Signatures of nematic quantum critical fluctuations in the Raman spectra of lightly doped cuprates

We consider the lightly doped cuprates ${\mathrm{Y}}_{0.97}{\mathrm{Ca}}_{0.03}{\mathrm{BaCuO}}_{6.05}$ and ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (with $x=0.02$, 0.04), where the presence of a fluctuating nematic state has often been proposed as a precursor of the stripe or, more generically charge density wave phase, which sets in at higher doping. We phenomenologically assume quantum critical longitudinal and transverse nematic, and charge-ordering fluctuations, and investigate their effects in the Raman spectra. We find that the longitudinal nematic fluctuations peaked at zero transferred momentum account well for the anomalous Raman absorption observed in these systems in the $B_{2g}$ channel, while the absence of such an effect in the $B_{1g}$ channel may be due to the overall suppression of Raman response at low frequencies, associated with the pseudogap. While in ${\mathrm{Y}}_{0.97}{\mathrm{Ca}}_{0.03}{\mathrm{BaCuO}}_{6.05}$ the low-frequency line shape is fully accounted for by longitudinal nematic collective modes alone, in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, also charge-ordering modes with finite characteristic wave vector are needed to reproduce the shoulders observed in the Raman response. This different involvement of the nearly critical modes in the two materials suggests a different evolution of the nematic state at very low doping into the nearly charge-ordered state at higher doping.

Figure 1

Diagrammatic representation of the Raman response due to the excitation of two CMs. The grey dots represent the ${\gamma }_{B1g}$ or ${\gamma }_{B2g}$ form factors. The solid lines are the propagators of the fermion quasiparticles in the fermionic loops, the wavy lines represent the NCM or CO CM propagators [Eqs. (2, 3, 4)], which are coupled to the quasiparticles by the coupling functions $g_{\lambda }(\mathbf{k},\mathbf{q})$ (solid dots).

Figure 2

Schematic representation of the AL-like Raman response of the three different CMs: longitudinal L-NCM (black curve), transverse (T) NCM (green curve), and CO CM (red curve). (Inset) AL-like Raman response from different CMs, but taken with the same nearly critical set of parameters ($m_{\lambda }=17 {\mathrm{cm}}^{-1}, {\mathrm{\Omega }}_{\lambda }=70 {\mathrm{cm}}^{-1}, c_{\lambda }=3.16$). The amplitudes are instead rescaled by factors of order one to bring all responses to a common maximal height for easier comparison. The coloring of the lines is the same as in the main panel.

Figure 3

Subtracted experimental Raman absorption spectra in the $B_{2g}$ channel, at various temperatures, for YBCO at $p\approx 0.015$ (symbols). The theoretical fits (solid lines) consider the contribution of the longitudinal NCM only. The fitting parameters are $c_{\ell }=0.63 {\mathrm{cm}}^{-1}, {\mathrm{\Omega }}_{\ell }=110{\mathrm{cm}}^{-1},A_{\ell }=5.0$ (a.u.). The inset reports the temperature dependence of the mass of the longitudinal NCM (black circles).

Figure 4

Subtracted experimental Raman absorption spectra in the $B_{2g}$ channel, at various temperatures, for LSCO at $x=0.02$ (symbols). The theoretical fits (solid lines) consider the contribution of the longitudinal NCM and of the CO CM, with $c_{\ell }=3.16 {\mathrm{cm}}^{-1}, c_c=333 {\mathrm{cm}}^{-1}, {\mathrm{\Omega }}_{\ell }=70 {\mathrm{cm}}^{-1}, A_{\ell }=8.3$ (a.u.). The other fitting parameters are reported in Fig. 6. The inset reports the temperature dependence of the mass of the longitudinal NCM (black circles) and of the CO CM (red squares).

Figure 5

Subtracted experimental Raman absorption spectra in the $B_{2g}$ channel, at various temperatures, for LSCO at $x=0.04$ (symbols). The theoretical fits (solid lines) consider the contribution of the longitudinal NCM and of the CO CM, with $c_{\ell }=3.16 {\mathrm{cm}}^{-1}, c_c=333 {\mathrm{cm}}^{-1}, {\mathrm{\Omega }}_{\ell }=50 {\mathrm{cm}}^{-1}$. $A_{\ell }=7.14$ (a.u.). The other fitting parameters are reported in Fig. 6. The inset reports the temperature dependence of the mass of the longitudinal NCM (black circles) and of the CO CM (red squares).

Figure 6

(a) High-frequency cutoff ${\mathrm{\Omega }}_c$ for the CO CM for a sample at $x=0.02$ (black empty squares) and at $x=0.04$ (red filled squares). (b) Amplitude coefficients $A_c$ for the CO CM for a sample at $x=0.02$ (black empty squares) and at $x=0.04$ (red filled squares).

Figure 7

Schematic comparison of the theoretical expectations and the experimental observation of an anomalous Raman absorption. The theoretically involved CMs are indicated with N in the case of the NCM, while for CO we also report the direction of the characteristic wave vector, as established by inelastic neutron scattering. The related symbols only appear in the box where they are expected to contribute on the basis of symmetry arguments. The experimental observation of an anomalous Raman absorption is depicted as a green case in the column of the corresponding channel. Red cases indicate instead the lack of anomalous Raman absorption in experiments. Our remarks and possible indications (in boldface) are contained in the comment boxes.

Figure 8

Schematic evolution of the nematic (blue region) and stripe CO (green region) phases in underdoped cuprates with doping (disregarding superconductivity). The pseudogap region, where the Raman response in the $B_{1g}$ channel is expected to be suppressed, is highlighted in red. Upon increasing doping, the nematic phase evolves into a CO phase, which in turn vanishes at a CO quantum critical point around optimal doping. The orientation of the segments and/or stripes may change with cuprate family and doping.

Figure 9

Subtraction procedure on the raw Raman data of a LSCO sample at $x=0.04$ and $T=52$ K (blue curve and symbols). The red curve and symbols correspond to the raw data at $T=331$ K. Once the latter are subtracted from the former, the purple curve and symbols are obtained. To eliminate the negative absorption, the parabola ${\chi }_b^{\prime \prime }=\omega (1000-\omega )/{600}^2$, with $\omega$ in ${\mathrm{cm}}^{-1}$, is added to finally yield the absorption reported with the black curve and symbols.

Figure 10

Square of the fermionic loops as a function of the angle $\varphi$ calculated according to Eq. (C1) in the $B_{1g}$ channel (solid lines) and in the $B_{2g}$ channel (dashed lines). The $F\left(\theta \right)$ function [Eq. (C2)] has been set to produce arcs shrinking the Fermi surface by a factor 1 (${\mathrm{\Theta }}_M=\pi /4$, whole Fermi surface, black curves), 0.75 (${\mathrm{\Theta }}_M=3\pi /16$, red curves), 0.5 (${\mathrm{\Theta }}_M=\pi /8$, green curves), and 0.25 (${\mathrm{\Theta }}_M=\pi /16$, blue curves). The parameter ${\mathrm{\Delta }}_{\theta }^2=0.05$, was set to smoothen the angular cutoff producing the arcs. The boson momentum $q$ has been chosen such that $q^2/2k_F^2=0.01$, while $M=1$.