[Paper Review] Low frequency Raman response near Ising-nematic quantum critical point: a memory matrix approach

Xiaoyu Wang, Erez Berg|arXiv (Cornell University)|Nov 3, 2020

Physics of Superconductivity and Magnetism5 citations

TL;DR

This paper develops a memory matrix approach to explain the low-frequency Raman response near an Ising-nematic quantum critical point, showing that the observed quasi-elastic peak arises from slow relaxation of Fermi surface nematic deformations. The peak frequency scales as $\Gamma(T) \propto \tau^{-1}\chi^{-1}$, where $\chi$ is the nematic susceptibility and $\tau^{-1}$ is the decay rate from critical fluctuations and impurities, with $\omega^{1/3}$ scaling at higher frequencies consistent with prior work.

ABSTRACT

Recent Raman scattering experiments have revealed a "quasi-elastic peak" in $\mathrm{FeSe_{1-x}S_x}$ near an Ising-nematic quantum critical point (QCP) \cite{zhang17}. Notably, the peak occurs at sub-temperature frequencies, and softens as $T^α$ when temperature is decreased toward the QCP, with $α>1$. In this work, we present a theoretical analysis of the low-frequency Raman response using a memory matrix approach. We show that such a quasi-elastic peak is associated with the relaxation of an Ising-nematic deformation of the Fermi surface. Specifically, we find that the peak frequency is proportional to $ τ^{-1}χ^{-1}$, where $χ$ is the Ising-nematic thermodynamic susceptibility, and $τ^{-1}$ is the decay rate of the nematic deformation due to an interplay between impurity scattering and electron-electron scattering mediated by critical Ising-nematic fluctuations. We argue that the critical fluctuations play a crucial role in determining the observed temperature dependence of the frequency of the quasi-elastic peak. At frequencies larger than the temperature, we find that the Raman response is proportional to $ω^{1/3}$, consistently with earlier predictions \cite{klein18a}.

Motivation & Objective

• To explain the experimentally observed quasi-elastic peak (QEP) in low-frequency Raman scattering of FeSe$_{1-x}$S$_x$ near an Ising-nematic quantum critical point (QCP).
• To understand the origin of the QEP's sub-temperature frequency and its softening as $T^\alpha$ with $\alpha > 1$ as temperature decreases toward the QCP.
• To reconcile the observed QEP with theoretical models by incorporating both impurity scattering and electron-electron scattering via critical nematic fluctuations.
• To derive the frequency dependence of the Raman response in the high-frequency regime, confirming $\omega^{1/3}$ scaling.

Proposed method

• The study employs a memory matrix approach to treat quasi-particle occupation numbers near the Fermi surface as slow variables, enabling a non-perturbative treatment of relaxation dynamics.
• The memory matrix $M_{\hat{Q},\hat{Q}}(\omega)$ is computed from the dynamical nematic susceptibility $D_{\text{nem}}(\mathbf{q}\rightarrow 0,\omega)$, incorporating both impurity scattering and critical nematic fluctuations.
• The approach accounts for Landau damping via electron-hole excitations near the Fermi surface and includes momentum diffusion effects in the high-frequency limit.
• The model uses a two-dimensional boson-fermion action with linear coupling between electrons and Ising-nematic order parameter fields, allowing for critical fluctuations near the QCP.
• The memory matrix is decomposed into contributions from nematic fluctuations ($M_{\text{nem}}$) and impurity scattering ($M_{\text{imp}}$), with $M(T) = M_{\text{nem}}(T) + M_{\text{imp}}(T)$.
• The dynamical response is derived from the imaginary part of the susceptibility, $\mathrm{Im}D_{\text{nem}}(\omega) \approx M_{\text{Q,Q}}(\omega)/\omega$, leading to $\omega^{1/3}$ scaling at $\omega \gg T$.

Experimental results

Research questions

• RQ1 What causes the quasi-elastic peak (QEP) in low-frequency Raman scattering of FeSe$_{1-x}$S$_x$ near the Ising-nematic QCP?
• RQ2 Why does the QEP frequency soften as $T^\alpha$ with $\alpha > 1$ as temperature decreases?
• RQ3 How do critical nematic fluctuations and impurity scattering jointly influence the relaxation dynamics of the nematic deformation?
• RQ4 What determines the frequency dependence of the Raman response at frequencies above the temperature scale?
• RQ5 Can the memory matrix approach reconcile the observed $\omega^{1/3}$ scaling with the sub-temperature QEP?

Key findings

• The quasi-elastic peak frequency $\Gamma(T)$ is proportional to $\tau^{-1}\chi^{-1}$, where $\tau^{-1}$ is the decay rate of the nematic deformation and $\chi$ is the thermodynamic nematic susceptibility.
• At low temperatures, impurity scattering dominates, leading to $\Gamma(T) \propto T$, consistent with the $T$-linear scattering rate observed in experiments.
• At higher temperatures, nematic fluctuations dominate, resulting in a stronger temperature dependence of $\Gamma(T)$, with $\alpha > 1$ in the $T^\alpha$ softening.
• In the high-frequency regime ($\omega \gg T$), the Raman response scales as $\omega^{1/3}$, confirming earlier perturbative predictions at zero temperature.
• The model quantitatively reproduces the experimental data for FeSe$_{1-x}$S$_x$ when the impurity scattering strength is tuned, without fine-tuning other parameters.
• The QEP is attributed to the slow relaxation of an Ising-nematic deformation of the Fermi surface, analogous to the Drude peak in optical conductivity.