Exact theory of superconductivity in a strongly correlated Fermi-arc model

Abstract

Because the normal state of underdoped cuprate superconductors is an enigmatic Fermi-arc metal, it is valuable to analyze an exactly solvable model that exhibits both Fermi arcs and $d$-wave superconductivity. Here, we focus on a recently proposed solvable model in which the emergence of Fermi arcs is especially transparent. Upon incorporating a $d$-wave pairing interaction, the model produces an asymptotically exact solution for the superconducting transition temperature $T_c$ that traces out a superconductivity dome as a function of hole doping, in qualitative agreement with experimental observations in cuprates. Crucially, we show analytically that the Fermi arcs generate an additional many-body effect that suppresses $T_c$ beyond the simple reduction expected from a shrinking Fermi surface. The many-body nature of the Fermi arcs further introduces the gap-to-$T_c$ ratio greatly surpassing the mean-field limit. These findings provide an analytic benchmark for understanding how Fermi-arc physics competes with $d$-wave superconductivity in high-$T_c$ superconductors.

Figures (4)

• Figure 1: (a) Normal-state Fermi surface and (d) single-particle excitation spectrum at a hole doping of $p=0.15$ for $U=2$ at $T= 0$. The dashed lines in (a) are the antiferromagnetic zone boundary. UHB and LHB are short for upper and lower Hubbard bands, respectively. Panels (b), (c), (e), and (f) are the same as (a) and (d) but for heavier doping levels of $p=0.35$ and $p=0.56$. An energy smearing of $0.04$ is used for visualization. The markers in the upper right corner denote the representative doping levels in Fig. \ref{['fig2']}(a) and Fig. \ref{['fig3']}(a).
• Figure 2: (a) Phase diagrams for various repulsion strengths at $J=0.8$. The gray line indicates the pseudogap (PG) onset temperature $T^*$, and the blue (orange) line represents the superconducting (SC) transition temperature $T_c$ calculated with (without) the many-body correction $A^{\prime}$. The momentum summation is carried out on a $512 \times 512$$\mathbf{k}$-grid. The markers correspond to underdoped, optimal doped and overdoped states, whose normal-state Fermi surfaces are shown in Fig. \ref{['figspectrum']}. (b) Relative difference between the superconducting transition temperatures calculated with and without $A^{\prime}$.
• Figure 3: Same as Fig. \ref{['fig2']} but for various pair interactions at a fixed repulsion strength $U=2$.
• Figure 4: Ratio of the zero-temperature superconducting gap to the transition temperature, $2\Delta(0)/T_c$, as a function of doping. The curves are the results from this model (MF: mean field) for $J=0.8$. The red circles and triangles are numerical results for the Hubbard model gull13superconductivity and experimental data for YBCO superconductors inosov11crossover, respectively. The doping level is normalized by the optimal doping level ($p_c$, dashed vertical line) for each system. The calculations are done for a $256\times256$$\mathbf{k}$-grid except for the mean-field case, for which a $512\times 512$$\mathbf{k}$-grid is used.