Efficient fluctuation-exchange approach to low-temperature spin fluctuations and superconductivity: From the Hubbard model to ${\mathrm{Na}}_x{\mathrm{CoO}}_2·{y\mathrm{H}}_2\mathrm{O}$

Superconductivity arises mostly at energy and temperature scales that are much smaller than the typical bare electronic energies. Since the computational effort of diagrammatic many-body techniques increases with the number of required Matsubara frequencies and thus with the inverse temperature, phase transitions that occur at low temperatures are typically hard to address numerically. In this work, we implement a fluctuation exchange (FLEX) approach to spin fluctuations and superconductivity using the “intermediate representation basis” (IR) [Shinaoka et al., Phys. Rev. B 96, 035147 (2017)] for Matsubara Green functions. This $\mathrm{FLEX}+\mathrm{IR}$ approach is numerically very efficient and enables us to reach temperatures on the order of ${10}^{-4}$ in units of the electronic bandwidth in multiorbital systems. After benchmarking the method in the doped repulsive Hubbard model on the square lattice, we study the possibility of spin-fluctuation-mediated superconductivity in the hydrated sodium cobalt material ${\mathrm{Na}}_x{\mathrm{CoO}}_2·{y\mathrm{H}}_2\mathrm{O}$ reaching the scale of the experimental transition temperature $T_{\mathrm{c}}=4.5\mathrm{K}$ and below.

Figure 1

Number of IR basis functions $N_{\mathrm{IR}}^{\alpha }$ needed to sufficiently expand (a) fermionic or (b) bosonic Green functions within an error bound $\delta$. The imaginary-time and Matsubara frequency grid sizes can be chosen equally large.

Figure 2

Comparison of static spin susceptibility (left column) at $T/t=0.03$ and eigenvalue of the Eliashberg equation as well as inverse magnetic susceptibility at the leading instability (right column) as calculated with our $\mathrm{FLEX}+\mathrm{IR}$ implementation with results from Ref. [28] (dashed lines). The rows show two different situations with dominant (a) antiferromagnetism (AF: $t^{\prime }/t=0$, $n=0.85$) and (b) ferromagnetism (F: $t^{\prime }/t=0.5$, $n=0.3$) for $U/t=4$.

Figure 3

Phase diagram of the Hubbard model with $t^{\prime }/t=0$ and $U/t=4$. (a) Comparison of different magnetic (AF) and superconducting (SC) eigenvalues calculated by the $\mathrm{FLEX}+\mathrm{IR}$ approach with results from Ref. [55]. (b) Comparison of calculated phase boundaries from $\mathrm{FLEX}+\mathrm{IR}$ to a variety of methods including $\mathrm{DMFT}+\mathrm{FLEX}$ [55], TPSC [59], DCA on a 16-site cluster [60], and ${\mathrm{DCA}}^{+}$ [61].

Figure 4

Crystal and electronic structure of the ${\mathrm{Na}}_x{\mathrm{CoO}}_2·{y\mathrm{H}}_2\mathrm{O}$ compound. (a) Vertical layered structure of ${\mathrm{CoO}}_2$ planes (light and dark gray) with intercalated ${\mathrm{Na}}^{+}$, ${\mathrm{H}}_2\mathrm{O}$, and ${\mathrm{H}}_3{\mathrm{O}}^{+}$. (b) Top view on ${\mathrm{CoO}}_2$ planes showing a triangular sublattice of Co ions with surrounding O ions. (c) ${\mathrm{CoO}}_6$ octahedron, which is trigonally deformed by the layered structure. (d) Electronic band structure with orbital character projections indicated by surrounding color patches. Model details are given in the Appendix. Energies are measured with respect to the chemical potential ${\xi }_{\mathit{k}}={\varepsilon }_{\mathit{k}}-\mu$. (e) Fermi surface corresponding to the band structure of panel (d).

Figure 5

Comparison of the largest eigenvalue of the static spin susceptibility to results from Ref. [14] at $T/t_3=0.02$ using a $32\times 32\mathit{k}$-mesh. The second-order correction used in the calculations is different between both panels [see the text and Eqs. (14) and (15)].

Figure 6

Evolution of the largest eigenvalues of static ($i{\nu }_0=0$) irreducible susceptibility ${\widehat{\chi }}^0$ and spin susceptibility ${\widehat{\chi }}^{\mathrm{s}}$ for two exchange interactions $J/U$.

Figure 7

(a) Possible triplet-pairing symmetries of the superconducting gap. Shown is the orbital trace of the converged order parameter $\mathrm{\Delta }(i{\omega }_1,\mathit{k})$ for $T/t_3=0.003$ and $J/U=0.2$. The line nodes (solid white) intersect differently with the Fermi surface (dashed black) depending on the gap symmetry. (b) Temperature dependence of the superconducting eigenvalue ${\lambda }_{\kappa }$ for different gap symmetries $\kappa =f_1,f_2,p$. The panels show different exchange interactions $J/U$. Note that the $T$-axis is logarithmic.

Figure 8

Dependence of magnetic fluctuations and eigenvalues of the superconducting Eliashberg equation on Fermi surface topology. Each row shows the noninteracting Fermi surface, maximal eigenvalue of irreducible and spin susceptibility, and superconducting eigenvalue at $T/t_3=0.003$ for the maximally convergable interaction parameters. The top row corresponds to a Fermi surface composed of the $a_{1g}$ pocket only (${\mathrm{\Delta }}_{\mathrm{CF}}=-1.2$), the middle row to both pocket types being present (${\mathrm{\Delta }}_{\mathrm{CF}}=0.4$), and the bottom row to only the $e_g^{\prime }$ pockets existing (${\mathrm{\Delta }}_{\mathrm{CF}}=9.0$).