Hyperscaling at the spin density wave quantum critical point in two-dimensional metals

The hyperscaling property implies that spatially isotropic critical quantum states in $d$ spatial dimensions have a specific heat, which scales with temperature as $T^{d/z}$, and an optical conductivity, which scales with frequency as ${\omega }^{(d-2)/z}$ for $\omega \gg T$, where $z$ is the dynamic critical exponent. We examine the spin density wave critical fixed point of metals in $d=2$ found by Sur and Lee [Phys. Rev. B 91, 125136 (2015)] in an expansion in $\epsilon =3-d$. We find that the contributions of the “hot spots” on the Fermi surface to the optical conductivity and specific heat obey hyperscaling (up to logarithms), and agree with the results of the large $N$ analysis of the optical conductivity by Hartnoll et al. [ Phys. Rev. B 84, 125115 (2011)]. With a small bare velocity of the boson associated with the spin density wave order, there is an intermediate energy regime where hyperscaling is violated with $d\to d_t$, where $d_t=1$ is the number of dimensions transverse to the Fermi surface. We also present a Boltzmann equation analysis which indicates that the hot-spot contribution to the dc conductivity has the same scaling as the optical conductivity, with $T$ replacing $\omega$.

Figure 1

(a) Hot spot geometry, labeling conventions, and choice of $x,y$-coordinate system in the Brillouin zone of the two-dimensional square lattice in which the fermions move. The boundary of the blue area denotes the Fermi surface separating the filled particlelike states (blue area) from the holelike states (white area). ${\mathbf{Q}}_{\mathrm{AF}}=(\pi ,\pi )$ is the (commensurate) antiferromagnetic ordering vector that intersects the Fermi surface at four pairs of hot spots. (b) (Top) Fermi surface patches from a hot-spot pair connected by ${\mathbf{Q}}_{\mathrm{AF}}$ centered at a common origin in momentum space. The two light colored regions are the regions occupied by fermions at the two hot spots of the pair, respectively, the dark colored region is occupied by fermions at both hot spots, and the white region is unoccupied. The arrows perpendicular to the Fermi surfaces denote the directions of the Fermi velocities. (Bottom) Under the RG flow, the Fermi surfaces are deformed as shown at the strange metal fixed point, and as indicated in Eq. (1.3). The Fermi velocities are exactly antiparallel only at the hot spot ($\mathbf{k}=0$).

Figure 2

Feynman graphs for the current-current correlator up to two loops. The black and grey boxes are current vertices for the $n$ and $\overline{n}$ hot-spot pairs, respectively. The wiggly lines are the boson propagators and the solid lines stand for fermion propagators. (a) One-loop contribution for free fermions. (b) Two-loop self-energy correction computed in Sec. 3b. There is also a partner diagram with the boson on the lower fermion line. (c) Two-loop vertex correction computed in Sec. 3c.

Figure 3

Key one-loop elements appearing in the two-loop self-energy correction (a) and two-loop (current) vertex correction (b).

Figure 4

The simplest interaction contribution to the free energy at $O\left(g^2\right)$.

Figure 5

Feynman rules and one-loop graphs for the self-energies of the fermions and bosons in the Keldysh formalism. (a) Fermion propagators. (b) Boson propagators. (c) Yukawa vertices. (d) Self energies. Here, $x=(t,\mathbf{r})$, hot-spot indices $(\ell ,\pm )$ are suppressed and the external legs on the self-energy diagrams are amputated. The external momentum and frequency on the self-energy diagrams is on shell. The $2-1$ and $q-q$ propagators are zero and hence are omitted.