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arXiv:2403.01933 · The European Physical Journal C

Non-relativistic conformal field theory in momentum space

R. K. Gupta, Meenu

Published Apr 15, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract Non-relativistic conformal field theory describes many-body physics at unitarity. The correlation functions of the system are fixed by the requirement of the conformal invariance. In this article, we discuss the correlation functions of scalar operators in non-relativistic conformal field theories in momentum space. We discuss the solution of conformal Ward identities and express 2,3, and 4-point functions as a function of energy and momentum. We also express the 3- and 4-point functions as the one-loop and three-loop Feynman diagram computations in the momentum space. Lastly, we generalize the discussion to the momentum space correlation functions in the presence of a boundary.

DOI
10.1140/epjc/s10052-025-14119-8

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Abstract Non-relativistic conformal field theory describes many-body physics at unitarity. The correlation functions of the system are fixed by the requirement of the conformal invariance. In this article, we discuss the correlation functions of scalar operators in non-relativistic conformal field theories in momentum space. We discuss the solution of conformal Ward identities and express 2,3, and 4-point functions as a function of energy and momentum. We also express the 3- and 4-point functions as the one-loop and three-loop Feynman diagram computations in the momentum space. Lastly, we generalize the discussion to the momentum space correlation functions in the presence of a boundary.
Abstract We study non-relativistic conformal field theory on a flat space in the presence of a planar boundary. We compute correlation functions of primary operators and obtain the expression for the boundary conformal block. We also discuss the non-relativistic conformal field theory on a general curved background in the presence of a boundary. As an example, we discuss the spectrum of boundary primary operator and compute scaling dimensions in a fermionic theory near one and three spatial dimensions.
Momentum space Ward identities are derived for the amputated n-point Green’s functions in 3 + 1 dimensional non-relativistic conformal field theory. For n = 4 and 6 the implications for scattering amplitudes (i.e. on-shell amputated Green’s functions) are considered. Any scale invariant 2-to-2 scattering amplitude is also conformally invariant. However, conformal invariance imposes constraints on off-shell Green’s functions and the three particle scattering amplitude which are not automatically satisfied if they are scale invariant. As an explicit example of a conformally invariant theory we consider non-relativistic particles in the infinite scattering length limit. 1 Poincaré invariant theories that are scale invariant usually have a larger symmetry group called the conformal group1. A similar phenomena happens for 3+1 dimensional nonrelativistic systems. These are invariant under the extended Galilean group, which consists of 10 generators: translations (4), rotations (3), and Galilean boosts (3). The largest space-time symmetry group of the free Schrödinger equation is called the Schrödinger or nonrelativistic
The fundamentals of the conformal symmetry transformations in two dimensions are introduced. From the adapted \textit{Ward identities under conformal constraints}, the \textit{operator product expansion} (OPE) is defined for free fields and the energy momentum tensor (EM). After demonstrating how the energy momentum tensor transforms under conformal invariance, the notion of the \textit{central charge c} is defined. Conformal fields in two dimensions are \textit{radially} quantized on the cylinder within the framework of the second quantization of quantum field theory and are then mapped onto complex plane via a conformal mapping. The \textit{radial ordering} of conformal fields is adopted. The \textit{Virasoro algebra} is derived from the identification of the \textit{conformal generators}, which follow from relations between equal-time commutators and contour integrals of conformal fields. The radial quantization is applied to the free fermionic and bosonic fields, where for the latter the \textit{Vertex operator} is defined and specified. The Fock space of the free boson is constructed through the \textit{operator state correspondence}. Various boundary conditions are applied to free fields, and their effects on the ground-state energy density are detailed, leading to the notion of the \textit{Casimir energy}. The OPEs are regularized for the \textit{interacting} conformal fields, namely the E-M tensor, and the related definition of \textit{normal ordering} is revisited. Utilization of the Virasoro algebra led first to the \textit{descendant states} and subsequently to the \textit{conformal families}. The OPEs are generalized for the conformal families, and it is demonstrated how the three-point function coefficients may be obtained in terms of the two-point functions. The same procedure is employed generically through the definition of the \textit{conformal blocks}. The \textit{conformal Bootstrap method} is introduced to calculate the coefficients of nnn point functions. The Hilbert space of the conformal fields in two dimensions is formally constructed through the \textit{representation theory of Virasoro algebra}. Accordingly, the reducible and irreducible \textit{Verma modules} are specified. Due to the unitarity requirement of the Hilbert space, the famous \textit{Kac determinant} is introduced to obtain constraints on the inner products of the Verma modules, whence on the \textit{conformal weights} and the central charge. The decoupling of the \textit{null states} from rest of the their conformal families is used to obtain \textit{fusion rules} of the families. The \textit{Kac table} for the constrained conformal weights is constructed, and its analysis under the requirement of unitarity leads to the identification of the \textit{minimal models}. The correspondence between the discrete statistical systems at criticality and the minimal models is studied, where special attention is given on the \textit{Ising model}.
We explore the consequences of conformal symmetry for the operator product expansions in nonrelativistic field theories. Similar to the relativistic case, the OPE coefficients of descendants are related to that of the primary. However, unlike relativistic CFTs the 3-point function of primaries is not completely specified by conformal symmetry. Here, we show that the 3-point function between operators with nonzero particle number, where (at least) one operator has the lowest dimension allowed by unitarity, is determined up to a numerical coefficient. We also look at the structure of the family tree of primaries with zero particle number and discuss the presence of conservation laws in this sector.
We explore the consequences of conformal symmetry for the operator product expansions in nonrelativistic field theories. Similar to the relativistic case, the OPE coefficients of descendants are related to that of the primary. However, unlike relativistic CFTs the 3-point function of primaries is not completely specified by conformal symmetry. Here, we show that the 3-point function between operators with nonzero particle number, where (at least) one operator has the lowest dimension allowed by unitarity, is determined up to a numerical coefficient. We also look at the structure of the family tree of primaries with zero particle number and discuss the presence of conservation laws in this sector.
Correlations in asymptotic fluxes provide one of the primary experimental means of studying physical systems. In the context of relativistic QFT, the formalization of such experiments in terms of detector operators has led to significant progress. Motivated by these successes, in this paper we initiate a study of detector operators in nonrelativistic conformal field theories (NRCFTs). In NRCFTs, fluxes are distributed in both angle and velocity, making the natural detector operator the energy flux differential in velocity, Ev(n^)\mathcal{E}_v(\hat{n})Ev​(n^). We provide non-perturbative definitions of these operators in NRCFTs, and study their symmetries and algebra. We calculate one-point functions of these detector observables in a variety of few and many body states of representative NRCFTs, including free fermions, as well as fermions at unitarity. In interacting three-body states, they exhibit a remarkably rich behavior reflective of the Efimov three-body wave-function, including the formation of an enhanced "jet" structure due to the presence of resonant interactions. We use these detectors to derive a nonrelativistic generalization of the Hofman-Maldacena conformal collider bounds by arguing for positivity of Ev(n^)\mathcal{E}_v(\hat{n})Ev​(n^), and imposing this in spin 1 states. We outline a number of future directions for the study of detectors in NRCFTs, and discuss how they can be measured in a variety of experimental platforms from cold atoms to low energy neutron experiments.