Ultraviolet/infrared mixing in non-Fermi liquids

We study low-energy effective-field theories for non-Fermi liquids with Fermi surfaces of general dimensions and codimensions. When the dimension of the Fermi surface is greater than one, low-energy particle-hole excitations remain strongly coupled with each other across the entire Fermi surface. In this case, even the observables that are local in the momentum space (such as the Green's functions) become dependent on the size of the Fermi surface in singular ways, resulting in a ultraviolet/infrared (UV/IR) mixing. By tuning the dimension and codimension of the Fermi surface independently, we find perturbative non-Fermi-liquid fixed points controlled by both UV/IR mixing and interactions.

Figure 1

(a) A compact Fermi surface can be divided into two halves centered at $\pm K^{*}$. For each half, a separate fermionic field is introduced. (b) The compact Fermi surface is approximated by two sheets of noncompact Fermi surfaces with a momentum regularization that suppresses modes far away from $\pm K^{*}$.

Figure 2

For $m>1$, any two points on the Fermi surface have at least $(m-1)$ common tangent vectors.

Figure 3

As the high-energy modes away from the Fermi surface are integrated out, the ratio $k_F/\mathrm{\Lambda }$ grows. $k_F$ is the size of the Fermi surface, and $\mathrm{\Lambda }$ is the energy cutoff perpendicular to the Fermi surface. For $m>1$, the Green's functions are singular in the $k_F/\mathrm{\Lambda }\to \infty$ limit, which results in the UV/IR mixing.

Figure 4

The one-loop diagrams for (a) the boson self-energy, (b) the fermion self-energy, and (c) the vertex correction. Solid lines represent the bare fermion propagator, whereas wiggly lines in (b) and (c) represent the dressed boson propagator which includes the one-loop self-energy in (a).

Figure 5

The plots for $d_c$ and $d_c^{\prime }$ as functions of $m$.

Figure 6

A two-dimensional slice of an $m$-dimensional Fermi surface. The typical momentum carried by a boson is proportional to ${\tilde{\alpha }}^{1/3}{\mathrm{\Lambda }}^{(d-m)/3}\sim {\tilde{e}}^{1/(m+1)}{\left(\frac{k_F}{\mathrm{\Lambda }}\right)}^{(m-1)/\left[2\right(m+1\left)\right]}{\mathrm{\Lambda }}^{1/2}$. For $m>1$, this momentum is much larger than ${\mathrm{\Lambda }}^{1/2}$ in the low-energy limit. As a result, the momentum transferred from a boson takes a fermion near the Fermi surface outside the thin shell of the UV cutoff. This leads to a suppression of the virtual particle-hole excitations by powers of $\mathrm{\Lambda }/k_F$ for $m>1$.

Figure 7

The diagrams for the two-loop boson self-energy.

Figure 8

The diagrams for the two-loop fermion self-energy.

Figure 9

The diagrams for the two-loop vertex corrections.