Spin-freezing perspective on cuprates

The high-temperature superconducting state in cuprates appears if charge carriers are doped into a Mott-insulating parent compound. An unresolved puzzle is the unconventional nature of the normal state above the superconducting dome and its connection to the superconducting instability. At weak hole doping, a “pseudogap” metal state with signatures of time-reversal symmetry breaking is observed, which near-optimal doping changes into a “strange metal” with non-Fermi-liquid properties. Qualitatively similar phase diagrams are found in multiorbital systems, such as pnictides, where the unconventional metal states arise from a Hund-coupling-induced spin freezing. Here, we show that the relevant model for cuprates, the single-orbital Hubbard model on the square lattice, can be mapped onto an effective multiorbital problem with strong ferromagnetic Hund coupling. The spin-freezing physics of this multiorbital system explains the phenomenology of cuprates, including the pseudogap, the strange metal, and the $d$-wave superconducting instability. Our analysis suggests that spin/orbital freezing is the universal mechanism which controls the properties of unconventional superconductors.

Figure 1

Illustration of the mapping of the plaquette with nearest-neighbor hopping $t$ and diagonal hopping $t^{\prime }$ onto a coupled pair of two-orbital models. If $U$ is the on-site interaction on the plaquette, the two-orbital system has a Slater-Kanamori type interaction with $\tilde{U}={\tilde{U}}^{\prime }=\tilde{J}=U/2$. The diagonal hopping translates into a crystal-field splitting $\delta =2t^{\prime }$. The third panel shows the self-consistent embedding of the two-orbital system into a noninteracting bath described by ${\left(G_0\right)}_{ij}$, and the rightmost panel the simplification to a single-site two-orbital impurity problem.

Figure 2

Noninteracting DOS for the $c$ and $f$ electrons extracted from a noninteracting plaquette CDMFT calculation (symbols) and semicircular DOS with identical variance. (a) Model with $t^{\prime }=0$. Black lines show the fit functions used in the realistic-DOS simulations. (b) Realistic DOS for the model with $t^{\prime }=-0.3t$ (symbols) and semicircular DOS with a crystal-field splitting $\delta =0.075W$. The arrows indicate the centers of the bands at energy $-0.34t$ and $0.26t$, respectively.

Figure 3

Doping dependence of the self-energy revealing the non-Fermi-liquid behavior and spin-freezing crossover. (a) $f$-electron self-energies for $U/W=1.25, \delta =0$ and indicated inverse temperatures $\beta$. The low-frequency behavior of the self-energy indicates a crossover to a spin-frozen state around filling $n=0.98$ at the higher temperature. The black dashed line is proportional to ${\left({\omega }_n\right)}^{1/2}$. (b) Exponents $\alpha$ extracted from the fits $-\text{Im}\mathrm{\Sigma }\left(i{\omega }_n\right)=b{\left({\omega }_n\right)}^{\alpha }$ at the lowest two Matsubara frequencies. Both the results for the $f$ (full symbols) and $c$ electron (open symbols) are shown. For comparison, we also plot by dashed black lines the exponents obtained for the calculation with realistic DOS at $\beta W=240$. (c) Analogous exponents extracted from the CDMFT calculation.

Figure 4

Phase transitions and crossover lines in the doping-temperature phase diagram. Here, we assume a bandwidth of 2 eV to translate the temperature into K. (a) Results from single-site DMFT simulations of the effective 2-orbital model with solid lines corresponding to $\delta =0.075W$ and dashed lines to $\delta =0$. (b) Crossover lines extracted from the behavior of the $f$-electron self-energy (black) and $f$-electron $S_z-S_z$ correlation function (light blue) in CDMFT. For three fillings near the spin-freezing line (pink symbols) we plot the ratio of next-nearest-neighbor (nnn) and nearest-neighbor (nn) $d$-electron $S_z-S_z$ correlations in (c). The nn and nnn curves are multiplied by a factor of 4 for better visibility.

Figure 5

Local spin susceptibility and single-particle spectral function. (a) Dynamical contribution to ${\chi }_{\mathrm{loc}}^{(c,f)}$ as a function of filling for indicated values of the inverse temperature. (b) Spectral function of the dynamical contribution to the spin-spin correlation function, obtained by maximum entropy analytical continuation [37, 38], for $\beta W=400$ and different fillings. (c) Doping evolution of the $f$-electron single-particle spectral function at $\beta W=400$. (2-orbital results for $\delta =0.075W$, CDMFT results for $t^{\prime }=0$ and $\beta W=480.$)

Figure 6

Illustration of the lowest-order diagram which generates an attractive interaction between $(1,f,\uparrow )$ and $(2,f,\downarrow )$.