Perturbative non-Fermi liquids from dimensional regularization

We devise a dimensional regularization scheme for quantum field theories with a Fermi surface to study scaling behavior of non-Fermi liquid states in a controlled approximation. Starting from two space dimensions, the co-dimension of the Fermi surface is extended to a general value while its dimension is fixed. When the Fermi surface is coupled with a critical boson centered at zero momentum, the interaction becomes marginal at a critical space dimension $d_c=5/2$. A deviation from the critical dimension is used as a small parameter for a systematic expansion. We apply this method to the theory where two patches of the Fermi surface are coupled with a critical boson, and show that the Ising-nematic critical point is described by a stable non-Fermi liquid state slightly below the critical dimension. Critical exponents are computed up to the two-loop order.

Figure 1

The right- and left-moving modes near the Fermi surface can be combined into a two-component Dirac fermion.

Figure 2

(a) The one-dimensional Fermi surface embedded in the $d$-dimensional momentum space. (b) For each $k_d$, there is a Fermi point at $(k_1,k_2,\cdots ,k_{d-2},k_{d-1})=(0,0,\cdots ,0,-\sqrt{d-1}{k}_d^2)$, which is denoted as a (red) dot. Around the Fermi point, the energy disperses linearly like a two-component Dirac fermion in $(d-1)$-dimensional space.

Figure 3

Fermi lines in three dimensional momentum space can be obtained by turning on $p$-wave superconducting order parameter that gaps out the cylindrical Fermi surface except for the line node denoted by the thick (red) line.

Figure 4

The one-loop boson self-energy.

Figure 5

(a) The one-loop fermion self-energy. (b) The one-loop vertex correction.

Figure 6

The diagrams for two-loop boson self-energy.

Figure 7

The diagrams for two-loop fermion self-energy.

Figure 8

The diagrams for two-loop vertex corrections.

Figure 9

Aslamazov-Larkin-type contributions to boson self-energy. Diagrams (a) and (b) correspond to the particle-particle and particle-hole channels, respectively.

Figure 10

The one-loop diagrams that renormalize the $2k_F$ scattering amplitude $r$.