Two-point function of a $d=2$ quantum critical metal in the limit $k_F\to \infty$, $N_f\to 0$ with $N_f{k}_F$ fixed

We show that the fermionic and bosonic spectrum of $d=2$ fermions at finite density coupled to a critical boson can be determined nonperturbatively in the combined limit $k_F\to \infty ,N_f\to 0$ with $N_f{k}_F$ fixed. In this double scaling limit, the boson two-point function is corrected but only at one loop. This double scaling limit therefore incorporates the leading effect of Landau damping. The fermion two-point function is determined analytically in real space and numerically in (Euclidean) momentum space. The resulting spectrum is discontinuously connected to the quenched $N_f\to 0$ result. For $\omega \to 0$ with $k$ fixed the spectrum exhibits the distinct non-Fermi-liquid behavior previously surmised from the RPA approximation. However, the exact answer obtained here shows that the RPA result does not fully capture the IR of the theory.

Figure 1

Here we show the dominant scaling of fermion loops with different numbers of vertices in the limit of $N_f\to 0$ with $N_f{k}_F$ constant. This is the scaling after symmetrizing the external momenta. The two-vertex loop on the left does not get symmetrized and is the only loop that does not vanish in this limit.

Figure 2

Real and imaginary parts of the self energy obtained using the large-$k_F$ Landau-damped propagator. This plot shows the agreement between the numerics and the analytical solution, verifying that both solutions are correct. Notice the difference in magnitude between the real and imaginary part. The agreement of the real parts shows that the numerical procedure has a very small relative error. All plots are for the $k_x,\omega ={\lambda }^2$ slice with $v=1$.

Figure 3

Exact fermion spectral function based on the large-$M_D$ approximation for the exact boson propagator. Notice that the function is not positive everywhere. Here $k={\lambda }^2,v=1$ and $M_D=2\pi {\lambda }^2$.

Figure 4

(a) Density plots of the imaginary part of the exact (Euclidean) fermion Green's function $G(\omega ,k_x)$ for various values of $M_D$. In the quenched limit $M_D=0$ the three Fermi surface singularities are visible. For any appreciable finite $M_D$ the Euclidean Green's function behaves as a single Fermi surface non-Fermi liquid. (b) Real and imaginary parts of $G(\omega ,k_x)$ for very small $\omega =0.01{\lambda }^2$.

Figure 5

Real and imaginary parts of the fermion self-energy for (a) $\omega =0.01{\lambda }^2$ and for (b) $k_x=0.01{\lambda }^2$. Dashed lines show the $G_{\mathrm{IR}}$ approximation; dotted lines show the RPA result.

Figure 6

Momentum distribution function. The two plots show the same function for different ranges. The error bars of the lower figure show an estimate of the error due to using the free Green's function close to the UV. The error bars are exaggerated by a factor 100.

Figure 7

A sketch of the regimes of applicability of various approximations to the exact fermion Green's function of the elementary quantum critical metal in the double scaling limit. The $G_{\mathrm{IR}}$ approximation is applicable both in the deep yellow region and the red and green regions.

Figure 8

The left image shows part of the Riemann surface of the function $h$ and the right image shows part of the Riemann surface of the G function. The part in red of the left image can be expressed as the G function evaluated on its first sheet, the red part of the right image. The green arrow shows how points are mapped from one Riemann surface to the other by the mapping $u\mapsto -u^3/4$.

Figure 9

The region of convergence. In the shaded area the $L\to \infty$ ($N_f\to 0$) limit is not convergent while outside this area ${lim}_{L\to \infty }{G}_L=G_{\mathrm{quenched}}$.

Figure 10

The relative difference between $G_L$ and the quenched result as a function of $L$ for different frequencies and momenta. Left: for a point in the $(\omega ,k)$ plane which is outside of the shaded region of Fig. 9 the “error” goes to zero for large $L$. Middle: inside the shaded region the error is oscillating with diverging amplitude as $L$ goes to larger values. However, for larger values of $\omega$ there is an intermediate range of $L$, where the relative error is smaller than 0.3. Therefore the quenched approximation is qualitatively correct. Right: for small $\omega$ the amplitude of the oscillation is large.