Toward precision Fermi-liquid theory in two dimensions

The ultracold and weakly coupled Fermi gas in two spatial dimensions is studied in an effective field theory framework. It has long been observed that universal corrections to the energy density to two orders in the interaction strength do not agree with Monte Carlo simulations in the weak-coupling regime. Here, universal corrections to three orders in the interaction strength are obtained, and are shown to provide agreement between theory and simulation. Special consideration is given to the scale ambiguity associated with the nontrivial renormalization of the singular contact interactions. The isotropic superfluid gap is obtained to next-to-leading order, and nonuniversal contributions to the energy density due to effective range effects, $p$-wave interactions, and three-body forces are computed. Results are compared with precise Monte Carlo simulations of the energy density and the contact in the weakly coupled attractive and repulsive Fermi-liquid regimes. In addition, the known all-orders sum of ladder and ring diagrams is compared with Monte Carlo simulations at weak coupling and beyond.

Figure 1

(a) Two-body diagram with $L$ loops. Three-body diagrams at tree level (b) and at two-loop level (c) (fish slash). The black circle (gray square) corresponds to an insertion of the $C_0$ ($D_0$) operator.

Figure 2

Diagrams contributing to isotropic scattering. The black circle (red square) corresponds to an insertion of the $C_0$ ($C_2$) operator.

Figure 3

Running of the coupling with $\mu =1$ and $\left|\alpha \right(1\left)\right|=1$, as described in the text: the solid blue curve corresponds to repulsive coupling, $\alpha \left(\nu \right)=\left|\alpha \right(\nu \left)\right|$, while the dashed black curve corresponds to attractive coupling, $\alpha \left(\nu \right)=-\left|\alpha \right(\nu \left)\right|$. The vertical dotted (blue) line to the right corresponds to the position of the Landau pole at $\nu =e$ in the repulsive case, while the vertical dotted (black) line to the left corresponds to the bound state at $\nu =e^{-1}$ in the attractive case.

Figure 4

Leading $p$-wave contributions to scattering. The empty circle (square) corresponds to an insertion of the $C_2^{\prime }$ ($C_4^{\prime }$) operator.

Figure 5

Top: The gap equation diagrammatically, with the empty triangle denoting an insertion of the gap. Bottom: The shaded circle represents an insertion of the two-particle irreducible potential $-i{V}_{pp}$, with the dotted lines denoting a $C_0$ interaction. Crossed diagrams and diagrams which cancel are not shown.

Figure 6

Leading-order diagram (the bow tie) contributing to the energy density. The black circle corresponds to an insertion of the $C_0$ operator.

Figure 7

Next-to-leading-order diagrams contributing to the energy density. Only diagram (a), the beach ball diagram, is nonvanishing.

Figure 8

Next-to-next-to-leading-order diagrams contributing to the energy density. Only diagrams (a) and (b) are nonvanishing.

Figure 9

Effective range contribution to the energy density. The red square corresponds to an insertion of the $C_2$ operator.

Figure 10

Leading three-body contribution to the energy density. The gray square corresponds to an insertion of the $D_0$ operator.

Figure 11

$P$-wave contribution to the energy density. The empty circle corresponds to an insertion of the $C_2^{\prime }$ operator.

Figure 12

The contact density, $C_{\mathrm{FL}}$ in units of $k_F^4$, vs coupling strength. The gray dashed curve is derived from the $\mathit{O}\left({\alpha }^2\right)$ Fermi-liquid energy and the solid black curve is from $\mathit{O}\left({\alpha }^3\right)$. The gray band corresponds to varying the RG scale in the $\mathit{O}\left({\alpha }^3\right)$ Fermi-liquid energy and is a measure of the uncertainty associated with truncating the perturbative expansion, as discussed in the text. Note that the MC data have the gap contribution removed.

Figure 13

Schematic representation of ladder diagrams to all orders (left) and ring diagrams to all orders (right). The black circle corresponds to an insertion of the $C_0$ operator.

Figure 14

Resummed ladder and ring functions, $\tilde{\mathbf{L}}\left(\alpha \right)$ and $\mathbf{R}\left(\alpha \right)$, respectively, and their sum, vs coupling strength. The dashed black line is the asymptotic value of the ring function, $-1/6$.

Figure 15

Energy per particle with mean-field piece subtracted vs coupling strength. The gray dashed curve is NLO and the solid black curve is NNLO. The gray band corresponds to varying the RG scale at NNLO and is a measure of the uncertainty associated with truncating the perturbative expansion, as discussed in the text. The red curve is the complete ladder and ring resummation. The MC data are as described in the text.

Figure 16

Energy per particle with mean-field contribution subtracted vs coupling strength. Magnification of Fig. 15 near the origin of the attractive branch.

Figure 17

Energy per particle with mean-field contribution subtracted vs coupling strength. Magnification of Fig. 15 near the origin of the repulsive branch.

Figure 18

Energy per particle with mean-field piece subtracted vs coupling strength. MC data are included over all attractive coupling strengths from the BCS region to the BEC region. The data are as described in the text, with LO gap energy subtracted, and connected by the green curve for ease of viewing. The solid black curve is the NNLO Fermi-liquid prediction. The red curve is the complete ladder and ring resummation, and the blue curve is the [1,2] Padé approximant, as discussed in the text.