Raman scattering near a $d$-wave Pomeranchuk instability

Motivated by recent transport and neutron-scattering experiments suggesting an orientational symmetry breaking in underdoped cuprates we present a theoretical study of Raman scattering near a $d$-wave Pomeranchuk instability (PI). The $d$-wave component of Raman scattering from electrons and phonons allows one to study directly order parameter fluctuations associated with the PI. Approaching the PI from the normal state by lowering the temperature, a central peak emerges both in electronic and, as an additional low-frequency feature, in phononic scattering. Approaching the PI in the superconducting state at low temperature by decreasing the doping concentration the central peak is replaced by a soft mode with strongly decreasing width and energy and increasing spectral weight. These predicted low-energy features in Raman scattering could confirm in a rather direct way the presence of a PI in high-temperature cuprate superconductors and in Sr${}_3$Ru${}_2$O${}_7$.

Figure 1

(a) Graphical representation of ${\chi }^{B_{1g}}$. The vertex with a circle (square) indicates the form factor ${\gamma }_{\mathbf{k}}^{B_{1g}}$ ($d_{\mathbf{k}}$). (b) Effective electron-electron interaction driving the $d$FSD instability. (c) Full electronic Green’s function. The single solid line denotes the free-electron propagator with the dispersion ${\epsilon }_{\mathbf{k}}$. (d) Electronic self-energy originating from the coupling to some bosonic fluctuations represented by the sawlike line.

Figure 2

(a) Model of ${\alpha }^{2}F(\nu )$. (b) Imaginary part and (c) real part of the electronic self-energy for several choices of $T$ in the normal state for ${\nu }_0={\nu }_c/4$, ${\nu }_c=2/3$, and $a_0=1/4$. The energy unit is taken as $t$.

Figure 3

(a) $A(\mathbf{k},\omega )$ as a function of $\omega$ at the momentum $\mathbf{k}-{\mathbf{k}}_F=2\pi k_r(1,1)$ with $k_r$ ranging from zero to $-0.10$ with an interval of 0.01; ${\mathbf{k}}_F$ is the Fermi momentum along the $(0,0)-(\pi ,\pi )$ direction. (b) The renormalized electronic dispersion.

Figure 4

Graphical representation of the phonon propagator (double dashed line); the single dashed line denotes the noninteracting phonon propagator and the solid square denotes the form factor $g_{\mathrm{ph}}{d}_{\mathbf{k}}$. The rest of the notation is the same as Fig. 1.

Figure 5

Graphical representation of the $d$-wave charge compressibility. A spring denotes two interactions, the electron-electron interaction (wavy line) and the electron-phonon interaction (dashed line), and the corresponding form factors at the vertex (open circle) are $d_{\mathbf{k}}$ and $g_{\mathrm{ph}}{d}_{\mathbf{k}}$, respectively. The shaded vertex is defined by the equality of the first and second line; the rest of the notation is the same as in Fig. 1.

Figure 6

(a) $\omega$ dependence of Im$\hspace{0.5em}{\chi }^{B_{1g}}(\omega )$ for a sequence of temperatures $T$ close to the $d$FSD instability at $T_c=0.098$ in the normal state at $\delta =0.10$. (b) Low-energy region of $S(\omega )$ near the $d$FSD instability. (c) Im$\hspace{0.5em}{\Pi }^{\mathit{dd}}(\omega )$ and (d) Re$\hspace{0.5em}{\Pi }^{\mathit{dd}}(\omega )$ as a function of $\omega$ for several values of $T$.

Figure 7

(a) $\omega$ dependence of $S(\omega )$ for a sequence of doping concentrations in the superconducting state at $T=0.001$; the actual value of $S(\omega )$ is obtained by multiplication with the factor indicated near each peak except for $\delta =0.50$. (b) $\omega$ dependence of Re$\hspace{0.5em}{\Pi }^{\mathit{dd}}(\omega )$ (solid line) and Im$\hspace{0.5em}{\Pi }^{\mathit{dd}}(\omega )$ (dashed line) at $\delta =0.25$ and $0.50$. (c) The peak position of $S(\omega )$ (solid circles) and its peak height (solid line) as a function of doping; also shown are the energies ${\omega }_{\mathrm{res}}$ (open circles); the $d$FSD instability occurs at ${\delta }_c=0.207$.

Figure 8

$\omega$ dependence of $S_{\mathrm{ph}}$ (a) and Re$\hspace{0.5em}{\mathrm{\Pi ̃}}^{\mathit{dd}}(\omega )$ (b) for several choices of temperatures in the normal state at $\delta =0.10$; the renormalized critical temperature of the $d$FSD instability is $T_c=0.126$. (c) Peak positions ${\omega }^{\mathrm{ph}}$ and ${\omega }_s^{\mathrm{ph}}(<{\omega }^{\mathrm{ph}})$ of $S_{\mathrm{ph}}(\omega )$ and their peak heights as a function of $T$.

Figure 9

(a) $\omega$ dependence of $S_{\mathrm{ph}}(\omega )$ for several choices of $\delta$ in the superconducting state at $T=0.001$; the actual value at $\delta =0.50$ is obtained by multiplying with a factor of 2. (b) $\omega$ dependence of Res$(\omega )$ for different $\delta$. (c) Peak positions ${\omega }^{\mathrm{ph}}$ and ${\omega }_s^{\mathrm{ph}}(<{\omega }^{\mathrm{ph}})$ of $S_{\mathrm{ph}}(\omega )$ as a function of $\delta$ together with their peak heights. The $d$FSD instability occurs at ${\delta }_c=0.233$.