Optical conductivity of a two-dimensional metal at the onset of spin-density-wave order

We consider the optical conductivity of a clean two-dimensional metal near a quantum spin-density-wave transition. Critical magnetic fluctuations are known to destroy fermionic coherence at “hot spots” of the Fermi surface but coherent quasiparticles survive in the rest of the Fermi surface. A large part of the Fermi surface is not really “cold” but rather “lukewarm” in a sense that coherent quasiparticles in that part survive but are strongly renormalized compared to the noninteracting case. We discuss the self-energy of lukewarm fermions and their contribution to the optical conductivity $\sigma \left(\Omega \right)$, focusing specifically on scattering off composite bosons made of two critical magnetic fluctuations. Recent study [S. A. Hartnoll et al., Phys. Rev. B 84, 125115 (2011)] found that composite scattering gives the strongest contribution to the self-energy of lukewarm fermions and suggested that this may give rise to a non-Fermi-liquid behavior of the optical conductivity at the lowest frequencies. We show that the most singular term in the conductivity coming from self-energy insertions into the conductivity bubble ${\sigma }^{\prime }\left(\Omega \right)\propto {ln}^3\Omega /{\Omega }^{1/3}$ is canceled out by the vertex-correction and Aslamazov-Larkin diagrams. However, the cancellation does not hold beyond logarithmic accuracy, and the remaining conductivity still diverges as $1/{\Omega }^{1/3}$. We further argue that the $1/{\Omega }^{1/3}$ behavior holds only at asymptotically low frequencies, well inside the frequency range affected by superconductivity. At larger $\Omega$, up to frequencies above the Fermi energy, ${\sigma }^{\prime }\left(\Omega \right)$ scales as $1/\Omega$, which is reminiscent of the behavior observed in the superconducting cuprates.

Figure 1

A two-dimensional Fermi surface with hot spots denoted by 1, $\overline{1}$, 2, and $\overline{2}$. Hot spots 1 and 2 are connected by the spin-density-waver ordering vector ${\mathbf{q}}_{\pi }=(\pi ,\pi )$. Hot spots $\overline{1}$ and $\overline{2}$ are the mirror images of hot spots 1 and 2, correspondingly.

Figure 2

Imaginary part of the fermionic self-energy from ${\mathbf{q}}_{\pi }$ scattering, ${\Sigma }_{{\mathbf{q}}_{\pi }}^{\prime \prime }(\overline{k},\overline{\Omega })$ (left axis), and the quasiparticle residue, $Z_{\overline{k}}\left(\overline{\Omega }\right)$ (right axis), as a function of $\overline{\Omega }$. Left panel: $\overline{k}<1$; right panel: $\overline{k}>1$. Dimensionless variables are defined according to Eqs. (2.14) and (2.15).

Figure 3

Imaginary part of the fermionic self-energy from ${\mathbf{q}}_{\pi }$ scattering, ${\Sigma }_{{\mathbf{q}}_{\pi }}^{\prime \prime }(\overline{k},\overline{\Omega })$ (left axis), and the quasiparticle residue, $Z_{\overline{k}}\left(\overline{\Omega }\right)$ (right axis), as a function of $\overline{k}$. Left panel: $\overline{\Omega }<1$; right panel: $\overline{\Omega }>1$. Dimensionless variables are defined according to Eqs. (2.14) and (2.15).

Figure 4

Composite vertex with intermediate processes in the particle-hole (a) and particle-particle channel (b). Labels 1, $\overline{1}$, 2, and $\overline{2}$ correspond to hot spots in Fig. 1. The notation “$P,1$” means that the 2D component of $P=(\mathbf{p},{\Omega }_p)$ is near hot spot 1, and similarly for other $2+1$ momenta. In a forward-scattering event, the initial and final states belong to the same hot spot, e.g., spot 1. A $2k_F$ event involves fermions from opposite hot spots, 1 and $\overline{1}$. Double solid lines denote off-shell fermions at hot spots 2 (forward scattering) and $\overline{2}$ ($2k_F$ scattering). The total composite vertex ($C$) is a sum of vertices $A$ and $B$. The wavy line denotes the effective dynamic interaction carrying a momentum close to ${\mathbf{q}}_{\pi }$ [Eq. (2.8)].

Figure 5

(a) One-loop self-energy due to composite scattering. The hatched box is the composite vertex in Fig. 4. (b) Equivalent representations of diagram (a) in terms of the original interaction in Eq. (2.8). As in Fig. 4, a double line denotes an off-shell fermion.

Figure 6

Two-loop self-energy due to composite scattering. The hatched box is the composite vertex in Fig. 4.

Figure 7

A sketch of the imaginary part of the self-energy, ${\Sigma }^{\prime \prime }(\overline{k},\overline{\Omega })$, as a function of the dimensionless frequency, $\overline{\Omega }=\Omega /\overline{g}$, at fixed (dimensionless) distance from the hot spot, $\overline{k}=\delta k{v}_F/\overline{g}$. Abbreviations of the asymptotic regimes and asymptotic forms of ${\Sigma }^{\prime \prime }$ are given in Table 2.

Figure 8

A sketch of the imaginary part of the self-energy, ${\Sigma }^{\prime \prime }(\overline{k},\overline{\Omega })$ (solid line), and quasiparticle residue, $Z$ (dashed line), as functions of the dimensionless distance from the hot spot, $\overline{k}=\delta k{v}_F/\overline{g}$, at fixed (dimensionless) frequency, $\overline{\Omega }=\Omega /\overline{g}$. Abbreviations of the crossover regimes and asymptotic forms of ${\Sigma }^{\prime \prime }$ are given in Table 2.

Figure 9

A sketch of the imaginary part of the self-energy, ${\Sigma }^{\prime \prime }(\overline{k},\overline{\Omega })$ (solid line), and quasiparticle residue, $Z$ (dashed line), as functions of the dimensionless distance from the hot spot, $\overline{k}=\delta k{v}_F/\overline{g}$, at fixed (dimensionless) frequency, $\overline{\Omega }=\Omega /\overline{g}$. Abbreviations of the crossover regimes and asymptotic forms of ${\Sigma }^{\prime \prime }$ are given in Table 2.

Figure 10

Examples of the three-loop self-energy diagrams. (a) Particle-particle (Cooper) channel. (b) Particle-hole channel. The hatched box is the composite vertex in Fig. 4.

Figure 11

The real part of the conductivity as a function of frequency.

Figure 12

Diagrams for the conductivity. Labels 1 and $\overline{1}$ correspond to hot spots in Fig. 1. $Q_0=(\mathbf{0},\Omega )$ is the $(2+1)$ momentum of the external electric field. (a) Self-energy diagrams. (b) Vertex-correction diagram.

Figure 13

Aslamazov-Larkin diagrams for the conductivity. Notations are the same as in Fig. 12.