Low-frequency Raman response near the Ising-nematic quantum critical point: A memory-matrix approach

Recent Raman scattering experiments have revealed a quasielastic peak in ${\mathrm{FeSe}}_{1-\mathrm{x}}{\mathrm{S}}_{\mathrm{x}}$ near an Ising-nematic quantum critical point (QCP) [Zhang et al., PNAS 118, 20 (2021)]. Notably, the peak occurs at subtemperature frequencies, and softens as $T^{\alpha }$ when temperature is decreased toward the QCP, with $\alpha >1$. This temperature dependence is inconsistent with an impurity scattering scenario, and suggests that quantum critical fluctuations play an important role. In this work,we incorporate these effects in the framework of a memory matrix approach. The quasielastic peak is associated with the relaxation of an Ising-nematic deformation of the Fermi surface. We identify the dynamical scattering rate ${\tau }^{-1}$ of this deformation as the product of the quasielastic peak frequency $\mathrm{\Gamma }$ and the Ising-nematic thermodynamic susceptibility $\chi$. Over a broad temperature regime, we find that ${\tau }^{-1}\left(T\right)$ exhibits a quasilinear dependence on temperature, in qualitative agreement with experiments. This behavior reflects a crossover from quantum critical scaling to a regime where the lifetime is governed by scattering from quasielastic thermal fluctuations. At frequencies larger than the temperature, we find that the Raman response is proportional to ${\omega }^{1/3}$, consistently with earlier theoretical predictions.

Figure 1

Schematic plot of the imaginary part of the dynamical nematic susceptibility in the vicinity of an Ising-nematic QCP, featuring a low-frequency quasielastic peak, and ${\omega }^{1/3}$ dependence at frequencies higher than temperature. The Fermi energy $E_F$ is taken to be much larger than temperature.

Figure 2

Feynman diagrams for the thermodynamic susceptibility ${\chi }_{{\widehat{n}}_{\mathbf{k}},{\widehat{n}}_{\mathbf{k}}^{\prime }}$. The double-wiggly line represents the dressed propagator for nematic fluctuations.

Figure 3

Feynman diagrams for the memory matrix ${\widehat{M}}_{{\widehat{n}}_{\mathbf{k}\sigma },{\widehat{n}}_{{\mathbf{k}}^{\prime }{\sigma }^{\prime }}}$, adapted from Fig. 1 of Ref. [28]. (a) The vertex function for ${\dot{n}}_{\mathbf{k}\sigma }\left(\omega \right)$. The empty (solid) circle at the vertex denotes an incoming (outgoing) frequency $\omega$. (b)–(d) Class I and II Feynman diagrams under random phase approximation. The external momentum indices $(\mathbf{k},{\mathbf{k}}^{\prime })$ are colored red.

Figure 4

(a) Imaginary part of the dynamical nematic susceptibility showing a quasielastic peak feature for various temperatures. (b) QEP peak frequency $\mathrm{\Gamma }\left(T\right)$ as a function of temperature. Inset is the log-derivative plot $\alpha \equiv dln\mathrm{\Gamma }/dlnT$ showing the temperature variation of the power-law exponent. The dashed lines correspond to $T^2{\chi }^{-1}$ (Fermi liquid behavior) and $T^{4/3}{\chi }^{-1}$. (c) Inverse of the QEP peak height $A\left(T\right)$ (red circle) compared the thermodynamic susceptibility (blue solid line).

Figure 5

(a) Imaginary part of the dynamical nematic susceptibility in the presence of impurity scattering ($g_{\text{imp}}=0.1E_F$). (b) QEP peak frequency $\mathrm{\Gamma }\left(T\right)$ as a function of temperature. Inset is the log-derivative plot showing the temperature variation of the power-law exponent $\alpha$.

Figure 6

(a) Quasielastic peak frequency for $\mathrm{Fe}{\mathrm{Se}}_{\text{1-x}}{\mathrm{S}}_{\text{x}}$ at dopings $x=0.15$ (orange) and $x=0.2$ (magenta), extracted from Ref. [23]. The extrapolated Curie-Weiss temperature is zero at $x=0.15$, and negative at $x=0.2$. $E_F$ is taken to be 30 meV. The dashed lines are linear fits using $a+1.2T/E_F$, with $a=0.2$ and 0.4, respectively. (b) A replot of Fig. 5 with the $y$ axis being $\mathrm{\Gamma }\left(T\right)/T$.

Figure 7

Raman response for a two-pocket electronic model in the clean limit. Compared to Fig. 4, here the quasielastic peak frequency is reduced by a factor of 2 due to fermion band doubling, but the power-law temperature dependence remains unchanged, showing a universal behavior independent on band structure details.