Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

OPE convergence in non-relativistic conformal field theories

Walter D. Goldberger, Zuhair U. Khandker, Siddharth Prabhu

TL;DR

This paper analyzes the operator product expansion in non-relativistic conformal field theories with Schrödinger symmetry, focusing on charged operators. It shows that the OPE organizes into primary operators and descendant towers, with descendant Wilson coefficients fixed by recursion relations tied to $K_i$ and $C$ actions, while the primary coefficients depend on a model-specific function $F(v)$. By mapping to an oscillator frame, it establishes a Hilbert-space interpretation in which the OPE converges exponentially in operator dimensions, and provides a practical bound on the tail using four-point functions; consistency checks are performed in free theory, unitary bosons, and a holographic model. The results point toward a potential non-relativistic bootstrap program for constraining bound-state spectra and interactions in NRCFTs, though they do not yet address the zero-charge sector (stress tensor, currents).

Abstract

Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NRCFT, in particular the mapping between operators and states in a non-relativistic "radial quantization" Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2,d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.

OPE convergence in non-relativistic conformal field theories

TL;DR

This paper analyzes the operator product expansion in non-relativistic conformal field theories with Schrödinger symmetry, focusing on charged operators. It shows that the OPE organizes into primary operators and descendant towers, with descendant Wilson coefficients fixed by recursion relations tied to and actions, while the primary coefficients depend on a model-specific function . By mapping to an oscillator frame, it establishes a Hilbert-space interpretation in which the OPE converges exponentially in operator dimensions, and provides a practical bound on the tail using four-point functions; consistency checks are performed in free theory, unitary bosons, and a holographic model. The results point toward a potential non-relativistic bootstrap program for constraining bound-state spectra and interactions in NRCFTs, though they do not yet address the zero-charge sector (stress tensor, currents).

Abstract

Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NRCFT, in particular the mapping between operators and states in a non-relativistic "radial quantization" Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2,d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.

Paper Structure

This paper contains 18 sections, 106 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. OPE structure in the Galilean frame
  3. Preliminaries: Local operators, representations, and correlation functions
  4. Correlation functions
  5. OPE constraints
  6. Constraints in d=1
  7. Hilbert space structure and OPE in the oscillator frame
  8. Hilbert space structure in the oscillator frame
  9. Hilbert space interpretation of correlation functions
  10. Bounds on OPE convergence
  11. Estimating the convergence rate
  12. Consistency checks
  13. Free field theory
  14. Bosons at unitarity
  15. Holographic model
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Schematic depiction of how time-slices map under the transformation between Galilean and oscillator coordinates.
  • Figure 2: The three-point function involving two $\phi$ insertions and one insertion of $\Phi=(\phi^2)^\dagger$ is given by the sum of loop diagrams shown in the figure. The sum of diagrams is equivalent to the exchange of a charge $-2$ field of dimension $\Delta=2$.