Quantum critical response at the onset of spin-density-wave order in two-dimensional metals

We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface “hot spots” and the remainder of the “cold” Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\phi$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\phi$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\phi }^2$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.

Figure 1

The four pairs of hot spots and locations of the fermion fields ${\Psi }_a^{\ell }$, related to the fields in the figure by Eq. (2.2).

Figure 2

(From Ref. 13) Configuration of the $\ell =1$ pair of hot spots, with the momenta of the fermion fields measured from the common hot spot at $\vec{k}=0$, indicated by the dark filled circle at the center of the diagram. The Fermi velocities ${\vec{v}}_{1,2}$ of the ${\psi }_{1,2}$ fermions are indicated. The momentum components of the ${\psi }_2^1\left(\vec{p}\right)$ fermion parallel ($p_{\parallel }$) and orthogonal ($p_{\perp }$) to the Fermi surface are indicated.

Figure 3

(a) Fermion and (b) boson self-energies in the rainbow approximation. Solid lines are fermion Green's functions, while dashed lines are $\phi$ Green's functions. All internal lines are full self-consistent propagators. In this section we are dropping the hot-spot index $\ell$, and so ${\psi }_1\equiv {\psi }_1^1$ and ${\psi }_{\overline{1}}\equiv {\psi }_2^1$.

Figure 4

Diagrams for the current vertex ${\Gamma }_a^{\pm }$ in the rainbow approximation. The cross denotes the bare vertex and the solid circle denotes the full vertex. The Aslamazov-Larkin type diagrams in (c) and (d) cancel against each other due to pseudospin symmetry. As in Fig. 3, ${\psi }_1\equiv {\psi }_1^1$ and ${\psi }_{\overline{1}}\equiv {\psi }_2^1$.

Figure 5

Scattering processes involving fermions from the hot-spot pair $\ell =1$. The diagram in (a) corresponds to regular scattering, which conserves the current. The two diagrams in (b) correspond to umklapp scattering, which conserves the lattice momentum but not the current. Note that ${\psi }_2\equiv {\psi }_{\overline{1}}$.

Figure 6

Umklapp scattering processes in Fig. 5b. Such processes degrade the electric current and lead to a finite optical conductivity.

Figure 7

The exponent $r_0$ [Eq. (3.23)], associated with the current vertex ${\Gamma }^{-}$, as a function of the angle $2\varphi$ between the Fermi surfaces.

Figure 8

The constant contribution to the conductivity as a function of the angle $2\varphi$ between the Fermi surfaces.

Figure 9

Generating a local interaction between cold fermions and $\mathcal{O}={\phi }^2$. As in Fig. 4, a full line is a fundamental fermion $\psi$, and a dashed line is a “fundamental boson” $\phi$. A wavy line represents a propagator for the operator $\mathcal{O}$. The intermediate fermion on the left-hand graph is necessarily very off shell.

Figure 10

The three leading-order corrections in ${\lambda }_1$ to the conductivity due to cold fermions. The wavy lines denote propagators of the neutral bosonic quantum critical operator $\mathcal{O}$.

Figure 11

The leading contribution to the cross-correlator of hot and cold fermion currents $\langle J^{\text{hot}}{J}^{\text{cold}}\rangle$. The triangle denotes the three-point vertex $\langle J^{\text{hot}}\mathcal{O}\mathcal{O}\rangle$ in the critical theory.

Figure 12

Generating a local interaction of lukewarm fermions $\psi ={\psi }_1^3$ and $\tilde{\psi }={\psi }_1^1$ with the operator $\mathcal{O}={\psi }_2^{1†}{\psi }_2^3$. The intermediate “fundamental” boson on the left-hand side is necessarily very off shell. Wavy lines are propagators for the operator $\mathcal{O}$, as in Fig. 9.

Figure 13

The $2k_F$ operator generates the umklapp process shown here. The scattering is from the lukewarm regions near the hot spots and conserves total momentum up to a reciprocal lattice vector. Note that this process arises after integrating out high-energy degrees of freedom from the effective theory in Eq. (2.1).

Figure 14

The three leading-order corrections in ${\lambda }_3$ to the conductivity due to cold fermions. Tildes over momenta indicate that the fermion propagator in question corresponds to the opposite patch. The hot operator $\mathcal{O}$ is now complex and so its propagators carry an arrow.

Figure 15

The three leading-order corrections in ${\lambda }_2$ to the conductivity due to cold fermions. A few fermionic arrows are reversed compared to Fig. 14.

Figure 16

The one loop contribution to the correlation function of the $\mathcal{O}={\phi }^2$ operator.

Figure 17

Fermion self-energy in the $2k_F$ channel. The scattering processes are those shown in Fig. 13. The 1 fermion refers to ${\psi }_1^1$, the 2 to ${\psi }_2^1$, the $-1$ to ${\psi }_1^3$, and the $-2$ to ${\psi }_2^3$. These fermions are also referred to as ${\psi }_1$, ${\psi }_{\overline{1}}$, ${\psi }_{-1}$, and ${\psi }_{-\overline{1}}$ respectively. The momentum labels in (b) correspond to the expression in Eq. (D1).

Figure 18

Fermion self-energy in the $2k_F$ channel. The 2 and $-2$ fermions in Fig. 17b are assumed to be off shell and are integrated out. The resulting process is effectively one-dimensional. The interaction between the 1 and $-1$ fermions has a singular ($\sim$$1/k_{\parallel }^2$) dependence upon their distance ($k_{\parallel }$) from the hot spot.

Figure 19

Tree level (left) and leading correction (right) to the propagator for the operator ${\psi }_2^{1†}{\psi }_2^3$. The subscripts show the hot spot involved. 1 and $-1$ correspond to the $a=1$ hot spots of the $\ell =1$ and $\ell =3$ pair of hot spots, respectively. Similarly, 2 and $-2$ correspond to the $a=2$ hot spots of the $\ell =1$ and $\ell =3$ pair.

Figure 20

The two diagrams describing the contribution to the conductivity of the cold fermions due to critical fluctuations. The wavy lines denote propagators of the neutral bosonic quantum critical operator $\mathcal{O}$.

Figure 21

The two diagrams describing the contribution to the conductivity of the cold fermions due to neutral critical fluctuations carrying momentum $2k_F$. Tildes over momenta indicate that the fermion propagator in question corresponds to the opposite patch. The hot operator $\mathcal{O}$ is now complex and so its propagators carry an arrow.

Figure 22

The two diagrams describing the contribution to the conductivity of the cold fermions due to charged Cooperon-like critical fluctuations.

Figure 23

Numerical solution for the current vertex in the rainbow approximation. (Top) $f_0$, current vertex in the static limit. (Bottom) Function $A$ [Eq. (B4)], which enters the first frequency-dependent correction to the current vertex.