Nonrelativistic conformal field theories

TL;DR

The paper develops a nonrelativistic conformal field theory framework based on the Schrödinger algebra, establishing an operator–state correspondence that maps primary operators to harmonic-oscillator eigenstates. By performing controlled ε-expansions near d=4 and near d=2, it computes scaling dimensions for fermionic and anyonic composites and translates them into exact- or perturbative-energy results in a harmonic trap for up to six fermions and up to four anyons. Padé interpolation between the two near-dimensional limits yields estimates at d=3 that agree reasonably with known numerical values, illustrating the utility and limitations of the approach. The work provides a systematic, diagrammatically tractable route to few-body spectra in unitary and anyonic systems via nonrelativistic CFT techniques, with potential extensions to larger N and other interaction channels.

Abstract

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.

Figures (8)

• Figure 1: One-loop self-energy diagram to renormalize the wave function of $\phi$.
• Figure 2: One-loop diagram to renormalize the three-fermion operator $\phi{\psi_\uparrow}$.
• Figure 3: One-loop diagram to renormalize the four-fermion operator $\phi^2$.
• Figure 4: One-loop diagram to renormalize the four-point vertex coupling $\bar{g}^2$.
• Figure 5: One-loop diagram to renormalize the two-fermion operator ${\psi_\downarrow}{\psi_\uparrow}$.