The optical conductivity of a metal near a quantum critical point (QCP) is expected to depend on frequency not only via the scattering time but also via the effective mass, which acquires a singular frequency dependence near a QCP. On the other hand, the quasiparticle residue $Z$, no matter how singular, does not appear in the conductivity as the latter probes quasiparticles rather than bare electrons. In local theories of QCPs, however, the ratio of band and renormalized masses, $m^{*}/m_b$, coincides with $1/Z$, and it is not straightforward to separate the two quantities. In this work, we use a direct diagrammatic approach and compute the optical conductivity, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$, near two-dimensional (2D) nematic and spin-density wave (SDW) QCPs, using the local approximation in which $Z=m_b/m^{*}$. If renormalization of current vertices is not taken into account, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ is expressed via $Z=m_b/m^{*}$ and the transport scattering rate ${\gamma }_{\text{tr}}$ as ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)\propto Z^2{\gamma }_{\text{tr}}/{\mathrm{\Omega }}^2$. For a nematic QCP (${\gamma }_{\text{tr}}\propto {\mathrm{\Omega }}^{4/3}$ and $Z\propto {\mathrm{\Omega }}^{1/3}$), this formula suggests that ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ would tend to a constant at $\mathrm{\Omega }\to 0$. We explicitly demonstrate that the actual behavior of ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ is different due to strong renormalization of the current vertices, which cancels out a factor of $Z^2$. As a result, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ diverges as $1/{\mathrm{\Omega }}^{2/3}$, as earlier works conjectured. In the SDW case, we consider two contributions to the conductivity: from hot spots and from “lukewarm” regions of the Fermi surface. The hot-spot contribution is not affected by vertex renormalization, but it is subleading to the lukewarm one. For the latter, we argue that a factor of $Z^2$ is again canceled by vertex corrections. As a result, ${\sigma }^{\prime }\left(\mathrm{\Omega }\right)$ at a SDW QCP scales as $1/\mathrm{\Omega }$ down to the lowest frequencies, up to possible multiplicative logarithmic factors.

中文翻译：

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二维金属在量子临界点附近的光导率：扩展的Drude公式的状态

预计金属的量子临界点（QCP）附近的光导率不仅取决于频率，还取决于散射时间，还取决于有效质量，有效质量会在QCP附近获得奇异的频率依赖性。另一方面，准粒子残基$ž$，无论多么奇异，都不会在电导率中出现，因为后者探测的是准粒子而不是裸电子。但是，在QCP的局部理论中，能带与重归一化质量之比，${米}^{*}/{米}_b$，与 $\mathrm{1个}/ž$，并且将两个数量分开并不容易。在这项工作中，我们使用直接的图解方法并计算出光导率，${\sigma }^{\prime }（\mathrm{\Omega }）$，在二维（2D）向列和自旋密度波（SDW）QCP附近，使用局部逼近 $ž={米}_b/{米}^{*}$。如果不考虑当前顶点的重归一化，${\sigma }^{\prime }（\mathrm{\Omega }）$ 通过表示 $ž={米}_b/{米}^{*}$ 和传输散射率 ${\gamma }_{\text{TR}}$ 作为 ${\sigma }^{\prime }（\mathrm{\Omega }）\propto {ž}^{\mathrm{2个}}{\gamma }_{\text{TR}}/{\mathrm{\Omega }}^{\mathrm{2个}}$。对于向列型QCP（${\gamma }_{\text{TR}}\propto {\mathrm{\Omega }}^{4/3}$ 和 $ž\propto {\mathrm{\Omega }}^{\mathrm{1个}/3}$），此公式表明 ${\sigma }^{\prime }（\mathrm{\Omega }）$ 将趋于恒定 $\mathrm{\Omega }\to 0$。我们明确证明了${\sigma }^{\prime }（\mathrm{\Omega }）$ 由于当前顶点的强烈归一化而有所不同，这抵消了 ${ž}^{\mathrm{2个}}$。因此，${\sigma }^{\prime }（\mathrm{\Omega }）$ 发散为 $\mathrm{1个}/{\mathrm{\Omega }}^{\mathrm{2个}/3}$，就像早期的作品一样。在SDW情况下，我们考虑对电导率的两个贡献：热点和费米表面的“不冷”区域。热点贡献不受顶点重归一化的影响，但它导致了不冷不热的影响。对于后者，我们认为${ž}^{\mathrm{2个}}$被顶点校正再次取消。因此，${\sigma }^{\prime }（\mathrm{\Omega }）$ SDW QCP的规模为 $\mathrm{1个}/\mathrm{\Omega }$ 直至最低频率，直至可能的对数因子。