Nonrelativistic conformal field theories

We study representations of the Schrödinger algebra in terms of operators in nonrelativistic conformal field theories. We prove a correspondence between primary operators and eigenstates of few-body systems in a harmonic potential. Using the correspondence we compute analytically the energy of fermions at unitarity in a harmonic potential near two and four spatial dimensions. We also compute the energy of anyons in a harmonic potential near the bosonic and fermionic limits.

Figure 1

One-loop self-energy diagram to renormalize the wave function of $\varphi$.

Figure 2

One-loop diagram to renormalize the three-fermion operator $\varphi {\psi }_{\uparrow }$.

Figure 3

One-loop diagram to renormalize the four-fermion operator ${\varphi }^2$.

Figure 4

One-loop diagram to renormalize the four-point vertex coupling ${\overline{g}}^2$.

Figure 5

One-loop diagram to renormalize the two-fermion operator ${\psi }_{\downarrow }{\psi }_{\uparrow }$.

Figure 6

Energy of three fermions in a harmonic potential with $l=0$ (left panel) and $l=1$ (right panel) as functions of spatial dimension $d$. The four solid curves, although hard to distinguish, represent different Padé approximants interpolating the expansions around $d=2$ and $d=4$. The dashed (dotted) lines are extrapolations from the $ϵ=4-d$ ($\overline{ϵ}=d-2$) expansions. The symbols (×) indicate the exact values for each $d$ [9].

Figure 7

One-loop diagrams to renormalize the contact interaction coupling $v$.

Figure 8

One-loop diagrams to renormalize the two-anyon operators.