Path-Integral Fujikawa's Approach to Anomalous Virial Theorems and Equations of State for Systems with $SO(2, 1)$ Symmetry

TL;DR

This paper uses Fujikawa's path-integral method to derive anomalous virial theorems and equations of state for 2D systems with $SO(2,1)$ symmetry, highlighting the role of anomalies and Tan contact terms.

Contribution

It introduces a formal approach to derive anomalies in 2D many-body systems with $SO(2,1)$ symmetry using Fujikawa's method, applicable to both bosonic and fermionic systems.

Findings

Derived anomalous virial theorem for 2D systems

Established connection between anomaly corrections and Tan contact

Generalized methods for various 2D many-body systems

Abstract

We derive anomalous equations of state for nonrelativistic 2D complex bosonic fields with contact interactions, using Fujikawa's path-integral approach to anomalies and scaling arguments. In the process, we derive an anomalous virial theorem for such systems. The methods used are easily generalizable for other 2D systems, including fermionic ones, and of different spatial dimensionality, all of which share a classical $SO(2,1)$ Schrodinger symmetry. The discussion is of a more formal nature and is intended mainly to shed light on the structure of anomalies in 2D many-body systems. The anomaly corrections to the virial theorem and equation of state - pressure relationship - may be identified as the Tan contact term. The practicality of these ideas rests upon being able to compute in detail the Fujikawa Jacobian that contains the anomaly. This and other conceptual issues, as well as some recent developments, are discussed at the end of the paper.