Quantum phase transitions of metals in two spatial dimensions. II. Spin density wave order

We present a field-theoretic renormalization group analysis of Abanov and Chubukov’s model of the spin density wave transition in two dimensional metals. We identify the independent field scale and coupling constant renormalizations in a local field theory and argue that the damping constant of spin density wave fluctuations tracks the renormalization of the local couplings. The divergences at two-loop order overdetermine the renormalization constants and are shown to be consistent with our renormalization scheme. We describe the physical consequences of our renormalization-group equations, including the breakdown of Fermi liquid behavior near the “hot spots” on the Fermi surface. In particular, we find that the dynamical critical exponent $z$ receives corrections to its mean-field value $z=2$. At higher orders in the loop expansion, we find infrared singularities similar to those found by Lee [Phys. Rev. B 80, 165102 (2009)] for the problem of a Fermi surface coupled to a gauge field. A treatment of these singularities implies that an expansion in $1∕N$ (where $N$ is the number of fermion flavors) fails for the present problem. We also discuss the renormalization of the pairing vertex and find an enhancement which scales as logarithm squared of the energy scale. A similar enhancement is also found for a modulated bond order which is locally an Ising-nematic order.

Figure 1

Square lattice Brillouin zone showing the Fermi surface appropriate to the cuprates. The filled circles are the hot spots connected by the SDW wave vector $\vec{Q}=(\pi ,\pi )$. The locations of the continuum fermion fields ${\psi }_1^{\ell }$ and ${\psi }_2^{\ell }$ are indicated.

Figure 2

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 filled circle. The Fermi velocities ${\vec{v}}_{1,2}$ of the ${\psi }_{1,2}$ fermions are indicated.

Figure 3

Modification of the Fermi surfaces in Fig. 2 by SDW order with $⟨\varphi ⟩\ne 0$. The full lines are the Fermi surfaces, and the white, light shaded, and dark shaded regions denote momenta where 0, 1, and 2 of the bands are occupied. The upper and lower lines are boundaries of hole and electron pockets respectively.

Figure 4

The boson self-energy at $N=\infty$. The full lines represent the ${\psi }_{1,2}$ fermions, and the dashed lines represent the boson ${\varphi }^a$.

Figure 5

The leading contribution to the fermion self-energy.

Figure 6

The leading correction to the boson-fermion vertex.

Figure 7

The leading correction to the boson self-energy. A sum over both directions of the fermion loop is implied.

Figure 8

Modification of the Fermi surfaces in Fig. 2 at the SDW quantum critical point. As in Figs. 2, 3, the full lines are the Fermi surfaces, and the white, light shaded, and dark shaded regions denote momenta where 0, 1, and 2 of the bands are occupied. The equation of one of the Fermi surfaces is given in Eq. (3.43).

Figure 9

Three loop corrections to the boson-fermion vertex that are enhanced in $N$, scaling as $\mathcal{O}\left(N^0\right)$.

Figure 10

Double line representation for the boson-fermion vertex.

Figure 11

Double line representation for the boson-fermion vertex. The directions of momentum and particle flow need not coincide.

Figure 12

Double line representation applied to the diagrams in Fig. 9. The enhancement of the diagram in $N$ is related to the number of loops $n$ in the double line representation via Eq. (4.9).

Figure 13

A three loop vertex correction with no enhancement in $N$.

Figure 14

A diagram for the fermion self-energy that is of $\mathcal{O}(1∕N)$ as a result of enhancement.

Figure 15

A diagram for the boson self-energy that is of $\mathcal{O}\left(N\right)$ as a result of enhancement.

Figure 16

Converting vacuum energy diagrams into surfaces: a face is attached to each solid and dotted loop in the double-line representation (below). In the present case, the resulting surface is a sphere.

Figure 17

Producing a boson four-point function from a vacuum bubble by cutting two boson propagators. If the initial diagram is planar and only two solid lines are cut in the double-line representation then the resulting diagram is disconnected, as in (a). Diagrams of highest degree are obtained by starting with a planar diagram and cutting three solid line loops, as in (b), or starting with a diagram with $\chi =1$ and cutting two solid line loops, as in (c).

Figure 18

A diagram for the boson four-point function that diverges logarithmically and scales as $N^3$.

Figure 19

Pairing of the electrons at the $\ell =\pm 1$ hot spots of Fig. 1. Electrons at opposite ends of the arrows form spin-singlet pairs. The $\mu =+1$ $(\mu =-1)$ pairing amplitude in Eq. (5.1) has the same (opposite) sign on the two arrows. Only the $\mu =-1$ spin singlet pairing is enhanced near the SDW critical point.

Figure 20

The leading corrections to (a) BCS pairing vertex and (b) density-wave vertex.

Figure 21

Spin singlet density operators $(\sim {\psi }^{†}\psi )$ of the electrons at the $\ell =\pm 1$ hot spots of Fig. 1 (see also Fig. 19), shown with an arrow pointing from the Brillouin zone location of ${\psi }^{†}$ to that of $\psi$. The dashed arrows are the density operators in the first Brillouin zone. The full arrows are in an extended zone scheme which shows that these operators have net momentum ${\vec{Q}}_1=2K_y(-1,1)$, where $(K_x,K_y)$ is the location of the $\ell =1$, $i=1$ hot spot. The density operator with opposite signs $(\mu =-1)$ on the two arrows is enhanced near the SDW critical point. Similarly the $\ell =\pm 2$ hot spots contribute density operators at ${\vec{Q}}_2=2K_y(1,1)$.

Figure 22

(Color online) Plot of the bond density modulations in Eq. (6.10). The lines are the links of the underlying square lattice. Each link contains a colored square representing the value of $⟨c_{\vec{r}\sigma }^{†}{c}_{\vec{s}\sigma }⟩$, where $\vec{r}$ and $\vec{s}$ are the sites at the ends of the link. We chose the ordering wave vector ${\vec{Q}}_1=(2\pi ∕16)(1,-1)$. Notice the local Ising-nematic ordering, and the longer wavelength sinusoidal envelope along the diagonal.

Figure 23

(Color online) As in Fig. 22, but for orderings along both ${\vec{Q}}_1=(2\pi ∕16)(1,-1)$ and ${\vec{Q}}_2=(2\pi ∕16)(1,1)$.