Extracting the dynamical effective interaction and competing order from an analysis of Raman spectra of the high-temperature La${}_{2-x}$Sr${}_x$CuO${}_4$ superconductor

We discuss the problem of the glue mediating the effective electron-electron interaction and the related order underlying superconductivity in the cuprates. Assuming as mediators spin and charge fluctuations with different characteristic wavevectors, we calculate the Raman response function including self-energy and vertex corrections. In this way, exploiting the related cancellations, we are able to individuate the charge and spin contributions in the Raman response function. Raman scattering experiments in ${\mathit{La}}_{2-\mathrm{x}}{\mathit{Sr}}_{\mathrm{x}}{\mathit{CuO}}_4$ single crystals have been systematically carried out at various doping levels $x$. By fitting the experimental spectra within the above theoretical framework, we find a charge and spin nearly ordered state, whose fluctuations give rise to a spin dominated glue function at intermediate doping and a charge dominated one above optimal doping.

Figure 1

(a) Experimental Raman response ${\chi }_{\mu }^{\prime \prime }(\omega ,T)$ of LSCO at $x=0.20$ on a broad frequency range up to 8000 cm${}^{-1}$ at intermediate $T\approx 100$ K in the $B_{1g}$ channel (solid triangles, red online) and in the $B_{2g}$ channel (empty circles, green online). (b-f) Raman response ${\chi }_{\mu }^{\prime \prime }(\omega ,T)$ of LSCO at $x=0.15,0.17,0.20,0.25$, and $0.26$ on a reduced frequency range up to 4000 cm${}^{-1}$. Data in the $B_{1g}$ channel are given by solid triangles (red online), while the spectra in the $B_{2g}$ symmetry are given by empty circles (green online). The various spectra are taken at similar temperatures around $T\approx 100$ K. The results of the fits obtained within the “nearly-critical” charge/spin theory (see text) are given by the black solid lines.

Figure 2

(a) Raman response ${\chi }_{\mu }^{\prime \prime }(\omega ,T)$ of LSCO at $x=0.20$ in the $B_{1g}$ (solid triangles) and $B_{2g}$ (empty circles) at three different temperatures above $T_c$ and the related fits (solid curves) obtained within the nearly critical scheme described in the text: $T=56$ K (blue online), $T=104$ K (green online), $T=201$ K (red online). (b) Temperature dependence of the inverse square correlation length ${\xi }_c^{-2}$ (we use a unit lattice spacing $a=1$) of the charge modes at different dopings [$x=0.15$ (squares, black online), $x=0.17$ (circles, blue online), $x=0.20$ (up triangles, green online), $x=0.25$ (down triangles, orange online), $x=0.26$ (diamonds, red online)]. The lines are a guide to the eyes. (c) Same as in (b) for the inverse square correlation length ${\xi }_s^{-2}$ of the spin collective modes. (d) Temperature dependence of the low-frequency typical energy scale ${\omega }_1=\sqrt{m\overline{\Omega }}$ of the charge (solid lines) and spin (dashed lines) collective modes.

Figure 3

Hot spot structure for both charge (with ${\mathbf{q}}_c$) and spin (with ${\mathbf{q}}_s$) scattering. In panels (a) and (b) the sign structures of the form factors for the $B_{1g}$ and $B_{2g}$ channels, ${\gamma }_{1g}\left(\mathbf{k}\right)=\cos \left(k_y\right)-\cos \left(k_x\right)$ and ${\gamma }_{2g}\left(\mathbf{k}\right)=\sin \left(k_x\right)\sin \left(k_y\right)$ (we use a unit lattice spacing), respectively, are reported.

Figure 4

Self-energy diagrams (S${}_1$ and S${}_2$) and vertex diagram (V) correcting the Raman response function with the exchange of a single collective mode. The black dots represent the ${\gamma }_{\mathbf{k}}$ form factor, while the shaded dot represents ${\gamma }_{\mathbf{k}+\mathbf{q}}$. The dotted lines represent collective mode Green’s functions, Eq. (2).

Figure 5

(a-e) Glue functions due to the “allowed” modes in the fitted Raman spectra [for example, Fig. 1 and Fig. 2a]. The solid lines are the glue functions due to charge CM, while the dashed lines are the spin CM glue functions. From right to left the curves refer to high ($T\sim 150-200$ K, red online), intermediate ($T\sim 100$ K, green online) and low ($T\sim 50$ K, blue online) temperatures. (a) $x=0.15$, $T=$ 42 K, 95 K, 195 K; (b) $x=0.17$, $T=$ 54 K, 104 K, 254 K; (c) $x=0.20$, $T=$ 56 K, 104 K, 201 K; (d) $x=0.25$, $T=$ 53 K, 104 K, 201 K; (e) $x=0.26$, $T=$ 46 K, 102 K, 149 K; (f) Total weight of the glue functions mediated by charge $W_c$ (solid lines) and spin $W_s$ (dashed lines) as a function of doping $x$. Circles (blue online) for low ($T\approx 50$ K), squares (green online) for intemediate ($T\approx 100$ K), and diamonds (red online) for high ($T\approx 200$ K) temperatures.

Figure 6

Imaginary (a-c) and real (d-f) parts of the optical memory functions obtained summing the charge and the spin glue functions reported in Fig. 5a-5c. From top to bottom the curves refer to high ($T\sim 150-200$ K, red online), intermediate ($T\sim 100$ K, green online) and low ($T\sim 50$ K, blue online) temperatures. (d-f) From bottom to top at low frequency, the curves refer to high ($T\sim 150-200$ K, red online), intermediate ($T\sim 100$ K, green online) and low ($T\sim 50$ K, blue online) temperatures. For details of the calculations see Appendix .

Figure 7

Fermi surfaces for LSCO samples at $x=0.15$ [(a) and (b)] and $x=0.25$ [(c) and (d)]. The left and right panels display the nodal lines of the $B_{1g}$ and $B_{2g}$ form factors respectively, with $+$ and $-$ signs indicating the regions where the form factors are positive or negative. The solid dots (blue online) are the hot spots connected by spin fluctutions, while the empty circles (red online) are connected by the charge CM wavevectors. Solid (dot-dashed) segments indicate wavevectors which connect regions with form factors of opposite (the same) sign. Therefore solid-line scatterings represent CMs with larger couplings, while dot-dashed wavevectors are related to CMs with vanishing or small effective coupling $g_{\lambda }$.

Figure 8

(a) Glue functions for two CMs having the same dynamical range between ${\omega }_1=330$ cm${}^{-1}$ and ${\omega }_2=1500$ cm${}^{-1}$, but different diffusive-propagating crossover scale: $\overline{\Omega }=200$ cm${}^{-1}$ for the more propagating mode (solid curve, red online), $\overline{\Omega }=1000$ cm${}^{-1}$ for the more diffusive mode (dashed black curve). (b) Imaginary part of the memory function (scattering time) corresponding to the two glue functions in (a). (c) Raman response obtained for the two glue functions in (a) with $g_{\lambda }=0.5$.