi arcs is quantitatively described by two emergent characteristic length scales of the system: one is the finite superconducting correlation length $ξ(T)$, and another the nodal BCS coherence length $ξ_\text{BCS}(k)$. To support our theoretical findings, we numerically report the observation of Fermi arcs in a Hubbard-like model, proposed originally by X. Y. Xu and T. Grover in Phys. Rev. Lett. $\textbf{126}$, 217002 (2021), with sign-problem-free determinant quantum Monte Carlo (DQMC) calculations. As far as we noticed, it is the first time in a correlated model that phase-fluctuating Fermi arcs are identified with unbiased simulations. The numerical results for the scattering rate $Γ_\text{pf}$ of Cooper pairs exhibit excellent agreements with our theoretical predictions, where $Γ_\text{pf}$ is expected to scale linearly with the inverse superconducting correlation length $ξ(T)^{-1}$. This convergence of theory and numerics thereby strongly validates the universal connection between phase fluctuations and Fermi arcs in 2D nodal superconductors.

Figures (7)

• Figure 1: (a)-(c) Theoretical illustration of pseudogap and Fermi arcs. (a) Evolution of spectral weights $A(k_F,\omega)$ as a competition of $\xi$ and $\xi_\text{BCS}(k)$. (b) Schematic plot of the Fermi arcs. (c) Evolution of Fermi arcs with increasing temperature. (d)-(g) Observation of Fermi arcs in the Hubbard-like model with DQMC on a $L=20$ lattice. (d) Green's function $\beta G(\bm{k},\beta/2)$ as an estimation of $A(\bm{k},\omega=0)$ at $T/T_c=1.25$. The node/antinode is labeled as $K$/$X$. (e) $d$-wave gap function $\abs{\Delta_{\bm{k}}}$ extracted directly from $A(\bm{k},\omega)$ at $T/T_c=0.75$. (f) $A(\bm{k},\omega)$ along the momentum path perpendicular to Fermi surface. The momentum paths intersect with the Fermi surface at $K_1$ and $K_2$ respectively, as marked in (d) where $K_1$ belongs to the Fermi arc while $K_2$ is not. (g) $A(\bm{k},\omega)$ along the node-antinode line $X$-$K$-$X$ with varying $T$. The red curves outline the shape of Fermi arcs.
• Figure 2: (a)(c) Static $d$-wave pairing correlation $P_d$ and $\alpha^2$ for varying inverse temperature $\beta$ and system size $L$. We have fixed the critical exponent $\eta(T_c)=1/4$ for the BKT transition. (b)(d) Data collapse of $P_d$ and $\alpha^2$. (e) Antinodal pseudogap $A_{(0,\pi)}(\omega)$ extracted by combining DQMC and SAC on a $L=20$ lattice. (f) Self-energy evaluated from the spectrum in (e). Note that the y-axis is log-scaled for better visualization. (g)(h) Pair-scattering rate $\Gamma_\text{pf}$ extracted by (g) fitting $\Sigma(\omega)$ to obtain $\Gamma_{\text{pf},\Sigma}$, and (h) identifying the half-width of $A(|\omega|\sim\abs{\Delta_k})$ as $\Gamma_{\text{pf},A}$. The dashed lines are linear fittings of $\Gamma_\text{pf}$ with respect to $\xi^{-1}$. And the momenta are chosen near the antinodes as $K_n=K_0 + n\Delta k$ with $K_0=(0,\pi)$ and $\Delta k=(2\pi/L,-2\pi/L)$.
• Figure S1: Feynman diagram for the second-order self-energy correction. The arrowed solid lines denote the bare propagators of band electrons, while the dashed red line represents the disorder average over the fluctuating pairing correlations.
• Figure S2: Benchmark of self-energies in Eq. \ref{['eq:self_energy_s4']} and Eq. \ref{['eq:self_energy_s6']} by varying both the correlation length $\xi$ and the Fermi momentum $\bm{k}_F[\theta]$. $\theta$ denotes the angle to the $d$-wave node, as illustrated in the mini charts of the rightmost figures. The exact self-energies, Eq. \ref{['eq:self_energy_s4']}, are plotted in solid lines while the approximated ones, Eq. \ref{['eq:self_energy_s6']}, are in dashed lines. It is found that Eq. \ref{['eq:self_energy_s6']} produces similar real and imaginary components as those of Eq. \ref{['eq:self_energy_s4']}, especially when $k_F\xi$ grows up. We have also tested Eq. \ref{['eq:self_energy_s6']} for momenta away from the Fermi surface, which only causes a frequency shift as compared to the results above, and hence we do not show them separately.
• Figure S3: Single-particle spectral function $A(\bm{k}_F,\omega)$ at the Fermi momentum for varying $\gamma/\abs{\Delta_{\bm{k}}}$. We outline the spectral peaks at $\omega=\Delta_\text{pg}=\sqrt{\abs{\Delta_{\bm{k}}}^2-2\gamma^2}$ using the dashed line with circles. (a) The crossover from the BCS physics $(\gamma\lesssim\abs{\Delta_{\bm{k}}})$ to the Fermi liquid $(\gamma\gtrsim\abs{\Delta_{\bm{k}}})$ with $\gamma/\abs{\Delta_{\bm{k}}}$ ranging from $10^{-1}$ to $2$. The pseudogap develops in the intermediate regime of this crossover, and the energy gap is fully closed when $\gamma/\abs{\Delta_{\bm{k}}}=1/\sqrt{2}$, noted as the grey solid curve. (b) Broadening of the BCS quasiparticle peak, e.g. $A(\omega\sim\abs{\Delta_{\bm{k}}})$, in the weakly fluctuating regime for $\gamma/\abs{\Delta_{\bm{k}}}$ ranging from $2\times10^{-2}$ to $10^{-1}$. The two-sided arrow implies that the spectral peak acquires a half-width proportional to $\gamma$.